Question

How do you find the voltage gain from Z-parameters?

Answer 1

To find the voltage gain from Z-parameters (impedance parameters or network parameters), you need to calculate the ratio of the output voltage to the input voltage in a two-port network. The Z-parameters are represented in a matrix form as follows:

| V1 | | Z11 Z12 | | I1 |
| | = | | | |
| V2 | | Z21 Z22 | | I2 |

Where:
V1 = Input voltage at Port 1
I1 = Input current at Port 1
V2 = Output voltage at Port 2
I2 = Output current at Port 2
Z11 = Input impedance with Port 2 short-circuited (V2 = 0)
Z12 = Forward transfer impedance
Z21 = Reverse transfer impedance
Z22 = Output impedance with Port 1 open-circuited (V1 = 0)

The voltage gain, Av, is defined as the ratio of the output voltage to the input voltage, considering the output and input currents:

Av = V2 / V1

To find the voltage gain using Z-parameters, follow these steps:

Step 1: Set the input current, I1, to zero (open-circuit the input):
I1 = 0

Step 2: Calculate the output voltage, V2, when I1 is set to zero. To do this, you can use the first row of the Z-parameters matrix:

| V1 | | Z11 Z12 | | I1 |
| | = | | | |
| V2 | | Z21 Z22 | | I2 |

Since I1 = 0, the equation becomes:
| V1 | | Z11 Z12 | | 0 | => | V1 | | Z11 Z12 | | 0 |
| | = | | | | | | = | | | |
| V2 | | Z21 Z22 | | I2 | | V2 | | Z21 Z22 | | I2 |

Simplify the equation:
V1 = Z11 * 0 + Z12 * I2
V2 = Z21 * 0 + Z22 * I2
V2 = Z22 * I2

Step 3: Calculate the input voltage, V1, when I2 (output current) is zero. To do this, use the second column of the Z-parameters matrix:

| V1 | | Z11 Z12 | | I1 | => | V1 | | Z11 Z12 | | I1 |
| | = | | | | | | = | | | |
| V2 | | Z21 Z22 | | 0 | | V2 | | Z21 Z22 | | 0 |

Simplify the equation:
V1 = Z11 * I1 + Z12 * 0
V1 = Z11 * I1

Step 4: Now, calculate the voltage gain, Av:
Av = V2 / V1
Av = (Z22 * I2) / (Z11 * I1)

Note: The voltage gain obtained this way is dependent on the load impedance connected at Port 2 (Z_load). If you assume an ideal load (Z_load = Z22), then the voltage gain becomes independent of the load impedance:

Av = Z22 / Z11

Keep in mind that Z-parameters are typically used for linear circuits, and this approach assumes small-signal behavior.