ELEC 477L
Topics in Wireless System Design
Spring 2007
Lab #4: Single-Stub Impedance Matching
Introduction
There are many ways to accomplish the important task of impedance matching in wireless
systems. At relatively low frequencies (around 1 GHz and below) the usual method is to employ
LC networks. At microwave frequencies, however, the effects of parasitic impedances generally
preclude the use of this technique. Instead, the impedance transformation properties of
transmission lines are exploited. Quarter-wave matching sections are sometimes used, but they
usually require a line with a nonstandard characteristic impedance. An alternative approach is
shown in Figure 1. Here a “stub” made from a line with a standard characteristic impedance is
connected in parallel to the main transmission line at an appropriate distance from the
mismatched load. Although a shorted stub is shown, an open-circuited stub or a stub with a
purely reactive load could also be used. If the stub’s length and location are properly chosen, the
input impedance Zin seen at the junction of the stub and the main line can be made to match
exactly the system impedance Zo. In this lab exercise you will design and construct a single-stub
matching system.
lmain
Zo
Zo
ZL
Zo
Zin
lstub
Figure 1. A single-stub matching system.
Theoretical Background
The input impedance Zin of a lossless, terminated transmission line is given by
Z in  Z o
Z L  jZ o tan  l
,
Z o  jZ L tan  l
where Zo is the characteristic impedance of the line; ZL is the terminating impedance;  is the
phase constant (equal to 2/l, where l is the wavelength along the line); and l is the distance
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along the line from the load. The variation of the input impedance with line length is cyclical; an
impedance measured at a given distance from the load is repeated every half-wavelength further
away from the load. It is also interesting to note that the real part of the input impedance varies
over a range of values that always straddles the characteristic impedance of the line. That is, the
value of Zo always lies between the maximum and minimum values of the real part of Zin. Also,
the imaginary part of Zin cycles through positive and negative values over each half-wavelength
of transmission line. For example, consider a 50- line terminated in a load impedance of 25 .
The resulting input impedance vs. line length is shown in Figure 2. The frequency of operation
is 30 MHz, and the dielectric constant of the line’s insulation is 1.5, so the wavelength along the
line is 8.2 m (0.5l ≈ 4 m). The input impedance clearly repeats every half wavelength in this
example, and the system impedance (50 ) lies between the maximum (100 ) and minimum
(25 ) values of the real part.
real part
imaginary part
Figure 2. Variation of input impedance with line length for a 50- line
terminated in a load impedance of 25 .
Using this example as a guide, one strategy is to locate a position along the line where the real
part of the input impedance equals the system impedance. One place this occurs is at a distance
from the load of 1.24 m. The input reactance at this distance is approximately +35 . To
achieve a match one could insert a short- or open-circuited stub (or a capacitor) tuned to −j35 
in series with the line at this location as shown in Figure 3. The negative reactance of the stub
would cancel the positive imaginary part of the input impedance, yielding an overall input
impedance of 50 , which is a perfect match to the remainder of the line lying to the left of the
stub’s location. Although this is an effective technique in theory, it is not very practical because
it is usually difficult to break a conductor in a transmission line without affecting the
performance of the line. Imagine trying to break the center conductor of a coaxial cable.
Nevertheless, if a series stub is used, its distance from the load can be determined using
l main 
l
 r  cos 1   ,
4
2
where || and r are the magnitude and phase, respectively, of the reflection coefficient
associated with the load ZL. The input reactance of the main line at this location is given by
X in  
2
1 
2
Zo ,
where the upper sign is associated with the upper sign in the formula for lmain. The series stub
would have to have a reactance equal to the negative of this value in order to provide a match to
the system impedance.
−j35  stub
Zo
1.24 m
Zo
Zo
Zin = 50 
ZL
25 
50 + j35 
Figure 3. A stub inserted in series with a transmission line to match a 25- load
to a 50- system impedance.
Because a series stub is usually an impractical solution, the approach taken by most designers is
to connect a stub in parallel with the line at a strategic location as shown in Figure 1. In this case
it is necessary to find the location where the input admittance of the main line is equivalent to a
conductance of value Yo (equal to 1/Zo) in parallel with a susceptance. If the stub is made to have
an input susceptance exactly equal in magnitude and opposite in sign to the input susceptance of
the main line at this location, then an equivalent parallel RLC resonant circuit is formed that has
an overall admittance of simply Yo. See Figure 4 for the equivalent lumped-element circuit.
Recall that the input admittance of a loaded lossless transmission line is given by
Yin  Yo
YL  jYo tan  l
.
Yo  jYL tan  l
From this expression the distance from the load at which the shunt stub should be located can be
found using
l main 
l

