ELE-6256 Active RF CircuitsA Seminar Report on Conventional Linear Two-port Network Parameters Author: Bijaya Shrestha Student Id: 217370 Date: 10.11.2010 Table of Contents Page Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Network Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 One-port Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Multiport Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Linear Two-port Network Parameters . . . . . . . . . . . . . . . . . . . . 3 2 z-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Derivations of z-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Reciprocal Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 y-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 Derivations of y-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 π-Equivalent Reciprocal Model . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 h-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.1 Derivations of h-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ii 4.4.1 Terminated Equivalent Two-port . . . . . . . . . . . . . . . . . . . 16 4.4.2 Parameters of Common Emitter BJT . . . . . . . . . . . . . . . . . 17 5 ABCD-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.1 Derivation of ABCD-parameters . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3 Applications & Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3.1 Cascaded Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3.2 Finding Length of Microstrip Line . . . . . . . . . . . . . . . . . . . 22 6 Two-port Parameter Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.1 Expressing y-parameters in Terms of z-parameters . . . . . . . . . . . . . . 24 6.2 Expressing h-parameters in terms of z-parameters . . . . . . . . . . . . . . 25 6.3 Conversion Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 iii Chapter 1 Introduction 1.1 Network Basics An electrical circuit or device can be referred to as a network because it consists of different electrical components or devices interconnected to each other. The devices are made to make our lives easier. An information is provided to the device and gets processed to produce the required result. There are different types of devices for fulfilling varieties of applications. The most commonly used device is the amplifier; it is highly used in the communications circuits to overcome the losses during signal propagation. One-port and multiport network concepts are used to simplify the complicated circuits and determine their performance in a convenient way. A device can be treated as a black box and its properties can be obtained without knowing its internal structure by determining the input and output port parameters. Port means a pair of terminals carrying equal currents in opposite directions. In this paper, impedances are represented by resistors in all the figures. 1.2 One-port Network If only the relationship between port voltage and current is of interest then a one-port [1] network model is used. Such networks are therefore used for finding only the input- output properties of a device. A resistor, capacitor, and inductor are the one-port devices satisfying the current-voltage relationships v R = Ri R , i C = Cdv C /dt, and v L = Ldi L /dt respectively. A one-port device may contain any number of resistors, capacitors, inductors, and other devices interconnected to each other. Thevenin and Norton equivalent circuits 1 [1] are used to determine one-port models. ' I 1 , , - V 1 + E I 1 One-port Figure 1.1: One-port Model. 1.3 Multiport Network A network having more than one pairs of terminals is called the multiport network. Two- port networks are linear models and widely used to characterize different active and passive devices; transformers and amplifiers are the typical examples. Power dividers and circula- tors consist of more than two ports. ' I N−1 , , - V N−1 + E I N−1 Port N −1 ' I 1 , , - V 1 + E I 1 Port 1 Multiport E I N , , - V N + ' I N Port N E I 2 , , - V 2 + ' I 2 Port 2 , , , , , , Figure 1.2: Multiport Network with N Ports [2]. 2 1.4 Linear Two-port Network Parameters The following figure is the two-port linear model comprising of two ports. V 1 and I 1 are respectively voltage and current of port 1 and V 2 and I 2 are respectively voltage and current of port 2. The conventional directions and polarities of voltages and currents are as shown in the figure below. ' I 1 , , - V 1 + E I 1 Two-port E I 2 , , - V 2 + ' I 2 Figure 1.3: Two-port Network. Modeling a two-port means definining a relationship among these variables. The net- work is linear because this model gives any two of the variables i.e., dependent variables as the linear combinations of the other two variables i.e., independent variables. Different parameters are defined according to the choice of currents and voltages being dependent or independent as tabulated below. Table 1.1: Two-port Network Parameters. Dependent Variables Independent Variables Description V 1 , V 2 I 1 , I 2 z-parameters I 1 , I 2 V 1 , V 2 y-parameters V 1 , I 2 I 1 , V 2 h-parameters I 1 , V 2 V 1 , I 2 g-parameters V 2 , I 2 I 1 , I 1 ABCD-parameters V 1 , I 1 V 2 , I 2 inverse t-parameters There are six conventional linear two-port network parameters as listed in Table 1.1. 3 Inverse hybrid-parameters or g-parameters and inverse ABCD parameters or inverse t- parameters are generally not used from applications point of view. z-parameters, y- parameters, h-parameters, and ABCD-parameters are extensively used and are the major topics to be discussed in this report. 4 Chapter 2 z-parameters For determining z-parameters of a two-port linear network V 1 and V 2 are written as the linear combinations of I 1 and I 2 . The coefficients of the resulting equations are called the z-parameters or impedance parameters because they all have the units of impedance. V 1 = z 11 I 1 + z 12 I 2 (2.1) V 2 = z 21 I 1 + z 22 I 2 (2.2) In matrix form, they can be written as V 1 V 2 = z 11 z 12 z 21 z 22 I 1 I 2 . 