Improving The Bandwidth Of Simple Matching Networks
High Frequency Design
From March 2008 High Frequency Electronics
Copyright © 2008 Summit Technical Media, LLC
MATCHING NETWORKS
Improving the Bandwidth
of Simple Matching Networks
By Gary Breed
Editorial Director
L
C
2
2
This tutorial describes
methods for broadbanding
Impedance matching
is probably the most
engineering task in
matching networks using
RF/microwave design.
R
C
R
R
L
R
1
1
2
1
1
2
cascaded sections and
This tutorial is intended
compensating reactance
to demonstrate the first
steps from simple two-
and three-element networks that are designed
ω C = Q/R
ω L = R /Q
1
1
1
1
for a specific center frequency, to larger net-
ω L = Q·R
ω C = 1/(Q·R )
2
2
2
2
works that provide an acceptable match over a
wider bandwidth. These wider bandwidth net-
Q = √ R /R – 1
1
2
works are important for modern communica-
R > R
1
2
tions systems that have operating bandwidths
that are much wider than older technologies
Figure 1 · The two basic L-network topolo-
using FM and BPSK modulation. Even at nar-
gies and their design equations.
rower bandwidths, many digital modulation
formats require flat amplitude and linear
phase response, which can be achieved by
large impedance transformations. Also note
using wideband matching networks, which
that the equation for Q requires the shunt
have much smaller variation over a signal’s
reactance to be located adjacent to the higher
occupied bandwidth.
impedance.
Two L-network sections can be connected
Classic L, T and Pi Matching Networks
back-to-back, as shown in Figure 2. The inter-
The simplest impedance transformation
mediate impedance at the center of the net-
network is the L-network, which requires just
work is virtual (no actual load is present) and
two reactive components. Like a filter, the L-
is selected by the user, usually to achieve a
network can have a highpass or lowpass fre-
particular value of Q. Back-to-back connection
quency response characteristic. Even if the
requires this virtual impedance to be either
particular response is unimportant, it means
that two topologies are available—which is
L
L
1a
2b
important when matching reactive loads, as
we will see later.
Figure 1 shows the two L-network configu-
R
C
C
R
1
2a
1b
2
rations and their design equations for resis-
RInt
tive sources and loads. Note that Q is deter-
mined by the ratio of the impedances to be
R
< R
> R
1
Int
2
matched and cannot be chosen by the design-
er. Thus, L-networks are low-Q for small
Figure 2 · Back-to-back L-networks with a
impedance transformations and high-Q for
user-selected intermediate impedance.
56
High Frequency Electronics
High Frequency Design
MATCHING NETWORKS
quency response than a simple L-network. If wider band-
width is the primary objective, L-networks can be cascad-
ed in series rather than back-to-back, such as the 850
MHz 5-ohm to 50-ohm network shown in Figure 4 [2].
With cascaded sections, the lowest Q (and widest
bandwidth) is achieved when the intermediate impedance
is the geometric mean of the source and load impedances.
For example, with source and load impedances of 5 and 50
ohms, a single L-network would have a Q of 3. When two
Two L-networks
Pi-networks
sections are cascaded with √(50 × 5), or 15.81 ohms, as the
intermediate impedance, each section has a Q of 1.47.
With ideal, lossless components, the lower Q results in
more than 3 times greater bandwidth near the center fre-
quency (between the 0.1 dB points) [2].
Figure 5 is a plot of the impedance at the 50-ohm port
for the cascaded network of Figure 4, compared to a sin-
gle lowpass L-network. It is easy to see that the
impedance deviation of the cascaded networks is far less
Two L-networks
T-networks
than the single L-network section over this 35% band-
width range.
Figure 3 · T- and Pi-networks combine the center com-
Wider bandwidths and flatter impedance curves can
ponents of back-to-back L-networks. Additional topo-
be achieved by cascading more sections with the addi-
logies are illustrated in Ref. [1).
tional intermediate impedances creating smaller
impedance ratios, and correspondingly lower Q for each
section.
higher (as shown in Fig. 2) or lower than both the source
and load impedance.
Incorporating Reactances
The center components of Figure 2 can be combined
The above discussion was for resistive loads, but most
into a single component, with the result being the T-net-
practical applications involve loads that include reac-
work. When the intermediate impedance is lower than
tance. The usual design procedure is to cancel the load
the source and load, the result of combining the center
reactance, then match the remaining resistive component
components is the Pi-network. Figure 3 shows some of the
to the system impedance. Figure 6 shows the two ways
ways that two L-networks can create T- and Pi-networks
that reactance can be cancelled: (a) an equal but opposite
[1]. The choice among these topologies is determined by
sign reactance in series, and (b) a parallel reactance that
such factors as DC continuity and highpass, lowpass or
resonates with the load reactance. A 5 –j5 ohm load is
bandpass frequency response. Practical component values
shown in this example.
are a major consideration in some cases, especially at
The manner in which the reactance of the load is
high power levels.
