Conversion of Z parameters to Y parameters

For a two-port, the Z (open-circuit impedance) parameters express the port voltages in terms of the port currents, and the Y (short-circuit admittance) parameters express the currents in terms of the voltages. Since both describe the same network, $[Y]$ is simply the inverse of $[Z]$.

Z-parameter equations

$$\begin{aligned} V_1 &= z_{11} I_1 + z_{12} I_2 \\ V_2 &= z_{21} I_1 + z_{22} I_2 \end{aligned}$$

Derivation

We want $I_1$ and $I_2$ in terms of $V_1$ and $V_2$. Solve the two equations by Cramer's rule. Let

$$\Delta_z = \begin{vmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{vmatrix} = z_{11}z_{22} - z_{12}z_{21}$$

Then

$$I_1 = \frac{\begin{vmatrix} V_1 & z_{12} \\ V_2 & z_{22} \end{vmatrix}}{\Delta_z} = \frac{z_{22}}{\Delta_z}V_1 - \frac{z_{12}}{\Delta_z}V_2$$ $$I_2 = \frac{\begin{vmatrix} z_{11} & V_1 \\ z_{21} & V_2 \end{vmatrix}}{\Delta_z} = -\frac{z_{21}}{\Delta_z}V_1 + \frac{z_{11}}{\Delta_z}V_2$$

Comparing with the Y-parameter equations $I_1 = y_{11}V_1 + y_{12}V_2$ and $I_2 = y_{21}V_1 + y_{22}V_2$:

In matrix form:

$$[Y] = [Z]^{-1} = \frac{1}{\Delta_z}\begin{bmatrix} z_{22} & -z_{12} \\ -z_{21} & z_{11} \end{bmatrix}$$

Remarks

• The conversion exists only if $\Delta_z \neq 0$.
• For a reciprocal (passive RLC) network $z_{12} = z_{21}$, which gives $y_{12} = y_{21}$.
• Note $y_{11} \neq 1/z_{11}$ in general: $y_{11}$ is measured with port 2 shorted, $z_{11}$ with port 2 open.

Example. For a T-network with series arms $Z_a, Z_b$ and shunt arm $Z_c$: $z_{11} = Z_a+Z_c$, $z_{22} = Z_b+Z_c$, $z_{12} = z_{21} = Z_c$. With $Z_a = Z_b = Z_c = 1\ \Omega$: $\Delta_z = 2\times2 - 1 = 3$, so $y_{11} = y_{22} = 2/3$ S and $y_{12} = y_{21} = -1/3$ S.