Double-stub for inductive antenna load on 78 Ω line

Assumptions: an inductive antenna load ZL = 117 + j78 Ω on a Z0 = 78 Ω line at f = 1 GHz (air-filled coax, λ = 30 cm). Short-circuited 78 Ω stubs, the first stub at the load, and spacing d = 3λ/8.

Step 1: Normalise and convert to admittance

zL = 1.500 + j1.000
yL = 1/zL = 0.462 − j0.308
ΓL = 0.415∠41.6°,  VSWR = 2.420

On the chart, zL is at 0.1922λ WTG and yL is diametrically opposite, at 0.4422λ WTG.

Step 2: First stub at the load

The first stub is placed at the load, so y1 = yL = 0.462 − j0.308.

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

βd = 360° × 0.375 = 135°
g must be ≤ 1/sin²βd = 2.000
g1 = 0.4615  → 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 = 0.462 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 = 0.7101, √ = 0.8427
Sol A: b1 = −1.5350
Sol B: b1 = +0.1503

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

Sol A: y after stub 1 = 0.462 − j1.843
   (0.3260λ) → after 0.375λ: y2 = 1.000 + j2.826
   b2 = −2.8257
Sol B: y after stub 1 = 0.462 − j0.157
   (0.4688λ) → after 0.375λ: y2 = 1.000 − j0.826
   b2 = +0.8257

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 = 0.0919λ, l2 = 0.0541λ
Sol B short: l1 = 0.2737λ, l2 = 0.3599λ

Physical lengths (λ = 30.000 cm):

d1 = 0.000 cm, spacing = 11.250 cm
Sol A short: l1 = 2.757 cm, l2 = 1.624 cm
Sol B short: l1 = 8.212 cm, l2 = 10.796 cm

Answer (choice): solution A (l1 = 0.0919λ = 2.76 cm, l2 = 0.0541λ = 1.62 cm). These are the shortest stubs, so the design has the lowest loss and the widest bandwidth.

Check: y at stub 2 = 1.000 + j0.000, so the line is matched.

Electrical connection

 Z0=78Ω          3λ/8 = 11.25 cm          
 o------+---------------+--------[117+j78Ω]
        |               |
     stub 2          stub 1
     l2=1.62 cm       l1=2.76 cm
       SC              SC

Physical connection

Two coaxial T-junctions are placed 11.25 cm apart on the main 78 Ω coax. The first is at the antenna terminals. Each side arm is a 78 Ω coaxial stub closed by an adjustable sliding short, set to l1 = 2.76 cm and l2 = 1.62 cm.

Physical connection (coaxial tuner)

        sliding short   sliding short
             ┃               ┃
  stub 2  ═══╋═══   stub 1 ══╋══
             ║               ║
 gen ════════╩═══════════════╩══════ antenna
      main coax     d            (load)