S-matrix of an amplitude modulator

Importance of S-parameters at microwave frequencies

• Voltage and current cannot be measured directly at microwave frequencies (and are not unique in waveguides); incident and reflected power waves can.
• Z, Y and h parameters need open or short terminations, which are hard to build at high frequency and may make active devices oscillate. S-parameters are measured with matched loads.
• S-parameters give reflection (S11, S22), gain/loss (S21) and isolation (S12) directly, and are measured with a vector network analyzer.
• Network properties are simple to test: reciprocal → [S] = [S]ᵀ, lossless → [S] unitary.

S-matrix of an amplitude modulator

A microwave amplitude modulator is a two-port device whose transmission coefficient changes with a low-frequency modulating signal. A common form is a PIN diode mounted in shunt across a matched line (or waveguide). The diode's RF resistance R is controlled by its bias current, so the bias carries the modulating signal.

  a1 ->                       -> b2
  o-------------+-------------o
  port 1 (Z0)   |        port 2 (Z0)
               [R]  PIN diode, R(t)
                |   (bias = modulating signal)
  o-------------+-------------o

Assumption: the diode acts as a pure resistance with normalized shunt admittance y = Z0/R; ports are matched lines of Z0.

Step 1 – S11. With port 2 matched, port 1 sees Z0 in parallel with R, i.e. normalized admittance 1 + y:

 S11 = (1 - (1 + y))/(1 + (1 + y)) = -y/(2 + y)

Step 2 – S21. For a shunt element, the voltage across both ports is the same, so V2 = V1 = V1+(1 + S11):

 S21 = 1 + S11 = 2/(2 + y)

Step 3 – symmetry. The element is symmetric and reciprocal, so S22 = S11 and S12 = S21:

          1     | -y   2 |
 [S] = ------- · |       |
        2 + y   |  2  -y |

Step 4 – modulation. If the bias makes y vary with the modulating signal, the output is b2 = S21(t)·a1 = [2/(2 + y(t))]·a1. Choosing the bias so that |S21(t)| = k[1 + m·cos ωₘt] gives an AM wave:

 a1 = A cos ωc t
 b2 = kA [1 + m cos ωm t] cos ωc t

with modulation index m.

Limiting cases:

• y → 0 (R very high, diode reverse biased): S11 = 0, S21 = 1, full transmission (carrier peak).
• y → ∞ (R very low, diode forward biased): S11 = −1, S21 = 0, full reflection (carrier off).

Properties of the matrix:

• Reciprocal: S12 = S21.
• Not lossless: |S11|² + |S21|² = (y² + 4)/(2 + y)² < 1 for 0 < y < ∞; the missing power is absorbed in the diode resistance.
• Not matched: S11 ≠ 0 except at y = 0, so an isolator is often placed before the modulator to protect the source from the reflected power.

An ideal (matched) modulator is therefore often modeled simply as [S] = [[0, S21(t)], [S21(t), 0]] with S21(t) = k[1 + m cos ωₘt]·e^(−jθ).