 r  cos 1    .
4
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Bstub
Zin = Zo
Yin = Yo
Gin
= Yo
Bin
equivalent
circuit of
input of
terminated
line
Bstub = −Bin
Figure 4. Equivalent circuit of a shunt stub matching system.
Note that this formula is identical to the one for the series stub, except for the minus sign in the
argument of the inverse cosine function. The corresponding input susceptance of the main line is
Bin  
2
1 
2
Yo ,
where again the upper sign is associated with the upper sign in the formula for lmain. The shunt
stub should have an input susceptance equal to the negative of this value in order to provide a
match.
Matching stubs are usually made as short as possible in order to limit cost and to reduce the
effects of attenuation. Shorted stubs are usually preferred over open-circuited stubs, since a true
open circuit condition is difficult to achieve. Stubs are sometimes terminated with variable
capacitors in order to make them tunable.
Experimental Procedure
Keep a written record of the results of the following procedures. Turn in one set of notes for
your group at the end of the lab session.

Design a single-stub matching system to match a load of 100  in parallel with 100 pF to a
52- system impedance at an operating frequency of 30 MHz. Use a shorted stub. The
important parameters of the cable, such as its characteristic impedance and velocity factor,
that you need to make your calculations will be provided. Clearly describe your design
approach. You may use any computational tools that you find helpful, but remember to
include a print-out of your Matlab session or Mathcad worksheet, if relevant.

Using Mathcad, Matlab, Excel, or any other software of your choice, calculate the input
VSWR of the matching system (i.e., load, main line, and stub) over the frequency range of
20-40 MHz. Plot the VSWR vs. frequency. Later you will sketch the VSWR vs. frequency
obtained from the hand-held impedance analyzer on this same plot, so you should make sure
the graph has sufficient grid lines and labels to make that task easier.
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
Construct the single-stub matching system using the two coaxial cable sections that will be
provided to you. One cable will serve as the “main” line between the load and the shunt stub,
and the other will serve as the stub itself. Each cable has a BNC connector on one end.
These cable ends will be attached to a BNC “Tee” connector to allow the tuning stub to be
connected in shunt with the main transmission line. A diagram of the assembly is shown in
Figure 5. One advantage of BNC connectors is that they are designed to have characteristic
impedances close to 50 . Thus, there is essentially no impedance transition experienced by
a wave inside a coaxial cable when it encounters a BNC connector; hence, the connector can
be considered an extension of the cable to which it is attached. If the stub has been designed
correctly, the input impedance looking into the free port (male connector) of the BNC Tee
should be close to 52 .
lstub
lmain
coaxial cable
coaxial cable
ZL
BNC “T”
connector
to impedance analyzer
(continuation of main line)
Figure 5. Coaxial single-stub matching system using a shorted stub and a BNC
“Tee” connector.

Cut one cable to the length lmain. Separate approximately 0.5-1 cm of the center conductor
and braid at the cut end, and solder the “load” (a 100- resistor in parallel with a 100-pF
capacitor) between the two exposed conductors. You will have to decide whether or not
some or all of the half-length of the Tee connector and the length of the exposed conductors
should be part of the line length lmain.

Cut the other cable to the stub length lstub. Again, separate the center conductor and braid at
the cut end, and solder one conductor to the other to form the short-circuit load for the stub.
As with the first cable section, you will have to decide whether or not to include the Tee
connector and the conductor lengths in the overall stub length.

Connect the stub assembly to the handheld impedance analyzer, and plot the VSWR vs.
frequency over the range of 20-40 MHz. Carefully sketch the plot displayed on the
analyzer’s screen onto the same graph you generated using software for the ideal stub.
Comment on the results. What factors might account for any differences you see?
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