Example B.III.1.3 Single Stub Matching
a) Consider a series RC circuit as shown in Figures 1.1.a and 1.1.b (R =100 W, C = 1.326
pF). We want to match this load at 2 GHz to a 50 W transmission line by using the single
stub matching method. Either short circuit or open circuit stub can be used. Show the
matching process on the Smith chart and find d and l in each case.
Figure 1.1.a
Figure 1.1.b
b) Consider the design of Figure 1.1.a. For 1.8 GHz to 2.2 GHz, locate the impedance
seen at the load plane, before the stub, and at the input, on the Smith chart.
c) Replace the short stub in Figure 1.1.a with an inductor and the open stub in Figure
1.1.b with a capacitor (Figures 1.1.c and 1.1.d). Show the reflection coefficient on the
Smith chart from 1.8 GHz to 2.2 GHz for all of the four possible matching networks.
Draw the magnitude of reflection coefficient as a function of frequency.
Figure 1.1.c
Figure 1.1.d
d) Now, assume that the transmission line is not loss less. For the attenuation of a= 0, 1,
2, 3, 4, 5 dB per meter, draw the magnitude of reflection coefficient as a function of
frequency.
Solution
a) For the impedance of the load at the center frequency we have (see Figure 1.1.e):
The first step is to locate this load on the admittance Smith chart (point M1). By moving
on a constant SWR circle toward load, this circle intersects the 1+jb circle at two points.
The admittance at the first point (M2, Y=1+j1.1) can be moved to the center of the Smith
chart (M3) by an inductive parallel stub. From the chart:
d/l = 0.1256 (bd =45.2°) and l/l = 0.1171 (bl = 42.16°).
figure_1.1.f illustrates the matching process on the Smith chart for this design. In a
similar manner, for figure 1.1.b, capacitive stub is used to bring the admittance on the
constant b circle, from the point Y=1-j1.1 to the center of the Smith chart and we have:
d/l = 0.2953 (bd =106.3°) and l/l = 0.133 (bl = 47.87°).
b) For the matching network of Figure 1.1.a, the loci of impedance at the load plane,
before the stub, and the input plane are shown at figure_1.1.g.
c) In Figure 1.1.c, the short stub is replaced with an inductor and we have:
Similarly, for the capacitor of Figure 1.1.d:
The loci of input impedance on the Smith chart for 1.8 to 2.2 GHz and the magnitude of
the reflection coefficient as a function of frequency are illustrated in Figures 1.1.h and
1.1.i.
d) For the matching network of figure 1.1.a, we replace the ideal transmission line with a
lossy line with attenuation rate a (dB/m). figure_1.1.j shows the reflection coefficient for
1.8 to 2.2 GHz for different values of a.
Figure 1.1.e Schematics of the designs.
Figure 1.1.f Single stub matching; From M1 to M2 (M4) on a constant SWR circle. From
M2 (M4) to M3 on a constant conductance circle.
Figure 1.1.g Smith chart representation of frequency variations.
Figure 1.1.h Variation with the frequency of the four different designs, shown on the
Smith chart.
Figure 1.1.i Frequency variations of the reflection coefficient.
Figure 1.1.j Reflection coefficient for differential values of the attenuation of the line.