Definition of Z-parameters

Z-parameters describe the relation between voltage and current of an N-port network. In case of two-port network, the first number in the subscript refers to port for voltage, while the second number refers to port number for current. Thus Z11 means impedance of port 1 (V1/I1).

In case of two-port network:

Remark:

- Z_ij is a complexe impedance: Z_ij = R_Zij + jX_Zij

- Z_s, Z_c and Z_L are real impedance.

Z11 : input impedance

(z_ref is usual selected at 50Ω)

Remark:

No confuse Z11 with Zin. Z11 = Zin only if I₂=0 (Z_L = ∞)

Z22 : output impedance

Remark:

No confuse Z22 with Zout. Z22 = Zout only if I₁=0 (Z_s = ∞)

Z12 : Reverse transfert impedance

Z21 : Direct transfert impedance

Voltage and Current relations

Voltage Gain

Current Gain

Impedance relations

The input impedance of a two-port network is given by:

Similarly, the output impedance is given by:

- Input matched impédance:

Z_e = Z_c^* (if Zc is a real impedance : Z_e = Z_c)

- output matched impédance:

Z_L = Z_c^* (if Zc is a real impedance : Z_L = Z_c)

- No adaptated line:

The impedance bring back at input of line is given by:

Remark:

Skin effect is not taking into acount.

Input reflection coefficient:

Output reflection coefficient:

Return Loss:

= -10 log₁₀ (Pr/Pi) = -20 log₁₀ (|S₁₁|)

Mismatch Loss:

ROS: Rapport des ondes stationnaires

TOS: Taux d’onde stationnaire

VSWR: Voltage Standing Waves Ratio

SWR: Standing Waves Ratio

Converting Two-Port Z-Parameters to S-Parameters

The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions.

or with Z-parameter normalized by Z0.