Question

What is the relationship between transmission parameters (ABCD) and impedance parameters (Z-parameters)?

Answer 1

The transmission parameters (also known as ABCD parameters) and the impedance parameters (Z-parameters) are two different representations of the same two-port network in electrical engineering. These parameters are used to describe the behavior of linear two-port networks, which are circuits with two input terminals and two output terminals.

Transmission Parameters (ABCD Parameters):
The transmission parameters are also known as the chain parameters or the cascade parameters. They are commonly denoted as A, B, C, and D. These parameters describe the relationship between the voltage and current at the input and output of the two-port network. Here's the general representation of ABCD parameters:
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[ V1 ]   [ A  B ] [ I1 ]
[    ] = [      ] [    ]
[ V2 ]   [ C  D ] [ I2 ]


Where:

V1 is the voltage at the input terminal
V2 is the voltage at the output terminal
I1 is the current at the input terminal
I2 is the current at the output terminal
A, B, C, and D are constants that depend on the network's characteristics.
Impedance Parameters (Z-Parameters):
The impedance parameters, also known as the open-circuit parameters or the driving-point impedance parameters, are commonly denoted as Z11, Z12, Z21, and Z22. These parameters describe the relationship between the input and output voltages and currents when one port of the two-port network is terminated with an impedance while the other port is open-circuited. Here's the general representation of Z-parameters:
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[ V1 ]   [ Z11  Z12 ] [ I1 ]
[    ] = [          ] [    ]
[ V2 ]   [ Z21  Z22 ] [ I2 ]


Where:

Z11 is the input impedance seen when the output port is open (V2 = 0)
Z12 is the transfer impedance from port 2 to port 1 when the output port is open (V2 = 0)
Z21 is the transfer impedance from port 1 to port 2 when the input port is open (V1 = 0)
Z22 is the output impedance seen when the input port is open (V1 = 0)

Relationship between ABCD and Z-Parameters:
The ABCD parameters and Z-parameters are related to each other through the following equations:

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A = (Z11 * Z22 + Z12 * Z21) / det(Z)
B = Z12 / Z22
C = (Z11 * Z21) / det(Z)
D = Z11 / Z22


Where det(Z) is the determinant of the Z-parameters matrix, given by det(Z) = Z11 * Z22 - Z12 * Z21.

Conversely, you can also derive Z-parameters from ABCD parameters using the following equations:

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Z11 = A / C
Z12 = (A * D - B * C) / C
Z21 = 1 / C
Z22 = D / C


It's important to note that both ABCD and Z-parameters are convenient representations for analyzing and designing linear two-port networks. The choice of representation depends on the specific application and the type of analysis required.