EEL 4140 Lab Manual

LABORATORY MANUALDEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING UNIVERSITY OF CENTRAL FLORIDA EEL 4140 ANALOG FILTERS DESIGN Revised September 2005 TABLE OF CONTENTS SAFETY RULES AND OPERATING PROCEDURES LABORATORY SAFETY INFORMATION EXPERIMENT # 1 Study Guide A Study Guide B Effect of Op Amp Frequency Dependence on Finite Gain Amplifiers and Bandwidth Extension Techniques using Composite Op Amps (CNOA) Composite Operational Amplifiers: Generation and Finite-Gain Applications Inverting Integrator and Active Filter Applications of Composite Operational Amplifiers EXPERIMENT #2 Real Zero and Pole Synthesis EXPERIMENT # 3 Sallen-Key Filters EXPERIMENT # 4 State-Variable Biquads EXPERIMENT # 5 Single Op Amp Band-Pass Filters EXPERIMENT # 6 Two Op Amps Current Generalized Immittance Structure (CGIC) Based Biquad Biquads II: The current Generalized Immittance (CGIC) Structure Study Guide C EXPERIMENT # 7 Study Guide C High-Order Low-Pass Filter Design Biquads II: The current Generalized Immittance (CGIC) Structure EXPERIMENT # 8 Butterworth Filter Approximation APPENDIX LIST OF COMPONENTS RESISTOR COLOR CODE TUTORIAL ii Safety Rules and Operating Procedures 1. Note the location of the Emergency Disconnect (red button near the door) to shut off power in an emergency. Note the location of the nearest telephone (map on bulletin board). 2. Students are allowed in the laboratory only when the instructor is present. 3. Open drinks and food are not allowed near the lab benches. 4. Report any broken equipment or defective parts to the lab instructor. Do not open, remove the cover, or attempt to repair any equipment. 5. When the lab exercise is over, all instruments, except computers, must be turned off. Return substitution boxes to the designated location. Your lab grade will be affected if your laboratory station is not tidy when you leave. 6. University property must not be taken from the laboratory. 7. Do not move instruments from one lab station to another lab station. 8. Do not tamper with or remove security straps, locks, or other security devices. Do not disable or attempt to defeat the security camera. 9. ANYONE VIOLATING ANY RULES OR REGULATIONS MAY BE DENIED ACCESS TO THESE FACILITIES. I have read and understand these rules and procedures. I agree to abide by these rules and procedures at all times while using these facilities. I understand that failure to follow these rules and procedures will result in my immediate dismissal from the laboratory and additional disciplinary action may be taken. ________________________________________ Signature Date ________________ Lab # iii contact the Department Office for help or call 911. Skin that is broken. If resistors or other components on your proto-board catch fire. and necklaces. Be quick! Some of the staff in the Department Office are certified in CPR. If able. If the victim is unconscious or needs an ambulance. Avoid using fire extinguishers on electronic instruments. These small electrical fires extinguish quickly after the power is shut off. press the Emergency Disconnect (red button located near the door to the laboratory). Electrolytic capacitors can explode and cause injury. Skin beneath a ring or watch is damp.Laboratory Safety Information Introduction The danger of injury or death from electrical shock. 100ma of current passing through the chest is usually fatal. This shuts off all power. iv . fire. or damp with sweat has a low resistance. or explosion is present while conducting experiments in this laboratory. This reduces the likelihood of an accident that results in current passing through your heart. turn off the power supply and notify the instructor. Muscle contractions can prevent the person from moving away while being electrocuted. if needed. If the victim isn’t breathing. lowering the skin resistance. be careful to observe proper polarity and do not exceed the voltage rating. press the Emergency Disconnect (red button). A first aid kit is located on the wall near the door. The resistance of dry. Proceed to Student Health Services. Shoes covering the feet are much safer than sandals. Make sure your hands are dry. Explosions When using electrolytic capacitors. it is important that you understand the prudent practices necessary to minimize the risks and what to do if there is an accident. If electronic instruments catch fire. except the lights. Electrocution has been reported at dc voltages as low as 42 volts. find someone certified in CPR. the victim should go to the Student Health Services for examination and treatment. When working with an energized circuit. wet. Be cautious of rings. Do not touch someone who is being shocked while still in contact with the electrical conductor or you may also be electrocuted. keeping your left hand away from all conductive material. Fire Transistors and other components can become extremely hot and cause severe burns if touched. unbroken skin is relatively high and thus reduces the risk of shock. watches. To work safely. Instead. Electrical Shock Avoid contact with conductors in energized electrical circuits. work with only your right hand. II. RI and RO are the input and output resistance.1.1 The simplified model of a practical Op-amp . respectively. a Va + + RI b Vb − RO − Vo o AOL ( jω )[Va − Vb ] Fig. In this model. and ω 0 is the dominant-pole frequency. The open-loop gain AOL ( jω ) can be written as: AOL ( jω ) = A0 (1) ω 1+ j ω0 Where A0 is the DC open-loop gain. and to study the concept of Composite Op Amps (CNOA). Introduction The simplified model of a practical Op Amp is shown in Fig. Objective To understand the effect of finite gain bandwidth product of practical Op Amps in finite gain applications.EEL 4140 ANALOG FILTERS LABORATORY 1 Effect of Op Amp Frequency Dependence on Finite Gain Amplifiers and Bandwidth Extension Techniques using Composite Op Amps (CNOA) I. In this amplifier. 3. is used to illustrate the bandwidth shrinkage by the voltage gain K . the input and output resistances of the Op Amp are assumed to be infinite and zero.2. Equation (3) can be simplified as: (4) . respectively. In this figure. The transfer function of this amplifier can be derived as: T ( jω ) = Vo ( jω ) Vi ( jω ) = A0 × A0 1+ j 1+ K 1 ⎛ ⎝ (3) ω ω 0 ⎜1 + A0 ⎞ ⎟ K⎠ Normally.The relationship between AOL ( jω ) and the frequency ω is shown in Fig. it is true that A0 >> K Therefore. or the unity-gain bandwidth. the gain-bandwidth product ω G . is defined as: ω G ≈ A0ω 0 (2) AOL ( jω ) (dB) A0 (dB) 0 (dB) log(ωG ) log(ω 0 ) log(ω ) Fig.2 The magnitude response of the open-loop gain The positive finite gain amplifier. as shown in Fig. CNOA has other important . or 3dB frequency.1 T ( jω ) ≈ K × 1+ j (5) ω ω0 A0 K and the cutoff frequency. it is clear that the cutoff frequency is inversely proportional to the gain of the amplifier. is given by: ⎛ A0 ⎞ ⎟ ⎝ K ⎠ ω cutoff = ω 0 ⎜ (6) From Equation (6). The CNOA is versatile since it has three external terminals that correspond to those of the regular Op Amps. K . each stage having the gain of N K to realize an overall gain K . of this amplifier. Consequently. by 1 N K . 1 2 N −1 . cascading N stages introduces another shrink factor of In total. These resistor ratios can be used advantageously to reduce the deviation of the overall active realization and to guarantee the system stability. Therefore. resulting in (N-1) resistor ratios. the bandwidth of a single Op Amp amplifier realization shrinks by a factor of 1 K . N K Composite Op Amp (CONA) The CNOA is constructed using N Op Amps and 2(N-1) resistors. Vi ( jω ) + Vo ( jω ) R1 = R R2 = (K − 1)R Fig 3 The positive finite gain amplifier One method to increase the bandwidth is to use the N-stage amplifier. the bandwidth of each stage shrinks In addition. the bandwidth of the N-stage amplifier shrinks by a factor of 1 N 2 − 1 [1]. The open-loop gain of the Op Amps used in the modeling of the C2OAs (assuming a single-pole model) is Ai ( jω ) = Ao. are found to meet the good performance criteria. m = 1.i jω + ω 0. and wide dynamic range. C2OA-3. For Four different C2OA structures. the bandwidth of the amplifier constructed by CNOA shrinks only by a factor of N 1 K . As shown in References [2. referred as C2OA-1. Compared with the shrinkage factor of 1 K for the 1 single Op Amp amplifier. i are the DC open-loop gain and the cutoff frequency of the ith Op Amp. and ω 0. 4. Here. the general case of CNOA. the CNOA considerably extends the useful bandwidth.4 (8) where for C2OA-1 Vo1 = Va A2 ( jω )(1 + A1 ( jω ) )(1 + α ) A ( jω ) A2 ( jω )(1 + α ) − Vb 1 A1 ( jω ) + (1 + α ) A1 ( jω ) + (1 + α ) (9) for C2OA-2 Vo 2 = Va A1 ( jω ) A2 ( jω )(1 + α ) A ( jω ) A2 ( jω )(1 + α ) − Vb 1 A2 ( jω ) + (1 + α ) A2 ( jω ) + (1 + α ) (10) for C2OA-3 Vo 3 = Va A1 ( jω ) A2 ( jω ) A ( jω )(1 + A1 ( jω ) ) − Vb 2 (1 + α ) (1 + α ) (11) . and C2OA-4.3. respectively.and two-pole Op Amps model. and 2 N −1 N K for the N-stage amplifier. see References [2. low sensitivity to component and Op Amps mismatch. such as stability with one. C2OA-2.iω 0. 3] attached. and are shown in Fig.properties. we only discuss C2OA. The output voltage of the C2OAs is given by: Vom = Va Aam ( jω ) − Vb Abm ( jω ). α is the resistor ratio. composed of two Op Amps and 2 resistors.i i = 1 or 2 (7) where Ao. i . In this experiment.2. 3]. (11). (a) C2OA-1.and for C2OA-4 Vo 4 = Va a A2 ( jω )( A1 ( jω ) + α ) A ( jω )[A1 ( jω ) + (1 + α )] − Vb 2 (1 + α ) (1 + α ) a + C2OA1 b + o C2OA2 b - a a (12) o - + A1 + αR R A2 A2 - R - A1 b o + o - αR b + (a) (b) b b - C2OA3 a o - C2OA4 a + o + b b R - A1 - A2 αR - o αR - + A1 A2 o + + + a a (c) R (d) Fig 4. From this table. (c) C2OA-3. The Composite Operational Amplifiers (C2OAs). it is clear that the 3-dB . The applications of the four proposed C2OA’s in the positive and negative finite-gain amplifiers are summarized in Table 1. (b) C2OA-2. the transfer functions of the circuits using C2OA’s can be derived. (d) C2OA-4. and (12). From Equations (9). (10). Positive Finite Gain Amplifiers a. 1+α 1+ k Calculate the resistor ratio as: α = QP 1 + k − 1 = 2.1 . Each stage has a gain of 10 to realize the c. b. 2. Design a negative finite-gain amplifier employing C2OA-1. The gain of this amplifier k is That is ω1 = ω 2 (13) A1 = A2 (14) and The quality factor can be simplified as: QP = 3.707 ) with close-loop gain of . Design a positive finite gain amplifier using C2OA-1 ( Q p = 0. overall gain of 100. Assume the two Op Amps in the C2OA-1 are identical.32 4. 5. Use LM471 Op Amps. and the quality factor Qp of C2OA is 1. 10. Design a single-stage positive finite gain amplifier with an overall gain of 100 using LM471 Op Amps. K bandwidth of the finite amplifier shrinks by a factor of Design Procedure for Negative Finite-Gain Amplifiers Employing C2OA-1 1.2kΩ (17) Calculate the resistor value as: III. Design a two-stage positive finite gain amplifier. (15) (15) Choose the resistor value R in the C2OA-1 as: R = 10kΩ (16) αR = 23. Design 1. Repeat for an overall gain of 25. Negative Finite Gain Amplifiers a. Positive Finite Gain Amplifiers a. and comment on the useful bandwidth in each realization. b. Simulate the positive finite gain amplifier with finite gain of 100 using C2OA-1. and plot the magnitude response of the amplifiers. 2. magnitude response. magnitude response. Record and compare these two cutoff frequencies. From the formulas and the graphs compare the bandwidth shrinkage to the amplifier closed-loop gain. Plot Compare these cutoff frequencies of the three realizations of finite overall gain of 100. 2. Determine the cutoff frequency.707 ) with an overall gain of 100. Determine the cutoff frequency. b. Plot the Compare these cutoff frequencies of the three realizations of finite overall gain of 100. Each stage has a gain of 10 to realize the c. c. Simulate the negative finite gain amplifier with close-loop gain of 100 using C2OA-1. Repeat for an overall gain of 25. Use LM471 Op Amps. . Record and compare these two cutoff frequencies. Computer Simulations 1. the magnitude response. Relate the bandwidth shrinkage to the amplifier close-loop gain. Design a negative finite gain amplifier by using C2OA-1 ( Q p = 0. Repeat for an overall gain of 25. Design a single-stage negative finite gain amplifier with an overall gain of 100 using LM471 Op Amps. Repeat for an overall gain of 25. Negative Finite Gain Amplifiers a. Simulate the single-stage positive finite gain amplifier with overall gain of 100. b. IV. Determine the cutoff frequency. Use LM471 Op Amps. overall gain of 100. Simulate the two-stage negative finite gain amplifier with overall gain of 100.100. Design a two-stage negative finite gain amplifier. Determine the cutoff frequency. and plot the magnitude responses of the amplifier. Use LM471 Op Amps. Plot the c. and show the useful bandwidth improvement. Simulate the single-stage negative finite gain amplifier with closed-loop gain of 100. Simulate the two-stage positive finite gain amplifier with close-loop gain of 100. Plot the magnitude response. pp. No. Inc. 1972. May 1987. References [1]. “Composite Operational Amplifiers: Generation and Finite-Gain Applications. on Circuits and Systems. Vol. “Integrated Electronics. 5. and C. pp.” IEEE Trans. May 1987. Millman. Experiments This lab is a computer simulation lab.” IEEE Trans. Sherif Mickael. No. Mikhael. 461-470. 34. [3]. No actual experiment.386 [2].V. and Sherif Mickael. . on Circuits and Systems. and Wasfy B. J. “Inverting Integrator and Active Filter Applications of Composite Operational Amplifiers. 449-460. 5.” Mcgraw-Hill. Wasfy B. Mikhael. Halkias. 34. pp. Vol. Analog and Digital Circuits and Systems. Table 1 Negative and Positive Finite Gains Vo V Using the C2OA’s i C2OA-i Negative Finite Gain Positive Finite Gain The 3-dB The quality factor Qp Transfer Function Transfer Function bandwidth ωp C2OA-1 1 Ti 1+ 1+ 2 s s + 2 ω pQ p ω p C2OA-2 1 Ti 1+ C2OA-3 Ti 1+ C2OA-4 Ti 1+ s ω1 1+ (1 + α )s Ti 2 s s + ω p Q p ωw 2p C2OA s s + 2 ω pQ p ω p 2 s s + 2 ω pQ p ω p 1+ ω1 + R 2 1 Ti s s2 + 2 ω pQ p ω p 1+ 1 1+ s s + 2 ω pQ p ω p 1+ Vo ω1 ω1ω 2 1+α 1+ k 1+ k ω1ω 2 1+α 1+ k ω2 ω1 s s2 1+ + ω p Q p ω 2p Ti 2 1+ Vi Ti s Vi R αs ω1 1+ k ω1 ω2 ω1ω 2 (1 + k )(1 + α )ω1 ω1ω 2 (1 + k )ω1 (1 + α )ω 2 (1 + k )(1 + α ) ω2 (1 + k )(1 + α ) 2 s s + 2 ω pQp ω p + C2OA Vo - - kR Vo = −k = Ti Vi kR Vo = (1 + k ) = Ti Vi is the ideal transfer function Ti . Study Guide A EEL 4140 ANALOG FILTERS DESIGN Composite Operational Amplifiers: Generation and Finite-Gain Applications . MEMBER. such as wider gain-bandwidth product (GBWP). Generally. integrators. negative. MIKHAEL. the CNOA concept has proved to be useful in nonlinear and high-speed.00 01987 IEEE * . and wide dynamic range. West Virginia University. and differential finite-gain amplifiers are given and shown theoretically and experimentally to compare favorably with the state-of-the-art realizations using the same number of OA’s.its nullor representation. such as fast A/D and D/A conversion. Naval Postgraduate School. stability with one. :@p. the passive components restrict operation in a higher frequency range relative to that of the OA. W. This results in increased degreesof freedom in choosing the companion network: In the third approach. the generation of the CNOA’s 0098-4094/87/0500-0449$01. are shown to satisfy . [34]. This has resulted in many contributions. the procedure for generating the CNOA’s is presented. each OA is replaced by a composite operational amplifier (CNOA) [17]-[20] without modifying the topology of the companion network. The objective of this paper is to present a comprehensivetreatment of the CNOA method and its application in linear active signal processing. In addition. AND Abstract-A practical and effective general approach is presented for extending the useful operating frequencies and improving the performance of linear active networks realized using operational amplifiers (OA’s).fOl (b) A--t-m (a) An operational amplifier (VCVS) and (b). each dealing with the solution of this frequency limitation in specific applications [l]-[14]. and differential finite-gain amplifiers. Four of the C20A’s are found to meet useful performance criteria.1985.(s).the suggested performance criterion. FELLOW. Michael is with the Department of Electrical and Computer Engineering. each OA in a given configuration is simply replaced by an OA that has improved characteristics.IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. NO. revised August 26. namely.and two-pole OA models. In Section II. SHERIF MICHAEL. low sensitivity to the components and OA mismatch. the authors suggestedan approach using socalled composite operational amplifiers (CNOA’s) that achieved a considerable performance improvement and bandwidth extension of almost all linear active networks used in signal processingand amplification for audio and video communications as well as instrumentation [17]-[20]. and 4. 3. This is achieved by replacing each OA in the active network by a composite operational amplifier (CNOA) constructed using N OA’s.’ The technique of generating the CNOA’s for any given N is proposed. In this technique. IEEE Log Number 861?469. for N = 2. The operating frequencies are defined to be those frequencies for which the deviation of the actually obtained transfer function T’(s) of an active realization from its ideal value q(s) (due to the OA’s finite gain and frequency dependence) falls within a predetermined acceptable range. B. The frequency limitations due to the passive components are not addressed here. S. Recently. First. In this contribution. positive. and switched capacitor filtering [23]-[26]. hfikhael is with the Electrical Engineering Department. In the first approach. 1986. In the secondapproach. Monterey. these are retained for use in design.+“+ . the CNOA’s applications in inverting. (4 Fig. a general technique for the generation of a number of C20A’s (N = 2) using imllator-norator pairing [27]-[30] is described. CAS-34.These active elements have frequency-dependentgains which restrict the operating frequencies of the linear active circuits. WV 26506.To further illustrate the generality of the proposed technique. the passive configuration in which the OA’s are embedded (called the companion network) is carefully designed. three approacheswere considered to minimize the dependenceof the realization on the active Manuscript received October 29. L INEAR ACTIVE INTRODUCTION circuits. MAY 449 1987 Composite Operational Amplifiers: Generation and Finite-Gain Applications WASFY B. Several families of CNOA’s. [16]. In practice. an increasednumber of OA’s are used to realize a given 7). Applications of the CNOA in inverting integrator and active filter realizations are presented in a companion contribution (321. This has been verified by other researchers[21]. and active filters are usually realized with operational amplifiers (OA’s) as the active elements. extending the useful bandwidth (BW) of the most commonly used linear active circuits has received the attention of many researchersin this field. IEEE. [22]. noninverting. The realizations employing the CNOA are examined according to a stringent performance criterion satisfying such important properties as extended bandwidth. I. IEEE ~qfjj&!+!F~ 1)N”ll. CA 93943. 5. Morgantown. VOL. for a given fixed number of OA’s.digital communications.high-accuracyapplications. For practical reasons. device parameters and consequently its variations [15]. 1. the input impedance Z. NO. like the infinite-gain controlled source. l(a) is a voltage-controlled voltage source (VCVS). MAY 1987 . the nullor. II. (4 (H-I)R Fig. This subject is well documented in the literature [27]-[30]. In general.the C20A’s is as follows.450 IEEE TRANSACTIONS ON CIRCUITS 0) (4 7h AND SYSTEMS. (e) -H and (f) +H finite-gain amplifier realizations used in Fig. Although the nullator (or norator) alone is not an admissible element for modeling a physical network. 