Single stub for 78.27+j60.93 Ω on 75 Ω line

Given: Z0 = 75 Ω, ZL = 78.27 + j60.93 Ω. Design (a) a shunt short-circuited stub and (b) a shunt open-circuited stub.

Step 1: Normalise and plot the load

zL = ZL/Z0 = 1.044 + j0.812
ΓL = (zL − 1)/(zL + 1) = 0.370∠65.2°
VSWR = (1 + |Γ|)/(1 − |Γ|) = 2.174

Draw the constant-VSWR circle through zL (zL is at 0.1594λ 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 = 0.597 − j0.464   (0.4094λ 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 + j0.796  at 0.1552λ WTG
y2 = 1.000 − j0.796  at 0.3448λ WTG
d1 = 0.2458λ from the load
d2 = 0.4355λ 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 = −0.7964
   short stub l = 0.1430λ
   open stub  l = 0.3930λ
Sol 2: bstub = +0.7964
   short stub l = 0.3570λ
   open stub  l = 0.1070λ

(a) Shunt short-circuited stub design

Solution 1 is preferred: stub at d = 0.2458λ from the load, with a short-circuited stub of length l = 0.1430λ.

(b) Open-circuited stub design

Solution 2 is preferred: stub at d = 0.4355λ, with an open stub of length l = 0.1070λ. (Solution 1 with an open stub would need l = 0.3930λ.)

Check: y(d) = 1 ± j0.796. The stub supplies ∓j0.796, so y = 1 and VSWR = 1 (it was 2.17 before matching).

Physical diagram (microstrip)

(a) short stub              (b) open stub
     ||  l=0.1430λ               ||  l=0.1070λ
     ● via to ground             ○ open end
     ||                          ||
=====##== d=0.2458λ ==[ZL]  =====##== d=0.4355λ ==[ZL]
 75 Ω microstrip             75 Ω microstrip

The stub is a 75 Ω strip of the same width as the main line, joined at a T-junction.