2.1 Derivations of z-parameters The parameters can be determined by open circuiting the ports one at a time. When port 2 is open circuited and port 1 is provided an excitation, I 2 becomes zero. From equation 2.1 z 11 = V 1 I 1 I 2 = 0 (2.3) and from equation 2.2 z 21 = V 2 I 1 I 2 = 0. (2.4) Similarly, when port 1 is open circuited and port 2 is excited I 1 becomes zero. From equation 2.1 z 12 = V 1 I 2 I 1 = 0 (2.5) 5 and from equation 2.2 z 22 = V 2 I 2 I 1 = 0. (2.6) Since all the z-parameters are obtained either by open-circuiting port 1 or port 2 they are also called open-circuit impedance parameters. Moreover, z 11 is called the driving-point input impedance, z 22 the driving-point output impedance, and z 12 and z 21 the transfer impedances. 2.2 Equivalent Circuit Model Equations 2.1 and 2.2 can be realized by an equivalent circuit model [1] consisting of two dependent current-controlled voltage sources as shown below. ¡ ¡ ¡ e e e z 11 d d z 12 I 2 + − , , + - V 1 E I 1 d d z 21 I 1 + − ¡ ¡ ¡ e e e z 22 , - V 2 + , ' I 2 Figure 2.1: Equivalent Circuit Modeled by z-parameters. Considering input section of the above figure and applying Kirchhoff’s Voltage Law (KVL), voltage V 1 is the sum of voltage drop across z 11 and current-controlled voltage source z 12 I 2 i.e., V 1 = z 11 I 1 +z 12 I 2 as given by equation 2.1. This equation thus models the input port of the network in terms of z-parameters. Similarly, the output port is modeled by equation 2.2. This equivalent circuit helps to find the voltage gains, input and output impedances of terminated two-port networks. 6 2.3 Reciprocal Networks A network is said to be reciporcal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2. Networks are reciprocal if they contain only linear passive elements (i.e., resistors, capacitors, and inductors) and the presence of dependent or independent sources makes them non-reciporcal [3]. In terms of z-parameters, the networks can be treated as reciporcal if z 12 =z 21 . And, the network can be represented by an equivalent T model. Since z 12 =z 21 , equations 2.1 and 2.2 can be written as V 1 = z 11 I 1 + z 12 I 2 (2.7) V 2 = z 12 I 1 + z 22 I 2 (2.8) Adding and subtracting the right hand side of equation 2.7 by z 12 I 1 and equation 2.8 by z 12 I 2 , V 1 = (z 11 −z 12 )I 1 + z 12 (I 1 + I 2 ) (2.9) V 2 = z 12 (I 1 + I 2 ) + (z 22 −z 12 )I 2 . (2.10) And, the equivalent T network is shown in Figure 2.3. , - V 1 + , E I 1 ¡ ¡ ¡ e e e z 11 −z 12 ¨ ¨ ¨ r r r z 12 ¡ ¡ ¡ e e e z 22 −z 12 , - V 2 + , ' I 2 Figure 2.2: T-Equivalent Circuit Modeled by z-parameters for a Reciprocal Two-port. 7 2.4 Examples Let z-parameters of a two-port network be available. What are the expressions for input impedance, output impedance, and gains of the following terminated network? ¡ ¡ ¡ e e e Z s ` _¸ + − V s ¡ ¡ ¡ e e e z 11 d d z 12 I 2 + − , , + - V 1 E I 1 d d z 21 I 1 + − ¡ ¡ ¡ e e e z 22 , - V 2 + , ' I 2 ¨ ¨ ¨ r r r Z L E Z in ' Z out Figure 2.3: Terminated Two-port Network Modeled by z-parameters. Input Impedance: For the input section, V 1 = z 11 I 1 + z 12 I 2 . And, for the output section, V 2 = z 21 I 1 +z 22 I 2 . Also, V 2 = −I 2 Z L . So, z 21 I 1 +z 22 I 2 = −I 2 Z L . After simplifying few steps for the last expression, I 2 = −z 21 z 22 +Z L I 1 . Substituting I 2 in the first expression results V 1 = I 1 z 11 − z 12 z 21 z 22 +Z L . Thus input impedance is found to be Z in = V 1 I 1 = z 11 − z 12 z 21 z 22 + Z L (2.11) Output Impedance: For determining output impedance, input voltage source is short circuited so that KVL in input section gives 0 = (Z s +z 11 )I 1 +z 12 I 2 , or I 1 = −z 12 Zs+z 11 I 2 . KVL in output section results V 2 = z 21 I 1 + z 22 I 2 . By substituting I 1 in this expression, output impedance can be obtained as Z out = V 2 I 2 = z 22 − z 12 z 21 z 11 + Z s . (2.12) Gain: Voltage gain for the given network can be expressed as G v = V 2 V 1 = V 2 V 1 V 1 V s (2.13) By using voltage division rule, V 1 V s = Z in Z in + Z s (2.14) 8 and, V 2 = Z L Z L + z 22 z 21 I 1 (2.15) = Z L Z L + z 22 z 21 V 1 Z in . Therefore, V 2 V 1 = Z L Z L + z 22 z 21 Z in . (2.16) Finally, using equations 2.13, 2.14, and 2.16, the voltage gain of the network is obtained as G v = Z in Z in + Z s Z L Z L + z 22 z 21 Z in (2.17) = Z L Z L + z 22 z 21 Z in + Z s . Now the above derived formulas can be used to find input and output impedance and voltage gain of an amplifier or of any circuit if the z-parameters are known. 2.5 Limitations The impedance parameters can not be defined for all kinds of two-port networks. For examples, an ideal transformer and the following circuit don’t have z-parameters. ¡ ¡ ¡ e e e R Figure 2.4: A Circuit Having No z-parameters . It is obvious from this circuit that when any port is open-circuited, both the port currents must be zero. When supply is provided at any port current will flow which violates the port condition for determining z-parameters. For an ideal transformer, voltages V 1 and V 2 can not be expressed as functions of I 1 and I 2 [1]. Consequently z-parameters can not be defined. 9 Chapter 3 y-parameters The y-parameters are determined by short circuiting the input and output ports one at a time. Therefore, they are also called short-circuit parameters. Currents I 1 and I 2 are expressed as I 1 = y 11 V 1 + y 12 V 2 (3.1) I 2 = y 21 V 1 + y 22 V 2 (3.2) In matrix form, I 1 I 2 = y 11 y 12 y 21 y 22 V 1 V 2 where, the coefficients y 11 , y 12 , y 21 , and y 22 are called the y-parameters or the short-circuit admittance parameters. 3.1 Derivations of y-parameters When port 2 is short circuited and port 1 is provided an excitation, V 2 becomes zero. From equation 3.1 y 11 = I 1 V 1 V 2 = 0 (3.3) and from equation 3.2 y 21 = I 2 V 1 V 2 = 0. (3.4) Similarly, when port 1 is short circuited and port 2 is excited V 1 becomes zero. From equation 3.1 y 12 = I 1 V 2 V 1 = 0 (3.5) 10 and from equation 3.2 y 22 = I 2 V 2 V 1 = 0. (3.6) 3.2 Equivalent Circuit Model The equations 3.1 and 3.2 can be modeled by an equivalent circuit [1] with two dependent voltage controlled current sources and two admittances as shown in figure below. , - V 1 + , E I 1 ¨ ¨ ¨ r r r y 11 c d d y 12 V 2 c d d y 21 V 1 ¨ ¨ ¨ r r r y 22 , - V 2 + , ' I 2 Figure 3.1: Equivalent Circuit Modeled by y-parameters. 3.3 π-Equivalent Reciprocal Model If the network is reciprocal, then y 12 = y 21 . Equations 3.1 and 3.2 can be rewritten as I 1 = y 11 V 1 + y 12 V 2 (3.7) I 2 = y 12 V 1 + y 22 V 2 (3.8) Adding and subtracting the right hand side of equation 3.7 by y 12 V 1 and equation 3.8 by y 12 V 2 , I 1 = y 11+y 12 V 1 −y 12 (V 1 −V 2 ) (3.9) I 2 = y 12 (V 2 −V 1 ) + (y 22 + y 12 )V 2 . (3.10) These equations lead to the equivalent π-network as shown in Figure 3.3. 