2. VOL. a redundant amplifier of finite gain + H . replacing each OA by a nullor results in a nullor equivalent network. In any physical circuit that contains N OA’s. Each of these N! nullor networks yields a physical realization which has a different dependence on the nonideal active elements. which uses nullator and norator singular elements [27]-[30]. Use of the proposed CNOArs in inverting and noninverting finite-gain applications is given in Section III. The nullors can further be split into nullators and norators to yield a nullator-norator equivalent network.a nullor equivalent network containing N nullors correspondsto 2” physical networks. are presented. The procedure to generate. (g) The composite operational amplifier (C20A-i) symbol.cc. 4(a)-(d). In the first step. Similarly. a nullator-norator equivalent network containing N nullators and N norators yields N! nullor equivalent networks. which meet the above performance criteria. can be used for this purpose. This corresponds to the model shown in Fig. l(b). (a)-(d) Four different networks fbr generating the composite operational amplifiers using two single OA’s (C20A’s). 5. GENERATION OF COMPOSITEOPERATIONAL AMPLIFIERS (CNOA’s) USING N SINGLE OA’s A. -It is shown theoretically and experimentally that appreciable performance improvements are realized over the present state-of-the-art designs which utilize the same number of OA’S. The equivalence established is valid whether A + cc or A + . (1) The matrix in (1) is called the nullor chain transmission matrix of an ideal OA. Generationof the C2OA’s TN = 2) An operational amplifier. since nullators and norators can be paired into nullors in an arbitrary manner. for N > 2 is described. shown in Fig. The ideal OA is replaced by a nullor which is describedby [:]=[: :I*[-q. In the ideal case. Sample results of C30A’s and C40A’s. and so in practice a nullor can be replaced by a high-gain differential controlled source in two ways. CAS-34. -+ cc. (c) C20A-3.= K’. each constructed using two single OA’s. as given in (1). one for + H and the other for .AAl A. This satisfies the necessary(but not sufficient) conditions for stability. (d) C20A-4 (Fig. 3.. are examined according to the following performance criteria. It is interesting to note that a special case of C20A-3 can be derived from the transistor Darlington pair [31].A.Aai(s)-GAbi where for C20A-1 v = 01 v Ad+ a Ad(l+ A. That is. Also. The resulting C20A’s.. 3. The improvement should be sufficient to justify the increased number of OA’s. The open-loop gain of the single OA’s used in the modeling of the C20A’s (assuminga single-pole model) is Ai=-== AoitiLi wi WLi+S i=lor2 s + WLi’ (4 where Aoi. This results in 136 composite operational amplifiers (C20A’s).MIKHAEL AND MICHAEL: COMPOSITE OPERATIONAL I//’ . i.(s) and Ah(s) should be realized through differences.e.+(l+a) a) _ v A.+(l+a) bA2+(1+a) (4 . none of the numerator or denominator coefficients of A.. This eliminates the need for single OA’s with matched GBWP’s and results in low sensitivity of the C20A with respect to its components.H. The denominator polynomial coefficients of A. where the norators are both at ac ground in the Darlington pair enabling an OA realization. and are shown in Fig. Two topologies are obtained. It can be easily shown that the open-loop input-output relationships for the C20A-1 to C20A-4 are given by (i=l. iv) The resulting input-output relationship T. (b) C20A-2. i) Let A.Fig. for each position of the three-way switch. and the GBWP of the ith single OA. the overall two-port network realizes one VCVS. 2(a)-(d).-4) V. (a) C20A-1.(s) and Ah(s) denote the noninverting and inverting open-loop gains of each of the 136 C20A’s examined. ii) The external three-terminal performance of the C20A should resemble as closely as possible that of the single OA.(s) in the applications considered should have extended frequency operation with minimum gain and phase deviation from the ideal q(s).(s) and Ah(s) should show no change in sign. C20A-3 and C20A-4. Eight possible OA realizations can be obtained from each of these 17 topologies (nullor networks). shown in Fig. C20A-2. the 3-dB bandwidth. leading to six topologies per network. 2(e) and (f)) is combined with a single OA such that the chain matrix of the resulting two-amplifier network (assuming ideal amplifiers) correspondsto that of a nullor.(l+ bAI+(l+ci) a) (3) for C20A-2 +a>_v AAl +4 Yd=v. Four C20A’s referred to as C20A-1. respectively. iii) No right-half s-plane (RHS) zeros due to the single OA pole should be allowed in the closed-loop gains of the C20A’s (for minimum phase shifts). It is easy to show that 17 of the 24 topologies realize true nullors. 2(g). wLi and wi are the dc open-loop gain. no special stipulation on network elements or signals is required. 451 AMPLIFIERS (4 i /’ I/ ’ (4 The composite operational amplifiers (C20A’s). Six possible topologies can be obtained for each of the four networks shown in Fig. of the 136 examined are found to meet these performance criteria. although each network contains two VCVS’s. 5. even for relatively large common-mode signal applications. (d) C30A-4.+(l+a)l b (1+4 (6) (7b) where QIis a resistor ratio. as illustrated in Fig. i. common-mode rejection ratio (CMRR) problems should not be encountered using C20A-2. Also. (a) C3OA-1. where the second pole occurs. As (Y increases. A.. I ’ I cl& I I Fig. i. the CMRR of the C20A-1 and C20A-3 is (A. to w. (53. both the C20A-3 and the C20A-4 have an open-loop dc gain given by and for C20A-4 4(4+d (l+a) -v A. for (l+ a) <<A. Assuming identical OA’s. the location of the secondpole.l=A.z=A. (c) C30A-3. for C20A-3 T/od=K AND A&+a> = A.. + l/2). (74 From (2) and (7b).(1 + a) Ocl= 1+ (1 -I-a)/A.452 IEEE TRANSACTIONS ON CIRCUITS a I. the composite amplifier has a single-pole rolloff from q/A.. from (5) and (6). and (6). ‘\ From (2) and (7a).e. The composite operational amplifiers (C30A’s). For singleended applications (small common-mode signal). the dc gain increases while the frequency of the secondpole decreases. VOL. Thus. 3.. (b) C30A-2. when v. while that of the C20A-4 is (A. and as (Yincreasesthe dc gain decreaseswithout affecting. 4. From (3). MAY 1987 0 C30Ab ‘\ a- ‘\ ‘\ -\ ‘. and A. = 0..e.. AoC2has double poles (12-dB/octave) at wi/A. and 01=w2=wi it is interesting to examine the open-loop gains given by (3)-(6) in the single-endedinverting application. Only the C20A-2 has identical expressionsfor the positive and negative open-loop gains A. For C20A-1 and C20A-2.. + (Y+ l/2). (f) C30A-6. (e) C30A-5./(l+ or). NO. no prob- .[A. CAS-34. the open-loop dc gain Aocl is given by A + SYSTEMS. as well as others. Generationof CNOA’s (N > 2) Following an analogous approach. CNOA’s for N > 2 can be generated for extending the operating frequencies at the expense of additional amplifiers.) output. C30A’s. are shown in Figs. C20A-3.it will be shown to yield excellent results. . which is used here. or harmonic distortion problems should arise. The CNOA’s can be obtained in two different ways. the increasedcomplexity is expected to give rise to practical problems in spite of the advantagesof an extended operating range. The first approach starts from the basic single OA with additional redundant amplifiers. (a) C40A-1. 5. can be found elsewhere[33].MIKHAEL MICHAEL: COMPOSITE OPERATIONAL 453 AMPLIFIERS Fig. or C20A-4. Consequently. Hence. The composite operational amplifiers (C40A’s). In the second approach. Similarly. This results in many possible combinations of C40A’s. 4 and 5. Thirty-two possible combinations of C30A’s can be obtained using the four proposed C20A’s. C40A’s are generatedhere by replacing each of the single OA’s in a C20A with any of the C20A’s or by replacing one of the single OA’s in C30A’s with a C20A. Samplesof C30A and C40A novel designs. nullator-norator pairing is used as described in Section II-A. Hence. as verified experimentally later. (c) C40A-3. The process can be continued (by using C20A’s. no dynamic-range reduction of V.. N should be limited in practice. lems are anticipated in using the C20A-1.of the C20A’s in Fig.. Then. the C20A’s are used as single OA replacementsin the C20A structure. 3. The open-loop expressionsof these C30A’s and C40A’s. (b) C40A-2.which meet the performance criteria described in Section II-A. and C40A’s) to obtain CNOA’s for any number N. It is easy to show that the voltage swing at the first OA(A. the dynamic range is determined by the voltage swing of the output voltage V. is always less than the output voltage V. (4 C40A-4 (e) C40A-5. C30A’s are obtained by starting with one of the proposed C20A’s and replacing one of its single OA’s by any . B. which is an internal node in each of C20A-1 to C20A-4. Although this second approach is not exhaustive. Hence. w2. N/D indicates the amplitude and phasedeviation of T.) + (S%.” v. CAS-34. from q. and consequently oZ. wP. has the form T. + cc) in somecases) *z D=1+b.c= vi f--x-& (W ‘fl+k 1 + (S/wpQp) + (S%..l+k 1 + WwpQp) + (S'/$) F l+k 1 T.). + VI (9) (a is zero ( w. o. 6. For the differential finite-gain realization.3) AND POSITIVE Function I GAINS FINITE Positive Finite 1 C20A-1 Ti + (S%. which is necessaryfor the stability of the transfer function.) ji$$ 1 + (VwPQP) + (S'/$) vow igz *va kR kR !!. Application N=l+a~=l+~ (1+ Lx).k=Ti (ltk) = Ti Ti(idea1 Transfer Function) *aR1 e kR (for maximum up).. the actual input-output relationship T. REALIZATION OF POSITIVE. of the C20A-2 as a differential fier. 1 + (S/opQp) * Ti + (S*/$) Ti + (S*/w. The network in Fig.s+b2s2=1+(s/WpQp)+(s2/~. From Table I.On the other hand..) w2 i. 1 + (ShPQP) C20A-2 V. and Q. determine the stability of T. can be shown to have the input-output relationship given by 1 where q1 = x(1 + k)/(l+ q2=-k Qp= 020+ (8) x) k) For the differential gain application given above and the finite-gain applications in Table III. finite-gain ampli- where q = the transfer function realized assuming ideal OA’s and 1 1 + s/wPQP + s ‘/o. Also. the b coefficients are always positive (assuming a single-pole . C20A-2 is used. DIFFERENTIAL FINITE-GAIN LR AND AMPLIFIERS V2 A. (1.IEEE TRANSACTIONS 454 ON CIRCUITS AND SYSTEMS. NEGATIVE.and QP.=q. NO.) I. o. and (Y. a mismatch of k 5 percent in w1 and w2 results in a +Spercent change in wP and a *2..OA model). MAY 1987 TABLE NEGATIVE CZOA-i Negative Finite Gain Trans..) Function wP QP WlW2 1 + (S/upQ. single OA’s with mismatched gain-bandwidth products within practical ranges can be used without appreciably affecting the stability or the sensitivity of the finite-gain realizations. are functions of the circuit parameters or. III./ F= 1 .wy WI yyaz (l+Ww1) Ti 1 + (S/wPQP) + (S%. Thus.WI (14 -f-K l. which guaranteesthe low sensitivity of T. while a.. b. and b. when each single OA is modeled by (2). often referred to as the instrumentation amplifier shown in Fig 6.T 1 (l+S/u) Ti 1 + (ShPQP) C20A-4 Gain Trans.‘a.5-percent change in QP. 5..P to the circuit parameters. Finite Gain Amplifiers Using the ProposedC2OA’s The application of the four proposed C20A’s in positive and negative finite-gain amplification is given in Table I. 6. (Ta) THE C20A’s (1Ww) 1 CZOA-3 v] USING T. and b. VOL.None of the a and b coefficients is realized through differences. b. T2 Fig.) Cl+(l+~wwl Ti w1w. w1 J 1 1 + s/oPQP + s ‘.. . I v. open-loop model of the single OA’s is studied. [5].1 and the existing two-OA realizations (assuming OA GBWP = 1 MHz). -70 the stability properties of the positive and negative finite-80 gain amplifier realizations using a two-pole. respectively) are asFi 7. (Y realize an overall gain k. In practice.It can be easily where 114 is given by shown that the differential finite-gain amplifier in Fig. [6].707 a:4 . from 7. see Table II.or C20A-2 [33] is found to be amplifiers using C20A’s have been reported in the litera0+9<(1+W (11) ture [l]-[3]. the C20A’s require only two accurate gain- .to its unity gain 3-dB BW (1+X) >4(1$k).“Table II C20A-2 circuit BW’s can be designedto shrink by only a gives the values of (Y required to yield QP= l/a and factor of =l/fi for QP= 0. Also. and Table I I.455 MIKHAEL AND MICHAEL: COMPOSITE OPERATIONAL AMPLIFIERS TABLE II VALUESOF~ FORMAXIMALLYFLATANDFOR Q. [8] and Qted for their improved performance.It the necessary‘and sufficient condition for stability using should be noted that some -special cases of finite-gain the C20A-i’ .f= (1+. 6 . 40 50 60 70 80 90 100 110 120 130 140 I50 (4 I Unsatisfied 9 0 m C20A-4 -10 1 2(l+k) 9 -zI IO 20 ( Proposed design I I Unsatisfied -20 &-(ltk) I -30 I I -40 -50 1) Effect of the Single OA’s SecondPole on the Stability -60 of the C2OA’s Finite-Gain Realizations: In the following.The resulting BW shrinks by should be chosen in the stable range for a given . both amplitude and phasedeviations of T. gain applications from SW aid stability considerations. In conclusi?n. Let comparing the up’s in Table II. The usefui BW’s of greater than l/G for QP= 1 (k’> 1). The C20A-1 and results in” the best realizable value of QP and ‘o.707 (maximally flat) and Q. .of the conclusions. = 1 for the realizations in Table I. This is Phase response of negative finite-gain amplifiers. 03) (wi). . one finds that e cascade of two (single-OA realization) finite-gain ampliimposed hy the stability conditions (not necessarily BW fiers is obtained when each-am+fier has a gain fi to conditions): is physically realizable for all k. & ( I I (l+k) ! Unsatisfied I 30 30 for op= . In the different finite-gain amplifiers can be obtained by addition... Q.k’ that m/G = 0. (b) without affeCting the reliability.)( --&+k).=A... As oP increasesfor a fixed A=A. due to the absenceof gain difference terms in all the gain expressions obtained. -f kh. . and C2OA-2 are the most attractive configurationsin finiteBy invoking the Routh-Hurwitz stability criterion [16].it is clear that the C2OA-I and wh >> wL. r . for the C20A-4 the condition [33] is given by multiplying factor lik relative. at a given frequency o (w < wP) decrease. as seenfrom (3)-(6) (8). (a) Frequency response of negative finite-gain amplifiers. (14 Reizlizationswith Others: The SW of a finite-gain amplifier realized using a single OA shrinks approximately by a Finally. (12). while for the C20A-3 the condition [33] is found to be 2) Comparisons of the Proposed C2OA’s Finite-Gain (l&)>/m. .66/G relative’to ‘wl’[31]. If the dc gains of the (b) (b) first and second OA’s (A. =l C20A’s ANDTHE FINITE-GAINREALIZATIO~S~SING CORRESPONDINGBANDWIDTHAND STABILITYCONDITIONS dBb 1 CZOA-i 1% c2oA-1 PP 1 m Stability Condition for (I used -OF! Satisfied -2 fm Satisfied (ind$ment C20A-I 32 - 1 C20A-3 ( I I 0 1 OPmin = m(. (10) has similar excellent bandwith and stability properties as those obtained for C2OA-2 in’pbsitive and negative finite1 gain applications. and (13). Theoretical responses of the negative finite-gain amplifiers using sumed to be equal. the optimum n@paily flat 3-dB BW using a Upon examining (ll). and A. . this greatly simplifies the analysis 8 20A . 100 (LM747 OP AMPS).f 20 a= -10 20 40 80 60 100 30 40 50 120 140 160 180 200 220 c .25.VOL. &MAY 456 20 LOG Vo/Vi k I +2 I dB 0 = 0. and differential composite amplifiers can be designed using the CNOA structures proposed in Section II-B and shown in Figs. its low sensitivity to circuit elements and power supply variations.707 5 % 5 % --_ -6- -6 I . (a) (Qp = 0.707 I d0 100 k a = 100 =6 I -4- 10 20 30 40 50 60 70 80 so 100 (4 (4 Fig. and versatility. stability. 7 show clearly the excellent gain and phase performance of the proposed realizations. NO. 4 and 5. negative finite-gain amplification. 8 with those obtainable using single-amplifier realizations illustrates the considerable’improvement in the useful BW without sacrificing any of the single-OA desirable features. CAS-34. 8(c) and (d). so that extending the range of operating frequenciesbecomesmore important. 3) Experimental Results Using C2OA’s in Finite-Gain Applications: Experimental results of negative finite-gain amplifier realizations are given in Fig. (d) Effect of power supply variation from T 9 V to T 15 V on the closed-loop gain for k = 100 (LM 747 OP AMPS). (b) Effect of compensation resistor-ratio variation by T 5 percent (LM747 OP AMPS). determining components. negative. . The stability and low sensitivity to the power supply and to the active compensation resistor variations are examined as shown in Fig. Finite-Gain Applications Using C3OA’s and C4OA’s Wide-band positive. 7 with some of the most recently published negative finite-gain realizations which utilize a similar number of OA’s.707) Maximally flat closed-loop gain = . The performance of the C20A-1 in this application is illustrated and compared in Fig. namely. 8. . . LM747’s with a GBWP ranging from 1 to 1. 3(a). In the finite-gain expressionsof these C30A’s and C40A’s. (c) Effect of active compensation on extending the bandwidth (LM747 OP AMPS). In fact. the realizations of [2] and [8]. in contrast with the proposed ones.5 MHz were used to implement the C2OA’s in this section as well as in the experiments throughout this work. can be easily shown in theory to be unstable for all useful values of closed-loop gains. To further illustrate the usefulnessof the C20A’s. This has been verified experimentally as well. one may erroneously conclude that.50. one of the common applications considered in this paper is chosen. Indeed. 7 are for nominal gains > 1 for practical reasonssince an increase in k decreasesthe useful bandwidth. compared with four in the cascaderealization. they have better phase response. Positive and negative finite-gain expressionsfor C30A-1 through C30A-6 are given in Table III. due to the second Ok pole. namely. Upon examining those of [2] and [8]. the results in Fig. Comparing the results in Fig.1987 IEEE TRANSACTIONS ON CIRCUITS AND SY~TEMS. Experimental results using C2OA-1 in negative gain applications. in spite of their inferior amplitude characteristics. The proposed realizations are seen to be far superior in both amplitude and phaseresponsesrelative to those reported in [7] and [9]. The results shown in Fig. 8(a) and (b) using the C20A-1 of Fig. B.f LOG 70 80 90 100 II0 120 130 kHz IhO kHr (4 20 60 6 (b) Vo/Vin k I dB I 20LOG I k’ b/Vi k Qp = 0. while those for C40A-1 through C40A-4 can be found elsewhere[33]. and y while still satisfying the stability conditions.. each with gain k113(k114)to realize an overall gain k.p. 52 L & (l+k)>a(lw) ""i V . The BW of the new proposed C30A (C40A) circuits are found to shrink by only a factor 1/k’/3(1/k”4) (k B 1). 9 and 10 for gains of 100. From Figs.) . it is seenthat the 3-dB BW available from these C30A (C40A) finitegain amplifiers implemented using l-MHz single OA’s corresponds to the BW attainable from a single OA with zero-dB BW in excessof 25 (35) MHz! The performance of the C30A-1 is compared with the performance of recently published three-OA realizations.+ CL+ w* l+k ol(l+B) 1 s‘+ f&g+ (l+kJ$g (l'tk)> 1+ [+ + s 1 S + #$ & + (I+@& 9 Negative Finite Gain Trans. For practical reasons. ++ ($1 St (l+k) ?- 4 w2w3 Neqative Finite Gain Trans. J W+&- I -K C30A-4 l+k +(K)w a(l+B) Positive Finite Gain Trans.7. Maximally flat -response (Butterworth) as well as Chebyshev characteristics. with positive finite gain of 38./ y USING THE C30A’s Function -k(l+ kc "i AND i3. 9 and 10. thus. Also. both ZSD amplifiers are designed such that the stability condition. Fig. which is necessaryfor stability. Func. It is interesting to note that the condition for stability of the amplifier shown in Table III using the C30A-1 is l+k a< 1+p where k = 38.7. -K V z 3 l+(-$)$+(&&+(l+U& C30A-3 (l+k)>(l+a)(l+B) L V o= "i (l+K)( l+(s)+ 1 + & g+ + ( -$ & ) L )$q+ Positive Finite / Gain Trans./3.w:. (rl . using CNOA’s.MIKHAEL AND MICHAEL: COMPOSITE OPERATIONAL 457 AMPLIFIERS TABLE III NEGATIVE C30A-i Finite Gain Transfer C30A-1 1 t (1%) POSITWE FINITE GAINS V. Applying the same technique used in Sections II and III-A.