75
25
Broadbanding with Cascaded L-Networks
70
20
Although T- and Pi-networks represent great flexibili-
65
15
ty in design parameter choices, they have narrower fre-
60
10
55
5
C
L
2a
2b
50
0
8.05 pF
1.37 nH
45
–5
40
–10
35
–15
L
R = 50Ω
1a
R
C1b
R = 5Ω
Resistance (ohms) – solid lines
30
–20
S
int
6.36 nH
L
17.4 pF
Reactance (ohms) – dotted lines
25
–25
700
750
800 850 900 950 1000
Frequency (MHz)
f = 850 MHz R
= √ 50 × 5 = 15.81Ω
0
int
Figure 5 · Impedance at the 50 ohm port of a 5- to 50-
Figure 4 · Cascaded L-network example, with each
ohm matching network: Example of Fig. 4 (red); Single
section having a lower Q for improved bandwidth.
section lowpass L-network (blue).
58
High Frequency Electronics
High Frequency Design
MATCHING NETWORKS
70
20
65
15
X = +j5Ω
L
R
R
R
60
10
S
eff
L
50Ω
5 ± j0Ω
5 – j5Ω
55
5
50
0
45
–5
(a) Series reactance (cancellation)
40
–10
35
–15
30
–20
Resistance (ohms) – solid lines
25
–25
Reactance (ohms) – dotted lines
R
R
X =
R
S
eff
L
L
20
–30
+j10
700
750
800 850 900 950 1000
50Ω
10 ± j0Ω
Ω
5 – j5Ω
Frequency (MHz)
Figure 7 · Impedance at R port for the two matching
(b) Parallel reactance (resonating)
S
options of Fig. 6(a) (blue) and Fig. 6(b) (red), imple-
Figure 6 · Two methods for cancelling the load reac-
mented at f = 850 MHz.
0
tance, combined with matching the resulting non-
reactive impedance to a 50-ohm system.
to replace components with non-optimum values.
Trading loss for bandwidth—Often, a very low resis-
incorporated into the matching network affects band-
tance or high reactance load, such as the gate of some
width. The finished network can save a component by
power FET devices, can be more easily matched for wide
combining the series cancelling reactance with the adja-
bandwidth by adding a series resistance to raise the effec-
cent matching component, but the resonating solution
tive impedance. When the additional loss can be tolerat-
has a wider bandwidth.
ed, this technique can greatly simplify broadband match-
In Fig. 6(a), note that the effective load is 5 ohms for
ing network design.
the series cancelling method, but is 10 ohms for the res-
Non-standard impedances—Many matching tasks
onating inductor method of Fig. 6(b). This reduces the
involve interfacing between devices or circuits that are
magnitude of the impedance transformation by the L-net-
both higher or lower than the typical 50-ohm system
work, which was previously shown to result in a wider
impedance. Unless there is a need to test individual mod-
matching bandwidth. While this circuit has significant
ules, or to separate the modules for isolation, matching
variation in impedance away from the center frequency,
the actual impedances will result in the simplest or easi-
this has a much smaller effect on bandwidth than the
est to implement network.
increased effective load impedance.
Practical component values—With several options for
The results for this example, using a center frequency
matching topologies, the choice between them will often
of 850 MHz, is shown in Figure 7, which is a comparison
be based on the component values. The considerations
of the impedances at the source port for the two options of
include loss (e.g., large inductors with relatively low Q),
Fig. 6. The smaller impedance deviation for the resonat-
impractical component values, and high currents or volt-
ing solution of Fig. 6(b) is clearly illustrated.
ages at the various points in the network.
Reactance of the load—The magnitude of the load
Some Additional Considerations
reactance and the slope of reactance change across the
This tutorial has presented some of the “first steps” in
desired band must be accommodated in any broadband
the path from narrowband to broadband matching. There
matching network design. This will affect the choice of
are many additional techniques that are involved in fur-
topology and complexity of the network.
ther bandwidth improvement, as well as for implement-
ing matching networks with practical component values.
References
Below are a few notes on some of those techniques, along
1. Chris Bowick, et al, RF Circuit Design, 2nd ed.,
with notes on other issues that arise practical matching
Newnes imprint, Elsevier 2008, Chapter 4.
network design.
2. Les Besser, Rowan Gilmore, Practical RF Circuit
Transmission line equivalents—All designs using
Design for Modern Wireless Systems, Artech House 2003,
lumped elements may use transmission line elements, as
Chapter 5.
well. The choice depends on the physical construction
3. Andrei Grebennikov, RF and Microwave Power
method, frequency of operation and, in some cases, is used
Amplifier Design, McGraw-Hill 2005, Chapters 4 and 8.
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High Frequency Electronics