11 , - V 1 + , E I 1 ¨ ¨ ¨ r r r y 11 + y 12 ¡ ¡ ¡ e e e −y 12 ¨ ¨ ¨ r r r y 22 + y 12 , - V 2 + , ' I 2 Figure 3.2: π-Equivalent Circuit Modeled by y-parameters for a Reciprocal Two-port. 3.4 Examples If y-parameters of a two-port network are considered then the formulas of input and output admittance and voltage gain for the terminated case can be derived. Let us consider voltage V s with source admittance Y s be applied at port 1 and port 2 be terminated by load admittance Y L as shown in Figure 3.3. This is the configuration used in the real practice. ¡ ¡ ¡ e e e Y s ` _¸ + − V s , - V 1 + , E I 1 ¨ ¨ ¨ r r r y 11 c d d y 12 V 2 c d d y 21 V 1 ¨ ¨ ¨ r r r y 22 , - V 2 + , ' I 2 ¨ ¨ ¨ r r r Y L Figure 3.3: Terminated Equivalent Circuit Modeled by y-parameters. By proceeding the same way as done in section 2.3 the following properties [1] can be obtained. Input Admittance: Y in = y 11 − y 12 y 21 y 22 + Y L . (3.11) Output Admittance: Y out = y 22 − y 12 y 21 y 11 + Y s . (3.12) 12 Voltage Gain: G v = Y s Y s + Y in −y 21 y 22 + Y L . (3.13) These formulas can be applied for any two-port network defined by y-parameters. 3.5 Limitations In an ideal transformer currents I 1 and I 2 can’t be expressed as the linear combinations of voltages V 1 and V 2 . Therefore, ideal transformer doesn’t have y-parameters. The following circuit also doesn’t have admittance parameters. When one of the ports is short-circuited ¨ ¨ ¨ r r r R Figure 3.4: A Circuit Having No y-parameters . both the port voltages V 1 and V 2 are zero. Connecting a source in any port means non zero terminal voltages. Therefore y-parameters can’t be defined here as well. 13 Chapter 4 h-parameters Hybrid parameters or h-parameters are determined by expressing voltage V 1 and current I 2 as the linear combinations of current I 1 and voltage V 2 as given by the following equations. V 1 = h 11 I 1 + h 12 V 2 (4.1) I 2 = h 21 I 1 + h 22 V 2 . (4.2) In matrix form, they can be written as V 1 I 2 = h 11 h 12 h 21 h 22 I 1 V 2 where, the coefficients are called the h-parameters. 4.1 Derivations of h-parameters Short-circuiting port 2, V 2 is zero. Then from equations 4.1 and 4.2, h 11 abd h 21 can be obtained. h 11 = V 1 I 1 V 2 = 0 (4.3) and, h 21 = I 2 I 1 V 2 = 0. (4.4) Similarly, open-circuiting port 1, I 1 = 0. Then, h 12 = V 1 V 2 I 1 = 0 (4.5) 14 and, h 22 = I 2 V 2 I 1 = 0. (4.6) Here, h 11 is the ratio between input voltage and input current and is determined when port 2 is short-circuited. Therefore, it is known as the short-circuit input impedance. h 21 is the ratio between output current and input current and is determined by short-circuiting port 2. So, h 21 is termed as the short-circuit forward current gain. h 12 is given by the ratio between input voltage and output voltage when port 1 is open-circuited. It is hence termed as the reverse open-circuit voltage gain. The last parameter h 22 it the ratio between output current and output voltage when port 1 is open-circuited. So, h 21 is referred to as the open-circuit output admittance. All the parameters are not of same kind. They include different properties: impedance, admittance, current gain, and voltage gain. Also, they are obtained only when both open-circuit and short-circuit conditions are applied. That’s the reason why they are called hybrid parameters. 4.2 Equivalent Circuit Model The mathematical expressions given by equations 4.1 and 4.2 can be realized by an equiv- alent circuit [1] as shown below. , - V 1 + , E I 1 ¡ ¡ ¡ e e e h 11 d d h 12 V 2 + − c d d h 21 I 1 ¨ ¨ ¨ r r r h 22 , - V 2 + , ' I 2 Figure 4.1: Equivalent Circuit Modeled by h-parameters. Actually, this is the simplified model of a common emitter configuration of a bipolar junction transistor (BJT). The h-parameters are therefore extensively used for character- 15 izing the transistors at low frequencies. At high or microwave frequencies, scattering or s-parameters are used which is out of scope for this topic. 4.3 Reciprocity In Chapter 2, the equivalent T-model for a reciprocal two-port was discussed in terms of z-parameters whereas the equivalent π-model was discussed in terms of y-parameters in Chapter 3. This section only presents the condition for reciprocity in terms of h-parameters. If h 12 = −h 21 , then the two-port can be said reciprocal. 4.4 Examples 4.4.1 Terminated Equivalent Two-port In real life, a device is terminated in both the ports. One port is connected to a voltage source (V s ) or a current source (I s ) having internal impedance of Z s and the other port is terminated with a load as shown in figure below. 16 ¡ ¡ ¡ e e e ` _¸ + − V s Z s E Z in , - V 1 + , E I 1 ¡ ¡ ¡ e e e h 11 d d h 12 V 2 + − c d d h 21 I 1 ¨ ¨ ¨ r r r h 22 , - V 2 + , ' I 2 ¨ ¨ ¨ r r r Y L ' Y out Figure 4.2: Terminated Equivalent Circuit Modeled by h-parameters. From this circuit one can easily derive expressions [1] for input impedance, output admittance, and voltage gain which are directly written here. Input Impedance: Z in = h 11 − h 12 h 21 h 22 + Y L . (4.7) Output Admittance: Y out = h 22 − h 12 h 21 h 11 + Z s . (4.8) Voltage Gain: G v = − 1 Z in + Z s h 21 h 22 + Y L . (4.9) These formulas can be used to characterize a two-port network if its h-parameters are given. 4.4.2 Parameters of Common Emitter BJT If h-parameters of a BJT in the following configuration are h 11 = 1.6 kΩ, h 12 = 2e −4 , h 21 = 110, and h 22 = 20 µS, find the input impedance, output impedance, and voltage gain of the given circuit? [4] Solution: The hybrid equivalent circuit of the given circuit is given in Figure 4.4. The Thevenin equivalent in the input section gives Z Th = Z s ||470 k ≈ Z s and V Th ≈ V s . 17 ` _¸ + − V s ¡ ¡ ¡ e e e 1 kΩ _¸ d d ¨ ¨ ¨ r r r 470 kΩ ¨ ¨ ¨ r r r 4.7 kΩ , V CC ¨ ¨ ¨ r r r 3.3 kΩ Figure 4.3: Common Emitter Amplifier [4]. ¡ ¡ ¡ e e e ` _¸ + − V s Z s E Z in ¨ ¨ ¨ r r r 470 kΩ , - V 1 + , E I 1 ¡ ¡ ¡ e e e h 11 d d h 12 V 2 + − c d d h 21 I 1 ¨ ¨ ¨ r r r h 22 , - V 2 + , ' I 2 ¨ ¨ ¨ r r r 4.7 k||3.3 k ' Y out Figure 4.4: Hybrid Equivalent Circuit. Now, by using equation 4.7, input impedance is s Z in = 1.6e 3 − 2e −4 .110 20e −6 + 5.15e −4 = 1.6 kΩ Using equation 4.8, output admittance is Y out = 20e −6 − 2e −4 .110 1.6e 3 + 1e 3 = 1.2e −5 S. And hence, Z out = 1 Y out = 86.7 kΩ. 18 Using equation 4.9, voltage gain is G v = − 1 1.6e3 + 1e3 110 20e −6 + 5.15e −4 = −79. 