ELEC 477/677L
Topics in Wireless System Design
Spring 2006
Lab #2: LC Impedance-Matching Networks
Introduction
Most wireless systems are comprised of subsystems, often called stages, that are connected
together to accomplish a specific task. For example, a wireless receiver typically consists of
amplifiers, filters, mixers, and other circuits all operating together as a unit. An important
consideration in RF system design is to ensure that the input of each stage presents the proper
load impedance to the preceding stage and that it has the correct output impedance to feed the
following stage. This might be necessary to ensure maximum power transfer between stages.
Also, most filters require very specific source and load impedances (sometimes very different
from the standard 50 Ω) in order to operate properly. The process of changing an effective source
or load impedance from one value to another one is called impedance transformation, or
impedance matching. One way to match impedances is to insert a simple network of inductors
and capacitors between stages. In this lab exercise, you will design a basic L network to
transform a load impedance to 50 Ω.
Theoretical Background
An L network consists of a capacitor and an inductor arranged in one of the four topologies
shown in Figure 1. This type of network gets its name from the L-shaped arrangement of the
network components (“L” has nothing to do with the inductor). If the inductor and capacitor
L
L
Rin
C
RL
Rin
C
C
Rin
RL
C
L
Rin
RL
L
Figure 1. L network topologies used for matching source impedances (Rin) to
load impedances (RL).
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RL
values are chosen properly, the input impedance seen by the source at the design frequency
equals some real value greater than or less than the value of the load resistance RL, depending on
the network configuration used. That is, the value of RL can be transformed to any other desired
resistance Rin. At frequencies other than the design frequency, the input impedance will have
some reactance, and its real part (the input resistance) will be different from the intended value.
As shown in Figure 2 below, the underlying principle behind the operation of L networks is that
a parallel combination of a reactance and a resistance can be made to have the same impedance
as an equivalent series combination of a reactance and a resistance, and vice versa. The parallel
combination has the equivalent impedance
Z eq  jX p R p 
R p  jX p 
R p  jX p
.
To find the real and imaginary parts of this (in order to establish its equivalence to Rs and Xs),
multiply the numerator and denominator by the complex conjugate of the denominator to obtain
Z eq 
jR p X p
R p  jX p

R p  jX p

R p  jX p
R p X p2  jR p2 X p
R p2  X p2

R p X p2
R p2  X p2
 j
R p2 X p
R p2  X p2
.
Thus,
Z eq  Rs  jX s 
R p X p2
R p2  X p2
 j
R p2 X p
R p2  X p2
.
Using this result, the equivalent series combination can be found for a parallel combination. An
equivalent parallel combination can be found given a series combination by solving for Rp and
Xp. To do this, first note that
Rs 
R p X p2
R p2  X p2
and
Xs 
R p2 X p
 R X 
2
p
R p2  X p2
2
p
R p X p2
Rs
and
R X 
2
p
2
p
R p2 X p
which implies that
R p X p2
Rs

R p2 X p
Xs
.
jXs
Zeq
jXp
Rp
Zeq
Rs
Figure 2. Equivalence between a parallel combination and a series combination
of a resistance and a reactance.
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Xs
,
This result proves, by the way, that Xs and Xp must have the same sign (i.e., both inductive or
both capacitive), because the left-hand side is always positive. Both sides have the common
factor RpXp, which can be cancelled to yield
Xp
Rs

Rp
Xs
or
Rp
Xs

.
Rs
Xp
These two ratios are highly useful in the design of L networks and in other applications as well.
They form the basis of the definition of a quantity known as the quality factor, or Q, of the
network. The formal definitions use the magnitudes of the reactances (so that Q is always
positive) and are given by
Q
Q
Xs
Rs
Rp
, for a reactive component and a resistor in series, and
Xp
, for a reactive component and a resistor in parallel.
We will see shortly how these definitions lead to a very straightforward L network design
approach.
Proceeding with the derivation of the conversion formula from a series combination to a parallel
combination, we see that
Xp
Rs

Rp
Xs
 Xp 
Rp
Xs
Rs
 X 
2
p
R p2
X
2
s
Rs2 .
Substituting this into the formula for the series equivalent resistance leads to
Rs 
R p X p2
R p2  X p2
 R p2

R p  2 Rs2 
X

 s

 Rs 
2
 Rp

R p2   2 Rs2 
X

 s

 Rs 
which reduces to
Rp 
X s2  Rs2
.
Rs
A similar derivation for the reactance leads to
X s2  Rs2
Xp 
.
Xs
3
R 3p Rs2
R p2 X s2  R p2 Rs2