-) $+ Negative Finite Gain Trans.o _ q C30A-2 !!L "i (ltk)(l+ l+(l+# ' T+a TlFi w. it can be shown that the resistor ratios (Y. I. all the denominator coefficients are positive. for different compensation values. proposed in [lo]. The best theoretical results using the ZSD are obtainable with the minimum stability margin to allow for maximum bandwidth. Func.K (W. (l+K)(l+S/w2) AZ= "i Positive Finite Gain Trans.7 is chosen to permit direct comparison with the theoretical results previously published in [lo].e. The overall BW shrinks by a multiplying factor 0. .435/k”4) relative to wi [31]. Func. Func 5 SL + (l+k+-19 w1w2w3 1 (l+k)>(l+a)(l+6) Positive Finite Gain Trans. low coefficient sensitivities are obtained and reasonable OA mismatch is tolerated. 11(a) and (b) shows the theoretical magnitude and phase characteristics of the ZSD amplifiers. h Negative Finite Gain Configuration no terms containing differences are encountered. The figure depicts the extended frequency range of operation attainable with these C30A and C40A designs over the ZSD realization [lo]. . as well as the C30A-1 and C40A-1 amplifiers (which satisfy the stability constraints). can be achieved by controlling the resistor ratios OL. or a realization which uses three (four) cascaded single-OA finite-gain stages. and i can be chosen to extend the BW and to’ satisfy the necessary stability conditions assumingsingle-poleOA models.51/k’/3(0. Computer plots of the C30A-1 and C40A-1 transfer functions in the positive and negative finite-gain configurations are given in Figs. called the zero second derivative (ZSD) amplifiers.1) = 1. i. (E) &+ (l+k) s-+-. is exceeded by a margin of 10 percent. Func. Func. Func. using Routh’s test on the third-order denominator coefficients./w. A positive finite gain of 38. '1 Neaative Finite1 Gain Trans. Func. 1) Comparisons of the Proposed C3OA’s and C4OA’s Finite-Gain Realizations with Others: The optimum maximally flat 3-dB BW using three (four) single-CiAfinite-gain building blocks is obtained by cascadingthree (four) identical blocks.1 pk. for both the maximally flat response and the Chebyshev. simple resistor ratios can be used advantaeeouslvto reduce the deviatidh of the .8 p= I2 k= 100 .C30A-1 responses in Fig.99 :l43.26 26.14i. CONCLUSIONS A new approach is presented for extend& the useful operating frequency range in a wide variety *oflinear active networks which utilize OA’s.6 40. 10 closely agreewith the experimental results of Fig. 10.71.458 IEEE‘TRANSACTIONS ON CIRCUITS Ah-SYSTEMS. Several of the CNOA’s. to C20A-4. = 1 MHz). CAS-34.GBWP = 1 MHz). NO.8 i0. This is satisfied by a wide .8 = 2. In these CNOA’s. namely the C2OA:J. 9.8 k 34.6 31. 13.43 ?I5 . which makes them suitable in differential gain applications.71 -215. A systematic procedureis given for the generation of the CNOA’s. IV. Computer plots (magnitude and phase) of the. Exhaustive test results are documented in [33]. (b) Chebyshev computed frequency response (amphtude and phase) of the negative finite-gain amplifier (k = 100) using C40A-1 (single OA GBWP = 1 MHz).8 0.0 i3. and the GBWP mismatch effect of single OA’s. C40A-3.:CMRR.54 (I i .mancecriterion that considers stability. The computer frequency responseplots of the C40A-1 negative finite-gain realization in Fig. C40A-2.16 .C40A-1 (single C)A.o 25. The CNOA is versatile since it has three external terminals that correspond to those of a single OA. gain 100.0 32. The extended SW is.0 28.6 .35 34..9 k = 100 -71. CNOA’s%aregeneratedand examined according to a stringent perfor. Each CNOA is constructed using. 12 gives the experimental results using the C30A-1 in positive finite-gain applications of 38. The stability and low sensitivity to power supply as well as to the active compensation resistor variations were verified [33].6 23.C40A-1 transfer function for.margin in.81 a= 1.1) active compensating low-spread and lowaccuracy resistors. and C40A-5) have identical N/D multiplying factors in the positive and negative gain applications. it is interesting to note that some of the finite-gain designs presented here (C30A-2. (a) ‘Maximally flat computed frequency response (amplitude and phase) of the negative finite-gain .achieved by replacing each of the single OA’s in theactive realization by a composite OA (CNOA). sensitivity. 11(a) and (b).aihplifier (k = 100) using . .11 (3 = Il. C30A-4.50 0 60 160 240 320 4 0 (CHEBYCAEV) @I :. C30A-1 to C30A-6.29 - 27 0 60 160 240 ( MAXIMALLY FLAT 320 400 2 P = 2 v = 3. The application of the CNOA’s in finite-gain amplifiers is also given. The suggestedgeneration method gives rise to a large number of QJOA’s for a given N.. the . and 4. (a) Computed frequency response (amplitude and phase) of the positive finite-gain amplifier (k =lOO) using C30A-1 (single OA GBWP = 1 MHz).71. stability.C4OA’s in Finite-Gain Applications: Only sample experimental results using the C30A-1 and C40A-1 are given to illustrate the performance. resulting in (N . Computer plots (magnitude and phase) of the C30A-1 transfer function for gain 100.8 143. and low-sensitivity properties as the C20A designs in Section III-A.27 9’ 71. VOL. BW. Also.response. N-single OA’s and 2(N . 32. MAY 1987 .21 - i = 3. Fig. (b) Computed frequency response (amphtude and phase) of the positive finite-gain amplifier (k =lOO) using C30A-1 a-2 p=3 = 100 (single OA GBWP (CHEBYCHEV) (b) Fig.73 - 30. Fig. For N = 2. and C40A-1 to C40A-5.meet the performance criterion and have been found to be very useful in practice.1) resistor ratios.7. dB 37. 5.9 k = 100 .All of: these finitegain designs have the same attractive dynamic-range.09 aI0 dB 3i. 3. dynar$ range.20 -1 a 36. 2) Experimental Results Using C3OA’s and .85 kHz 200 f kHr 300 MAXIMALLY FLAT (4 1 (a) 143. represent ideal nullors independent of the absolute values of the compensating resistors. employing elements other than resistors for active compensation and using different types of OA’s in the same CNOA (e.7).4. the chip area and the power consumption of a CNOA are much less than N times those of a single OA.4. /3 =10m8 (stable with min. a high-accuracy OA for the input OA and a high-speed one for the output OA) are promising and challenging topics for further research and are presently under investigation. In addition. p = 6. 0: [lo] r =l. and [lo] for positive finite-gain applications (gain k = 38.e.6. and [lo] in positive finite-gain applications. overall active realization’s response from the ideal while guaranteeing stability.7)1/3. of the overall active realization. 11.g.l.5. This is the approach taken here for simplicity while still yielding excellent results. 100 e 200 - -4 - -6 - -3 dB __-------. and the effect of variation of the compensating resistor ratios a and j3 (LM747 OP AMPS). 30 40 d0 -2 I (experimental) 50 IO 20 200 4 - 400 (4 100 d0 +2 vi k I 100 150 200 250 300 350 400 450 f kHz e 500 The effect of active compensation on extending the bandwidth of the C40A-1 (using LM747 OP AMPS). Although the examplesgiven which use CNOA’s are for high-gain applications. 13. C40A-5. Moreover.-----. they can be made negligible by the proper choice of the impedance level of the appropriate compensatingresistors. It is to be noted that the CNOA’s. Theoretical results for C30A-5. @ : C30A-5 OL=i. p = 2. when implemented using ideal OA’s. (a) Theoretical amplitude responses of C30A-5.7. 0 : [lo] r = 2. each of gain (38. (b) Theoretrcal phase responses of C30A-5. L 33. @ : cascade of three single-OA finite-gain stages. C40A-5. Also. This is because in a CNOA there is one output OA only that drives an external load and that may be required to have power handling capability. Comparisons with the the state-of-the-art realizations using similar numbers of OA’s in these applications show the appreciable improvement of the realizations obtained -_- -IO.7.2 (unstable): @ : C30A-5 (Y=1. In certain configurations using CNOA’s implemented using frequency-dependent OA’s. B = 0. Novel composite dependent sources with considerable performance improvements are expectedto result when the procedure described is applied to the other dependentsources. using the proposed CNOA’s with respect to stability and useful BW. y = 12.MIKHAEL AND MICHAEL: COMPOSITE OPERATIONAL 459 AMPLIFIERS -vo/ 20 Log f kHz 200 300 50 60 70 00 90 f ‘KHz m 300 400 (b) Fig. p = 5. in an integrated implementation. Finite-gain applications utilizing the proposed CNOA’s are shown both theoretically and experimentally to be stable and to exhibit wide dynamic range and low sensitivity. Other researchersmay find the impedance level as an added degree of freedom that may be used advantageously. f kHz 300 Fig. @ : C40A-5 OL= 0. margin). . each of gain (38. 0 : c ascade of four single-OA finite-gain stages. 27. for a gain of 38. 12. Fig.i.9. and [lo] for positive finite-gain applications. it is applicable to any of the four types of dependentsources:voltage (and current) controlled voltage (or current) sources. it is worthwhile to mention that the method used to generate the CNOA’s is actually a composite dependent source generation technique. it is easyto show that the deviation in amplitude and phasefrom the ideal is much lower than other existing realizations.. even for closed-loop gains as low as unity. Experimental results of C30A-1 in positive finite-gain application.. another attractive feature of the proposed technique is that.4. the ratios of one or more of the compensating resistors to other resistors outside the CNOA appear in the high-order parasitic terms of T.7)‘i4. C40A-5. Although such terms are small.7. Dr. Brackett. WI CLC103. pp.D. Rao. Antomou. M. Bhattacharyya. 395-405. R. norators and nullors in active-network theory. Egypt. C. parallel transconductance implementation HA2. “High-speed high-accuracy integrated operational amplifiers. 1979. “Generation of actively compensated operational amplifiers and their use in extending the operating frequencies of linear active networks.” J. IEEE Int. L.” IEEE Proc. WV. CAS-26. He served as a First Lieutenant in an Engineering Corps specializing in water well drilling. West Virginia University.VOL. “A systematic general approach for the generation of composite OA’s with some useful applications in linear active networks. M. D.. Michael and W. Alpha Pi Mu. B. A. [61 A. 1311 M. 963-965. Michael and W. where he conducted. CAS-28. 141 B. Mikhael. OR: Matrix. vol. degree in electrical engineering from West Virginia University. June 1979. [31 A. pp. CAS-23. 749-757. degree (honors) in electronics and communications from Assiut University. Englewood Cliffs. Circuits Syst. Fleming. Quebec. 1984. vol. vol. 25th Midwest Symp. S. 1981. He received the BSc. Laker. and K. 1980. in 1982 and 1983. 1982. CA). radiation hardening. rIc ’ Wasfy B.. Circuits Syst. and R. Canada. of Electrical Engineering. 414-415. 6. vol. Michael is a member of Eta Kappa Nu. Jan. as a Field Engineer. “The significance of nullators. u71 W. Circuits Syst. Calgary. Michael.” IEEE Trans. Geiger and A. Soliman and M. vol. (Morgantown. no.. 299-300. June 1978. . B. 1241 M. on November 3. Feb. 454-463. Electronic Devices and Circuits: Discrete and Integrated. Montreal. pp.I. M.” IEEE Proc. Sankar. 4 ompensation of some operational-amplifier based RC-active networks. pp. His research interests are active networks. Eindhoven. L. Cairo. 67..Sc. Dec. pp. Eng.1944. WI M. Aug.” IEEE Trans. 1174-1177. Circuits Syst. 1251 S. and R. Monterey. vol.D. Japan). he joined the faculty of West Virginia University. FL. 1981. CAS-30. “Classification and generation of active compensated non-inverting VCVS building blocks. Circuits Syst. [331 S. Nov. he taught in the Computer Science Department at Sir George Williams University. 1271 A. Michael worked as a research and teaching fellow at West Virginia University.E. Mikhael. Circuits Syst.. vol. Circuits Syst. “A simple inverting-noninverting IEEE Proc. 34. he was an Engineer with the Telecommunications Organization. 1984. Egypt. W. Frank/in Inrt. “On nullators and norators. J. Oct. “Equivalent N. Circuit Theory. Rinehart and Winston. 320-341. From 1970 to 1973.” in IEEE Znt. amp. 116. Ravishankar. pp.” in Proc. 690-691. 1341 T. He has several patents and publications in the area of communication networks and active filters. 44-47. CO. “A PI high-quality double-integrator building-block for active-ladder filters. F. Mar. 290-294. Michael. vol. Reddy. pp. Canada. pp. respectively. Reddy. Schaumann. P81 A. Nov. Ontario.” IEEE Proc. pp. vol. R. Budak.. “Active filters with zero amplifier sensitivity. May 1983. UOI R. WI W. “Application of composite op-amps in nonlinear circuits. West Virginia University. pp.” IEEE Trans. July 1983. He received the MSc. pp. 191 A. “Active compensation of op-amps. “Precision high speed op. pp. Holland. Filter Theory and Design: Active and Passive. Apr. 1838-1850. Natarajan and B. Ismail. Ramamurthy. “An active-compensated double-integrator filter without matched operational amplifiers. June 1984.” in 27th Midwesf Symp. u41 R. no. CT-13. Morgantown. Circuit Theory.. M. From 1965 to 1968. B. 112-117. degree in industrial engineering and the Ph. K. June 1981. He received technical training at Philips Industries.” Harris Semiconductor. 1.N0. he has been a Member of the Scientific Staff at Bell-Northern Research. pp. 305. Dawson College. Soliman. Nov. WV. Mi ki ael and S. he worked on designing and implementing a new digital communication system for the Morgantown Personal Transit System (MPRT). Budak. Mikhael (S’70-h4’73-SM’83-F’86) was born in Manfalout. WI A. Since May of 1973. B. Stephenson. Wullink. (Kyoto. Ghausi. In August 1978. and adaptive signal processing. Antoniou.” in 27th Midwest Symp. Soliman. Ramamurthy. Egypt. 1970. (Morgantown. Symp.. 1979. Sherif Michael (S’78-M’83) was born in Alexandria.” IEEE Trans. P51 A. 1211 R. Apr. Ismail. WV). R. Canada.” Ph.” Znt. 534-538.. “Realization of gyrators using operational amplifiers. “Design of active filters independent of first. 1965. Circuits Syst. CAS-26. 171 S.MAY 1987 1291 J. As a Research Engineer with the National Transportation Research Center. M. 792-795. 5. (Albuquerque. pp. 1980. and 1973. B. Mikhael and S. 3.. 1983. 1980. B. he has been an Assistant Professor with the Department of Electrical and Computer Engineering at the Naval Postgraduate School. vol. degree from the University of Calgary. the M. Beaverton. CAS-27. B. J. “Instrumentation amplifiers with improved bandwidth. Myers. 277-288. “Composite operational amplifiers and their applications in active networks. Morgantown. B. Melbourne.” in IEEE Inf. WI G. CT-12. 1981. Dr. NM). “Active filter design for high frequency operation. Braun. “High frequency filtering and inductance simulation using new composite generalized immittance converters. Aug.. Dr. 461-470. vol. [51 K. He served as Chairman of the 1984 Midwest Symposium on Circuits and Systems. (Houghton. Mikhael. P91 W. this issue. G. Nandi and A. Since 1983. Circuit Theory and Ap lications. 1979. P21 W. 67. and the D. vol. Geiger and A. Canada.” 1131 A. B. R. 1231 T. networks with nullators and norators. Symp. and Tau Beta Pi and is a registered Professional Engineer. pp.Eng.” in Midwest Symp. June 1985. Michael and W.E. Rao. pp. Apr. pp. 1985. 68. Mikhael. Assiut. pp. Egypt. NJ: Prentice-Hall. S.C. Ottawa.. New York: Holt. 1981. Tellegen. switched-capacitor circuits.” Comlinear Corp. 573-576. pp. (Newport Beach. where he is now a Professor.” Harris Application Note #538.. His present research interests are in the area of analog integrated circuits and active networks design. L. as well as an adjunct Associate Professor in Electrical Engineering at Sir George Williams University (now known as Concordia University). “Inverting integrators and active filter applications of composite operational amplifiers. pp. vol. 466-469.” pp. CAS-28. Circuits Syst. Wilson. 1983. Budak. vol. Circuits Syst. Magazine.” IEEE Trans. Circuit Theory. “Active filters with zero transfer function sensitivity with respect to the time constant of operational amplifiers. 1978. Bandyopadhyay. Modern Filter Design Active-RC and Switched Ca acitor. and A. CAS-31. WI S. 420-423. “Monolithic sample/hold combines speed and precision. Quebec. “Design and some applications of extended bandwidth finite gain amplifiers. A.” IEEE Proc. CAS-34. pp.” IEEE Circuits Syst. PI R. B. voltage amplifier.548. Geiger. Zarabadi. Michael. 1980. in 1965. 68. WV).” IEEE Trans. CA. 1966. Davies. Nedungadi. Michael. Soliman. 849-854.” IEEE Trans. “Two-amplifier active-RC biquads with minimized dependence on op-amp parameters..” Radio Electron. vol. respectively. Ravichandran. Mikhael is presently Associate Editor of the Letters Section of the IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS. June 1984.. and their use in RC-active-network synthesis. 1973.” IEEE Trans. 1969. dissertation. Mikhael and S.” IEEE Trans. 443-446. 1290-1293. pp. pp. the Outstanding Researcher Award from the College of Engineering. Ghausi and K. “Design of RC-active networks by using generalized immittance converters. Circuits Syst..460 IEEE TRANSACTIONS REFERENCES PI A. degree from Sir George Williams University. pp.and second-order operational amplifier time constant effects. Apr. vol. He received the B. 259-267. 1967. Alberta. electronic measurements and supervised drilling operations in cooperation with Schlumberger Co. Montreal. Bhattacharyya. ON CIRCUITS AND SYSTEMS.. in 1974. July 1976.. ~321 S. Groom. pp. 7-9. Dr. vol. Circuits Sysr. Sedra and P. pp. and in 1973 he taught in the Mathematics Department. and G.Sc. Symp.” in Proc. Egypt. Circuits Syst. Apr. in 1980’and 1983. M. and the Halliburton Best Researcher Award in 1984. vol. Mikhael was the recipient of the Bell Northern Research Outstanding Contribution Patent Award in 1978. “A eneralized active compensated noninverting VCVS with reduced pfl ase error and wide bandwidth. Cairo. “A high-input impedance inverting/noninverting active gain block.” IEEE Trans. solar cells and space power applications. Morgantown. B. 8. MI). “Composite amplifier structures for use in active RC biquads. R. 411-412. degree in electrical engineering (electronics and communications) from Cairo University. “Fast settling wideband operational amplifiers. 