Single stub matching for ΓL = 0.5∠51°

Given: ΓL = 0.5∠51° (the "Ohm" in the question is a slip; Γ has no unit), Z0 = 50 Ω. Design a shunt single-stub match with (a) a short-circuited and (b) an open-circuited stub.

Step 0: Find the load from Γ

On the Smith chart, plot the point at radius 0.5 and angle 51°. It reads:

zL = (1 + ΓL)/(1 − ΓL) = 1.208 + j1.252
ZL = 50 × zL = 60.42 + j62.60 Ω

Step 1: Normalise and plot the load

zL = ZL/Z0 = 1.208 + j1.252
ΓL = (zL − 1)/(zL + 1) = 0.500∠51.0°
VSWR = (1 + |Γ|)/(1 − |Γ|) = 3.000

Draw the constant-VSWR circle through zL (zL is at 0.1792λ WTG).

Step 2: Convert to admittance (shunt stub)

Move zL by 180° on the VSWR circle, which is the same as moving λ/4:

yL = 1/zL = 0.399 − j0.414   (0.4292λ WTG)

Step 3: Move toward the generator to the g = 1 circle

The VSWR circle cuts the g = 1 circle at two points:

y1 = 1.000 + j1.155  at 0.1667λ WTG
y2 = 1.000 − j1.155  at 0.3333λ WTG
d1 = 0.2375λ from the load
d2 = 0.4042λ from the load

Step 4: Stub susceptance and length

The stub must cancel the susceptance: bstub = −b.

• 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).
• For an open-circuited stub: y = j tan βl, so l = (1/2π)·tan⁻¹(b). Measure toward the generator from the open-circuit point (y = 0; the left end, 0λ WTG).

Sol 1: bstub = −1.1547
   short stub l = 0.1136λ
   open stub  l = 0.3636λ
Sol 2: bstub = +1.1547
   short stub l = 0.3864λ
   open stub  l = 0.1364λ

Answer (choice): solution 1 with the short stub (d = 0.2375λ, l = 0.1136λ). With an open stub, solution 2 (d = 0.4042λ, l = 0.1364λ) is the shorter choice.

(a) Short stub            (b) Open stub
 Z0  d=0.2375λ             Z0  d=0.4042λ
 o---+--------[ZL]         o---+--------[ZL]
     |                         |
    l=0.1136λ                 l=0.1364λ
     |                         |
   short                      open

Check: y(d) = 1 ± j1.155. The stub supplies ∓j1.155, so y = 1, Γ = 0 and VSWR = 1 (it was 3.0 before matching).