R p Rs2
X s2  Rs2
,
Careful inspection of the conversion formulas we have just derived show that the series
equivalent resistance Rs is always smaller than the parallel equivalent resistance Rp. A more
direct conversion formula between these two quantities can be obtained by making use of the
quantity Q. For example, starting with the formula for Rp and substituting
Q
Xs
Rs

yields
X s2  Q 2 Rs2


X s2  Rs2 Q 2 Rs2  Rs2 Rs2 Q 2  1
Rp 


 Rs Q 2  1 .
Rs
Rs
Rs


Again, this shows that Rp is always greater than Rs. This new relationship in turn leads to the
formula
Rp
Q
1.
Rs
In a typical L network design procedure, the values of RL, Rin, and fo (the operating frequency)
are specified at the beginning. The values of RL and Rin are first compared to determine where the
“shunt” (parallel) reactance should be placed. Since a resistor in parallel with a reactance always
transforms to a lower series equivalent resistance, the shunt element should always be placed
next to (in parallel with) the larger resistance. Thus, if RL > Rin, where Rin is the source
resistance, then the shunt element should be in parallel with the load. Conversely, if RL < Rin,
then the shunt element should be in parallel with the source. The other reactive element (the
“series” element) should therefore be placed in series with the remaining resistance. It does not
matter whether the shunt element is a capacitor or an inductor, but the series element must be of
the opposite type in order to cancel the series equivalent reactance Xs. Next, the larger resistance
is designated as Rp and the smaller as Rs, and the Q of the network is found using the formula
just derived above. The reactances of the series and shunt elements are next determined with the
aid of
Rp
,
X s  QRs and X p 
Q
and the actual inductor and capacitor values are found from their respective reactances at the
operating frequency.
Thus, if the values of L and C are chosen carefully, we can make a resistance of one value RL
look like another value Rin at one particular resonant frequency. At frequencies above and below
resonance the input impedance is complex; however, over a narrow range of frequencies the
reactance will remain small and the resistance close to Rin. The extent of this range (often called
the matching bandwidth) is dictated by the intended resistance transformation ratio (i.e., Rin/RL or
RL/Rin, whichever is greater than one), or, by extension, the network Q. The range over which a
“good” impedance match is achieved is usually sufficient to accommodate all but the widest RF
signals. For example, consider the case of three L networks designed to match a source
impedance of 1000 Ω to load impedances 1 Ω, 50 Ω, and 500 Ω at a frequency of 2 MHz. As
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shown in Figure 3, the match is “good” over a wider bandwidth if the load resistance is closer to
the desired input resistance. That is, the 500-Ω curve is much flatter than the other curves,
because 500 Ω is closer to 1000 Ω than are 50 Ω and 1 Ω.
Figure 3. VSWR vs. frequency obtained with L networks designed to match
1000 Ω to 1 Ω, 50 Ω, and 500 Ω at 2 MHz.
Experimental Procedure
Record the results of the following procedures on a separate sheet(s) of paper, and turn them in at
the end of the lab session. Only one set of notes is required from each group. Please remember
that neatness counts.

Design an L network to match a resistance of 10 Ω to the source impedance of the bench top
function generators (50 Ω) at an operating frequency of 3.5 MHz.

Calculate the input impedance of the matching network/load resistor design over the
frequency range 3-4 MHz using Mathcad, Matlab, or Excel, and use the results to plot the
VSWR (with respect to 50 Ω) over that range.

Build the L network using the components available to you. The inductor should be wound
on a toroidal iron powder core. Information on these cores is available at the Amidon
Associates web site, which is linked on the laboratory web page. The required capacitance
can be approximated using series and/or parallel combinations of standard capacitor values.
Use a quarter-watt 10-Ω resistor to serve as the load. The components should be soldered
together for best results.
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
Connect the L network/10- load combination to the output of the function generator.
Monitor the input voltage to the matching network using the oscilloscope, and verify that it is
close to the correct magnitude for a 50- load at 3.5 MHz. Unfortunately, there is no way to
measure the phase shift between the source and load voltages. Remember that the cables
commonly used in the lab can introduce a significant amount of capacitance into the circuit
and that the input to the oscilloscope itself has a capacitance of approximately 13 pF. These
loading effects can be reduced significantly if the ×10 probes are used.

Use the handheld impedance analyzer to plot the VSWR vs. frequency over the same range
for which you calculated the VSWR. Sketch the curve displayed on the analyzer’s screen,
and compare it to your mathematically predicted curve (i.e., plot both on the same graph).
Besides measurement error, what other factors might account for any differences you see?
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