797-803. S. Loveland. [301 B. Circuits Syst. vol. Study Guide B EEL 4140 ANALOG FILTERS DESIGN Inverting Integrator and Active Filter Applications of Composite Operational Amplifiers . low sensitivity to the compensating elements. The extension in bandwidth (BW) is achieved by replacing each of the single OA’s in the active realization by a suitable composite OA (CNOA) that has been constructed using N OA’s. the theory for extending the active filters’ operating frequencies in the second category is presented. CA. In Section III-C. WASFY B. In Section II. The use of the CNOA’s to realiie inverting integrators and active filters is presented here. IEEE Log Number 8613470. In [l]. An application is given in Section III-B by employing the C20A’s in the two-iniegrator-loop filters as an example. The application of the theory. FELLOW.especially if the active networks are designed to operate at high frequencies and/or with high Q ‘s. a great deal of attention has been directed toward designing high-performance integrators and active RC filters. namely. It is well known that the poles and zeros actually realized are displaced from their nominal positions because of the frequency-dependent characteristics of the operational amplifier gains. MEMBER. Active filters in the first category are shown. Also. 1986. Their comparison with state-of-the-art designs is also given. Michael is with the Department of Electrical and Computer Engineering. that the members of these new C20A and C30A families extend the operating frequency range of inverting integrator realizations considerably beyond the presently available state-of-the-art designs. it is shown. B. while the second category consists of filters in which the OA’s are embedded in the passivenetwork.461 IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS. S. Severalcontributions using a variety of techniques have been reported to extend the useful. revised August 26. The extension in BW is achieved while maintaining low sensitivity to passive and active elements. wide dynamic range. theoretically in Section III-A and experimentally in Section III-B. IEEE In this paper. is given in Section III-D. West Virginia University. A general procedure is described where N OA’s are combined to form a new active device that resembles externally an OA and is referred to as a composite operational amplifier CNOA.as is demonstratedby experiment later in Section III. AND Abstract -A new approach for extending the useful operating frequency range of linear active networks realized using operational amplifiers (OA’s) has been reported [l]. ref. both theoretically and by computer simulations. I Manuscript received October 29. WV 26508. The considerable performance improvement of these realizations is demonstrated both theoretically and experimentally. [21]] successfully demon- 009%4094/87/0500-0461$01. comparison with one of the state-of-the-art designsis given which shows considerableimprovements in bandwidth and stability of the proposed technique. the active filters are considered to belong to two categories. 1985. and stable operation.and signal handling capacity. CAS-34. so that functional building blocks cannot be identified from the filter structure. Here. The technique generatesa very large number of CNOA’s for a given N. where a suitable C30A is used in one of the well-known multiple-feedback (MFB) structures. Naval Postgraduate School. operating frequency range of thesenetworks and to reduce their sensitivities with respect to the active elements. the use of the CNOA families in the realization of inverting integrators and active filters is investigated. to provide significant improvement at high operating frequencies when the improved inverting integrators proposed here and finite-gain amplifiers proposed in [l] are used. I. MAY 1987 Inverting Integrator and Active Filter Applications of Composite Operational Amplifiers SHERIF MICHAEL. This results in both phase and magnitude errors in the response. the authors propose a technique for extending the operating frequency range of linear active networks by using composite OA’s. the operational amplifiers (OA’s) [3]-[24]. supported by experiments. Mikhael is with the Department of Electrical Engineering. The resulting filter maintains the practical useful features such as stability. the proposed technique results in practically useful designs.00 01987 IEEE . It is worthwhile to note that Schaumann [l. IEEE. MIKHAEL. INTRODUCTION N RECENT YEARS.The first category consists of filters which are realized with functional building blocks (finite-gain amplifiers and integrators). 5. Appreciable improvements in the performance of different types of modern active filters using the CNOA’s are shown in Section III. NO. Morgantown. W. The effect of using suitable CNOA’s as one-to-onereplacements of the single OA’s in finite-gain realizations is found to result in an extended operating frequency range relative to that of existing realizations that use a similar number of OA’S. Monterey. In addition. VOL. UJ. 5.MAY1987 462 TABLE I INVERTINGINTEGRATORSV.IEEE TRANSACTIONSON CIRCUITSAND SYST!2MS. c 1 l+S/wpQp+wwpZ & Transfer I* Function (T. '1 + S/wpQp Function Ti 1 1 + (. _’ Fig./ I: USINGTHE C20A’s Actual CZOA-i CZOA-1 Negative Integrator 1 T.CAS-34. . constant = RC = 11 wt ) "i I i i/ .VOL. YW2 -lx + SLl~p~l v Ideal (Where Transfer Tt is the integrator = F= i time $ = &.+a) I-- s/w..N0.) 'W. I’ The composite operational amplifiers C20A’s [l]. 1.(l+al C20A-2 CZOA-3 Ti * [ CZOA-4 Ti .(l+a) l+T. . using the C2OA-1 to C20A-4 in inverting integrator applications. which x guaranteesthe stability of the transfer function.. (jp. Also. This results in stable realizations down to 0-dB closed-loop gain. as wh increases. all the integrators become stable.. l/Ai is given by where oL..2). In C20A-1 and C20A-2 integrators. APPLICATIONOFTHEPROPOSEDCNOA'SIN INVERTING INTEGRATORS A. 02. N/D determines the amplitude and phase deviation of T. the b coefficients are always positive (assuming a single-pole OA model). and (5) that. and consequently wZ.) 0 ( c-iis zero (0. 1 for convenience.. Here. the authors and are repeated in Fig. is the transfer function realized assuming ideal OA’s and N=l+a. is seen to slightly exceed wi [2]. For the C20A-3..463 MICHAEL AND MIKHAEL: APPLlCATIONSOFCOMPOSlTEOPERATIONALAMPLIFIERS strated the drastic active sensitivity improvement of the Deliyannis filter... from T. and Qp (as defined in Table I) are functions of the circuit parameters which are the single-OA GBWP’s ot. = wj = qfi Wh ' + Wh > 3"' I 0. and Qp with respect to the circuit parameters. for the C20A-4. = oh and oi= w2= wi). 3-dB bandwidth 2 wi/A. as can be seen from Table I. The stability conditions for particular values of (Y are summarized in Table II. Now let us assumea two-pole open-loop gain Ai of the ith single OA (i = 1. In general.The actually realized transfer functions T.). the condition is given by (4) Also. =. b. The Ta’s have the samegeneralform found in [l]: r a ity for (I used 0 CZOA-1 Unrealizable 1 0 CZOA-2 1 Unrealizable z? 1 0 CZOA-3 where T.. a. since the two-pole OA model reducesto a single-pole model (Ai = w/(si + oL )). wZ. TABLE11 VALUESOF~EC~OA'~RESISTORCOMPENSA~ONRATIO &FOR SELECTEDVALUESOFQ .ANDT~CORRESPONDINGSTABXLITY CONDITIONSFORTHE PNVERTINGINTEGRATOFSUSINGTHE CZOA-i II. since wh increases as (Yincreases. second pole frequency (w. + w) in somecases) CZOA-4 Thus. None of the a and b coefficients are realized through differences. wp. which belongs to category 2.. = o. w. On the other hand.there is a minimum value of wh/oi which satisfies the stability condition.. From physical considerations. w.. for a given (Y.The C20A-2 integrator requires wh/wi to be greater than 3/2 for stable operation when (Y equals zero and wh/wi to be greater than 2 when a tends toward infinity. are given in Table I.1 A 3w. for identical single-OA models (wh. The C20A-4 integrator has an advantage. [33]] that. Aoi dc gain of the ith OA.666 wi is the GBWP of each single OA and. -+ cc. oi gain-bandwidth product (GBWP) of the single-pole model of the ith OA. (4). since for wh = oi. and b. the stability condition is given by It is clear from (3). determine the stability of T. and the C20A compensation ratio (Y(as well as TV. For . Z+ wL. the condition a = 0 results in the minimum value for wh.the integrator time constant).JOA’s The amplifiers C20A-1 through C2OA-4 were previously proposed by. by using C20A-4. It is easyto show [l. w.the stability improves. assuming a single-pole OA model. Inverthg Integrators Using C. a value of OLexists (a = l/3) for which the integrator is stable with excellent frequency response. and b. is tile single-OA second pole frequency. if o. b..s=l+(s/o. ref. thus guaranteeing the low sensitivity of T.. From graphical data sheets of internally compensatedOA’s such as’741’s and 747’s. the necessary and sufficient stability conditions (the parasitic poles are in the left half of the s plane) for the C20A-1 and C20A-2 integrator realizations are given by 1 A Unrealizable 1 : 1 . (b) Percentage deviation from ideal of the transfer function phase versus normalized frequency for the proposed C20A-4 integrator and several other integrators (l/r.835) with existing negative integrators proposed in [3]-[5]. Comparison of the C20A-4 negative integrator for (Q. = 0. with (Y= l/3. are given in [l] and [l. = q. 3(a) for convenience. yields the following integrator 04 Fig.is the integrator time constant). externally compensated OA’s such as the 702 and 709.w.464 IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS.707. together with their actually realized open-loop transfer functions. Qp = 0. very nearly matches the excellent performance of the integrator in [3]. [33]]. (6) Using the C30A-5. Thus. (a) Percentaee deviation from ideal of the transfer function magnitude versus no&lized frequency for the proposed C20A-4 i&g&r and several other integrators (l/r. the integrator in [3] suffers from stability problems while the proposed one does not. 2 shows that the percentage deviation in magnitude and phase from the ideal for the proposed C20A-4 negative integrator. (Y can be chosen in the range that guaranteesstability while yielding the most desirable T.05). The theoretical results in Fig. C30A-2.wi = 0.) In practice. NO. MAY 1987 . The different structures of these C30A’s. Inverting Integrators Using C3OA’s In this section.=T.(s). =l/rt. T.05). 2. C30A-5. (7) Using the C30A-5. Using the C30A-2. which are employed here. C30A-5.internally compensated OA’s with a phasemargin of less than 60” at 0-dB closed-loop gain (wh/wi < 3/2) are used.05 (w. T. three high-frequency integrators which use actively compensated multiple OA’s (CNOA’s) for N = 3 are introduced. CAS-34. ref. % deviation from ideal IHI 6% t CBOA-4 7% 6% 5% 4 % 3 % 2 % I % % deviofion from ideal B. = 0.=q. (8) . T. 2 for ot/oi = 0. (The C20A-4 integrator with (Y= 0 is identical to the integrator in [3]. and C30A-6. 5. where rr . and C30A-6 in the inverting integrator circuit transfer functions. A theoretical comparison of the C20A-4 negative integrator with state-of-the-art negative integrator realizations [3]-[5] is given in Fig. VOL. Using the C30A-2. as verified experimentally by the authors in active filters (two-integrator-loop biquad-Section III-B) when. the importance of the controlling parameter (Y is apparent since it guarantees stable operation using commercial internally compensated OA’s without sacrificing performance. are given in Fig. the low coefficient sensitivities are achieved and the necessarycondition for stability is satisfied without the need for matched amplifiers. [ l+. (b) EXTENDING THE ACTIVE FILTER OPERATING FREQUENCY RANGE USING CNOA’s Active filters have been designedusing a wide variety of approaches [6]-[lo]. 20 40 60 60 ‘100 120 I40 I60 I60 200 220 240 260 260 f kHz (4 Fig. In this contribution.1). = 0. the two-integrator-loop filter.1). Thus. 3. In (6)-(8)./w. it can be shown that no difference terms appear in any of the numerator and denominator coefficients when OA’s with different gains are used. I (1+0)(1+~) 51. the necessary and sufficient conditions for stability.)]>l. Fig. while allowing a wide range of (Yand p variations. The performance improvement in both magnitude and phaseis obvious. 4% 3 % 2% I % f ktiz 20 40 60 80 100 120 140 160 180 200 220 240 260 280 III. For illustration.465 MICHAEL AND MIKHAEL: APPLICATIONSOF COMPOSITEOPERATIONALAMPLIFIERS where wi is the GBWP of the OA’s used. b I +[.(1”. C30A-5. Examples’ are the positive-gain Sallen and Key filter. active filters are considered to belong to one of two categories.e. and the SFG filter [6].][1+. The second category contains those filters whose OA’s are embeddedin the passive network. i. inverting integrators and finite-gain amplifiers. [91- 5 % 4 % . T. namely. and functional building blocks cannot be isolated in the filter structure. This can be explained as follows. A. (c) Percentage deviation from ideal of the transfer function phase versus normalized frequency for the proposed C30A-6 negative integrator and the one proposed in [3] (at/w. tsa) For the C30A-5 integrator.1)./s (T. Examples are the multiple-feedback (MFB) filter and the generalizedimmittance converter (GIC) filters WI. for the roots of D to be strictly in the left half of the s plane. the ideal integrator transfer function is equal to . . (b) Percentage deviation from ideal of the transfer function magnitude versus normalized frequency for the proposed C30A-6 negative integrator and the one proposed in [3] (o. 3(b) and (c) shows sample theoretical results using the C30A-6 and those obtained using the integrator in [3] that employs two OA’s. extended operating frequencies are obtained. I It can be easily shown that these new integrators can be designed to satisfy the above stability conditions. and C30A-6 [l]. = 0. For the C30A-2 integrator. --) 7J as the OA Ai’s -+ co). Comparison of C30A-6 negative integrator and the one proposed in [3] (0.. the behavior of the active functional building blocks approaches the ideal over a wider frequency range. as . (a) The composite operational amplifiers C30A-2.w.. are as follows./w.f+)][(~+@+~+. Assuming a single-pole OA model. Improving the Performance of Active Filters in the First Category It is easy to show that for filters in the first category.(l+P)]>l. from ideal (9b) For the C30A-6 integrator. The first category includes active filters that are realized using functional building blocks. = 0. since the higher order derivatives of G may be increased in a manner that offsets the reduction in AG obtained by nulling the lower order derivative terms. .. Similarly. Note that. G. = G and Gii = Gi. i. for a given number of OA’s. to minimize AT. where 7i =l/GBWP =l/bi). . Also. The first may occur if one attempts to choosea structure for realizing G which results in zero lower order derivatives G.IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS.‘rf + G. The minimization can be carried out in many different ways. T.. VOL. Also. . must be appropriately minimized using the CNOA’s compensating resistor ratios in the expansion of G given by (15).r11[1) 01) where j=l. several techniques are possible for error minimization and bandwidth extension in this category of filters. The Taylor’s seriesexpansion of T. but independent of the nonidealnessof G. This has been examined by researchers and referred to as stability during activation [25]-[27].+ dG G=C.). T ‘.Gi) Tl=%=a. Then.) of the OA’s is cc (7i = 0. G.‘. let G..-. can be written as Ta=f(G1. 5. This can be easily seen by comparing the performance of the C20A-4 with the C20A-1 in finite-gain applications [l]... from a Maclaurin series expansion of (11) in ri and r2 about their ideal values of ri = r2 = 0. -. . Gii.. . both the phase and gain deviations in G of the functional building block... ..) denote functional dependence. G. i = T’AG T”(AG)2+ . 1 a’T. .‘rz + 2G$r1r2] + +. since it leads to local instabilities. MAY 1987 466 Let us assume an active filter that realizes a transfer function T. The following argument can be shown to be valid for any integer values of n and N. . G. The second pitfall may result if one controls the phase angle of G by moving some of the parasitic poles of G to the right half of the s plane to cancel the phase shift due to the other poles and zeros. are dependent on the topology In this subsection. . there may exist a structure that fails to satisfy the zero low-order sensitivity property but results in a smaller AT. let q. of G with respect to its parameters. . two pitfalls are pointed out. .--*. . In both approaches. A simple but less exact approach is to minimize each ]AG] individually. As mentioned above. Another more exact but more involved approach is to substitute for AG from (15) in AT given by (13).-T. n.This is intolerable in finite-gain applications and very undesirable even when the active building block is embedded in filter structures. it follows from (13) that if the active building blocks in a filter are replaced by the proposed building blocks which have a smaller AG. can be expressedas Gj=g(r1j. T “. I for AG very small (13) where AT=T. a realization that is characterized by zero low-order sensitivities may not be optimum. f (. To conclude this subsection. it is shown how the operating frequencies of the first category of filters can be extended through the use of the functional building blocks that are presented in [l] and in Section II. Gni be the corresponding ideal transfer functions when the active elements (OA’s) used are frequency independent. A less global and more straightforward approach is employed in this subsection for demonstrating where the bandwidth of each block is individually extended. GpGz. particularly as the frequency increases. about Gi is given by (G. Here. (15) It is to be noted that. (G-GJ*+ (12) Equation (12) can be rewritten as AT=T’(AG)+ . T. the jth gain Gj realized using the jth CNOA.) and g( . from (13) and (15). These blocks allow the minimization of AG and the necessary tradeoff between gain and phase deviations to achievestable high-frequency operation with low sensitivity to the active compensation elements. the gain-bandwidth product (GBWP. . Great care should be exercised in attempting this. and AG=G-Gi. B. whose OA ri’s are r.j to rNj. the compensating resistor ratios are chosen to minimize AT as desired. it can be shown that of the active filter. . let n = 1 and N = 2. . using functional active building blocks whose individual transfer functions are G. is the actually realized transfer function using nonideal (frequency-dependent) active building blocks. . the filter’s performance can be improved. Here.GZ. AG is complex in general. In addition.. Thus.2. (10) Similarly.. are the actually realized (nonideal) transfer functions of the building blocks using frequency-dependent active elements(OA’s)..AT may be minimized at a critical frequency or over a band of frequencies. Improving the Performance of an Active Filter in the First Category (A Multiple-Amplifier Biquad) Using C2OA’s Equation (14) can be rewritten as AG = Girl + G$r2+ i [G . At this point.. For simplicity. G... corresponding to AG’s real and imaginary parts. CAS-34.e. This is found useful in practice and is applied in Section III-B.. NO.r2j.. evenif Gi is real.G. =R. The filter that is chosen is the well-known state-variable filter [lo]. (b) Percentage results obtainable with regular OA’s.=R. It usestwo inverting integrators and a differential finite-gain amplifier. and building blocks.=R. thesedesign values correspond to output voltage swing of 12 V peak to peak.tors.12 V. A biquadratic active filter.. where v-power supply =-.. using the C20A’s are ‘compared to those utilizing the WhP = RJR.1 = 0. It is very interesting to note that when (Yis set to zero (which results X=2Q*. be less than 50 dB below the fundamental at fPf. In Fig. from Sect& II and Table II. which are constructed using the C20A-4 integrators proposed in Section II and From [l]. (c) Percentage variation of Qp/ as a function of Qp. and dyThe elements are chosen as namic range are also verified experimentally.C.-1+4 (17) in a previously reported integrator [3]. which uses the functional where QPr is the complex pole-pair selectivity factor. for an Referring to [lo]. (filter) as a function of Q for bandpass filter (wpf = 50 krad/s). - 150.g./R. 4(b)-(d). _ s equal to unity (maximally flat) in the C20A-4’s integra. W.differential amplifier.! the useful operating frequency range demonstratedin Fig.C./R. (d) Percentage variation of Qp.=R any of the filter outputs is measuredover a wide range of and aPr (the complex pole-pair resonant frequency) = frequencies and signal levels. and QP. ‘i 300 CZOA-2) 250 C20Ad’rl 200. l+R. is designed and tested. 4. for maximally flat response(Q.f = 16 kHz. for ban“dpass filter (o . = l/a) of the the C20A-2 differential amplifier [l]. Total harmonic distortion (THD) of much less than 1 percent at C. r. r&I I single OA’s. the experimental results for the filter 1+&/R. 1y= 9 . e. =-IO. ' R. excellent theoretical sensitivity. OQf 20 40 60 80 (4 100 120 (4 Fig. [ll]. stability. = 30.J I .8 krad/s) (LM747 OP AMP).467 MICHAEL AND MIKHAEL: APPLICATIONSOF COMPOSITEOPERATIONALAMPLIFIERS Cl CP 350. The biquad’s trans. the THD was found to 1/RC. a! is set l+R. 4(b)-(d). Also./R.