Question

Derive the formulas to convert ABCD parameters to Z-parameters.

Answer 1

To convert ABCD parameters to Z-parameters, we need to consider a two-port network. The ABCD parameters describe the relationship between the voltage and current at the input and output ports of the network, while the Z-parameters describe the impedance seen at the input and output ports. Let's derive the formulas for the conversion:

Consider the two-port network with the following ABCD parameters:

A B
C D

The input and output voltages and currents can be represented as follows:

Vin = A * Iin + B * Vout
Iout = C * Iin + D * Vout

Similarly, we can represent the Z-parameters (Z11, Z12, Z21, and Z22) as the input and output currents and voltages:

V1 = Z11 * I1 + Z12 * I2
V2 = Z21 * I1 + Z22 * I2

To find the Z-parameters in terms of the ABCD parameters, we need to express the currents I1 and I2 in terms of Vin and Iout using the equations mentioned earlier:

Expressing Iin in terms of Vin and Vout:
Iin = (D * Vin - Vout) / (B * D - A * C)

Expressing Iout in terms of Vin and Vout:
Iout = (A * Iin + B * Vout)

Now, we can substitute the expressions for Iin and Iout into the Z-parameter equations:

V1 = Z11 * I1 + Z12 * I2
V2 = Z21 * I1 + Z22 * I2

V1 = Z11 * [(D * Vin - Vout) / (B * D - A * C)] + Z12 * [(A * (D * Vin - Vout) / (B * D - A * C)) + B * Vout]
V2 = Z21 * [(D * Vin - Vout) / (B * D - A * C)] + Z22 * [(A * (D * Vin - Vout) / (B * D - A * C)) + B * Vout]

Now, we can rearrange the equations to solve for the Z-parameters:

Z11 = V1 / I1 = (Z12 * B - Z22 * A) / (A * D - B * C)
Z12 = V1 / I2 = (Z11 * D - Z21 * B) / (A * D - B * C)
Z21 = V2 / I1 = (Z22 * C - Z12 * A) / (A * D - B * C)
Z22 = V2 / I2 = (Z21 * B - Z11 * C) / (A * D - B * C)

These are the formulas to convert ABCD parameters to Z-parameters for a two-port network. Please note that the resulting Z-parameters depend on the values of the ABCD parameters of the network.