Electric Circuit Theory

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Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320

Chapter 18 Two-Port Circuits Nam Ki Min nkmin@korea.ac.kr 010-9419-2320

Contents and Objectives 3 Chapter Contents 18.1 The Terminal Equations 18.2 The Two-Port Parameters 18.3 Analysis of the Terminated Two-Port Circuit 18.4 Interconnected Two-Port Circuits Chapter Objectives 1. Be able to calculate any set of two-port parameters with any of the following methods: Circuit analysis; Measurements made on a circuit; Converting from another set of two-port parameters using Table 18.1. 2. Be able to analyze a terminated two-port circuit to find currents, voltages, impedances, and ratios of interest using Table 18.2. 3. Know how to analyze a cascade interconnection of two-port circuits.

The Terminal Equations 4 Definitions Port : a pair of terminals through which a current may enter or leave a network. - A port is an access to the network and consists of a pair of terminals; the current entering one terminal leaves through the other terminal so that the net current entering the port equals zero. - Two-terminal devices or elements (such as resistors, capacitors, and inductors) result in one-port networks. - Most of the circuits we have dealt with so far are two-terminal or one-port circuits, represented in Fig.(a). Two-port network - Fig.(b) shows a four-terminal or two-port circuit. - In general, a network may have n ports. (a) one-port network (b) two-port network

The Terminal Equations 5 Terminal Equations In viewing a circuit as a two-port network, we are interested in relating the current and voltage at one port to the current and voltage at the other port. Figure 18.1 shows the reference polarities of the terminal voltages and the reference directions of the terminal currents. The references at each port are symmetric with respect to each other; that is, at each port the current is directed into the upper terminal, and each port voltage is a rise from the lower to the upper terminal. This symmetry makes it easier to generalize the analysis of a two-port network and is the reason for its universal use in the literature. The most general description of the two-port network is carried out in the s domain.

The Terminal Equations 6 Of these four terminal variables, only two are independent. Thus for any circuit, once we specify two of the variables, we can find the two remaining unknowns. There are six different ways in which to combine the four variables: V 1 = z 11 I 1 + z 21 I 2 V 2 = z 21 I 1 + z 22 I 2 (18.1) V 1 = h 11 I 1 + h 12 V 2 I 2 = h 21 I 1 h 22 V 2 (18.5) I 1 = y 11 V 1 + y 12 V 2 I 2 = y 21 V 1 + y 22 V 2 (18.2) I 1 = g 11 V + g 12 I 2 V 2 = g 21 V 1 + g 22 I 2 (18.6) V 1 = a 11 V 2 a 12 I 2 I 1 = a 21 V 2 a 22 I 2 (18.3) V 2 = b 11 V 1 b 12 I 1 I 2 = b 21 V 1 b 22 I 1 (18.4)

The Terminal Equations 7 The coefficients of the current and/or voltage variables on the right-hand side of Eqs.18.1 18.6 are called the parameters of the two-port circuit. z parameters : z 11, z 21 z 21, z 22 h parameters : h 11, h 12 h 21, h 22 y parameters : y 11, y 12 y 21, y 22 g parameters : g 11, g 12 g 21, g 22 a parameters : a 11, a 12 a 21, a 22 b parameters : b 11, b 12 b 21, b 22

The Two-Port Parameters 8 Definitions and Meanings of Two-port Parameters z-parameters z 11 = V 1 I 1 I2 =0 Ω (18.7) : the impedance seen looking into port 1 when port 2 is open. z 12 = V 1 I 2 I1 =0 Ω (18.8) : a transfer impedance. It is the ratio of the port 1 voltage to the port 2 current when port 1 is open. z 21 = V 2 I 1 I2 =0 Ω (18.9) : a transfer impedance. It is the ratio of the port 2 voltage to the port 1 current when port 2 is open. z 22 = V 2 I 2 I2 =0 Ω (18.10) : the impedance seen looking into port 2 when port 1 is open.

The Two-Port Parameters 9 y-parameters y 11 = I 1 V 1 V2 =0 S y 12 = I 1 V 2 V1 =0 S y 21 = I 2 V 1 V2 =0 S y 22 = I 2 V 2 V1 =0 S (18.11) Admittance a-parameters y 11 = I 1 V 1 V2 =0 y 12 = I 1 V 2 V1 =0 Ω (18.12) y 21 = I 2 V 1 V2 =0 S y 22 = I 2 V 2 V1 =0

The Two-Port Parameters 10 b-parameters y 11 = I 1 V 1 V2 =0 y 12 = I 1 V 2 V1 =0 Ω y 21 = I 2 V 1 V2 =0 S y 22 = I 2 V 2 V1 =0 (18.13) h-parameters y 11 = I 1 V 1 V2 =0 Ω y 12 = I 1 V 2 V1 =0 y 21 = I 2 V 1 V2 =0 y 22 = I 2 V 2 V1 =0 S (18.14)