=C. In addition to the appreciableimprovement in R. (LM747 OP AMP). as a function of fp. [12]).=R. Experimental results of the two-integrator-loop BP filter using the proposed C20A-2 and C20A-4 and the theoretical using CZOA-2 and C20A-4.C.s l+R._. (a) Bandpass filter (two-integrator-loop) variation of Qp. for bandpass filter (LM747 OP ARIP). 4(a). the filter 3 &=l. and is (18) shown in Fig. independent of fer function 7. ' (16) 7 .=C R.(s) (at the bandpassoutput) is given by X and QP.R. . SINGLE OA 30 SlNGLE OA 30 2s 20 15 I oo- IO MEASURED “SING C2OA’l 5 50. (4 I oo50 -% 100 ')Pf 4 / Lop. . MFB active BP filter using C30A-1. R. as predicted from the stability strate the usefulness of the proposed technique. namely.. Hence. R.s {l+s[zCR. = Ri//R2.2 zy(t)‘+ ... but also difference terms do not appear in any of the . for OA used in constructing the jth n =1 and N = 2. MAY 1987 468 From (24) and (26). + CR.by combining (24) and (26)."'.‘+ *-. +‘ (23) t=O *t+Yi =T’t+ . the filter transfer function Ti tively. tj can be expressedas . 2 CR1 <‘+ -+ A o. let CNOA. has to be minimized appropriately using the CNOA’s compensating resistor ratios (OL’S. t. NO. 0 "0 D. Equation (23) can be rewritten as AT23 1[ wi w. . Improving the Performance of an Active Filter in the Second Category (A Single-timplifier Biquad Using C3OA) For a given filter in this category. .. is given by t. In the following results. In the following subsection. let simplicity.}/ and t = &I. VOL.OA’s.t. For ideal of the first.I 1+2RCs+R tj=g(71j. e.. n th composite amplifier.(l+P) 0.) 09) illustrate the improved filter performance using one of the where t. A multiple-feedback (MFB) biquadratic bandpass acfunction of the t's of the CNOA’s as follows: tive filter is considered here (Fig.) 72 + 7. i Also.CsH $L (27) V. t. + CR.ltzO+ aT. not only is OA gain matching not required. (25) I is given by t=- at aT1 T1+ 71 . In this subsection. (a’s and R’s) are chosen using the search method in a straightforward computer program that minimizes the C. a simpler but equally useful approach is adopted to demonbecame unstable in practice. the Maclaurin series expansion of (21) about its ideal value of t = 0 is given by +wi(l+P) T.2. as an example to T. Also. (24) +S4 t-o **-. to Fig. For simplicity. at +s2 C2R. are reciprocals of the open-loop gains proposed C30A’s [l]. . Assuming. (22) 1 Thus. which is complex in general. C2R.+~ ( -sCR3H(1+. it can be shown that the dynamic range of this filter is identical to that obtained if a single OA replaces the C30A.--. 72). uPr = CNOA and 7ij is the reciprocal of the GBWP of the i th l/(Cm). .) 0.7NI) (20) 1 1R 3C2s2 where tj is the reciprocal of the open-loop gain of the jth where H = R2/(Ri + R2).3). at (t=o+)+( f)Jgq=o*(‘. and 721= r2. Also. can be expresseddirectly as a stability. in Section III-A. second.T2j.. respec. Ai + cc (i = 1.t. For a composite amplifier constructed using N is given by OA’s.. error minimization can be done in many different ways depending on the cost function and the optimization criterion. i.. identical frequency-dependent OA’s and R’ X- ~ii = 7i. CAS-34.=f(t. etc).R3 w?(1+ a) 1 f&l + a) 1 C2R. .(t)2+ . a way of improving the performance of a filter in the second category by employing CNOA’s is given. it is feasible to derive closed-form expressions. the C30A-1.R2 + (2CR.R’s.=f(t) (21) T.= T. 5). minimize the error appropriately.R. 5.IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS. to minimize AT..R. = (1/2)-/m.(1+/3) + 1 a2T. the CNOA’s compensatingresistor ratios analysis in Section II..(l+ a) +- 1 w.t. T. the actually realized transfer function T. T. I + (28) Again.R. = t. As stated before.7* = 0 at Jr2 (2CR. Improving the Performance of Active Filters in the filter’s response deviation from the ideal and satisfies Second Category In the second category. = T2= 0 (26) C’R. and Qp.e. t at. 5. In the design in [ 131. 6. Ideal and experimental results are shown in Fig. = 10 is designed and tested. coefficients.. and deviation from the ideal response. as a function of fPf for Qpr = 5. tolerance of OA mismatch. 10 20 30 40 50 60 70 90 90 100 (a> AAx I 100 QPf I % fp. 20.which requires the same number of OA’s as C30A-1. IDEAL MFE FILTER (ideal OA’s) QPpxpsrimsntol = 9. sensitivity. 5 are given in Fig. (+$dooI % t 10 _ C3OAl 9- II31 a=o. = 1.. even when different OA’s are used.. 7 5. the compensation remains fixed for all the results in Fig. 7(a) shows the percentage deviation in &. 6J4. = 5. In this paper. an integrated technique for alleviating the problem of BW limitation due to OA frequency dependence has been presentedwhich extends the useful BW of linear active networks using OA’s without impairing other important properties such as stability. Experimental results of the MFB filter designed using the proposed C30A-1 and the design proposed in [13]. The design values are C = 1 nF. (b) Percentage variations in Q. = 37. p = 1. Experimental amplitude response of the MFB bandpass filter using C30A-1. Applying the Performance comparisons are made with the recently reported high-frequency filter design given in 1131. Rather.2.0 = 0. R.2 fpfhHz c 1 42 46 Fig. 7(b) shows the percentage deviation in Qpr as a function of jp. it can be easily shown numerically that the necessaryand sufficient conditions are well satisfied with a wide margin for the component values (Yand p. for Q. It is worthwhile . 20. dB 46 W[101. namely three. R.while other important properties such as stability and low sensitivity are maintained. and its variations due to the compensating element variations. = 84 kQ. (a) Percentage variatrons in center frequency of the MFB filter for Q. kH* IO 20 30 40 50 60 70 80 90 -%- (b) Fig. while in the second category the OA’s are embedded in the passivenetwork so that functional building blocks cannot be isolated from the filter structure. 10. A BP filter with j”.. 7..p=m 9= 0. applications of the CNOA’s in inverting integrators and active filters are investigated. The comparisons in Fig. Filter examples are consideredfrom both categoriesand are constructed using the proposed CNOA’s and the improved functional building blocks. p = cc (compensating resistors R' and /3R are open. 7.9 kHz and Qp.7 fptexperimental = 37. To analyze the effectiveness of this approach. The circuits thus attained are shown to perform as well as or better than the best available state-of-the-art designs with respect to .f for Q. sensitivity. Theoretical and experimental results of utilizing the proposed CNOA’s (1v = 2. P = I 87..3 k& and R. 10. 7. CONCLUSIONS A general technique for the synthesisof composite OA’s (CNOA’s) and for using them in extending the BW of finite-gain amplifiers has been presentedin [l].. No error minimization is attempted at each operating aPr and Q. 6. * Trapezoidal oscillations were encountered. 7 show clearly the performance improvements in the proposed filter design. The first category consists of filters realized using func‘tional building blocks (finite-gain amplifiers and inverting integrators). 10. In conclusion. Rj = 250 Cd. The OA’s used are LM747’s with a GBWP ~1 MHz. In the C30A-1 design IX= 0. and dynamic range. while aR' and R resistorsare short).extension of BW. and 20. stability.MICHAEL AND MIKI-IAEL: APPLICATIONSOF COMPOSITEOPERATIONALAMPLIFIERS 469 Routh-Hurwitz stability criterion. 6 illustrate the appreciable improvement over a single-OA realization in the MFB structure under consideration at this QP. Fig. ---Limiting diodes were necessary.2.. transfer function T. ’ The experimental results using the proposed C30A-1 design and the design in [13] of the MFB BP filter in Fig. active filters are considered to belong to one of two categories. and 20. 10. = 210 a./ = 5.3) in inverting integrator applications show that the BW is extended considerably beyond present stateof-the-art techniques. The experimental results in Fig. IV. Fig. and aP. 1976. pp. Natarajan. Dr. “Synthesis of actively compensated double-integrator filter without matched operational amplifiers. WV. 449-460. 1980. 116. Circuits Syst. 1161 S. Hilberman.. 1547-1548. 670-676. Alberta.N0. Eng. NJ: Prentice-Hall. in 1974. pp. 67. Circuits Syst. His research interests are active networks. vol. 0.. 961-963.” IEEE Trans. vol. “Star: An active biquadratic filter section. Montreal. [26] [27] A.” Electron. Michael is a member of Eta Kappa Nu. He has several patents and publications in the area of communication networks and active filters. June 1980. Montreal. R. “Frequency limitations of active ]241 filters using onerational amnlifiers. Feb.” IEEE Trans.Sc. &MAY 1987 470 mentioning that the technique described here and in [l] is general and is applicable to a wide range of linear active realizations using OA’s. vol. where he is now a Professor of Electrical Engineering. 83-86. He received the B. A. as well as an adjunct Associate Professor in Electrical Engineering at Sir George Williams University (now known as Concordia University). CAS-29. pp. CAS-28. Mikhael was the recipient of the Bell Northern Research Outstanding Contribution Patent Award in 1978. Switched CaDacitor. Michael worked as a research and teaching fellow at West Virginia University. Since May of 1973. and K. In August 1978. Canada. 122-130.” IEEE Trans. West Virginia University. Assiut.. Aatre. the Outstanding Researcher Award from the College of Engineering. CAS-23. degree from the ‘University of Calgary. 141 A. 1969. Elee. WJI A. and D. [I81 S. Nov. Canada. July 1976. S. Ravichandran. degree (honors) in electronics and communications from Assiut University. Symp. 1968. Soliman and M. 1191 A. and the Halliburton Best Researcher Award in 1984. Circuits Syst.‘circuits with applications to active-RC filters. He served as-a First Lieutenant in an Engineering Corps specializing in water well drilling. .” Electron. pp. A. J. Texas Instruments.. Circuits Syst. 1174-1177. He received the M.and second-order operational amplifier time constant effects. pp. respectively. B. pp. Ravichandran and K. B.” IEEE Trans.Sc. pp. pp. “A note on frequent and Q limitations of active filters.. “Low sensitivity high-frequency active R filters. PI The Linear and Interface Circuit Data Book for Design Engineers. 1838-1850. electronic measurements and supervised drilling operations in cooperation with Schlumberger Co. Ismail. CAS-23. New York: Wilev 1071 Modern Filter Design Active-RC and [91 M.Sc. on November 3.” IEEE Proc. Jul-y 1972: A. 288-289. Ismail. Mitra.Eng. Apr. Brackett and A. Friend? A. “Stability properties of some gyrator circuits. Ravishankar. Morgantown. Bhattacharyya. Circuits Syst.” IEEE Trans. as a Field Engineer. Antoniou. CT-19. Michael.. the M.. PI G. Egypt. Dec. 1976. June 1977. Modern Filter Theoty and Design. Brackett. June 1979. As a Research Engineer with the National Transportation Research Center. pp. “On the stability of the phase-lead integrator. WI M. He served as Chairman of the 1984 Midwest Symposium on Circuits and Systems. he was an Engineer with the Telecommunications Organization. Sedra an B J. vol. vol.4nalysis and Synthesis bf Linear Active Networks. CAS-28. vol. 1231 J.E. R.. Circbits gyst. Feb. Aatre.” IEEE Int. Dr. vol. degree in electrical engineering from West Virginia University. Sedra and P.i’KT’L”&er.-p. 87-90. 1980. Mitra.&g “An active-compensated double-integrator filter without matched operational amplifiers. and WI E: R. pp: 115-121. Geiger and A. and Tau Beta Pi and is a registered Professional Engineer. vol. CA. 151 P. Budak. K. pp. Nov. “An insensitive active-RC filter for high Q and high frequencies. pp. Holland.. “High-quality double-integrator building-block for active-ladder filters. M. S. Circuits Syst. degree in industrial engineering and the Ph. 1977. pp. Dec. 1982.. degree from Sir George Williams University. “Design of active filters independent of first. CAS-22. “A universal variable phase 3Lport VCVS and its application in two-integrator loop filters. 1981. Ramamurthy. Let?. Dr. Rao.” IEEE Trans. Monterey. 322-328. Cairo. 239-245. vol. WI M. Reddy. 68. M. vol.E. CAS-23.” IEEE Trans. 321-324. Inc. Eindhoven. 1978. he has been a Member of the Scientific Staff at Bell-Northern Research..” pp. P. Rao. Morgantown. Enelewood Cliffs.” IEEE Proc. pp.” IEEE Trans. respectively. Cairo. vo Y CAS-24. vol. Bailey and R. no. VI A. 11. Egypt. S. . WV. 1981. New York: Wiley. 215-218. Reddy. Intioduction to the Theory and Design of Actibe Filters.” IEEE Trans. Calgary. Dr. as well as other controlled sources. radiation hardening. Geiger. Mitra and V. in 1965. 161 A. 1980. Egypt. Circuits Syst.. Aug.. A. M. OR: Matrix. L. vol. Canada. “Stability properties of some RC-active realizations. His present research interests are in the area of analog integrated circuits and active networks design. From 1970 to 1973. amp. C. he worked on designing and implementing a new digital communications system for the Morgantown Personal Transit System (MPRT). 4. Inst. “A novel active compensation scheme for active-RC filters. “Composite operational amplifiers: Generation and finite-gain applications. 217. Temes and S. Reddy. 749-757. Mikhael (S’70-M’73-SM’83-F’86) was born in Manfalout. June 1972. in 1982 and 1983. Harris. vol. 1970. and in 1973 he taught in the Mathematics Department. vol. and the D. 1980: [I31 R. Quebec. degree in electrical engineering (electronics and communications) from Cairo University. Mikhael and S. vol. Left.” Proc. Dawson College. Sedra. L. B. where he conducted. pp. Beaverton. Since 1983.. R. B..1975. “Active sensitivity minimization in SAB’s with active compensation and optimization. A r. 743-744.CAS-34. Wasfy B. CAS-22. Ghausi an .. 131 G. vol.. Ramamurthy. 8. K. 429-433. W. Egypt. p. Canada. [I41 K. . 68. “A new integrator with reduced amplifier dependence for use in active RC-filter synthesis. Petrela. He received the B. K. in 1980 and 1983. Huelsman and P. 68. Ottawa. P71 S. Alpha Pi Mu. Circuits Syst. Sedra. “On the active compensation of noninverting integrators. Circuits Syst. “Sensitivitv and freauencv limitations of biquadratic act&e filters. A. S. 510-512. Mikhael is presently Associate Editor of the Letters Section of the IEEE TRANSACTIONSONCIRCUITSAND SYSTEMS. and 1973. 68-73. Budak and D. “Realization of gyrators using operational amplifiers ]251 and their use in RC-active network synthesis. pp. New York: McGraw-Hill. Apr. 171 S. Natarajan. “Active compensation for highfrequency effects in op.VOL.IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS.Sc. Circuit Theorv. Mitra and V. REFERENCES PI W. K. Quebec. B.” IEEE Proc. R. and adaptive signal processing.. 1975.“E. Mikhael and B. CAS-24. Esoinoza.D. Ontario.” IEEE T>ans. he taught in the Computer Science Department at Sir George Williams University.. P21 A. he joined the faculty of West Virginia University. solar cells and space power applications. pp. 534-538. Circuits Syst.1944. Circuits Syst. 1981. pp. Symp. Martin and A. this issue. vol..” IEEE Proc. Allen. pp.” in IEEE Int. 1969. Sherif Michael (S’78-M’83) was born in Alexandria. He received technical training at Philips Industries. Antoniou. Filter Theory and Design: Active and Passive. K. Egypt. [lOI L.” IEEE Trans. Soliman and M. he has been an Assistant Professor with the Department of Electrical and Computer Engineering at the Naval Postgraduate School.. switched-capacitor circuits. From 1965 to 1968.. can be either real number or complex number.. + b1s + b0 (1) where the numerator coefficients a0 . Introduction The general form of the transfer functions can be written as the ratio of two polynomials as follows: a s m + am −1s m −1 + . Objective To study real zero and pole synthesis. That is m≤n (2) The numerator and denominator polynomials can be factored.(s − z m ) T (s ) = m (s − p1 )(s − p2 )..... Complex zeros and poles... must occur in conjugate pairs.2 . and the poles...EEL 4140 ANALOG FILTERS LABORATORY 2 Real Zero and Pole Synthesis I.. II.... and T (s ) can be expressed as: a (s − z1 )(s − z 2 ). z m .(s − pn ) (3) where the zeros. however..... b1 .. pn . z1 . a m and denominator coefficients b0 . the transfer function T (s ) is factored as: T (s ) = T1 (s )T2 (s ).. (4) . and cascade design of first-order circuits. only real zeros and poles are discussed. In this case. i = 1...n are bilinear transfer functions. and n is the order of the filter... a1 .Tn (s ) where Ti ( s )' s... + a1s + a0 T (s ) = m s n + bn −1s n −1 + . bn −1 are real numbers... p1.. z 2 . p2 ... In this experiment. The degree of the numerator polynomial must be less than or equal that of the denominator polynomial for causality reasons. Second. Third. we connect these circuits in the sense that each successive circuit does not load the previous circuit. and unknown resistor and capacitor values. other components can be calculated. One of the designer’s main functions is to choose reasonable capacitor values. Capacitors remain the most expensive and sensitive components in the discrete. T1 ( s ) and T2 ( s ) . capacitor and Op Amp circuits. (5) . we synthesize the Ti ( s )' s using first-order circuits. this requirement consists of limiting size or value of capacitors. we find a set of relations between zeros and poles. the design procedure begins with a selection of reasonable capacitors. This property does not represent a problem but rather an opportunity. For resistor. we factor the transfer function T (s ) in terms of bilinear transfer functions Ti ( s )' s as Equation (4). the task is to synthesize each stage using first-order circuits. Usually. The transfer function T (s ) is given by: T (s) = (s + 5 × 10 )(s + 10 ) (s + 2 × 10 )(s + 10 ) 3 4 3 5 Equation (5) is expressed as the product of two bilinear functions. According to pole and zero locations. Subsequently. The number of the unknown values is generally more than the number of equations. Then. and sensitivity to environment changes. we choose suitable circuits to synthesize these poles and zeros.Once the transfer function is factored into multiple stages. Cascade Design of First-Order Circuits First. parasitic sensitivity. hybrid. tolerance allocation. Cost implies not only dollars but also size. In this way. the transfer function of this cascading circuit is T (s ) . and integrated circuit design. weight. Real Zero and Pole Synthesis Synthesizing real poles and zeros is based on the cost of the energy storage elements. Design Procedure for Real Zero and Pole Synthesis 1. Synthesize the transfer functions T1 ( s ) and T2 ( s ) . as shown in Fig. 1 is given by: Tcir ( s ) = − R2 ( sC1 R1 + 1) R1 ( sC2 R2 + 1) C ( s + 1 ( C1 R1 ) ) =− 1 C2 ( s + 1 ( C2 R2 ) ) 3. (8) For both circuits. 4 5 For both transfer functions T1 ( s ) and T2 ( s ) .1 = C1.1 = 50kΩ (11) and In the second circuit. the resistor values are computed as: R1.1 = C 2. the same first-order circuit is used. The transfer function of the circuit in Fig. 2 = C 2. the capacitance values are chosen as: C1.01uF (9) In the first circuit.T1 ( s ) = (s + 5 × 10 ) (s + 2 × 10 ) (6) (s + 10 ) (s + 10 ) (7) 3 3 and T2 ( s ) = 2.1 = 20kΩ (10) R2.1. 2 = 0. which synthesizes T1 ( s ) . 2 = 10kΩ and (12) . the resistor values are computed as: R1. Choose proper circuits to realize poles and zeros. which synthesizing T2 ( s ) . 01uF R1.01uF Vi (s ) R2. 2 = 0. Since the output of the first-order circuit is just the output of Op Amps.R2. these two circuits can be connected directly.1 = 0. ( )( ) (s + 2 × 10 )(s + 10 ) 3 4 function is T ( s ) = s + 5 × 10 s + 10 3 5 Vo (s ) The resulting transfer . Cascade the designed two circuits to realize the transfer function T (s ) . 2 = 0. 2 = 1kΩ (13) 4. 