Double-stub microstrip network and its S-matrix

Sketch: double-stub matching network in microstrip

Two shunt stubs are etched in the same microstrip as the main line. They are spaced d = 3λ/8 apart (λ is the guided wavelength in the microstrip). A shorted stub ends in a via hole to the ground plane, and an open stub simply ends.

Top view (microstrip, ground plane below)

   stub 2 (l2)        stub 1 (l1)
      ||                 ||
      ||                 ||
      ●via (short)       ●via (short)
      ||                 ||
=====##======== d =======##=====[ ZL ]
port 1 (Z0)    3λ/8      d1 to load

Electrical equivalent
 Z0 o---+-------- d --------+-- d1 --[ZL]
        |                   |
      jb2 (stub 2)        jb1 (stub 1)

Preparing its S-matrix

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.