Double-stub on 75 Ω coax, load 0.4+j0.85

Given: Z0 = 75 Ω coaxial line, normalised load zL = 0.4 + j0.85 (ZL = 30 + j63.75 Ω). The stubs are short-circuited 75 Ω coaxial lines. Assumptions (not given): the first stub is at the load, and the stub spacing is d = 3λ/8.

Step 1: Normalise and convert to admittance

zL = 0.400 + j0.850
yL = 1/zL = 0.453 − j0.963
ΓL = 0.635∠94.0°,  VSWR = 4.483

On the chart, zL is at 0.1195λ WTG and yL is diametrically opposite, at 0.3695λ WTG.

Step 2: First stub at the load

The first stub is placed at the load, so y1 = yL = 0.453 − j0.963.

Step 3: Forbidden-region check and rotated g = 1 circle

βd = 360° × 0.375 = 135°
g must be ≤ 1/sin²βd = 2.000
g1 = 0.4533  → matchable

Rotate the g = 1 circle by 0.375λ (270° on the chart) toward the load. The point just after stub 1 must lie on this rotated circle, so move from y1 along the constant g = 0.453 circle until it meets the rotated circle.

Step 4: Susceptance of stub 1

Analytically (Pozar), with t = tan βd:

t = tan 135° = −1.0000
b1 = −B + [1 ± √((1+t²)g − g²t²)]/t
(1+t²)g − g²t² = 0.7011, √ = 0.8373
Sol A: b1 = −0.8741
Sol B: b1 = +0.8005

Step 5: Move 0.375λ to stub 2 and find b2

Sol A: y after stub 1 = 0.453 − j1.837
   (0.3263λ) → after 0.375λ: y2 = 1.000 + j2.847
   b2 = −2.8473
Sol B: y after stub 1 = 0.453 − j0.163
   (0.4681λ) → after 0.375λ: y2 = 1.000 − j0.847
   b2 = +0.8473

Both y2 values lie on the g = 1 circle, as required.

Step 6: Stub lengths

• For a short-circuited stub: y = −j cot βl, so l = (1/2π)·cot⁻¹(−b). Measure toward the generator from the short-circuit point (y = ∞; with y plotted on an ordinary Smith chart, this is the right end, 0.25λ WTG).

Sol A short: l1 = 0.1357λ, l2 = 0.0538λ
Sol B short: l1 = 0.3574λ, l2 = 0.3619λ

Answer (choice): solution A (l1 = 0.1357λ, l2 = 0.0538λ) has the shorter stubs, so it is preferred.

Check: the input admittance at stub 2 is 1.000 + j0.000, so Γ = 0 and VSWR = 1.

Sketch using microstrip

In microstrip, each "short-circuited" stub is a strip that ends in a via hole to the ground plane. The coax line is replaced by a 75 Ω microstrip trace.

Top view (ground plane underneath)

     stub 2           stub 1
   l2=0.0538λ       l1=0.1357λ
       ||               ||
       ● via            ● via
       ||               ||
=======##====== 3λ/8 ===##[ZL]
in (75 Ω microstrip)       load