The Two-Port Parameters 11 g-parameters y 11 = I 1 V 1 V2 =0 S y 12 = I 1 V 2 V1 =0 y 21 = I 2 V 1 V2 =0 y 22 = I 2 V 2 V1 =0 Ω (18.15) Immittance Parameters The two-port parameters are also described in relation to the reciprocal sets of equations. The impedance and admittance parameters are grouped into the immittance parameters. The term immittance denotes a quantity that is either an impedance or an admittance

The Two-Port Parameters 12 Transmission Parameters The a and b parameters are called the transmission parameters because they describe the voltage and current at one end of the two-port network in terms of the voltage and current at the other end. The immittance and transmission parameters are the natural choices for relating the port variables. In other words, they relate either voltage to current variables or input to output variables. Hybride Parameters The h and g parameters relate cross-variables, that is, an input voltage and output current to an output voltage and input current. Therefore the h and g parameters are called hybrid parameters.

The Two-Port Parameters 13 Relationships Among the Two-Port Parameters Because the six sets of equations relate to the same variables, the parameters associated with any pair of equations must be related to the parameters of all the other pairs. In other words, if we know one set of parameters, we can derive all the other sets from the known set. Because of the amount of algebra involved in these derivations, we merely list the results in Table 18.1. Here, we do derive those between the z and y parameters and between the z and a parameters. These derivations illustrate the general process involved in relating one set of parameters to another. To find the z parameters as functions of the y parameters, we first solve Eqs.(18.2) for V 1 and V 2. We then compare the coefficients of I 1 and I 2 in the resulting expressions to the coefficients of I 1 and I 2 in Eqs.(18.1). From Eqs.(18.2), V 1 = I 1 y 22 I 2 y 11 y 22 y 12 = y 21 y 22 y 22 y I 1 y 12 y I 2 (18.16) V 2 = y 11 I 1 y 21 I 2 y = y 21 y I 1 + y 11 y I 2 (18.17)

The Two-Port Parameters 14 Relationships Among the Two-Port Parameters Comparing Eqs.(18.16) and (18.17) with Eqs.(18.1) shows z 11 = y 22 y (18.17) z 21 = y 21 y (18.20) z 12 = y 12 y (18.17) z 22 = y 11 y (18.21) To find the z parameters as functions of the a parameters, we rearrange Eqs.(18.3) in the form of Eqs.(18.1) and then compare coefficients. From the second equation in Eqs.(18.3), V 2 = 1 a 21 I 1 + a 22 a 21 I 2 (18.22) Therefore, substituting Eq. 18.22 into the first equation of Eqs. 18.3 yields V 1 = a 11 a 21 I 1 + a 11a 22 a 21 a 12 I 2 (18.23)

The Two-Port Parameters 15 Relationships Among the Two-Port Parameters From Eq.(18.23), z 11 = a 11 a 21 (18.24) z 21 = 1 a 21 (18.26) From Eq.(18.23), z 12 = a a 21 (18.25) z 22 = a 22 a 21 (18.27)

18.2 The Two-Port Parameters 16 Reciprocal Two-Port Circuits If a two-port circuit is reciprocal, the following relationships exist among the port parameters: z 12 = z 21 (18.28) y 12 = y 21 (18.29) a 11 a 22 a 12 a 21 = a = 1 (18.30) b 11 b 22 b 12 b 21 = b = 1 (18.31) h 12 = h 21 (18.32) g 12 = g 21 (18.33) A two-port circuit is reciprocal if the interchange of an ideal voltage source at one port with an ideal ammeter at the other port produces the same ammeter reading.

The Two-Port Parameters 17 Reciprocal Two-Port Circuits A reciprocal two-port circuit is symmetric if its ports can be interchanged without disturbing the values of the terminal currents and voltages. Figure 18.6 shows four examples of symmetric two-port circuits. In such circuits, the following additional relationships exist among the port parameters: z 11 = z 22 y 11 = y 22 a 11 = a 22 b 11 = b 22 h 11 h 22 h 12 h 21 = h = 1 g 11 g 22 g 12 g 21 = g = 1 (18.38) (18.39) (18.40) (18.41) (18.42) (18.43) For a symmetric reciprocal network, only two calculations or measurements are necessary to determine all the twoport parameters.

Analysis of the Terminated Two-Port Circuit 18 Figure 18.7 shows the s-domain circuit diagram for a typically terminated two-port model. Here, Z g represents the internal impedance of the source, V g the internal voltage of the source, and Z L the load impedance. Analysis of this circuit involves expressing the terminal currents and voltages as functions of the two-port parameters, V g, Z g and Z L. Six characteristics of the terminated two-port circuit define its terminal behavior: the input impedance Z in = V 1 I 1, or the admittance Y in = I 1 V 1 the output current I 2 the Thevenin voltage and impedance (V Th, Z Th ) with respect to port 2 the current gain I 2 I 1 the voltage gain V 2 V 1 the voltage gain V 2 V g

Analysis of the Terminated Two-Port Circuit 19 The Six Characteristics in Terms of the z Parameters To illustrate how these six characteristics are derived, we develop the expressions using the z parameters to model the two-port portion of the circuit. Table 18.2 summarizes the expressions involving the y, a, b, h, and g parameters. The derivation of any one of the desired expressions involves the algebraic manipulation of the two-port equations along with the two constraint equations imposed by the terminations. If we use the z-parameter equations, the four that describe the circuit in Fig. 18.7 are V 1 = z 11 I 1 + z 12 I 2 (18.44) V 2 = z 21 I 1 + z 22 I 2 V 1 = V g I 1 Z g (18.45) (18.46) V 2 = I 2 Z L (18.47) Equations(18.46) and (18.47) describe the constraints imposed by the terminations.