2 = 10kΩ R2. 2. which synthesizes real poles and zeros.1 The noninverting first order operational amplifier circuit C2.2 The cascading design example. as shown in Fig. 2 = 1kΩ - - + + Fig. Thus.1 = 20kΩ C2.1 = 50kΩ R1. Vi (s) C1 C2 R1 R2 Vo (s ) + Fig.1 = 0.01uF C1.01uF C1. the output resistance of the first-order circuit can be considered as zero. Discuss any discrepancies. the phase response of the cascading circuit should equal the summation of the phase responses of two first-order circuits V. Experiments 1. Synthesize real zeros and poles using the circuit shown as in Fig. Follow the above procedure. Simulate the designed band-pass filter. Use LF 351 Op Amps with a split 2. if any. Plot the magnitude and phase responses of the overall circuit in the frequency range from 30Hz to 40kHz. Compare the simulation results with the experiment results. Comment on deviations from expected results. . you need present the experiment results and compare them with the simulation results. power supply voltage of ±15V. and compute the resistance and capacitance values. 2. 2.1. Your report should include the following: 1. and make comments. and the reasons for these deviations. In addition. 3. Lab Report In the report. The design steps and results. The experiment results. The simulation results. Plot the magnitude and phase responses of each first-order circuit in the frequency range from 30Hz to 40kHz. Note that the magnitude response of the cascading circuit should equal the product of the magnitudes responses of two first-order circuits. IV. 4. Design The transfer function of the band-pass filter is given by: (s + 10 )(s + 2 × 10 ) T (s ) = (s + 2 × 10 )(s + 10 ) 3 4 3 4 (14) Decompose the transfer function in the factor format. 3. Computer Simulations 1. Build this band-pass filter using two first-order circuits. VI. Measure the magnitude responses of the overall circuit in the frequency range from 30Hz to 40kHz.III. 3. Measure the magnitude response of each first-order circuit in the frequency range from 30Hz to 40kHz. 1997. [2]. 1982. M.References [1].” University of Central Florida. Van Valkenburg. E. . “EEL 4140: Lab Manual for the Design of Analog Filters. Oxford University Press. Dr. Robert Janes Martin. Analog Filter Design. and V ( s ) is the amplifier input voltage. band-pass. is defined as: TFF ( s ) = V ( s) Vi ( s ) V ( s ) =0 o and the feedback gain of the passive network. Thomas C. 1. is defined as: (2) . There are several circuits that can implement the biquad transfer functions. is the output voltage of an amplifier having gain K . Sallen-Key circuit is one of these circuits. All realizable polynomials used in analog filters can be factored to second order forms and generally are of quadratic nature. TFF ( s ) . high-pass. and notch. If we apply superposition to the circuit. we obtain V ( s ) = TFF ( s )Vi ( s ) + TFB ( s )Vo ( s ) (1) where the feedforward gain of the passive network. Sallen-Key Filters Sallen and Key proposed a class of circuits. which means the ratio of two quadratic polynomials. TFB ( s) . the box labeled “second-order passive RC network” contains resisters and two (or sometime three) capacitors. named as Sallen-Key filters.EEL 4140 ANALOG FILTERS LABORATORY 3 Sallen-Key Filters I. II. Thus. This circuit incorporates a single amplifier embedded in a passive RC network to generate any type of second-order transfer functions: low-pass. In Fig. Lee named this title several decades ago. The basic Sallen-Key structure is shown in Fig. Introduction The term biquad is an edited form of the word biquadratic. Vo (s ) Vi (s ) is the input voltage. and it is now in common usage. Objective To study design and implementation of Sallen-Key filters.1. the biquad is a useful and universal building block. in 1955. we have the amplifier relation as: Vo ( s ) = KV ( s ) (4) We obtain the relationship between the input voltage Vi (s ) and the output voltage Vo (s ) in term of TFF (s ) and TFB (s ) as: Vo ( s ) = KTFF ( s ) Vi ( s ) 1 − KTFB ( s ) (5) Unless the RC network is degenerate. where Vi ( s) is the input voltage. we can express TFF ( s ) and TFB ( s) as: TFF ( s ) = N FF ( s ) D( s ) (7) . TFF ( s ) and TFB ( s) have the same denominator polynomial D(s) as: D( s ) = s 2 + b1s + b0 (6) Therefore. Vo ( s) is the output voltage of an amplifier having gain K . 1 The basic Sallen-Key topology. Also. and V ( s ) is the amplifier input voltage.TFB ( s ) = V (s) Vo ( s ) V ( s )=0 (3) i Vi (s) Second-order passive RC network Vo (s ) V (s) K Fig. and (8). we can . Otherwise. if the amplifier gain K < 0 . Low-Pass Positive Sallen-Key Filter C1 Vi (s ) R1 R2 + Vo (s ) C2 Ra Rb Fig. this kind of circuit is called as the positive Sallen-Key filter. we can derive the transfer function of the Sallen-Key filter as: V ( s) T ( s) = o Vi ( s ) (9) KN FF ( s ) = D( s ) − KN FB ( s ) KN FB (s ) can modify the coefficients of the denominator polynomial of T (s ) . (7). 2 The low-pass positive Sallen-Key filter The low-pass positive Sallen-Key filter is shown in Fig. Using Equations (5). 2. the circuit is referred to as the negative Sallen-Key filter. we can classify two kinds of Sallen-Key filters.and TFB ( s ) = N FB ( s ) D( s) (8) where N FF (s ) and N FB (s ) are polynomials with degree of at most two. If the amplifier gain K > 0 . It is evident that the poles of T (s ) can be placed anywhere in the complex plane by appropriately choosing K . According to the different K values. From this configuration. we can derive the transfer function T (s) as: K R1 R2 C1C 2 T (s) = ⎡ 1 − K R1 + R2 ⎤ 1 s2 + ⎢ + ⎥s + R1 R2 C1C 2 ⎣ R2 C 2 R1 R2 C1 ⎦ (12) where the amplifier gain K is given by: K= Ra + Rb Ra (13) From Equation (12). we can obtain the cutoff frequency as: ω0 = and the quality factor as: 1 R1 R2 C1C 2 (14) .compute the feedforward and feedback gains of the passive network as: TFF ( s ) = N FF ( s ) D( s) 1 R1R2C1C2 = ⎡ 1 R + R2 ⎤ 1 s2 + ⎢ + 1 ⎥s + R1R2C1C2 ⎣ R2C2 R1R2C1 ⎦ (10) and TFB ( s ) = N FB ( s ) D( s ) 1 s R2C2 = ⎡ 1 R + R2 ⎤ 1 s2 + ⎢ + 1 ⎥s + R1R2C1C2 ⎣ R2C2 R1R2C1 ⎦ (11) Substituting Equations (10) and (11) into Equation (9). we choose R1 = R2 = R and C1 = C 2 = C . we can easily obtain the value of Rb as: Rb = ( K − 1) Ra (18) Note that the DC gain is equal to K for the above design. and then compute the resistance value of R . 4. for instance. we can arbitrary choose three values. 3. a voltage divider can be used to replace R1 . 2. R2 . C 2 ) and the active parameter (the amplifier gain K = Thus. C1 .1 R1R2C1C2 Q= 1 − K R1 + R2 + R2C2 R1R2C1 = (1 − K ) (15) 1 R1C1 ⎛ R ⎞ RC + ⎜⎜1 + 2 ⎟⎟ 1 2 R2C2 ⎝ R1 ⎠ R2C1 Design Procedure for Low-Pass Positive Sallen-Key Filters 1. The first needs no elaboration. Although the element spread for the . The second means that resistance and capacitance values should not be spread too widely. There are five adjustable parameters in the circuit as shown in Fig. Here. Based on the above choice. We choose the proper capacitance value of C . we can rewrite the expression for ω 0 and Q as: 1 RC (16) 1 3− K (17) ω0 = and Q= Hence. 2: the four passive component values ( R1 . Each choice leads to a filter with different properties. we can calculate K and RC . If we decide the value of Ra . Ra + Rb Ra ). The rationale for this choice relies on two principal aims: mathematical convenience and low element spread. given values of ω 0 and Q . If we want the DC gain to be unity. 3 The low-pass negative Sallen-Key filter The low-pass negative Sallen-Key filter is shown in Fig. Thus. it turns out that the quality factor is strongly sensitive to variations in component values.equal-R and equal-C filter is excellent. it is of some interest to investigate an alternative design in practice. Low-Pass Negative Sallen-Key Filter R3 Ro Vi (s ) R1 R4 R2 Vo (s ) C1 + C2 Fig. We can use the same method to derive the transfer function of the low-pass negative Sallen-Key filter as: −K R1 R2C1C2 T ( s) = ⎡ 1 1 1 1 1 ⎤ ⎡1 + K 1 1 1 1 ⎤ 1 + + + + + + + + s2 + ⎢ ⎥s + ⎢ ⎥× ⎣ R3C1 R1C1 R2C1 R4C 2 R2C 2 ⎦ ⎣ R2 R3 R1 R2 R3 R4 R1 R4 R2 R4 ⎦ C1C2 (19) where the amplifier gain K is given by: K= Ro R4 (20) The cutoff frequency is given by: ⎛1+ K 1 1 1 1 ⎞ 1 ⎟⎟ × + + + + ω 0 = ⎜⎜ ⎝ R2 R3 R1R2 R3 R4 R1R4 R2 R4 ⎠ C1C2 and the quality factor is expressed as: (21) . 3. Equations (21) and (22) can be simplified as: K +5 RC ω0 = (23) and Q= K +5 5 (24) 2. We choose the resistance and capacitance values as C1 = C 2 = C . 5. R1 = R2 = R3 = R4 = R and Then. We solve the following equation for the value of RC . We compute the resistance value of Ro as: Ro = KR4 III. Design Design example 1: the low-pass positive Sallen-Key filter (27) . we obtain K as: K = 25Q 2 − 5 (25) 3. We choose the proper capacitance value of C . Given the Q value. RC = K +5 ω0 (26) 4.⎛1+ K 1 1 1 1 ⎞ ⎜⎜ ⎟⎟ + + + + R R R R R R R R R R 1 2 3 4 1 4 2 4 ⎠ ⎝ 2 3 Q= ⎛ 1 ⎛ 1 1 1 ⎞ C 1 ⎞ C ⎜⎜ + + ⎟⎟ 2 + ⎜⎜ + ⎟⎟ 1 ⎝ R1 R2 R3 ⎠ C1 ⎝ R2 R4 ⎠ C 2 (22) Design Procedure for Low-Pass Negative Sallen-Key Filters 1. and then compute the resistance value of R . and write conclusions. “Sallen-Key Filters. Simulate the above two designed low-pass filters. Robert Janes Martin. is from 10Hz to 30kHz. Oxford University Press. Your report should include the following: 1. in RC Active Filter Design Handbook. M. V. References [1]. The experiment results: the magnitude and phase responses of both filters and the recorded graphs. . and 10kHz. [2]. 4.” University of Central Florida. Plot the magnitude and phase responses of these two filters in the frequency range 10Hz-30kHz. Measure the magnitude and phase responses of two designed filters. 1kHz. Follow the above procedure to choose the appropriate values of capacitors and resistors. Discuss any discrepancies. Summary and conclusions. 2. Van Valkenburg. 3. and channel Two to show the output voltage waveform. you need present the experimental results and compare them with the expected results.” Chapter 6. 10Hz. IV. Lab Report In the report. Record the input and output voltage waveforms at the frequencies. 100Hz. Build two designed filters. The computer simulation results: the magnitude and phase responses of both filters. pp 171-179. The complete circuit design processes and results. make comments. Computer Simulations 1. E. Design example 2: the low-pass negative Sallen-Key filter The filter specifications are given by ω 0 = 2π * 4000 rad / s and Q = 1 . The frequency range VI. 2. Analog Filter Design.The filter specifications are given by ω 0 = 2π *1000 rad / s and Q = 5 . Use the Channel One of digital oscilloscope to show the input voltage waveform. using LF 351 Op Amps with a split power supply voltage of ±15V. 1997. Experiments 1. Artice M. Follow the above procedure to choose the appropriate values of capacitors and resistors. [3]. 3. Davis. 2. Dr. “EEL 4140: Lab Manual for the Design of Analog Filters. 1982. 1985. Inc.Edited by F. . W. John Wiley & Sons.. Stephenson. three basic active building blocks are generally used: the summer. A relatively high quality value can be achieved in this circuit. II. we can obtain the cutoff frequency as: (1) . 1. R5 . another biquad. while the second block is a summer amplifier. Sallen-Key filter. two kinds of state-variable biquads will be introduced: Tow-Thomas biquad and Kerwin-Huelsman-Newcomb (KHN) biquad. In this configuration. and low sensitivity. a single amplifier embedded in a passive RC network is used to generate a second-order transfer function. state-variable biquads provide flexibility. Tow-Thomas Biquad The structure of the Tow-Thomas biquad is shown as in Fig. the integrator. and the third one is an integrator. it is subject to great sensitivity to the constituent components’ values. and the lossy integrator.EEL 4140 ANALOG FILTERS LABORATORY 4 State-Variable Biquads I. In this experiment. R4 . In this experiment. one realization of biquads. However. good performance. Analyzing the Tow-Thomas biquad. all positive terminals of the Op Amps are grounded. State-variable methods of solving differential equations are employed in the development of the realization. Objective To study design and implementation of state-variable biquads. Introduction In the previous experiment. and R6 constitute feed forward paths to obtain the transmission zeros. will be studied. The first basic building block composed of the Op Amp U1 is a lossy integrator. In addition. The path composed of R3 is feedback. In the implementation of these realizations. we can derive the second-order transfer function from this realization as: ⎛ 1 1 R6 ⎞ R6 1 ⎟⎟ + s 2 + s⎜⎜ − − R8 ⎝ R1C1 R4C1 R7 ⎠ R7 R3 R5C1C2 T (s) = R8 1 R6 s2 + s + R1C1 R2 R3 R7 C1C2 From Equation (1). was discussed. The structure of the Sallen-Key filter is relatively simple. The implementation of the state-variable baud is based on the state-variable approach. the state-variable structure. In the Sallen-Key filter. For example.ω0 = R8 R2 R3 R7 C1C2 (2) R8C1 R2 R3 R7 C2 (3) and the quality factor as: Q = R1 Since the numerator is of general second-order form. we can realize the second–order band-pass filter by choosing R5 = R6 = ∞ (4) Where ∞ denotes the infinite value. substitute Equation (4) into Equation (1). we can achieve any filter type by choosing proper resistor values. Equation (4) implies that R5 and R6 are not present in the circuit. the transfer function specified in R8 R4 R7 C1 (5) R8 1 s +s + R1C1 R2 R3 R7 C1C 2 2 In Equation (5). Equation (1) is simplified as: s T (s) = We Therefore. In practice.1 The feed forward Tow-Thomas circuit U3 . the voltage gain at the center frequency ω 0 is given as: H BP = R1 R8 R4 R7 (6) R3 R1 C1 V1 ( s ) C2 R8 R4 R7 + R2 + U1 U2 + Vo (s ) R6 R5 Fig. Q . and H BP . Equations (2). and (6) can be simplified as following: ω0 = 1 αRC (10) R Q= 1 αR (11) R H BP = 1 R4 (12) 5. Determine the resistance value of R2 as: R2 = α 2 R (14) 7. Compute the positive number α as: α= 1 ω 0 RC (13) 6. Based on the above chosen parameter values. Define a positive constant α such that R2 = α 2 R3 =α R 2 (9) 4. (3). Given the filter specifications as ω . Choose the resistance and capacitance values C and R such that C1 = C2 = C (7) and R3 = R7 = R8 = R (8) 3. Compute the resistances values of R1 and R4 as: R1 = QαR and (15) . 2.Design Procedure for band-pass Tow-Thomas filters 1. these three transfer functions have R5 C1 R6 C2 R2 R1 V1 ( s ) R3 + U1 + V2 ( s ) V4 ( s ) + U2 R4 U3 V3 ( s ) Fig. the same poles. and low-pass filter. Moreover. This circuit is R4 and R5 form the feedback The three output terminals voltages. composed of a summer amplifier and two integrators.R4 = R1 H BP (16) Kerwin-Huelsman-Newcomb Biquad The Kerwin-Huelsman-Newcomb (KHN) biquad is shown in Fig.2. and V4 ( s ) . V2 ( s ) . band-pass. respectively.2 The Kerwin-Huelsman-Newcomb (KHN) circuit The high-pass transfer function is obtained by the ratio of V2 ( s ) and V1 ( s ) as: V2 ( s ) H HP s 2 = V1 ( s ) s 2 + ⎛ ω 0 ⎞ s + ω 2 ⎜ Q⎟ 0 ⎠ ⎝ (17) Where H HP is given by: R 1+ 6 R5 H HP = R 1+ 3 R4 the cutoff frequency ω 0 is given by: (18) . paths. achieve high-pass. V3 ( s ) . Q . (24) . and H BP .R6 R5 ω 02 = R1R2C1C2 (19) and the quality factor Q is given by: R 1+ 4 R3 Q= ⎛ R ⎞ R R C ⎜⎜1 + 6 ⎟⎟ 5 2 2 R5 ⎠ R6 R1C1 ⎝ (20) From the relationship between V1 ( s ) and V3 ( s ) . expressed as: R 1+ 5 R6 H LP = R 1+ 3 R4 Design Procedure for band-pass KHN filters 1. Given the design specifications: ω 0 . the band-pass transfer function is given as: ω − H BP ⎛⎜ 0 ⎞⎟ s V3 ( s ) ⎝ Q⎠ = V1 ( s ) s 2 + ⎛ ω 0 ⎞ s + ω 2 ⎜ Q⎟ 0 ⎝ ⎠ (21) where H BP is the voltage gain at the frequency ω 0 as: R H BP = 4 R3 (22) The low-pass filter is achieved by the relationship between V1 ( s ) and V4 ( s ) as: H LPω 02 V4 ( s ) = V1 ( s ) s 2 + ⎛ ω 0 ⎞ s + ω 2 ⎜ Q⎟ 0 ⎝ ⎠ (23) where H LP is the DC gain. 2. Choose the capacitance value C such that C1 = C 2 = C (25) 3. Let R2 , R3 , R5 , and R6 have the same resistance value as: R2 = R3 = R5 = R6 = R (26) 4. Define a positive constant number α such that R1 = α 2 R2 = α 2R (27) 5. Equations (19), (20), and (22) are simplified as: ω0 = Q= 1 αRC (28) R ⎞ ⎜⎜1 + 4 ⎟⎟ 2⎝ R3 ⎠ α⎛ H BP = 2 Q α −1 (29) (30) 6. Compute the positive constant number α as: α= 2Q H BP + 1 (31) 7. Compute the resistance value of R as: R= 1 ω 0αC (32) 8. Compute the resistance values of R1 and R4 as: R1 = α 2 R (33) and ⎛ Q ⎞ R4 = ⎜ 2 − 1⎟ R ⎝ α ⎠ (34) III. Design Design example 1: the band-pass Tow-Thomas filter Design a band-pass filter having the gain H BP = 3 , the quality factor Q = 5 , and the cutoff frequency ω 0 = 2π *1000 rad / s , using the Tow-Thomas circuit. Choose the appropriate values of capacitors and resistors, following the above procedure. Design example 2: the band-pass Kerwin-Huelsman-Newcomb (KHN) filter Design a band pass filter having the gain H BP = 3 , the quality factor Q = 10 , and the cutoff frequency ω 0 = 2π *1000 rad / s , using the KHN circuit. Choose the appropriate values of capacitors and resistors, following the above procedure. IV. Computer Simulations 1. Simulate the two designed band-pass filters. 2. Plot the magnitude and phase responses in the frequency range from 10Hz to 20 kHz. 3. Compare these two magnitude response plots, and understand the mechanism of the quality factor Q . V. Experiments 1. Build above two designed biquad circuits, using LF 351 Op-amps with a split power supply voltage of ±15V. 2. Use the Channel One of digital oscilloscope to show the input voltage waveform, and channel Two to show the output voltage waveform. Record the input and output voltage waveforms at the frequencies, 10Hz, 100Hz, 1kHz, and 10kHz. 3. Measure the magnitude response for two circuits. 20kHz. The frequency range is from 10Hz to VI. Lab Report In the report, you need present the experiment results and compare them with the simulation results. Discuss any discrepancies, make comments, and write conclusions. Your report should include the following: 1. The complete circuit design processes and results. 2. The computer simulation results: the magnitude and phase responses for both circuits. 3. The experiment results: the magnitude responses for both filters and the recorded graphs. Note that two band-pass filters have the same center frequency and the same voltage gain, but the different quality factors. Compare the magnitude responses of these filters, and understand the rule of the quality factor Q p . 