Matching techniques and double-stub S-matrix

Impedance matching makes the load look like Z0 to the line, so that Γ = 0. This gives maximum power transfer, no standing waves, a better signal-to-noise ratio, and protection for the source.

Impedance matching techniques

1. Lumped L-section (L-network): one series and one shunt reactance (L or C). It is simple, but works only up to a few GHz because real L and C have parasitics.
2. Quarter-wave transformer: a λ/4 line with Z1 = √(Z0·RL) matches a real load. It is narrowband. Multisection (binomial or Chebyshev) versions give wider bandwidth.
3. Single-stub tuning: one shunt (or series) short or open stub at a distance d from the load. Both d and l must be adjustable.
4. Double-stub tuning: two shunt stubs at fixed positions (spacing usually λ/8, 3λ/8 or λ/4). Only the stub lengths are adjusted, which suits adjustable coaxial tuners. Some loads cannot be matched (forbidden region g > 1/sin²βd).
5. Triple-stub tuner: removes the forbidden region.
6. Tapered lines (exponential, triangular, Klopfenstein): very wideband.
7. Slide-screw tuner, E-H tuner: used in waveguides.

L-section       λ/4 transformer      single stub
-[jX]-+-        -[ Z1 , λ/4 ]-       --+--d--[ZL]
      |                                |
     jB                                l (stub)

S-matrix of a perfectly matched double-stub network

For a two-port with both ports referenced to Z0, the S-parameters follow from the normalised ABCD matrix:

S11 = (A + B − C − D)/Δ     S12 = 2(AD − BC)/Δ
S21 = 2/Δ                  S22 = (−A + B − C + D)/Δ
Δ = A + B + C + D

A double-stub tuner is a cascade of shunt stub 2, a line of length d, and shunt stub 1:

[ABCD] = |1    0| |cos βd   j sin βd| |1    0|
         |jb2  1| |j sin βd  cos βd | |jb1  1|

Building blocks:

Shunt jb:  S = 1/(2+jb) · | −jb   2  |
                          |  2   −jb |
Line βd:   S = |   0      e^(−jβd) |
               | e^(−jβd)    0     |

Condition for a perfect match: with the load ΓL on port 2,

Γin = S11 + S12 S21 ΓL/(1 − S22 ΓL) = 0

For a lossless, reciprocal tuner this holds when S22 = ΓL* (conjugate match at the load side). Unitarity then gives |S21|² = 1 − |ΓL|², and S12 = S21.

If port 2 is referenced to the load itself (the reference impedance is ZL), the matched network becomes an ideal matched, lossless, reciprocal two-port:

S = |    0      e^(−jφ) |
    | e^(−jφ)      0    |

Here φ is the total phase delay. S11 = S22 = 0 (no reflection) and |S21| = 1 (all power reaches the load).

Numerical example (ZL = 110 + j110 Ω on a 100 Ω line, first stub at the load, d = 3λ/8, b1 = −1.384, b2 = −2.844, stub lengths 0.0996λ and 0.0538λ):

S11 = 0.466∠145.5°   S12 = 0.885∠−45.8°
S21 = 0.885∠−45.8°   S22 = 0.466∠−57.2°
ΓL  = 0.466∠+57.2°  → S22 = ΓL*  ✓
|S11|² + |S21|² = 1 (lossless) ✓
Γin = 0  (VSWR = 1) ✓

Before matching, the line sees S11 = ΓL = 0.466∠57.2° (VSWR = 2.75). After matching, S11 at the input = 0.