Importance of S-parameters and two-port derivation

Importance of S-parameters in microwave network analysis

• At microwave frequencies, voltage and current are difficult to measure (and not uniquely defined in waveguides), but power waves travelling in and out of a port are easy to measure.
• Z, Y, h parameters need open or short terminations; these are hard to realise at high frequencies and can make active devices oscillate. S-parameters are measured with matched (Z0) loads.
• S-parameters directly give reflection coefficient, VSWR, return loss, insertion loss, gain and isolation.
• They can be measured accurately with a vector network analyzer and cascaded (via T-parameters) for system design.

Derivation of S-parameters for a two-port network

        a1 ->           -> b2
  o-----------+-------+-----------o
  port 1      |  [S]  |      port 2
  o-----------+-------+-----------o
        <- b1           <- a2

At each port n the total voltage and current are the sum of incident (+) and reflected (−) waves on a line of characteristic impedance Z0:

 Vn = Vn+ + Vn-
 In = (Vn+ - Vn-)/Z0

Define the normalized incident and reflected waves (their squares are power):

 an = Vn+/√Z0 = (Vn + Z0·In)/(2√Z0)
 bn = Vn-/√Z0 = (Vn - Z0·In)/(2√Z0)
 Incident power   = |an|²/2
 Reflected power  = |bn|²/2

For a linear network the outgoing waves are linear combinations of the incoming waves:

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

 | b1 |   | S11  S12 | | a1 |
 | b2 | = | S21  S22 | | a2 |

Each parameter is found by making one incident wave zero, i.e. terminating that port in a matched load Z0 (a matched load reflects nothing, so aₖ = 0):

• S11 = b1/a1 |a2=0: input reflection coefficient with output matched.
• S21 = b2/a1 |a2=0: forward transmission coefficient (gain or loss).
• S22 = b2/a2 |a1=0: output reflection coefficient with input matched.
• S12 = b1/a2 |a1=0: reverse transmission (isolation).

Relation with impedance parameters

Starting from V = Z·I and substituting V = √Z0(a + b) and I = (a − b)/√Z0 gives

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

For a one-port this reduces to S11 = (ZL − Z0)/(ZL + Z0) = Γ, the familiar reflection coefficient.

Useful results

• Return loss = −20 log|S11| dB; insertion loss = −20 log|S21| dB.
• Input VSWR = (1 + |S11|)/(1 − |S11|).
• Reciprocal network: S12 = S21; lossless network: |S11|² + |S21|² = 1 and S11·S12* + S21·S22* = 0.
• Example: an ideal matched attenuator of 3 dB has S11 = S22 = 0, S12 = S21 = 1/√2 ≈ 0.707.