4. Summary and conclusions. References [1]. M. E. Van Valkenburg, “Analog Filter Design”, Chapter 5, Oxford University Press, 1982. [2]. Dr. Robert Janes Martin, “EEL 4140: Lab Manual for the Design of Analog Filters,” University of Central Florida, 1997. [3]. E. Sanchez-Sinencio, “Biquad I: The State-Variable Structure,” Chapter 8, in RC Active Filter Design Handbook, Edited by F. W. Stephenson, John Wiley & Sons, Inc., 1985. In early radio. two major differences. The sharper the transition from the pass band to the stop band. The circuit used in this experiment is the multi-loop feedback filter. The figure of merit for a band-pass filter is the quality factor which is defined as the ratio of the center frequency to the 3dB bandwidth. A larger center frequency to bandwidth ratio was associated with higher quality factor. First. electro-optical systems. multi-loop feedback filter. tuned radio frequency filters were used directly in front of the detector. high-pass. the better the rejection of the adjacent channels. and communications systems. The first difference is that the active element is an Op Amp in the multi-loop feedback structure rather than a finite-gain amplifier. In this experiment. This is the reason for the name. or band-pass filter by choosing the different RC networks. only the band-pass filter is discussed. Then. 1. II. This led to better separation of stations. The second difference is that there are two feedback paths (rather than one) from the output of the amplifier to the RC network. Introduction Band-pass filters find wide use in modems. There are. The design procedures for these two realizations are also given. Objective To design and implement single Op Amp band-pass biquad filters. another band-pass realization increases the quality factor by incorporating the positive feedback. This structure is similar to Sallen-Key filters. Its basic circuit configuration is shown in Fig. a basic band-pass structure without the positive feedback is introduced. radio receivers.EEL 4140 ANALOG FILTERS LABORATORY 5 Single Op Amp Band-Pass Filters I. however. The multi-loop feedback circuit can achieve the low-pass. Vi (s ) Passive RC network + - Vo (s) . The term quality factor was first coined for band-pass filters in the first public communication systems. These radios were very noisy and had poor selectivity by today’s standards. (1) From Equation (1). we obtain the quality factor Q and the cutoff frequency ω 0 as: 1+ Q= R3 R1 R3C1 R3C 2 + R2 C 2 R2 C1 (2) and R3 R1 R2 R3C1C 2 1+ ω0 = (3) In addition.Fig. 1 The basic multi-loop feedback filter configuration Band-Pass Multiple-Loop Feedback Filters without the Positive Feedback The band-pass multiple-loop feedback filter without the positive feedback is shown in Fig. 2. we can get the midband gain as: R2 R1 H BP = C 1+ 2 C1 (4) . The transfer function is given by: V (s) T ( s) = o Vi ( s ) −s R1C2 = ⎞ ⎛ 1 1 ⎞ ⎛ 1 1 ⎟⎟ ⎟⎟ s + ⎜⎜ + s 2 + ⎜⎜ + ⎝ C2 R2 C1R2 ⎠ ⎝ C1C2 R1R2 C1C2 R2 R3 ⎠ Notice that this band-pass transfer function is an inverting one. it is convenient to use the equal-value capacitor design. Given the design specifications. 2 The band-pass multi-loop feedback filter without positive feedback Design Procedure for Band-Pass Multiple-Loop Feedback Filters without the Positive Feedback 1. for the capacitors C1 and C 2 . Determine the resistance value of R2 as: R2 = 2Q ω 0C (7) 5. 2. as: C1 = C 2 = C (5) 3.C2 Vi (s ) R2 C1 R1 Vo (s ) R3 + Fig. In the band-pass multi-loop feedback filter. C . Determine the resistance value of R3 as: R3 = Q ω 0 C 2Q 2 − H BP ( ) (8) . and H BP . Q . Determine the resistance value of R1 as: R1 = Q ω 0CH BP (6) 4. Choose a suitable capacitance value. ω 0 . To simplify equations describing the circuit. 3. H BP must be less than 2Q 2 in order that R2 be finite and positive. the effect of these two resistors is presented by defining a constant K as: K= Ra Rb C2 Vi (s ) (9) R2 C1 R1 Vo (s ) + Ra Rb Fig.2 can be used to reduce the spread of element values for high Q realizations. The positive feedback is provided by the voltage divider consisting of two resistors Ra and Rb . we obtain the quality factor Q and the cutoff frequency ω 0 as: (10) .Note that in Equation (8). The modified circuit is shown in Fig. 3 The band-pass multi-loop feedback filter with positive feedback The transfer function for this circuit is given by: V (s) T ( s) = o Vi ( s ) − s (K + 1) R1C2 = ⎛ 1 K ⎞ 1 1 ⎟⎟ + s 2 + s⎜⎜ + − ⎝ R2C2 R2C1 R1C2 ⎠ R1R2C1C2 From Equation (10). Band-Pass Multiple-Loop Feedback Filters with Positive Feedback A modification of the band-pass filter shown in Fig. (15) . 2. and is summarized in following. C . Design Procedure for Band-Pass Multiple-Loop Feedback Filters with the Positive Feedback 1. As before. Determine the ratio parameter m0 that would be required if there are no positive feedbacks as: m0 = 1 4Q 2 4. The design procedure for this filter is based on the solution of these equations. Given the design specifications ω 0 and Q . which is less than one. Choose a suitable capacitance value. and greater than m0 . in which the capacitors C1 and C 2 have the same value. it is clear that the quality factor is improved by introducing positive feedback K . it is convenient to use an equal-value capacitor design.1 = Q R1 ⎛ C1 C2 ⎜ + −K R2 ⎜⎝ C2 C1 R2 R1 C2 C1 ⎞ ⎟ ⎟ ⎠ (11) and ω0 = 1 R1 R2 C1C 2 (12) The midband gain is given as: (K + 1) H BP = R1C1 1 1 K + − R2C 2 R2 C1 R1C1 (13) From Equation (11). Select the desired resistor ratio m . for the capacitors as: C1 = C 2 = C (14) 3. using the multi-loop structure without positive feedback. Determine the value of R1 using the relation R1 = mR2 (20) 8. and calculate Ra as: Ra = KRb (18) 6. Design Design example 1: the second-order band-pass multi-loop filter without positive feedback Design a second-order band-pass filter having the gain H BP = 2 . the gain at resonance. The value of H BP . Choose a convenient value for Rb . Determine the resistance value of R2 as: R2 = 1 ω 0C m (19) 7. H BP = K +1 2m − K (21) III. Design example 2: the second-order band-pass multi-loop filter with positive feedback Design a second-order band-pass filter having the quality factor Q = 10 and the center . Choose the appropriate values of capacitors and resistors. and the center frequency ω 0 = 10 4 rad / s . is determined by the relation. the quality factor Q = 2 . following the above procedure.R m= 1 R2 (16) Use m to determine the amount of positive feedback K as: K = 2m − m Q (17) 5. 2. and channel Two to show the output voltage waveform. The Computer simulation results: the magnitude and phase responses for both circuits. [2]. John Wiley & Sons. Stephenson. Simulate the above two band-pass filters with the designed resistance and capacity values. E. Computer Simulations 1. 4. Robert Janes Martin. make comments. Build above two designed band-pass filters. 30kHz. in RC Active Filter Design Handbook. you need present the experiment results and compare them with the simulation results.. 2. [3]. 3. IV. and write conclusions. Lab Report In the report. Dr. Analog Filter Design. Experiments 1. 1kHz. Choose the appropriate values of capacitors and resistors. . Huselsman. using the multi-loop structure with positive feedback. “EEL 4140: Lab Manual for the Design of Analog Filters. Plot the magnitude and phase responses. W. The frequency range is from 10Hz to VI. Measure the magnitude responses of two filters. Edited by F.” Chapter 7. Discuss any discrepancies. 2. Van Valkenburg.frequency ω 0 = 10 4 rad / s . V. Inc. References [1]. The complete circuit design processes. Your report should include the following: 1. Use the Channel One of digital oscilloscope to show the input voltage waveform. 100Hz. Record the input and output voltage waveforms at the frequencies. for the frequency range 10Hz-30kHz. P. 3. L. following the above procedure. and 10kHz. M. 1982. Oxford University Press.” University of Central Florida. 10Hz. 1997. The Experiment results: the magnitude responses for both filters and the recorded graphs. using LF 351 Op-amps with a split power supply voltage of ±15V. “Multiple-Loop Feedback Filters. 1985. Summary and conclusions. are given below: V ( s) T1 ( s ) = 3 Vi ( s ) ⎡ ⎛ Y ⎞ YY ⎤ Y5 + h( s ) ⎢Y7 ⎜⎜1 + 6 ⎟⎟ − 5 8 ⎥ ⎣ ⎝ Y2 ⎠ Y2 ⎦ = D( s) (1) V ( s) T2 ( s ) = 4 V1 ( s ) ⎛ Y ⎞ YY Y5 ⎜⎜1 + 8 ⎟⎟ − 6 7 + h( s )Y7 Y4 ⎠ Y4 = ⎝ D( s) (2) and V ( s) T3 ( s ) = 2 Vi ( s ) Y + h( s )Y7 = 5 D( s) (3) where YY h( s ) = 2 3 Y1Y4 (4) . and to functionally tune the above circuits to get the specified values of the cutoff frequency and the quality factor. V3 ( s ) .EEL 4140 ANALOG FILTERS LABORATORY 6 Two Op Amps Current Generalized Immittance Structure (CGIC) Based Biquad I. The transfer functions between the input and output terminals. II. assuming ideal Op Amps. and V4 ( s ) . Objective To study CGIC biquads. V2 ( s ) . to use this structure to design second-order low-pass and band-pass filters with given specifications. 1. Introduction The general CGIC structure is shown in Fig. and a band-pass filter using the number 7 circuit.D ( s ) = (Y5 + Y6 ) + h( s )(Y7 + Y8 ) (5) The most commonly used second-order transfer functions can be easily generated from the above equations. as shown in Table 2. By using minimum sensitivity constraints in circuit 1. two kinds of filters will be discussed in detail: a low-pass filter using the number 1 circuit. 1 The basic CGIC configuration Design Procedure for Low-Pass CGIC Filters 1. 7. Given the design specifications. Y1 Y3 - V3 ( s ) Y2 Y4 + + V4 ( s) Y7 Y5 V2 ( s ) V1 ( s ) Y6 Vi (s ) Y8 Fig. The resistor and capacitor are chosen as follows: G1 = G4 = G5 = G8 = G G3 = G Q (7) (8) and C 2 = C3 = C (9) . 10. as summarized in Table 1. 3. ω 0 and Q . possible sets of element values have been obtained. 2. Circuit 1 in Table 2 realizes a low-pass filter. and 12. In this experiment. Choose an appropriate capacitor value C . compute the resistor value R as: R= 1 Cω o (12) R3 = RQ (13) 5. 1 RC (11) Then. 2 The second-order low-pass CGIC filter .where G is the conductance define as G = 1 . Consequently. R 3. Then. C3 R3 R1 - - C2 R4 + Vi (s ) + R5 Vo (s ) R8 Fig. we obtain ω0 = 4. the transfer function is simplified as: 2 T2 ( s ) = 2 2 R C 1 s s2 + + 2 RCQ R C 2 (10) From the definition of ω 0 . compute the resistance value of R as: R= 1 Cω o (19) R7 = RQ (20) 5. 2. Circuit 7 in Table 2 realizes a band-pass filter. The transfer function is simplified as: 2 RCQ T1 ( s ) = 1 s + s2 + RCQ R 2 C 2 s 4.Design Procedure for Band-Pass CGIC Filters 1. The resistance and capacitance values are chosen as follows: C 3 = C8 = C (14) G1 = G2 = G4 = G6 = G (15) G Q (16) G8 = 0 (17) G7 = and G is the conductance define as G = 1 . R 3. Consequently. (18) Then. Given the design specifications ω 0 and Q . . Choose an appropriate capacitor value C . following the above procedure. The circuits need to be functionally tuned to yield the specified values of the quality factor .C3 R1 - Vo ( s ) R4 R2 + R6 + Vi ( s ) R7 C8 Fig. Plot the magnitude and phase responses in the frequency range from 50Hz to 20kHz. Design example 2: the second-order band-pass CGIC filter Design a second-order band-pass filter having a quality factor Q = 10 . Build above the CGIC filters designed in part III. Choose the capacitor value as C = 0. following the above procedure. Design Design example 1: the second-order Butterworth low-pass CGIC filter Design a second-order low-pass filter with a quality factor Q = 0.01uF . Compute the appropriate values resistors. Computer Simulations 1. Simulate both of the above design filter examples with the calculated resistance values. Choose the capacitance value as C = 0. Experiments 1. 3 The second-order band-pass CGIC filter III. 2.707 . V.01uF . and a cutoff frequency ω 0 = 2π * 2 *10 3 rad / s . IV. Compute the appropriate values resistors. and a cutoff frequency ω 0 = 2π * 3 *103 rad / s . 2. using LF 351 Op Amps with a split power supply voltage of ±15V. 100Hz. Record the input and output voltage waveforms at the frequencies. A sinusoidal input is used during the tuning processes. the desired value of ω 0 is realized by adjusting R2 until the phase angle between the output and the input voltage equals 0 degrees at the sinusoidal input of the frequency ω 0 . Compute the quality factor Q and the cutoff frequency ω 0 from the measured resistance . 3. the gain H LP (the ratio of the output voltage between the input voltage) at a low frequency. In the case of the low-pass filter. VI. Lab Report In the report. then adjusting R8 and monitoring the phase angle difference between the output and input voltages. is determined. and channel Two to show the output voltage waveform. and the reasons for these deviations. using the specified element values. 1kHz. present the experimental results and compare them with the simulation results. Then. and 10kHz. R3 is adjusted until the gain of this filter become Q times H LP . a sinusoidal input at the frequency ω 0 is applied. 4. The frequency range is from 10Hz to 30kHz. much lower than ω 0 (usually a few Hz but not to 0Hz). Applying a sinusoidal input of the frequency ω 0 . In case of the band-pass filter. Derive the transfer function for the above two circuits in the form supplied in the table from the general form. Use the Channel One of digital oscilloscope to show the input voltage waveform. Your report should include the following: 1. the circuit is first tuned for ω 0 . To tune the circuit to attain the specified Q . 10Hz. Q can be attained by adjusting R7 until the output voltage advances the input voltage by 45 degrees when the frequency of the sinusoidal input is the lower cutoff frequency ω1 . 2. Comment on the deviations from the expected results. The circuit is tuned for ω 0 when the output voltage lags the input voltage by 90 degrees. Measure the magnitude response for each of two circuits after functional tuning is complete.Q and the cutoff frequency ω 0 . The computer simulation results: the magnitude and phase responses for both circuits. Mikhael. CRC Press.” Lab Manual. pp 2495-2514. W.” Chapter 9. . John Wiley & Sons. 5. in Circuits and Filter Design Handbook. W. Stephenson. in RC Active Filter Design Handbook. Mikhael. The experiment results: the magnitude responses for both filters and the recorded graphs. 4. [2]. [3]. “2 OA Current Generalized Immittance Structure (CGIC) Based Biquad. 1985. 3.. Summary and conclusions. “Biquad II: The Current Generalized Immittance (CGIC) Structure. Inc. Edited by F. 2003. References [1]. University of Central Florida. Stephenson. Wasfy B. “The Current Generalized Immittance (CGIC Biquad). Wasfy B. Wasfy B.” Chapter 82. Mikhael. Compare Q and ω 0 with the desirable specifications.values after functional tuning is complete. Edited by F. Table 1 Element identification for realizing the most commonly transfer functions Circuit Number Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 1 G1 sC2 sC3+G3 G4 G5 0 0 G8 2 sC1 G2 G3 sC4+G4 0 G6 G7 sC8+G8 3 G1 G2 sC3 G4 G6 sC7 sC8+G8 4 G1 G2 sC3 G4 G6 sC 7 1 + sC 7 R7 0 0 0 Transfer Function T2 ( s ) = G1G5 (G4 + G8 ) s 2 C 2 C3 G8 + sC 2 G3G8 + G1G4 G5 ⎛ G ⎞ G2 G3 G7 ⎜⎜1 + 6 ⎟⎟ ⎝ G2 ⎠ T1 ( s ) = 2 s C1C 4 G6 + s (C 2 G4 G6 + C8 G2 G3 ) + G2 G3 (G7 + G8 ) T1 ( s ) = T1 ( s ) = s 2 C 3C 7 (G2 + G6 ) s 2 (C 7 + C8 )G2 C3 + sC3G2 G8 + G6 G1G4 s 2 C3C 7 (G2 + G6 ) s C3C 7 G2 + sC 7 G1G4 G6 R7 + G6 G1G4 Remarks Low-pass Low-pass High-pass High-pass 2 5 G1 sC2 sC3 G4 sC5 G6 0 G8 ⎛ G ⎞ sC5 G1G4 ⎜⎜1 + 8 ⎟⎟ ⎝ G4 ⎠ T2 ( s ) = 2 s C 2 C3 G8 + sC5 G1G4 + G1G4 G6 Band -pass 6 sC1 G2 G3 sC4+G4 0 G6 sC7 G8 ⎛ G ⎞ sC 7 G2 G3 ⎜⎜1 + 6 ⎟⎟ ⎝ G2 ⎠ T1 ( s ) = 2 s C1C 4 G6 + s(C 7 G2 G3 + C1G4 G8 ) + G2 G3G8 Band-pass 7 G1 G2 sC3 G4 0 G6 G7 sC8+G8 8 sC1 G2 G3 sC4 G5 0 G7 sC8 T1 ( s ) = sC3G7 (G2 + G6 ) s 2 C3C8 G2 + sC3G2 (G7 + G8 ) + G1G4 G6 T2 ( s ) = s 2 C1G5 (C 4 + C8 ) + G2 G3G7 s 2 C1C 4 G5 + sC8 G2 G3 + G2 G3G7 Band-pass Notch . 9 G1 G2 sC3 G4 G5 0 sC7 G8 10 G1 G2 sC3 G4 G5 G6 sC7 sC8+G8 11 sC1 G2 G3 sC4 G5 G6 G7 sC8 12 G1 G2 sC3 G4 G5 G6 sC7 G8 T2 ( s ) = s 2 C3C 7 G2 + G1G5 (G4 + G8 ) s C 3C 7 G2 + sC3 G2 G8 + G1G4 G5 Notch 2 ⎛ 2 G1G4G5 ⎞ C7 ⎜⎜ s + ⎟ C3C7 G2 ⎟⎠ C7 + C8 ⎝ T3 ( s ) = G8 G G (G + G6 ) s2 + s + 1 4 5 C7 + C8 C3G2 (C7 + C8 ) T1 ( s ) = s 2 C1G4 G5 − sG5 C8 G3 + G3G7 (G2 + G6 ) s 2 C1C 4 (G5 + G6 ) + sC8 G2 G3 + G2 G3 G7 T1 ( s ) = s 2 C 3C 7 (G2 + G6 ) − sC3G5 G8 + G1G4 G5 s 2 C3C 7 G5 + sC3 G2 G8 + (G5 + G6 )G1G4 Notch For all-pass: G6 = 0 G5 = G 2 For all-pass: G6 = 0 G5 = G 2 . Table 2 Design values and tuning procedure Circuit Number Design values Transfer function realized ωn (from Table 1) 1 G1 = G4 = G5 = G8 = G , G3 = G Q , C 2 = C3 = C , where C = G ω 0 3 G1 = G2 = G4 = G6 = G , G8 = G Q , C8 = 0 , C3 = C7 = C , where C = G ω0 7 G1 = G2 = G4 = G6 = G , G7 = G Q , G8 = 0 , C3 = C8 = C , where C = G ω0 10 G1 = G2 = G4 = G5 + G6 = G , G8 = G Q , C3 = C7 + C8 = C , where ω 0 = G C and ω n2 = ω 0G5 C 7 12 Tuning sequence G1 = G2 = G4 = G5 = G , G6 = 0 , C3 = C7 = C , G8 = G Q , where C = G ω0 Note: D( s ) = s 2 + (ω 0 Q )s + ω 02 ω0 Q T2 ( s ) = 2ω 02 D( s ) G8 G3 T1 ( s ) = 2s 2 D( s) G4 G8 G2 G7 G2 G6 G8 G4 G4 G8 T1 ( s ) = (2ω 0 Q )s D( s) ( C7 s 2 + ω n2 T3 ( s ) = C D( s) T1 ( s ) = D(− s ) D( s) ) Study Guide C EEL 4140 ANALOG FILTERS DESIGN Biquads II: The current Generalized Immittance (CGIC) Structure . . . . . . . . . . . . . . . . . The cascade design method usually consists of three steps: decomposing transfer functions into poles and zeros. clipping may occur at output of an internal Op Amp before the filter output voltage shows clipping. must occur in conjugate pairs. one design method for the high-order filter is introduced: the cascade design method. any form of transfer functions can be expressed in term of poles and zeros. II. and synthesizing these filter sections. Thus. the effect of component variations on the overall frequency response is large. However.” This refers to the phenomenon of the large-voltage buildup at certain frequencies at internal nodes of filters (usually at the output terminal of an Op Amp). This internal clipping will . when filters are produced in large quantities and are subjected to stringent specifications. It is based on cascading first-order and second-order filter sections in such a manner that each section does not interact with others. Poles and zeros can be either real numbers or complex numbers. Current Generalized Immittance Structure (CGIC) Biquads are employed to build high-order low-pass filters. Complex zeros and poles. Step 1: Decomposing transfer functions In general. Consequently. This method is the most commonly used method because of its simplicity. Thus. grouping poles and zeros into first-order or second-order filter sections. In this experiment. or where filter specifications are not particularly tight. Introduction The previous experiments have discussed the design of first-order and second-order filters. Objective To study the cascade design method for realizing high-order filters. For the cascade design method. however. the cascade design method is not recommended. it is important to consider the occurrence of “internal resonances. the cascade design method may be tolerated for the applications where only a few filters are constructed and manual tuning is used. Step 2: Grouping poles and zeros into first-order or second-order filter sections In determining the order of sections and pairing of poles and zeros.EEL 4140 ANALOG FILTERS LABORATORY 7 High-Order Low-Pass Filter Design I. the gain to one of these nodes from the input of the filter may be higher than the gain to the output of the filter. When we cascade these sections together. and will restrict the range of input signal levels that the filter can handle. which are closest in the S plate. 1. (The Q value of the first-order section is assumed to be zero. If the output impedance of the last section is not zero. the low-pass CGIC Biquad. In this experiment. Rule 3: Distribute the overall gain equally among the sections. is used as the build-up circuit block for the high-order low-pass filter. The following rules of thumb are useful in maximizing the dynamic range of cascaded design filters. which was studied in the last experiment. Step 3: Synthesizing these first-order and second-order filter sections For each filter section decided in step 2. In order to achieve maximum dynamic range. we must consider effects of the input and output impedance.) Rule 2: Group the pole and the zero. we can connect these sections directly.show up as a level-dependent change in the overall frequency characteristic. the resistor and capacitor values are chosen as: R1 = R4 = R5 = R8 = R (1) and C 2 = C3 = C (2) Based on these resistance and capacitance values in Equation (1) and (2). the filter should be designed such that the clipping will first occur at the output of the last Op Amp of the filter. we choose the appropriate circuit to synthesize it. In this Biquad. a buffer is needed to insert between these two sections. and the input impedance of the next section can not be assumed as infinite. Rule 1: Place the sections in the order of increasing Q values. we can obtain the cutoff frequency ω 0 and the quality factor Q as: (3) . The circuit of the CGIC low-pass Biquad is shown in Fig. Otherwise. the transfer function is given by: 2 R C2 TCGIC ( s ) = s 1 s2 + + 2 2 R3C R C 2 From Equation (3). we can compute the resistor value R using Equation (4).12471 ( 2 )( 2 )( ) 3. Then. C3 R3 R1 - - C2 R4 + Vi (s ) + R5 Vo (s ) R8 Fig.061415 (7) s + 0. Design a six-order Chebyshev low-pass filter. it can be seen that this high-order filter consists of three second-order sections. .12436s + 0. 