19 Chapter 5 ABCD-parameters ABCD-parameters are also called transmission parameters or t-parameters because they are normally used in transmission line analysis. These parameters are related by the following equations. V 1 = AV 2 −BI 2 (5.1) I 1 = CV 2 −DI 2 . (5.2) They are represented in matrix form by V 1 I 1 = A B C D V 2 −I 2 The negative sign associated with I 2 is for indicating that current in the second port is also directed along right side. It can be seen that all the ABCD-parameters are some kinds of transfer functions. They relate directly between input and output. These parameters are very helpful for cascaded networks. 5.1 Derivation of ABCD-parameters Open-circuiting port 2, A = V 1 V 2 I 2 = 0 (5.3) C = I 1 V 2 I 2 = 0. (5.4) Short-circuiting port 2, B = − V 1 I 2 V 2 = 0 (5.5) 20 D = − I 1 I 2 V 2 = 0. (5.6) 5.2 Reciprocity For ABCD-parameters of a two-port network, the reciprocity can be checked with the value of determinant (AB −CD) (i.e.,|AB −CD|) [5]. If it is 1, the network is reciprocal, otherwise non-reciprocal. 5.3 Applications & Examples 5.3.1 Cascaded Networks Let us consider two two-port networks in cascade as shown in Figure 5.1. First network has ABCD-parameters of A1, B1, C1, and D1 and second network has corresponding parameters A2, B2, C2, and D2. , , - V 1 + E I 1 N/W 1 , , - V 2 + E −I 2 - V 3 + E I 3 N/W 2 , , - V 4 + E −I 4 Figure 5.1: Cascaded Two-port Networks. For first network, V 1 I 1 = A1 B1 C1 D1 V 2 −I 2 (5.7) For second network, V 3 I 3 = A2 B2 C2 D2 V 4 −I 4 (5.8) 21 Since, V 2 = V 3 and −I 2 = I 3 , expression 5.7 can be written as V 1 I 1 = A1 B1 C1 D1 A2 B2 C2 D2 V 4 −I 4 . (5.9) From the last expression it is seen that the ABCD-parameters of the overall system is the product of matrices of individual’s ABCD-parameters. Therefore analysis is easy for cascaded networks with ABCD-parameters. 5.3.2 Finding Length of Microstrip Line When a small series inductance is needed along the transmission line, its small portion can be made narrower so that it behaves like an inductor. ABCD- or transmission parameters can be used to find the length of this portion for the required value of inductance. Z 0 Z 0 Z 1 , l, β ¸¸¸¸¸¸ L Figure 5.2: Microstrip and Corresponding Inductance Model. Where, Z 1 , l, and β are respectively the characteristic impedance, length, and phase con- stant of the transmission line portion shown in the Figure 5.2. And, L is the the corre- sponding inductance as shown in the right side of the figure. For a section of transmission line, ABCD-parameters are given as A = cos(βl), B = jZ 1 sin(βl), C = jsin(βl) Z 1 , D = cos(βl)[2]. And, for an inductor of inductance L Henry, A = 1, B = jωL, C = 0, D = 1. Both of them are equivalent; the parameter B can be equated. So, jωL = jZ 1 sin(βl) 22 Assuming that βl is very very small, sin(βl) ≡ βl. Therefore, jωL = jZ 1 βl. Hence, the required length of the microstrip for given L is l = ωL βZ 1 . 23 Chapter 6 Two-port Parameter Conversions One set of parameters can be converted to another set because of linear relationships. This chapter includes some of the conversions. 6.1 Expressing y-parameters in Terms of z-parameters Since z-parameters are defined by V 1 V 2 = z 11 z 12 z 21 z 22 I 1 I 2 (6.1) and, y-parameters are defined by I 1 I 2 = y 11 y 12 y 21 y 22 V 1 V 2 , (6.2) Equation 6.1 can be rewritten as I 1 I 2 = z 11 z 12 z 21 z 22 −1 V 1 V 2 . (6.3) By comparing equations 6.2 and 6.3, y 11 y 12 y 21 y 22 = z 11 z 12 z 21 z 22 −1 . Similarly, z-parameters can be expressed in terms of y-parameters, z 11 z 12 z 21 z 22 = y 11 y 12 y 21 y 22 −1 . Both sets of parameters exist if determinants z 11 z 22 −z 12 z 21 = 0 and y 11 y 22 −y 12 y 21 = 0. 24 6.2 Expressing h-parameters in terms of z-parameters Recalling z-parameter equations, V 1 = z 11 I 1 + z 12 I 2 (6.4) V 2 = z 21 I 1 + z 22 I 2 (6.5) Rearranging the equations by making left hand side with V 1 and I 2 terms and right hand side with V 2 and I 1 terms, V 1 −z 12 I 2 = z 11 I 1 (6.6) z 22 I 2 = −z 21 I 1 + V 2 (6.7) In the matrix form they can be written as, 1 −z 12 0 z 22 V 1 I 2 = z 11 0 0 −z 21 I 1 V 2 . Taking the first matrix as the inverse to the right hand side and doing some matrix manupulations [1], V 1 I 2 = 1 z 22 z 11 z 22 −z 12 z 21 z 12 −z 21 1 I 1 V 2 . And, z 22 should not be zero. This is the matrix equation of h-parameters and hence the h-parameters in terms of z-parameters. Same process can be applied to express again h-parameters in terms of y-parameters. 25 6.3 Conversion Table It is not possible to present here every conversion process. So a table is presented which has interraltions among two-port parameters. Table 6.1: Two-port Parameter Conversions [6]. 26 Chapter 7 Conclusions The two-port network parameters are helpful in analysing the complicated circuits. By measuring terminal voltages and currents the network parameters can be determined which then are used to find the characterisics of the circuit. The main concerns of a device are about input impedance, output impedance and gains. These are determined without dealing with internal components by using the network parameters. A device can therefore be treated as a black box if some sets of two-port network parameters can be defined for that device. The impedance, admittance, hybrid, and transmission parameters are the conventional linear two-port network parameters. All the networks don’t have all sets of parameters. Impedance and Admittance parameters don’t exist for an ideal transformer. The hybrid parameters are normally used in the analysis of transistors. Transmission parameters are extensively used in the transmission line analysis. Moreover, they are helpful in solving many two-ports in cascade because the overall parameters is the product of the parameters of the individuals. One set of parameters can be converted to another set of parameters because all the expressions are linear. 