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 two-port network may be voltage-driven as in Fig. 1(a) or current-driven as in Fig. 1(b). From either Fig. 1(a) or (b), the terminal voltages can be related to the terminal currents as $$\bbox[10px,border:1px solid grey]{\begin{array}{l}\mathbf{V}_{1}=\mathbf{z}_{11} \mathbf{I}_{1}+\mathbf{z}_{12} \mathbf{I}_{2} \\\mathbf{V}_{2}=\mathbf{z}_{21} \mathbf{I}_{1}+\mathbf{z}_{22} \mathbf{I}_{2}\end{array}} \tag{1}$$ or in matrix form as $$\left[\begin{array}{l}\mathbf{V}_{1} \\\mathbf{V}_{2}\end{array}\right]=\left[\begin{array}{ll}\mathbf{z}_{11} & \mathbf{z}_{12} \\\mathbf{z}_{21} & \mathbf{z}_{22}\end{array}\right]\left[\begin{array}{l}\mathbf{I}_{1} \\\mathbf{I}_{2}\end{array}\right]=[\mathbf{z}]\left[\begin{array}{c}\mathbf{I}_{1} \\\mathbf{I}_{2}\end{array}\right] \tag{2}$$ where the $ \mathbf{z} $ terms are called the impedance parameters, or simply $ z $ parameters, and have units of ohms. The values of the parameters can be evaluated by setting $ \mathbf{I}_{1}=0 $ (input port open-circuited) or $ \mathbf{I}_{2}=0 $ (output port open-circuited). Thus, $$\begin{array}{ll}\mathbf{z}_{11}=\left.\frac{\mathbf{V}_{1}}{\mathbf{I}_{1}}\right|_{\mathbf{I}_{2}=0}, & \mathbf{z}_{12}=\left.\frac{\mathbf{V}_{1}}{\mathbf{I}_{2}}\right|_{\mathbf{I}_{1}=0} \\\mathbf{z}_{21}=\left.\frac{\mathbf{V}_{2}}{\mathbf{I}_{1}}\right|_{\mathbf{I}_{2}=0}, & \mathbf{z}_{22}=\left.\frac{\mathbf{V}_{2}}{\mathbf{I}_{2}}\right|_{\mathbf{I}_{1}=0}\end{array} \tag{3}$$ Since the $ z $ parameters are obtained by open-circuiting the input or output port, they are also called the open-circuit impedance parameters. Specifically,
$ \mathbf{z}_{11}= $ Open-circuit input impedance
$\mathbf{z}_{12}= $ Open-circuit transfer impedance from port 1 to port 2
$ \mathbf{z}_{21}= $ Open-circuit transfer impedance from port 2 to port 1
$ \mathbf{z}_{22}= $ Open-circuit output impedance According to Eq. (3), we obtain $ \mathbf{z}_{11} $ and $ \mathbf{z}_{21} $ by connecting a voltage $ \mathbf{V}_{1} $ (or a current source $ \mathbf{I}_{1} $ ) to port 1 with port 2 open-circuited as in Fig. 2(a) and finding $ \mathbf{I}_{1} $ and $ \mathbf{V}_{2} $; we then get $$\mathbf{z}_{11}=\frac{\mathbf{V}_{1}}{\mathbf{I}_{1}}, \quad \mathbf{z}_{21}=\frac{\mathbf{V}_{2}}{\mathbf{I}_{1}}$$ Similarly, we obtain $ \mathbf{z}_{12} $ and $ \mathbf{z}_{22} $ by connecting a voltage $ \mathbf{V}_{2} $ (or a current source $ \mathbf{I}_{2} $ ) to port 2 with port 1 open-circuited as in Fig. 2(b) and finding $ \mathbf{I}_{2} $ and $ \mathbf{V}_{1} $; we then get $$\mathbf{z}_{12}=\frac{\mathbf{V}_{1}}{\mathbf{I}_{2}}, \quad \mathbf{z}_{22}=\frac{\mathbf{V}_{2}}{\mathbf{I}_{2}}$$ The above procedure provides us with a means of calculating or measuring the $ z $ parameters.
Sometimes $ \mathbf{z}_{11} $ and $ \mathbf{z}_{22} $ are called driving-point impedances, while $ \mathbf{z}_{21} $ and $ \mathbf{z}_{12} $ are called transfer impedances. A driving-point impedance is the input impedance of a two-terminal (one-port) device. Thus, $ \mathbf{z}_{11} $ is the input driving-point impedance with the output port open-circuited, while $ \mathbf{z}_{22} $ is the output driving-point impedance with the input port open-circuited.
When $ \mathbf{z}_{11}=\mathbf{z}_{22} $, the two-port network is said to be symmetrical. This implies that the network has mirrorlike symmetry about some center line; that is, a line can be found that divides the network into two similar halves.
When the two-port network is linear and has no dependent sources, the transfer impedances are equal $ \left(\mathbf{z}_{12}=\mathbf{z}_{21}\right) $, and the two-port is said to be reciprocal. This means that if the points of excitation and response are interchanged, the transfer impedances remain the same. As illustrated in Fig. 3, a two-port is reciprocal if interchanging an ideal voltage source at one port with an ideal ammeter at the other port gives the same ammeter reading. The reciprocal network yields $ \mathbf{V}=\mathbf{z}_{12} \mathbf{I} $ according to Eq. (1) when connected as in Fig. 3(a), but yields $ \mathbf{V}=\mathbf{z}_{21} \mathbf{I} $ when connected as in Fig. 3(b). This is possible only if $ \mathbf{z}_{12}=\mathbf{z}_{21} $. Any two-port that is made entirely of resistors, capacitors, and inductors must be reciprocal. For a reciprocal network, the T-equivalent circuit in Fig. 4(a) can be used. If the network is not reciprocal, a more general equivalent network is shown in Fig. 4(b); notice that this figure follows directly from Eq. (1). It should be mentioned that for some two-port networks, the $ z $ parameters do not exist because they cannot be described by Eq. (1). As an example, consider the ideal transformer of Fig. 5. The defining equations for the two-port network are: $$\mathbf{V}_{1}=\frac{1}{n} \mathbf{V}_{2}, \quad \mathbf{I}_{1}=-n \mathbf{I}_{2}$$ Observe that it is impossible to express the voltages in terms of the currents, and vice versa, as Eq. (18.1) requires. Thus, the ideal transformer has no $ z $ parameters. However, it does have hybrid parameters,