Admittance chart and single stub for 80+j100 Ω

Admittance chart

An admittance chart is the Smith chart read in normalised admittance y = g + jb instead of z. It is the impedance chart rotated by 180°. Constant-g circles touch at the short-circuit point, and the upper half then means inductive susceptance (−jb). It is used for shunt elements (stubs, shunt L or C), because shunt admittances simply add. A point on the Z-chart becomes its admittance when moved λ/4 (180°) around the VSWR circle.

Single-stub design

Assumption: the microstrip line has Z0 = 50 Ω (not stated). ZL = 80 + j100 Ω. Shunt stub, both short and open designs.

Step 1: Normalise and plot the load

zL = ZL/Z0 = 1.600 + j2.000
ΓL = (zL − 1)/(zL + 1) = 0.637∠35.7°
VSWR = (1 + |Γ|)/(1 − |Γ|) = 4.503

Draw the constant-VSWR circle through zL (zL is at 0.2004λ 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 = 0.244 − j0.305   (0.4504λ 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 + j1.651  at 0.1799λ WTG
y2 = 1.000 − j1.651  at 0.3201λ WTG
d1 = 0.2295λ from the load
d2 = 0.3697λ 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 = −1.6508
   short stub l = 0.0867λ
   open stub  l = 0.3367λ
Sol 2: bstub = +1.6508
   short stub l = 0.4133λ
   open stub  l = 0.1633λ

Answer (choice): short stub, solution 1 (d = 0.2295λ, l = 0.0867λ). Open stub, solution 2 (d = 0.3697λ, l = 0.1633λ).

 Microstrip:
          stub (l)
            ||
            ● via (short) / open end
            ||
 ===========##==== d ====[80+j100 Ω]

Check: y(d) = 1 ± j1.651. The stub cancels ±j1.651, so the line is matched (VSWR goes from 4.50 to 1).