27 References [1] Raymond A. DeCarlo, “Linear Circuit Analysis,” Time Domain, Pha- sor, and Laplace Transform Approaches, Oxford University Press, New York, 2001, pp. 800–836. [2] Reinhold Ludwig and Gene Bogdanov, “RF Circuit Design,” Theory and Applications, Pearson Prentice Hall, Second Edition, pp. 145–163. [3] http://en.wikipedia.org/wiki/Two-port network [4] Robert L. Boylestad and Louis Nashelsky ”Electronic Devices and Circuit Theory.” Pearson Education Low Price Edition, Eight Edition, pp. 429. [5] Willian H. Hayt, Jarck E. Kemmerly, “Engineering Circuit Analysis,” MCGraw-Hill, INC., Fifth Edition, pp. 459–486. [6] http://www.ece.ucsb.edu/Faculty/rodwell/Classes/ece2c/resources/two port.pdf 28 Table of Contents Page Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2 1.3 1.4 Network Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-port Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiport Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Two-port Network Parameters . . . . . . . . . . . . . . . . . . . . 1 1 1 2 3 5 5 6 7 8 9 10 10 11 11 12 13 14 14 15 16 16 ii 2 z-parameters 2.1 2.2 2.3 2.4 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivations of z-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reciprocal Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 y-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 Derivations of y-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . π-Equivalent Reciprocal Model . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 h-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 Derivations of h-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii 4.4.1 4.4.2 Terminated Equivalent Two-port . . . . . . . . . . . . . . . . . . . Parameters of Common Emitter BJT . . . . . . . . . . . . . . . . . 16 17 20 20 21 21 21 22 24 24 25 26 27 28 5 ABCD-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 Derivation of ABCD-parameters . . . . . . . . . . . . . . . . . . . . . . . . Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications & Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 5.3.2 Cascaded Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . Finding Length of Microstrip Line . . . . . . . . . . . . . . . . . . . 6 Two-port Parameter Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 6.2 6.3 Expressing y-parameters in Terms of z-parameters . . . . . . . . . . . . . . Expressing h-parameters in terms of z-parameters . . . . . . . . . . . . . . Conversion Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 2 One-port Network If only the relationship between port voltage and current is of interest then a one-port [1] network model is used. 1. A resistor. and other devices interconnected to each other. iC = CdvC /dt. In this paper. and inductor are the one-port devices satisfying the current-voltage relationships vR = RiR . Thevenin and Norton equivalent circuits 1 . capacitors. The devices are made to make our lives easier. Such networks are therefore used for finding only the inputoutput properties of a device. capacitor.1 Network Basics An electrical circuit or device can be referred to as a network because it consists of different electrical components or devices interconnected to each other. and vL = LdiL /dt respectively. it is highly used in the communications circuits to overcome the losses during signal propagation. The most commonly used device is the amplifier. impedances are represented by resistors in all the figures. A device can be treated as a black box and its properties can be obtained without knowing its internal structure by determining the input and output port parameters. inductors. Port means a pair of terminals carrying equal currents in opposite directions.Chapter 1 Introduction 1. A one-port device may contain any number of resistors. There are different types of devices for fulfilling varieties of applications. One-port and multiport network concepts are used to simplify the complicated circuits and determine their performance in a convenient way. An information is provided to the device and gets processed to produce the required result. Power dividers and circulators consist of more than two ports.3 Multiport Network A network having more than one pairs of terminals is called the multiport network. Twoport networks are linear models and widely used to characterize different active and passive devices. One-port ' 1.2: Multiport Network with N Ports [2]. transformers and amplifiers are the typical examples. IE 1 + V1 I1 Figure 1.[1] are used to determine one-port models. I1 Port 1 ' + V1 E I '2 I1 INE −1 + Port N − 1 VN −1 ' r r r Multiport E r I2 r r 'N I + V2 - Port 2 + VN - E Port N IN −1 IN Figure 1. 2 .1: One-port Model. IE 1 + V1 I1 I2 ' Two-port ' E + V2 - I2 Figure 1. I2 I1 . Modeling a two-port means definining a relationship among these variables. 3 . I1 Independent Variables I1 .4 Linear Two-port Network Parameters The following figure is the two-port linear model comprising of two ports. The network is linear because this model gives any two of the variables i. Different parameters are defined according to the choice of currents and voltages being dependent or independent as tabulated below.1: Two-port Network Parameters.1. V2 I1 .. I2 V1 .e. The conventional directions and polarities of voltages and currents are as shown in the figure below. Dependent Variables V1 .1. V2 V2 . V2 V1 . Table 1. I2 I1 . I1 V2 . I2 V1 . independent variables. V1 and I1 are respectively voltage and current of port 1 and V2 and I2 are respectively voltage and current of port 2..e.3: Two-port Network. V2 I1 . I2 V1 . I2 Description z-parameters y-parameters h-parameters g-parameters ABCD-parameters inverse t-parameters There are six conventional linear two-port network parameters as listed in Table 1. dependent variables as the linear combinations of the other two variables i. 4 . z-parameters.Inverse hybrid-parameters or g-parameters and inverse ABCD parameters or inverse tparameters are generally not used from applications point of view. y- parameters. h-parameters. and ABCD-parameters are extensively used and are the major topics to be discussed in this report. 