TwoPort Impedance Model and ZParameters
Key Takeaways

A twoport impedance model represents the voltages of a system as a function of currents.

The Zparameter matrix of a twoport model is of order 2 2. The elements are either driving point impedances or transfer impedances.

The condition of reciprocity or symmetry existing in a system can be easily identified from the Zparameters. Condition for symmetry: Z_{11}= Z_{22} and condition for reciprocity:Z_{12}= Z_{21}

The twoport network model and Zparameters can be applied in the analysis of power distribution networks, synthesis of filters, and design of impedance matching circuits.
Figure.1 Twoport network model
It is difficult to study the inputoutput behavior of large complex circuits in power systems, communication engineering, process controls, and electronic systems with physical modeling. It is more convenient to develop a twoport model for predicting the circuit behavior under a given input in large systems.
The twoport network model is a popular modeling technique used to characterize the electrical and electronic circuits. The twoport network approach simplifies a complex circuit into a twoport network model made of basic electrical elements, and the inputoutput behavior of this model exactly resembles the initial large system.
Among the various approaches in twoport modeling, the twoport impedance (Z) model reproduces the system behavior by exciting the model with currents. As illustrated in Figure.1, the model is excited by supplying input port and output port with currents I_{1}and I_{2}, respectively. The responses to the excitation are obtained as the input port and output port voltages V_{1 }and V_{2}, respectively. The inputoutput behavior of a large complex system can be easily characterized using the four variablesV_{1},V_{2}, I_{1}, and I_{2}, and mathematically represented using the excitationresponse variables and coefficients, called Z parameters.
TwoPort Impedance Model
Any linear circuit can be represented as a twoport network, defined by four variables V_{1},V_{2}, I_{1}, and I_{2}. The direction of currents and polarity of voltages in port 1 and 2 are as shown in Figure.1 Out of these four quantities, the input quantities are independent variables, and the outputs are dependent variables.
The mathematical expression of the twoport network model is one pair of equations defining the output variables in terms of inputs and a matrix. The twoport parameter matrix is of order 22 and the elements are called twoport network parameters. Sparameters, when only voltages are used, are also quite common. The details of six possible twoport network models and parameters are given in Table 1 below.
Table.1 The twoport network models and parameters
The values of the twoport network parameters completely characterize the behavior of the linear circuit. The twoport network parameters are calculated using circuit analysis methods, or derived from other known parameters. Most of the twoport parameters share a dual relationship with other parameters, such as [Y]=[Z]^{1}, [G]=[H]^{1}, [T']=[T]^{1}. Each twoport model differs from the other, and parameters are either impedances, admittances, or scalars depending on the inputoutput relationship. However, all twoport models give us the exact characterization of the original circuit without fail.
Z Parameters
We have already seen the excitation and response variables in Zparameter modeling; the voltage equations governing the twoport impedance model are:
The Zparameters denoted by Z_{11}, Z_{12},Z_{21}, and Z_{22} are the coefficients of the currents I_{1}, and I_{2}, in the two equations above. As each Zparameter gives the voltagecurrent relationship, the coefficients are impedance values given in ohms. The equations (1) and (2) can be electrically represented by the equivalent circuit given in Figure.2:
Figure.2 Equivalent circuit of twoport impedance model
From the matrix representation of equations (1) and (2), the Zparameter matrix can be derived as follows:
Calculation of ZParameters
Now you know that the Zparameter matrix describes the voltagecurrent relationship in a twoport impedance network. But how would you calculate the Zparameters of a given large complex circuit? Unless the knowledge about the internal connections of the circuit under test is limited, circuit analysis is the best method. If the circuit on your workbench is a ‘black box’, then you need to go through the following steps to determine each of the Zparameters. (Refer to Figure.1):
The Zparameters are determined by open circuiting port 1 and port 2, hence the name opencircuit impedance parameters. The input and output impedance of any complex system can be determined easily with Zparameters. The Zparameters are ideal for identifying the nature of the large systems as given in Table 2:
Table.2: Nature of the system and Zparameter relationship
Application of TwoPort Impedance Model
The application of Zparameters makes great strides in realizing filter circuits. By analyzing the driving pointimpedance and transfer impedances, the physical elements suitable for the filter can be picked. The conversion of Zparameters into S, Y parameters are also significant in expanding application to the design, synthesis, and analysis of impedance matching circuits and power distribution networks. When a complex system is an amalgamation of several other circuits, the impedance parameters can help you to decode the interconnection between each of the subsystems, and to form a simplified model for further extensive study.
If you want to describe the analog behavior of a complex circuit, choose the twoport impedance model for best analysis results. The Zparameters extracted from the system under consideration is advantageous in understanding the nature of the circuit and to design the filter and impedance matching circuits for the same.