Analysis of the Terminated Two-Port Circuit 20 The Six Characteristics in Terms of the z Parameters Z in : the impedance seen looking into port 1 Z in = V 1 I 1 V 2 = z 21 I 1 + z 22 I 2 (18.45) V 2 = I 2 Z L (From Fig.18.7) I 2 = z 21I 1 Z L + z 22 (18.48) V 1 = z 11 I 1 + z 12 I 2 (18.44) Z in = V 1 I 1 = z 11 z 12z 21 z 22 + Z L (18.49)

Analysis of the Terminated Two-Port Circuit 21 The Six Characteristics in Terms of the z Parameters I 2 : the terminal current V 1 = z 11 I 1 + z 12 I 2 (18.44) V 1 = V g I 1 Z g (18.46) I 1 = V g z 12 I 2 z 11 + Z g (18.50) I 2 = z 21I 1 Z L + z 22 (18.48) I 2 = z 21 V g z 11 + Z g z 22 + Z L z 12 z 21 (18.51)

Analysis of the Terminated Two-Port Circuit 22 The Six Characteristics in Terms of the z Parameters V Th : Thévenin voltage The Thévenin voltage with respect to port 2 equals V 2 when I 2 = 0 V 1 = z 11 I 1 + z 12 I 2 V 2 = z 21 I 1 + z 22 I 2 (18.44) (18.45) When I 2 = 0 But V 1 I2 = 0 = z 11 I 1 V 2 I2 = 0 = z 21I 1 = z 21 V 1 z 11 (18.52) V 1 = V g I 1 Z g V 1 = z 11 I 1 V g I 1 = Z g + z 11 V 2 I2 = 0 = V Th = z 21 Z g + z 11 V g (18.53)

Analysis of the Terminated Two-Port Circuit 23 The Six Characteristics in Terms of the z Parameters Z Th : Thévenin impedance or output, impedance The Thévenin impedance is the ratio V 2 I 2 when V g is replaced by a short circuit. V 1 = V g I 1 Z g (18.46) When V g = 0 V 1 = I 1 Z g (18.54) Substituting Eq.(18.54) into Eq.(18.44) gives I 1 = z 12I 2 z 11 + Z g (18.55) Substituting Eq.(18.55) into Eq.(18.45) gives V 1 = z 11 I 1 + z 12 I 2 (18.44) V 2 = z 21 I 1 + z 12 I 2 = z 21 z 12 I 2 z 11 + Z g + z 12 I 2 The Thévenin impedance is (18.45) V 2 I 2 V g = 0 = Z Th = z 22 z 12z 21 z 11 + Z g (18.56)

Analysis of the Terminated Two-Port Circuit 24 The Six Characteristics in Terms of the z Parameters I 2 I 1 : Current gain I 2 = z 21I 1 Z L + z 22 (18.48) I 2 I 1 = z 21 Z L + z 22 (18.57) V 2 V 1 : Current gain V 2 = I 2 Z L (18.47) I 2 = V 2 V 2 = z 21 I 1 + z 12 I 2 (18.45) = z 21 I 1 + z 22 V 2 Z L (18.58) Z L V 1 = z 11 I 1 + z 12 I 2 z 11 I 1 = V 1 z 12 V 2 Z L (18.44) I 1 = V 1 z 11 + z 12V 2 z 11 Z L (18.59) V 2 V 1 = z 21 Z L = z 21Z L (18.60) z 11 Z L + z 11 z 22 z 12 z 21 z 11 Z L + z

Analysis of the Terminated Two-Port Circuit 25 The Six Characteristics in Terms of the z Parameters Voltage ratio V 2 V g : V 1 = z 11 I 1 + z 12 I 2 (18.44) V 1 = V g I 1 Z g (18.46) V 2 = I 2 Z L (18.47) I 1 = z 12 V 2 Z L z 11 + Z g + V g z 11 + Z g (18.61) V 2 = I 2 Z L (18.47) I 2 = V 2 Z L V 2 = z 21 I 1 + z 12 I 2 (18.45) = z 21z 12 V 2 Z L z 11 + Z g + z 21V g z 11 + Z g z 22 Z L V 2 (18.62) The desired voltage ratio V 2 V g : V 2 V g = z 21 Z L z 11 + Z g z 22 + Z L z 12 z 21 (18.63)

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