4) V1 I2 = 0 I1 (2.3) Similarly. when port 1 is open circuited and port 2 is excited I1 becomes zero.1 z11 = and from equation 2. V1 = z11 I1 + z12 I2 V2 = z21 I1 + z22 I2 In matrix form. they can be written as V z z I 1 = 11 12 1 .1) (2.1 Derivations of z-parameters The parameters can be determined by open circuiting the ports one at a time. From equation 2. I1 (2. When port 2 is open circuited and port 1 is provided an excitation.1 z12 = V1 I1 = 0 I2 5 (2. The coefficients of the resulting equations are called the z-parameters or impedance parameters because they all have the units of impedance. From equation 2. I2 becomes zero.5) .2) 2. V2 z21 z22 I2 (2.2 z21 = V2 I2 = 0.Chapter 2 z-parameters For determining z-parameters of a two-port linear network V1 and V2 are written as the linear combinations of I1 and I2 . I2 (2. I1 z11 ¡e¡e¡e z22 ¡e¡e¡e E 2 ' I + V1 + z12 I2 d d − + z21 I1 d d − + V2 - Figure 2.2 can be realized by an equivalent circuit model [1] consisting of two dependent current-controlled voltage sources as shown below. z11 is called the driving-point input impedance. the output port is modeled by equation 2.2 z22 = V2 I1 = 0.1 and 2.1: Equivalent Circuit Modeled by z-parameters.2 Equivalent Circuit Model Equations 2. voltage V1 is the sum of voltage drop across z11 and current-controlled voltage source z12 I2 i. 6 . input and output impedances of terminated two-port networks.6) Since all the z-parameters are obtained either by open-circuiting port 1 or port 2 they are also called open-circuit impedance parameters.. 2.e. V1 = z11 I1 + z12 I2 as given by equation 2. Moreover. Considering input section of the above figure and applying Kirchhoff’s Voltage Law (KVL). This equation thus models the input port of the network in terms of z-parameters. Similarly. This equivalent circuit helps to find the voltage gains.and from equation 2.1. and z12 and z21 the transfer impedances. z22 the driving-point output impedance.2. And.8) Adding and subtracting the right hand side of equation 2. the networks can be treated as reciporcal if z12 =z21 .1 and 2.2: T-Equivalent Circuit Modeled by z-parameters for a Reciprocal Two-port.. the equivalent T network is shown in Figure 2. In terms of z-parameters. resistors.2 can be written as V1 = z11 I1 + z12 I2 V2 = z12 I1 + z22 I2 (2.10) I1 z11 − z12 ¡e¡e¡e r ¨ z12 r ¨ r ¨ z22 − z12 ¡e¡e¡e E + I ' 2 + V2 - V1 - Figure 2. (2. Networks are reciprocal if they contain only linear passive elements (i. the network can be represented by an equivalent T model.2. 7 .7 by z12 I1 and equation 2.7) (2. and inductors) and the presence of dependent or independent sources makes them non-reciporcal [3]. And.8 by z12 I2 .3 Reciprocal Networks A network is said to be reciporcal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2. equations 2. V1 = (z11 − z12 )I1 + z12 (I1 + I2 ) V2 = z12 (I1 + I2 ) + (z22 − z12 )I2 . Since z12 =z21 .9) (2.e. capacitors.3. So. Thus input impedance is found to be Zin = V1 z12 z21 = z11 − I1 z22 + ZL (2.12) Gain: Voltage gain for the given network can be expressed as Gv = By using voltage division rule.2. output impedance can be obtained as Zout = V2 z12 z21 = z22 − .13) .11) Output Impedance: For determining output impedance. And. Also.3: Terminated Two-port Network Modeled by z-parameters. By substituting I1 in this expression. z22 +ZL 1 Substituting I2 in the first expression . V2 = z21 I1 + z22 I2 . Input Impedance: For the input section. for the output section. z21 I1 + z22 I2 = −I2 ZL . output impedance.14) V2 V2 V1 = V1 V1 Vs (2. or I1 = −z12 I. After simplifying few steps for the last expression.4 Examples Let z-parameters of a two-port network be available. Zs +z11 2 KVL in output section results V2 = z21 I1 + z22 I2 . V1 Zin = Vs Zin + Zs 8 (2. What are the expressions for input impedance. I2 = results V1 = I1 z11 − z12 z21 z22 +ZL −z21 I. input voltage source is short circuited so that KVL in input section gives 0 = (Zs + z11 )I1 + z12 I2 . I2 z11 + Zs (2. and gains of the following terminated network? s 1 E ¡e¡e¡e Z I z11 ¡e¡e¡e z22 ¡e¡e¡e 2 ' I + Vs E − + V1 + z12 I2 d d − + z21 I1 d d − + V2 ' - r ¨ r ZL ¨ r ¨ Zin Zout Figure 2. V2 = −I2 ZL . V1 = z11 I1 + z12 I2 . For an ideal transformer. both the port currents must be zero. = ZL + z22 Zin + Zs (2. V2 = ZL z21 I1 ZL + z22 ZL V1 = z21 . an ideal transformer and the following circuit don’t have z-parameters. the voltage gain of the network is obtained as Gv = Zin ZL z21 Zin + Zs ZL + z22 Zin z21 ZL . 2. It is obvious from this circuit that when any port is open-circuited. ZL + z22 Zin (2. using equations 2.4: A Circuit Having No z-parameters .5 Limitations The impedance parameters can not be defined for all kinds of two-port networks.16) V1 ZL + z22 Zin Finally.14. and 2. When supply is provided at any port current will flow which violates the port condition for determining z-parameters. For examples.17) Now the above derived formulas can be used to find input and output impedance and voltage gain of an amplifier or of any circuit if the z-parameters are known. ZL z21 V2 = .and.16. (2. R ¡e¡e¡e Figure 2. Consequently z-parameters can not be defined.13. voltages V1 and V2 can not be expressed as functions of I1 and I2 [1]. 2. 9 .15) Therefore. 1 y11 = and from equation 3.Chapter 3 y-parameters The y-parameters are determined by short circuiting the input and output ports one at a time. From equation 3.2 y21 = I2 V2 = 0.4) I1 V2 = 0 V1 (3.1 y12 = I1 V1 = 0 V2 10 (3. I y y V 1 = 11 12 1 I2 y21 y22 V2 where. Therefore. they are also called short-circuit parameters. when port 1 is short circuited and port 2 is excited V1 becomes zero. y12 . V2 becomes zero. (3. and y22 are called the y-parameters or the short-circuit admittance parameters.3) Similarly.1 Derivations of y-parameters When port 2 is short circuited and port 1 is provided an excitation. Currents I1 and I2 are expressed as I1 = y11 V1 + y12 V2 I2 = y21 V1 + y22 V2 In matrix form. the coefficients y11 .5) . From equation 3.2) 3. V1 (3.1) (3. y21 . 3 π-Equivalent Reciprocal Model If the network is reciprocal.1 and 3.8 by y12 V2 .2 can be rewritten as I1 = y11 V1 + y12 V2 I2 = y12 V1 + y22 V2 (3. V2 (3.9) (3. I1 = y11+y12 V1 − y12 (V1 − V2 ) I2 = y12 (V2 − V1 ) + (y22 + y12 )V2 .2 Equivalent Circuit Model The equations 3.3. These equations lead to the equivalent π-network as shown in Figure 3.7) (3. (3.6) 3.8) Adding and subtracting the right hand side of equation 3.1 and 3.and from equation 3. then y12 = y21 .2 can be modeled by an equivalent circuit [1] with two dependent voltage controlled current sources and two admittances as shown in figure below.2 y22 = I2 V1 = 0.7 by y12 V1 and equation 3. I1 E ' I2 + + V1 - r ¨ y11 r ¨ r ¨ y12 V2 d c d y21 V1 d c d r ¨ y22 r ¨ r ¨ V2 - Figure 3. Equations 3. 