19.1 Introduction

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19.1 Introduction

A pair of terminals through which a current may enter or leave a network is known as a port. Two-terminal devices or elements (such as resis tors, capacitors, and inductors) result in one-port networks. Most of the circuits we have dealt with so f ar are two-terminal or one-port circuits, represented in Fig. 19.1(a). We have considered the v oltage across or current through a single pair of terminals—such as the tw o terminals of a resistor, a capacitor, or an inductor. We have also studied four-terminal or two-port circuits involving op amps, transistors, and transformers, as shown in Fig. 19.1(b). In general, a network may have n ports. A port is an access to the netw ork and consists of a pair of terminals; the current entering one terminal lea ves through the other terminal so that the net current entering the port equals zero.

In this chapter, we are mainly concerned with two-port networks (or, simply, two-ports).

A two-port network is an electrical network with two separate ports for input and output.

Thus, a two-port network has two terminal pairs acting as access points. As shown in Fig. 19.1(b), the current entering one terminal of a pair leaves the other terminal in the pair. Three-terminal devices such as transistors can be configured into two-port networks.

Our study of tw o-port networks is for at least tw o reasons. First, such netw orks are useful in communications, control systems, po wer systems, and electronics. F or example, they are used in electronics to model transistors and to facilitate cascaded design. Second, knowing the parameters of a two-port network enables us to treat it as a “black box” when embedded within a larger network.

Figure 19.1 (a) One-port network, (b) two-port network.

To characterize a tw o-port network requires that we relate the ter minal quantities V1, V2, I1, and I2 in Fig. 19.1(b), out of which tw o are independent. The various terms that relate these v oltages and currents are called parameters. Our goal in this chapter is to deri ve six sets of these parameters. We will sho w the relationship between these param eters and how two-port networks can be connected in series, parallel, or cascade. As with op amps, we are only interested in the terminal behavior of the circuits. And we will assume that the two-port circuits contain no independent sources, although the y can contain dependent sources. Finally, we will apply some of the concepts developed in this chapter to the analysis of transistor circuits and synthesis of ladder networks.

19.2 Impedance Parameters

Impedance and admittance parameters are commonly used in the synthesis of filters. They are also useful in the design and analysis of impedance-matching networks and power distribution networks. We discuss impedance parameters in this section and admittance parameters in the next section.

A tw o-port netw ork may be v oltage-driven as in Fig. 19.2(a) or current-driven as in Fig. 19.2(b). From either Fig. 19.2(a) or (b), the terminal voltages can be related to the terminal currents as

$$V_1 = z_{11}I_1 + z_{12}I_2$$

\n

$$V_2 = z_{21}I_1 + z_{22}I_2$$

(19.1)

Reminder: Only two of the four variables (V1, V2, I1, and I2) are independent. The other two can be found using Eq. (19.1).

or in matrix form as

$$\begin{bmatrix} \mathbf{V}_1 \\ \mathbf{V}_2 \end{bmatrix} = \begin{bmatrix} \mathbf{z}_{11} & \mathbf{z}_{12} \\ \mathbf{z}_{21} & \mathbf{z}_{22} \end{bmatrix} \begin{bmatrix} \mathbf{I}_1 \\ \mathbf{I}_2 \end{bmatrix} = [\mathbf{z}] \begin{bmatrix} \mathbf{I}_1 \\ \mathbf{I}_2 \end{bmatrix}$$

(19.2)

where the z terms are called the impedance par ameters, or simply z parameters, and have units of ohms.

The values of the parameters can be e valuated by setting I1 = 0 (input port open-circuited) or I2 = 0 (output port open-circuited). Thus,

The linear two-port network: (a) driven by voltage sources, (b) driven by current sources.

Figure 19.3

Determination of the z parameters: (a) finding z11 and z21, (b) finding z12 and z22.

Figure 19.4

Interchanging a voltage source at one port with an ideal ammeter at the other port produces the same reading in a reciprocal two-port.

Because the z parameters are obtained by open-circuiting the input or output port, they are also called the open-circuit impedance parameters. Specifically,

• z11 = Open-circuit input impedance z12 = Open-circuit transfer impedance from port 1 to port 2 **^z**21 = Open-circuit transfer impedance from port 2 to port 1 (19.4)
• z22 = Open-circuit output impedance