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Two-Port Impedance Model and Z-Parameters

Key Takeaways

  • A two-port impedance model represents the voltages of a system as a function of currents.

  • The Z-parameter matrix of a two-port 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 Z-parameters. Condition for symmetry: Z11= Z22 and condition for reciprocity:Z12= Z21

  • The two-port network model and Z-parameters can be applied in the analysis of power distribution networks, synthesis of filters, and design of impedance matching circuits.

V1,V2, I1, and I2 are the four variables in a two-port network model

Figure.1 Two-port network model

It is difficult to study the input-output 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 two-port model for predicting the circuit behavior under a given input in large systems. 

The two-port network model is a popular modeling technique used to characterize the electrical and electronic circuits. The two-port network approach simplifies a complex circuit into a two-port network model made of basic electrical elements, and the input-output behavior of this model exactly resembles the initial large system. 

Among the various approaches in two-port modeling, the two-port 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  I1and  I2, respectively. The responses to the excitation are obtained as the input port and output port voltages Vand  V2, respectively. The input-output behavior of a large complex system can be easily characterized using the four variablesV1,V2, I1, and I2, and mathematically represented using the excitation-response variables and coefficients, called Z- parameters. 

Two-Port Impedance Model

Any linear circuit can be represented as a two-port network, defined by four variables V1,V2, I1, and I2. 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 two-port network model is one pair of equations defining the output variables in terms of inputs and a matrix. The two-port parameter matrix is of order 22 and the elements are called two-port network parameters. S-parameters, when only voltages are used, are also quite common. The details of six possible two-port network models and parameters are given in Table 1 below.

 

Network Excitation

(Independent variables)

Network Response

(Dependent variables)

Two-port network parameters

Input  1

Input 2

Output1

Output 2

I1

I2

V1

V2

Open-circuit impedance or Z- parameters

V1

V2

I1

I2

Short-circuit admittance or Y-parameters

I1

V2

V1

I2

Hybrid or H-parameters

V1

I2

I1

V2

Inverse-hybrid or G-parameters

V2

I2

V1

I1

Transmission or T-parameters or ABCD parameters

V1

I1

V2

I2

Inverse-transmission orT’-parameters or A’B’C’D’ parameters

Table.1 The two-port network models and parameters 

The values of the two-port network parameters completely characterize the behavior of the linear circuit.  The two-port network parameters are calculated using circuit analysis methods, or derived from other known parameters. Most of the two-port parameters share a dual relationship with other parameters, such as [Y]=[Z]-1, [G]=[H]-1, [T']=[T]-1. Each two-port model differs from the other, and parameters are either impedances, admittances, or scalars depending on the input-output relationship. However, all two-port models give us the exact characterization of the original circuit without fail.

Z -Parameters

We have already seen the excitation and response variables in Z-parameter modeling; the voltage equations governing the two-port impedance model are:

 two port impedance model voltage equations

The Z-parameters denoted by Z11, Z12,Z21, and Z22 are the coefficients of the currents I1, and I2, in the two equations above. As each Z-parameter gives the voltage-current 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:

Impedances and current-controlled voltage source are used to model the two-port impedance network equations

Figure.2 Equivalent circuit of two-port impedance model

From the matrix representation of equations (1) and (2), the Z-parameter matrix can be derived as follows:

Z-parameter matrix equation

Calculation of Z-Parameters

Now you know that the Z-parameter matrix describes the voltage-current relationship in a two-port impedance network. But how would you calculate the Z-parameters 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 Z-parameters. (Refer to Figure.1):

The Z-parameters are determined by open circuiting port 1 and port 2, hence the name open-circuit impedance parameters. The input and output impedance of any complex system can be determined easily with Z-parameters. The Z-parameters are ideal for identifying the nature of the large systems as given in Table 2:

System

Z-parameter relationship

Symmetrical system

Z11= Z22

Reciprocal system

Z12= Z21

Reciprocal lossless system

All Z-parameters are purely imaginary

Table.2: Nature of the system and Z-parameter relationship 

Application of Two-Port Impedance Model

The application of Z-parameters makes great strides in realizing filter circuits. By analyzing the driving point-impedance and transfer impedances, the physical elements suitable for the filter can be picked. The conversion of Z-parameters 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 two-port impedance model for best analysis results.  The Z-parameters 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.