3.1: Equivalent Circuit Modeled by y-parameters.10) 11 . ¡e¡e¡e E Ys I 1 + Vs − + I ' 2 + r ¨ y11 r ¨ r ¨ V1 - y12 V2 d c d y21 V1 d c d r ¨ y22 r ¨ r ¨ V2 - r ¨ r ¨ r ¨ YL Figure 3.3: Terminated Equivalent Circuit Modeled by y-parameters.3 the following properties [1] can be obtained.12) y12 y21 .4 Examples If y-parameters of a two-port network are considered then the formulas of input and output admittance and voltage gain for the terminated case can be derived. y22 + YL (3. 3. This is the configuration used in the real practice.I1 −y12 ¡e¡e¡e r ¨ y11 + y12 r ¨ r ¨ r ¨ y22 + y12 r ¨ r ¨ ' E I2 + + V1 - V2 - Figure 3. y11 + Ys (3.11) 12 . Let us consider voltage Vs with source admittance Ys be applied at port 1 and port 2 be terminated by load admittance YL as shown in Figure 3.2: π-Equivalent Circuit Modeled by y-parameters for a Reciprocal Two-port.3. By proceeding the same way as done in section 2. Input Admittance: Yin = y11 − Output Admittance: Yout = y22 − y12 y21 . 3. ideal transformer doesn’t have y-parameters.Voltage Gain: Gv = Ys Ys + Yin −y21 y22 + YL .13) These formulas can be applied for any two-port network defined by y-parameters. both the port voltages V1 and V2 are zero. Connecting a source in any port means non zero terminal voltages. 13 .5 Limitations In an ideal transformer currents I1 and I2 can’t be expressed as the linear combinations of voltages V1 and V2 . (3. Therefore y-parameters can’t be defined here as well. Therefore. When one of the ports is short-circuited Rr ¨ r ¨ r ¨ Figure 3. The following circuit also doesn’t have admittance parameters.4: A Circuit Having No y-parameters . Then. h12 = V1 I1 = 0 V2 14 (4.4) Similarly.1 and 4. h21 = (4. I1 = 0.2. V1 V2 = 0 I1 I2 V2 = 0. open-circuiting port 1. h11 abd h21 can be obtained. I1 h11 = and. V2 is zero.2) 4.1 Derivations of h-parameters Short-circuiting port 2.5) . Then from equations 4. they can be written as V h h I 1 = 11 12 1 I2 h21 h22 V2 where.3) (4. the coefficients are called the h-parameters. (4.1) (4.Chapter 4 h-parameters Hybrid parameters or h-parameters are determined by expressing voltage V1 and current I2 as the linear combinations of current I1 and voltage V2 as given by the following equations. V1 = h11 I1 + h12 V2 I2 = h21 I1 + h22 V2 . In matrix form. 6) Here. h12 is given by the ratio between input voltage and output voltage when port 1 is open-circuited. h21 is the ratio between output current and input current and is determined by short-circuiting port 2.and. Also. it is known as the short-circuit input impedance.1: Equivalent Circuit Modeled by h-parameters. this is the simplified model of a common emitter configuration of a bipolar junction transistor (BJT). 4. h22 = I2 I1 = 0.2 can be realized by an equivalent circuit [1] as shown below. h21 is referred to as the open-circuit output admittance. They include different properties: impedance. That’s the reason why they are called hybrid parameters. So. V2 (4.2 Equivalent Circuit Model The mathematical expressions given by equations 4. current gain. admittance. h11 is the ratio between input voltage and input current and is determined when port 2 is short-circuited. I1 h11 ¡e¡e¡e E + I ' 2 + h21 I1 d c d r ¨ h22 r ¨ r ¨ V1 - + h12 V2 d d − V2 - Figure 4. Actually.1 and 4. The last parameter h22 it the ratio between output current and output voltage when port 1 is open-circuited. Therefore. The h-parameters are therefore extensively used for character15 . All the parameters are not of same kind. they are obtained only when both open-circuit and short-circuit conditions are applied. So. and voltage gain. h21 is termed as the short-circuit forward current gain. It is hence termed as the reverse open-circuit voltage gain. the equivalent T-model for a reciprocal two-port was discussed in terms of z-parameters whereas the equivalent π-model was discussed in terms of y-parameters in Chapter 3.4.izing the transistors at low frequencies. a device is terminated in both the ports. This section only presents the condition for reciprocity in terms of h-parameters.3 Reciprocity In Chapter 2. At high or microwave frequencies. 4. 16 .4 4. If h12 = −h21 .1 Examples Terminated Equivalent Two-port In real life. then the two-port can be said reciprocal. One port is connected to a voltage source (Vs ) or a current source (Is ) having internal impedance of Zs and the other port is terminated with a load as shown in figure below. 4. scattering or s-parameters are used which is out of scope for this topic. 17 .9) h12 h21 . 4. h12 = 2e−4 . The Thevenin equivalent in the input section gives ZT h = Zs ||470 k ≈ Zs and VT h ≈ Vs . output impedance.2: Terminated Equivalent Circuit Modeled by h-parameters.6 kΩ. find the input impedance. output admittance.4. and voltage gain of the given circuit? [4] Solution: The hybrid equivalent circuit of the given circuit is given in Figure 4. Input Impedance: Zin = h11 − Output Admittance: Yout = h22 − Voltage Gain: Gv = − 1 Zin + Zs h21 h22 + YL . h22 + YL (4. From this circuit one can easily derive expressions [1] for input impedance.2 Parameters of Common Emitter BJT If h-parameters of a BJT in the following configuration are h11 = 1.7) These formulas can be used to characterize a two-port network if its h-parameters are given.Zs ¡e¡e¡e I1 h11 ¡e¡e¡e ' E I2 r ¨ r ¨ r ¨ + Vs + V E 1 − - Zin + h12 V2 d d − + h21 I1 d dc r ¨ h22 r ¨ r ¨ V2 ' - YL Yout Figure 4.4. h21 = 110. and voltage gain which are directly written here. h11 + Zs (4.8) h12 h21 . and h22 = 20 µS. (4. 3: Common Emitter Amplifier [4].7 kΩ. Zs ¡e¡e¡e I1 470 r ¨ kΩ r ¨ r ¨ h11 ¡e¡e¡e E + Vs − + I ' 2 + h21 I1 d c d r ¨ h22 r ¨ r ¨ V1 E - + h12 V2 d d − V2 -' r ¨ r 4.2e−5 S. 1. by using equation 4.VCC r ¨ ¨ r ¨ ¨ 470 kΩ r 4.15e−4 Using equation 4.3 kΩ Figure 4.8. Yout 18 2e−4 .7 kΩ r ¨ ¨ r r 1 kΩ ¡e¡e¡e 1( d 0) d r ¨ r ¨ r ¨ + Vs − 3.6 kΩ 20e−6 + 5. Zout = 1 = 86. output admittance is Yout = 20e−6 − And hence.7. Yout Now.110 = 1.6e3 − 2e−4 .3 k Zin Figure 4. input impedance is s Zin = 1.110 = 1.4: Hybrid Equivalent Circuit.7 ¨ r ¨ k||3.6e3 + 1e3 . 9.6e3 + 1e3 110 + 5.Using equation 4. voltage gain is 1 1. 