1 The second-order low-pass CGIC Biquad Design Procedure for the High-Order Low-Pass Filter 1. we first choose the capacitor value C . we can compute the resistance value of R3 using Equation (5).0dB and the bandwidth 3979 Hz. From (7).33976s + 0. The normalized six-order Chebyshev low-pass transfer function with a maximum pass-band attenuation of 1.46413s + 0.55772 s 2 + 0. That is ω 0 = 25.99073 s + 0. which has a maximum pass-band attenuation of 1.000rad / s (6) 2.0dB is given as: Tnorm ( s) = 0.ω0 = 1 RC (4) Q= R3 R (5) and When we use this circuit to realize a second-order transfer function. Finally. 24 *10 9 T1(s) = 2 s + 3.97 *10 9 s 2 + 8.99073 s + 0.49 *10 8 ) (12) 1.12471 ( 2 ) To denormalize these three transfer functions.55772 Tnorm 2 ( s ) = ( 2 (8) ) (9) and Tnorm 3 ( s ) = 2 * 0.00 (15) In the same way. we replace s by s = (10) s ω0 in Equations (8).49 *10 3 s + 3.2 and the quality factor Q2 corresponding to the .11*10 3 s + 6.1 = 2.79 *10 7 ) (13) ( T2 ( s ) = ( and ( For the denormalized transfer function T1 ( s ) .48 * 10 4 rad / s (14) and Q1 = 8. we obtain the following denormalized transfer functions as: 1. the cutoff frequency ω 0.Tnorm ( s ) is decomposed into three second-order transfer functions as: 2 * 0.1 and the quality factor Q1 are given as follows: ω 0.99073 Tnorm1(s) = ( 2 ) 2 * 0.12471 s + 0. (9).46413s + 0.192 *10 8 ) (11) 6. Consequently.56 *10 8 T3 ( s ) = 2 s + 1. and (10).55772 s + 0.12436 s + 0.16 *10 4 s + 7. the cutoff frequency ω 0.33976 s + 0. Synthesize T1 ( s ) . Notice that According to Rule 1. we arrange these sections in the increasing order of the quality factor.883 (21) R3 = R * Q1 = 4. For the first section.01 * 24. The T2 ( s ) and T1 ( s ) are synthesized in the second section and the third section. T2 ( s ) . respectively.3 = 8.76 (19) Q1 > Q2 > Q3 (20) 4. and T3 ( s ) using the CGIC low-pass Biquad as Fig. first section is used to synthesize T3 ( s ) . the cutoff frequency ω 0. 2 = 1. .3 and the quality factor Q3 are given by: ω 0.87 *10 4 rad / s (16) and Q2 = 2.1. choose the capacitor value as C = 0.02k * 8 = 32.83 * 103 rad / s (18) and Q3 = 0.1 = and For the second section.16kΩ (22) R= 1 Cω 0.01uF . For all three sections. 5.02kΩ 0. 1 = 4.transfer function T2 ( s ) are given by: ω 0.20 (17) For the transfer function T3 ( s ) . and the low-pass CGIC Biquad as the build-up circuit block.33k * 0.R= 1 Cω 0.61kΩ (26) 6. the output of each section is the Op Amp output. Design Design example: the six-order Butterworth low-pass filter.93186s + 1) (27) The cutoff frequency of this filter is given as f 0 = 2kHz. Computer Simulations 1. using LF 351 Op Amps with a split . Consequently. V.829 (25) and R3 = Q p. Thus. IV. the following resistance values are chosen for the third section. Following the above design procedure. In these Biquads. III. 2. 2 = 1 = 5. 2. there is no need to insert buffer here.36k * 2.01 * 8.79kΩ (24) Similarly.2 = 11. for the frequency range 10Hz-10kHz. and compute the appropriate resistor values.33kΩ 0. The normalized transfer function is given as: T (s) = 1 (s 2 + 0. Simulate the above design filter with the calculated resistance values.36kΩ 0.3 = 1 = 11. The final schematic circuit of this high-order filter is shown in Fig.51764s + 1)(s 2 + 1. In this application.01uF .76 = 8. choose the capacitor value C = 0. Plot the magnitude and phase responses.01 * 18. Cascade these three filter sections into a six-order Chebyshev low-pass filter. Experiments 1. use the cascade method to design this filter.41421s + 1)(s 2 + 1. the output resistance is assumed to be zero.67 (23) and R3 = Q2 R = 5. Build the six-order Butterworth filter designed in part III. R= 1 Cω 0.1 * R = 11. 3. Stephenson. Record the input and output voltage waveforms at the frequencies. in RC Active Filter Design Handbook. “The Current Generalized Immittance (CGIC Biquad). Inc. The computer simulation results: the magnitude and phase responses for this high-order filter. 3. W. John Wiley & Sons. Stephenson. Comment on deviations from expected results. and 10kHz. Lab Report In the report. Wasfy B. John Wiley & Sons. if any. [3]. Nelin. VI. Bert D. Edited by F. Mikhael.” Chapter 82. W.” Chapter 10. 10Hz.. 100Hz. The frequency range is from 10Hz to 10kHz. Edited by F. Use the Channel One of digital oscilloscope to show the input voltage waveform. [2]. E. Stephenson. 1985.power supply voltage of ±15V. Oxford University Press. 1982. 2.” Chapter 9. 2003. . Van Valkenburg. 1kHz. in Circuits and Filter Design Handbook. Your report should include the following: 1. W. Measure the magnitude response of this high-order filter. pp 2495-2514. present the experiment results and compare them with the simulation results. CRC Press. and the reasons for these deviations. References [1]. The experiment results: the magnitude responses for the high-order filter and the recorded graphs. Wasfy B. Summary and conclusions. 1985. M. and channel Two to show the output voltage waveform. Edited by F. Inc. Mikhael. Analog Filter Design. in RC Active Filter Design Handbook.. 4. “Biquad II: The Current Generalized Immittance (CGIC) Stucture. “Design of High-Order Active Filters. [4]. The complete circuit design processes. 2. 36kΩ 0.02kΩ .16kΩ - 4.33kΩ 32.02kΩ - + - + 4.01uF 11.01uF 11.33kΩ 4.0.36kΩ - 0.36kΩ 11.61kΩ 11.79kΩ 5.01uF + + 11.36kΩ Fig.02kΩ 0.2 The six-order Chebyshev low-pass filter Vo (s) 4.01uF 5.33kΩ 0.33kΩ 0.01uF + Vi (s ) 8.02kΩ 5.01uF + 5. Study Guide C EEL 4140 ANALOG FILTERS DESIGN Biquads II: The current Generalized Immittance (CGIC) Structure . . . . . . . . . . . . . . . . . . EEL 4140 ANALOG FILTERS LABORATORY 8 Butterworth Filter Approximation I. Objective To study Butterworth approximations of low-pass, band-pass, and high-pass filters. II. Introduction The magnitude response of the low-pass Butterworth filter is expressed as: T ( jω ) = 1 1 + (ω ω 0 ) 2n (1) Where n is the filter order and ω 0 is the cutoff frequency. From Equation (1), poles of low-pass Butterworth filters can be derived. Poles of Low-Pass Butterworth Filters From Equation (1), the following equation can be derived as: T (s )T (− s ) = 1 2n 1 + (s jω 0 ) = 1 (2) n⎛ s ⎞ 1 + (− 1) ⎜⎜ ⎟⎟ ⎝ ω0 ⎠ 2n The poles of equation (2) are the roots of its denominator. ⎛ s ⎞ 1 + (− 1) ⎜⎜ ⎟⎟ ⎝ ω0 ⎠ 2n n =0 (3) The poles of Equation (2) are given by: j 2πk ⎧ ⎪⎪ω e 2n , n is odd 0 , s=⎨ jπ (2 k +1) ⎪ 2n , n is even ⎩⎪ω 0 e k = 0,1,...,2n − 1 (4) As well known, poles in the right half-plane correspond to an unstable system. in the left half-plane are selected to associate with T (s ) . j 2πk ⎧ ⎪ω e 2n , ⎪ 0 s=⎨ jπ (2k +1) ⎪ ⎪ω 0 e 2n , ⎩ n is odd, k = The poles of T (s ) is given by: n +1 n + 3 n + (2n − 1) , ,..., 2 2 2 n is even, k = Thus, poles (5) n n n , + 1,..., + (n - 1) 2 2 2 From Equation (5), it is can be seen that the poles of the low-pass Butterworth filter are located on the circle with the radius ω 0 , and are separated by φ n = π n . pole on the real axis. If n is even, there are poles at φ = π ± φ n 2 . If n is odd, there is a As examples, the pole locations of the 4th and 5th order low-pass Butterworth filters are shown in Fig. 1 and Fig.2, respectively. jω n=4 jω 0 φn = 45o φn 2 φn σ − jω 0 Fig. 1 The pole locations of the 4th order low-pass Butterworth filter Low-pass filter specifications: the attenuation at the pass-band (from ω = 0 to ω = ω p ) should be smaller the α max . Band-pass filter specifications: the attenuation at two stop-bands (from ω = 0 to . and Fig. The attenuation α (ω ) is defined as: α (ω ) = −20 log T ( jω ) dB (6) The specifications for low-pass. 1. the attenuation at the stop-band (from ω = ω s to ω = ∞ ) should be larger than the α min . 3. 4. High-Pass. 3.jω n=5 jω 0 φ n = 36 o φn φn σ − jω 0 Fig. high-pass. High-pass filter specifications: the attenuation at the stop-band (from ω = 0 to ω = ω s ) should be larger than the α min . respectively. and Band-Pass Filter Specifications Filter specifications are usually given in term of attenuation characteristics. Fig. and band-pass filters are shown in Fig. 2. the attenuation at the pass-band (from ω = ω p to ω = ∞ ) should be smaller than the α max . 2 The pole locations of the 5th order low-pass Butterworth filter Low-Pass. 5. ω = ω s.3 Low-pass filter specifications α α min pass-band stop-band α max ωs ωp ω Fig.1 to ω = ω p. α α min pass-band stop-band α max ωp ω ωs Fig. the attenuation at the pass-band (from ω = ω p.1 .2 to ω = ∞ ) should be larger than α min . and from ω = ω s.4 High-pass filter specifications .2 ) should be smaller than α max . 1 ω p.1 ω p .1 = First. given ω p = 1000rad / s .1rad / s (10) The actual cutoff frequency is the geometric average of the pass-band and stop-band cutoff frequencies as: . Compute the cutoff frequency ω 0 . calculate the stop-band cutoff frequency as: [10α ωs max ] 1 10 − 1 2n = 1263.α α min α min pass-band stop-band stop-band α max ω s .5 Band-pass filter specifications Design Procedure for Low-Pass Butterworth Filter Approximation 1. 2.8321 2 log(ω s ω p ) (7) n=5 (8) up the next integer. ω 0.2rad / s (9) Then. Decide the filter order n as: n= Round 3. α max = 0. n [( ] ) log 10α min 10 − 1 10α max 10 − 1 = 4. and ω s = 2000rad / s . calculate the pass-band cutoff frequency as: ω 0. The low-pass filter specifications are by α min = 20dB .2 ω s.2 ω Fig.5dB .2 = ωp [10α min ] 1 10 − 1 2n = 1234. 75rad / s .56 * 106 ) Design Procedure for High-Pass Butterworth Filter Approximation 1.08rad / s (15) and ω s _ low = 3.1ω 0.6566 * 1015 (s + 1248.6)(s 2 + 2013. 4. Obtain the normalized low-pass Butterworth transfer function. T (s) = 1 (s + 1)⎛⎜ s 2 + s s ⎞⎛ ⎞ + 1⎟⎜ s 2 + + 1⎟ 0.62ω 0 ⎠⎝ 0 ⎠ (13) 5. the excess attenuations are achieved at the frequency ω s and ω p .50rad / s . ω p = 43.62 ⎠⎝ 1.2 = 1248.62 ⎠ ⎝ 5. and ω s = 12. α max = 1dB . Change the high-pass specifications to the corresponding low-pass specifications.8s + 1.ω 0 = ω 0.56 * 106 )(s 2 + 770. and stop frequencies of the corresponding low-pass filter are given by: ω p _ low = 1 ωs The pass = 0. (12) Denormalize the transfer function by substituting s in Equation (12) with s ω 0 . 2.7 s + 1. The high-pass filter specifications are given by α min = 25dB .6rad / s (11) Using this way. 1 ωp Design the corresponding low-pass Butterworth filter. according to the filter order n .62ω 0 ⎟ ⎝ ω0 ⎠⎜⎝ ω 0 0.0229rad/s (14) = 0. The transfer function satisfying the given specifications is given by: T (s) = = 1 ⎞⎛ s 2 ⎞ ⎛ s ⎞⎛ s 2 s s ⎜⎜ + 1⎟⎟⎜ + + 1⎟⎜ + + 1⎟ 2 ⎟⎜ ω 2 1. Its specifications are as follows: the . s T (s) = = 1 1 ⎞ 1 1 ⎞⎛ ⎛ + + 1⎟⎟ + 1⎟⎜⎜ ⎜ ⎝ 0.0296 ⎠ ⎝ 0.1 = 250rad / s .0296 2 * s 2 0.0296 * s ⎠ (17) s3 (s + 33. The center frequency ω c is given by: ω c = ω p. ω p.0229rad / s . The band-pass filter specifications: α min = 25dB .1ω p.5dB .3) Design Procedure for Band-Pass Butterworth Filter Approximation 1. The denomalized corresponding low-pass filter transfer function is given by T ( s ) low _ mod el = 1 2 ⎞ s ⎛ s ⎞⎛ s + 1⎟⎜⎜ + + 1⎟⎟ ⎜ 2 0. pass and stop frequencies of the corresponding low-pass filter are given by: ω p _ low = ω p.attenuation should be at most α max = 1dB at the frequency ω p _ low = 0. Change the band-pass specifications to the corresponding low-pass specifications.2 = 707. ω p.78)(s 2 + 33. ω s. Then. and ω s.1rad/s 2.0296 * s ⎠⎝ 0. and the attenuation should be at least α min = 25dB at the frequency ω s _ low = 0.2 = ω s.1 = 500rad / s .0296 (16) 4.2 = 2000rad / s .78s + 1141. Change the corresponding low-pass transfer function into the high-pass transfer function by 1 substituting s in Equation (16) with . α max = 0.0296 ⎠⎝ 0.0296rad/s .1 = 500rad / s and (18) The (19) .2 − ω p.08rad / s . and the cutoff frequency is ω0 = 0. the order of the corresponding low-pass filter is n = 3 .1ω s.2 = 1000rad / s . α max = 0. s T (s) = 1 ⎛⎛ 2 ⎜ ⎜ s + 707.6 * 0. Design 1.6 2 744.12 ⎞ ⎟ 744.1 ⎟ + 744 6 . 2. Design a high-pass Butterworth transfer function. Find the poles of the normalized 9th and 10th order low-pass Butterworth transfer function ( ω 0 = 1rad / s ).6 2 + ⎜ ⎜ ⎟ s ⎝ ⎠ ⎞ (744.6 * 1.54 ⎟⎜ 744. Design the corresponding low-pass Butterworth filter.54) + 1⎟⎟ ⎟ ⎠ 1 2 ⎛⎛ 2 2⎞ ⎛ 2 ⎜ ⎜ s + 707.31 ⎟ ⎝ ⎠⎝ ⎠ (21) Change the corresponding low-pass filter model into the band-pass filter by substituting s in Equation (21) with s 2 + ω c2 .6* 0.6*1.31) + 1⎟⎟ ⎟ ⎠ III. and ω s = 1725rad / s . α max = 1dB . The specifications are given by α min = 20dB . ω p = 1000rad / s .ω s _ low = ω s. and ω s = 300rad / s .1 = 1750rad / s 3.2 − ω s. Compare your results with the poles given in the textbook. ⎜⎜ ⎟ ⎜ ⎟ s s ⎜⎝ ⎠ ⎝ ⎠ ⎝ × (22) ⎞ (744.5dB . .6 2 744. Design a low-pass Butterworth transfer function. ω p = 1500rad / s .5dB at frequency ω p _ low = 500rad / s .12 ⎞⎟ 2 ⎜ s + 707.6rad/s . 3. and the attenuation must be at least α min = 25dB at the frequency ω s _ low = 1750rad / s The order of the corresponding low-pass filter is n = 4 is ω0 = 744. The specifications are given by α min = 55dB . (20) Its specifications are as follows: the attenuation must be at most α max = 0. 1 ⎛ s2 ⎞⎛ s 2 ⎞ s s ⎜ + + 1⎟⎜ + + 1⎟ ⎜ 744.1 ⎟ ⎜⎜ ⎟ s ⎜⎝ ⎠ ⎝ 2 ⎞2 ⎛ s 2 + 707. and the cutoff frequency The denomalized corresponding low-pass filter transfer function is given by: Tlow _ mod el = 4. 3. ω p. Computer Simulations 1. plot the magnitude and phase responses according to their transfer functions (using MATLAB. Compare the specification attenuations α min = 20dB and α max = 1dB of the low-pass Butterworth filter with the corresponding attenuations of the approximated transfer function.1 = 333rad / s .1 = 333rad / s . For the band-pass filter. or other languages). . VI. and ω s . Comment on deviations from expected results. record the attenuation value at the frequencies ω p. MATHCAD.1 = 500rad / s . For the low-pass filter. if any.2 = 1000rad / s . For the high-pass filter.1 = 500rad / s . and α max = 0.5dB of the band-pass Butterworth filter with the corresponding attenuations of the approximated transfer function. Compare the specification attenuations α min = 55dB and α max = 0.2 = 1000rad / s . The specifications are given by α min = 22dB . Design a pass-pass Butterworth transfer function.2 = 1500rad / s . Lab Report In the report.5dB of the high-pass Butterworth filter with the corresponding attenuations of the approximated transfer function. 4. IV. record the attenuation values at the frequencies ω p = 1000rad / s . 2. and the reasons for these deviations. V. 4. The design steps. ω s.4. record the attenuation values at the frequencies ω p = 1500rad / s . and ω s = 1725rad / s . 3. 5. The computer simulation results: the magnitude and phase responses for all designed Butterworth filters. and ω s. present the simulation results. Experiments This lab is a computer simulation lab. and ω s = 300rad / s . To see if the designed transfer function satisfies the requirements. ω s . Compare the specification attenuations α min = 22dB .5dB . No actual experiment. To see if the designed transfer function satisfies the requirements. For three designed Butterworth filters. Your report should include the following: 1. 2. ω p. ω p. α max = 0.2 = 1500rad / s . . References [1]. Oxford University Press.To see if the designed transfer function satisfies the requirements. 1982. E. Analog Filter Design. Van Valkenburg. 6. Summary and conclusions. M. write down the number associated with that color. identify the first band . Now 'read' the next color. If the resistor has one more band past the tolerance band it is a quality band. In this example it is two so we get '6200' or '6. If the 'multiplier' band is Gold move the decimal point one to the left. it will typically be gold (5%) and sometimes silver (10%).200'. If the 'multiplier' band is Black (for zero) don't write any zeros down. (To get better failure rates.APPENDIX How to read Resistor Color Codes First the code Black Brown Red Orange Yellow Green Blue Violet Gray White 0 1 2 3 4 5 6 7 8 9 How to read the code First find the tolerance band. Read the number as the '% Failure rate per 1000 hour' This is rated assuming full wattage being applied to the resistors. here it is red so write down a '2' next to the six. If the 'multiplier' band is Silver move the decimal point two places to the left. resistors are typically specified to have twice the needed wattage dissipation that the circuit produces) 1% resistors have three . Starting from the other end. in this case Blue is 6. (you should have '62' so far.) Now read the third or 'multiplier' band and write down that number of zeros. Therefore the first two digits of the resistance value are 33. The silver stripe means the actual value of the resistor mar vary by 10% meaning the actual value will be between 297 ohms and 363 ohms. (Since 5% of 10. Step Four: The value of the resistance is found as 33 x 10 = 330 ohms.05 x 6.8 kilohms = 6. (Since 5% of 6. Step One: The gold stripe is on the right so go to Step Two. Step Two: The first stripe is orange which has a value of 3. Find the resistance value. Find the resistance value.800 = 0. Step Two: The first stripe is blue which has a value of 6. The gold stripe means the actual value of the resistor mar vary by 5% meaning the actual value will be somewhere between 6.500 ohms. Step One: The silver stripe is on the right so go to Step Two. The second stripe is orange which has a value of 3.460 ohms and 7.500 ohms and 10. Find the resistance value. Step One: The gold stripe is on the right so go to Step Two.000 = 500) Example 2: You are given a resistor whose stripes are colored from left to right as orange. Step Four: The value of the resistance is found as 68 x 100 = 6800 ohms (6. Step Four: The value of the resistance is found as 10 x 1000 = 10. orange.000 = 0.800 = 340) 2 . gold. brown. silver. red.140 ohms.10 x 330 = 33) Example 3: You are given a resistor whose stripes are colored from left to right as blue. gold. Step Three: The third stripe is red which means x 100. Step Three: The third stripe is orange which means x 1. They have a different temperature coefficient in order to provide the 1% tolerance.000 ohms (10 kilohms = 10 kohms).05 x 10. The second stripe is black which has a value of 0. gray. Therefore the first two digits of the resistance value are 10. Step Three: The third stripe is brown which means x 10. Examples Example 1: You are given a resistor whose stripes are colored from left to right as brown. (Since 10% of 330 = 0. Therefore the first two digits of the resistance value are 68.8 kohms).bands to read digits to the left of the multiplier. orange. The second stripe is gray which has a value of 8. black.000. The gold stripe means the actual value of the resistor mar vary by 5% meaning the actual value will be somewhere between 9. Step Two: The first stripe is brown which has a value of 1. 55 ohms.Example 4: You are given a resistor whose stripes are colored from left to right as green.5 3 . Step Three: The third stripe is black which means x 1. brown. (Since 5% of 51 = 0. black. gold. Find the resistance value. The gold stripe means the actual value of the resistor mar vary by 5% meaning the actual value will be somewhere between 48. The second stripe is brown which has a value of 1.05 x 51 = 2. Therefore the first two digits of the resistance value are 51.45 ohms and 53. Step Four: The value of the resistance is found as 51 x 1 = 51 ohms. Step One: The gold stripe is on the right so go to Step Two. Step Two: The first stripe is green which has a value of 5.