Two-port network model and parameters

A two-port network is any linear circuit with an input port (1) and an output port (2), such as an amplifier, filter, attenuator or transmission line section. It is modeled as a "black box" described only by the relations between the port voltages, currents or waves.

      I1 ->                     <- I2
  o---------+-------------+---------o
  +         |             |         +
  V1        |  Two-port   |        V2
  -         |  network    |         -
  o---------+-------------+---------o
   port 1                     port 2

Four variables (V1, I1, V2, I2) exist; any two can be written in terms of the other two, giving the parameter sets below.

1. Impedance (Z) parameters

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

Z11 = V1/I1 at I2 = 0 (output open); Z21 = V2/I1 at I2 = 0; Z12 = V1/I2 at I1 = 0; Z22 = V2/I2 at I1 = 0. Units: ohm. Reciprocal → Z12 = Z21.

2. Admittance (Y) parameters

 I1 = Y11·V1 + Y12·V2
 I2 = Y21·V1 + Y22·V2

Measured with the opposite port short-circuited (V = 0). [Y] = [Z]⁻¹. Units: siemens.

3. Hybrid (h) parameters

 V1 = h11·I1 + h12·V2
 I2 = h21·I1 + h22·V2

h11 = input impedance (output shorted), h21 = current gain, h12 = reverse voltage ratio (input open), h22 = output admittance. Used for BJT models.

4. Transmission (ABCD) parameters

 V1 = A·V2 + B·I2'      (I2' = -I2, current leaving port 2)
 I1 = C·V2 + D·I2'

Cascaded networks multiply: [ABCD]total = [ABCD]1 · [ABCD]2. Reciprocal → AD − BC = 1. Example: a series impedance Z has A = 1, B = Z, C = 0, D = 1.

5. Scattering (S) parameters – the microwave model

At microwave frequencies the opens and shorts needed above are hard to make, so the network is described by waves. With reference impedance Z0:

 an = (Vn + Z0·In)/(2√Z0)   incident wave
 bn = (Vn - Z0·In)/(2√Z0)   reflected wave

 b1 = S11·a1 + S12·a2
 b2 = S21·a1 + S22·a2

• S11 = b1/a1 at a2 = 0 (port 2 matched): input reflection coefficient.
• S21 = b2/a1 at a2 = 0: forward transmission coefficient.
• S22 = b2/a2 at a1 = 0: output reflection coefficient.
• S12 = b1/a2 at a1 = 0: reverse transmission coefficient.

Derivation of S from Z: substitute V = √Z0(a + b) and I = (a − b)/√Z0 into V = Z·I:

 Z0(a + b) = Z(a - b)
 (Z + Z0·U) b = (Z - Z0·U) a
 [S] = (Z - Z0·U)(Z + Z0·U)^-1

For a two-port with ΔZ = (Z11 + Z0)(Z22 + Z0) − Z12·Z21:

 S11 = [(Z11 - Z0)(Z22 + Z0) - Z12·Z21]/ΔZ
 S21 = 2·Z21·Z0/ΔZ

Properties used in design

Useful derived quantities: input reflection Γin = S11 + S12·S21·ΓL/(1 − S22·ΓL); return loss = −20 log|S11|; insertion loss = −20 log|S21|; transducer gain (matched) = |S21|².

Example: a series impedance Z between two Z0 lines gives S11 = S22 = Z/(Z + 2Z0) and S21 = S12 = 2Z0/(Z + 2Z0); for Z = 50 Ω and Z0 = 50 Ω, S11 = 1/3 and S21 = 2/3.