19 .15e−4 Gv = − 20e−6 = −79. They are represented in matrix form by V A B V 1 = 2 I1 C D −I2 The negative sign associated with I2 is for indicating that current in the second port is also directed along right side. It can be seen that all the ABCD-parameters are some kinds of transfer functions. V1 = AV2 − BI2 I1 = CV2 − DI2 . These parameters are related by the following equations.3) (5.1 Derivation of ABCD-parameters V1 I2 = 0 V2 I1 I2 = 0. These parameters are very helpful for cascaded networks. (5.Chapter 5 ABCD-parameters ABCD-parameters are also called transmission parameters or t-parameters because they are normally used in transmission line analysis.5) (5.2) 5. They relate directly between input and output. B=− (5. V2 V1 V2 = 0 I2 20 Open-circuiting port 2. A= C= Short-circuiting port 2.1) (5.4) . For first network.e. B1. B2.6) 5..3. I2 (5. First network has ABCD-parameters of A1.1. V1 A1 B1 V2 = I1 C1 D1 −I2 For second network.1 Applications & Examples Cascaded Networks Let us consider two two-port networks in cascade as shown in Figure 5. 5. C1. the reciprocity can be checked with the value of determinant (AB − CD) (i. - −I4E N/W 2 + V4 - Figure 5.1: Cascaded Two-port Networks.D=− I1 V2 = 0. the network is reciprocal.7) (5. and D2. C2. and D1 and second network has corresponding parameters A2. If it is 1.3 5. otherwise non-reciprocal.|AB − CD|) [5]. V A2 B2 V 3 = 4 I3 C2 D2 −I4 21 (5.2 Reciprocity For ABCD-parameters of a two-port network. IE 1 + V1 N/W 1 −I2E I3E + + V2 V3 .8) . ABCD. l. 5. B = jωL.Since. the parameter B can be equated. and phase constant of the transmission line portion shown in the Figure 5. (5.2. Therefore analysis is easy for cascaded networks with ABCD-parameters. A = 1. And. B = jZ1 sin(βl). For a section of transmission line. D = 1.2 Finding Length of Microstrip Line When a small series inductance is needed along the transmission line. C = 0. β Z0 Z0 L §¤¤¤ §§ Figure 5.7 can be written as A2 B2 V V A1 B1 4 1 = I1 C1 D1 C2 D2 −I4 . ABCD-parameters are given as A = cos(βl). Z1 . its small portion can be made narrower so that it behaves like an inductor. and β are respectively the characteristic impedance. So. Z1 . C = And.3. length. V2 = V3 and −I2 = I3 . Where.2: Microstrip and Corresponding Inductance Model.9) From the last expression it is seen that the ABCD-parameters of the overall system is the product of matrices of individual’s ABCD-parameters.or transmission parameters can be used to find the length of this portion for the required value of inductance. for an inductor of inductance L Henry. Z1 . Both of them are equivalent. expression 5. jωL = jZ1 sin(βl) 22 jsin(βl) . D = cos(βl)[2]. L is the the corresponding inductance as shown in the right side of the figure. l. Therefore. the required length of the microstrip for given L is l= ωL . sin(βl) ≡ βl. βZ1 23 . Hence. jωL = jZ1 βl.Assuming that βl is very very small. Chapter 6 Two-port Parameter Conversions One set of parameters can be converted to another set because of linear relationships. y-parameters are defined by I1 y11 y12 V = 1 . I2 z21 z22 V2 (6.2) (6. z-parameters By comparing equations 6.1 can be rewritten as −1 I z z V 1 = 11 12 1 .3) −1 y11 y12 z11 z12 = . Similarly. I2 y21 y22 V2 Equation 6.1) (6. 24 . 6. can be expressed in terms of y-parameters.3.2 and 6. This chapter includes some of the conversions. z21 z22 y21 y22 Both sets of parameters exist if determinants z11 z22 − z12 z21 = 0 and y11 y22 − y12 y21 = 0. y21 y22 z21 z22 −1 z11 z12 y y = 11 12 .1 Expressing y-parameters in Terms of z-parameters Since z-parameters are defined by V1 z11 z12 I = 1 V2 z21 z22 I2 and. 0 z22 I2 0 −z21 V2 Taking the first matrix as the inverse to the right hand side and doing some matrix manupulations [1].5) Rearranging the equations by making left hand side with V1 and I2 terms and right hand side with V2 and I1 terms. V1 = z11 I1 + z12 I2 V2 = z21 I1 + z22 I2 (6.6) (6. Same process can be applied to express again h-parameters in terms of y-parameters. V1 − z12 I2 = z11 I1 z22 I2 = −z21 I1 + V2 In the matrix form they can be written as.4) (6.7) 25 . (6.2 Expressing h-parameters in terms of z-parameters Recalling z-parameter equations. 1 −z12 V z 0 I 1 = 11 1 . z22 should not be zero. z z − z12 z21 z12 I V 1 . 1 = 1 11 22 z22 −z21 1 V2 I2 And. This is the matrix equation of h-parameters and hence the h-parameters in terms of z-parameters.6. So a table is presented which has interraltions among two-port parameters. Table 6.3 Conversion Table It is not possible to present here every conversion process. 26 .6.1: Two-port Parameter Conversions [6]. Moreover. admittance. The impedance.Chapter 7 Conclusions The two-port network parameters are helpful in analysing the complicated circuits. One set of parameters can be converted to another set of parameters because all the expressions are linear. The hybrid parameters are normally used in the analysis of transistors. 27 . Transmission parameters are extensively used in the transmission line analysis. hybrid. The main concerns of a device are about input impedance. These are determined without dealing with internal components by using the network parameters. output impedance and gains. and transmission parameters are the conventional linear two-port network parameters. A device can therefore be treated as a black box if some sets of two-port network parameters can be defined for that device. Impedance and Admittance parameters don’t exist for an ideal transformer. All the networks don’t have all sets of parameters. By measuring terminal voltages and currents the network parameters can be determined which then are used to find the characterisics of the circuit. they are helpful in solving many two-ports in cascade because the overall parameters is the product of the parameters of the individuals. Eight Edition.” Theory and Applications. Second Edition. “Linear Circuit Analysis. 429. Boylestad and Louis Nashelsky ”Electronic Devices and Circuit Theory. “Engineering Circuit Analysis. pp.pdf 28 .” Pearson Education Low Price Edition. [6] http://www. Pha- sor. DeCarlo. Fifth Edition.org/wiki/Two-port network [4] Robert L. pp. 145–163. Hayt.” Time Domain.ece. INC.wikipedia.. [5] Willian H.References [1] Raymond A. 459–486. “RF Circuit Design.ucsb. Oxford University Press.” MCGraw-Hill.edu/Faculty/rodwell/Classes/ece2c/resources/two port. Kemmerly. pp. Pearson Prentice Hall. [2] Reinhold Ludwig and Gene Bogdanov. [3] http://en. New York. 800–836. Jarck E. pp. 2001. and Laplace Transform Approaches.
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