Matching Networks
CCE 5220
RF and Microwave System Design
Dr. Owen Casha B. Eng. (Hons.) Ph.D.
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Maximum Power Transfer Theorem
To achieve maximum
power transfer, one
needs to match the
load impedance to that
of the source
ZS = ZL*
(Complex Conjugate)
RS = RL and Xs = -XL
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What should
be done if
ZS ≠ ZL* ?
2
Matching Networks
Maximum power transfer is generally achieved
by using additional passive matching networks
connected between source and load.
Not only designed to meet the requirement of
minimum power loss.
Minimise noise influence
Maximising power handling capabilities
Linearising the frequency response
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Passive Matching Networks
Discrete Passive Networks
(low gigahertz range)
Microstrip lines
Stub Sections
Discrete
Passive Network
Stub Section
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Microstrip Line
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Two-Component Matching Networks
L-sections: capacitors / inductors
Design:
Analytical Approach
Precise
Suitable for Computer Synthesis
Smith Chart
Intuitive
Easier to verify
Faster
Smith Chart
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Two-Component Matching Networks
Eight Possible Network Configurations
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Example 1: L-section Matching Network
The output impedance of a transmitter operating at a
frequency of 2 GHz is ZT = 150 + j75 Ω. Design an
L-section matching network, such that maximum power
is delivered to the antenna whose input impedance is
ZA = 75 +j15 Ω.
L = 6.12 nH
C = 0.73 pF
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Simulation: Input Impedance ZT
200
180
160
140
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
Magnitude = 168 Ω
Phase = -26.6 deg
1
1.5
2
2.5
3
Frequency (GHz)
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The Smith Chart
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The Smith chart, invented by
Philip H. Smith is a graphical aid
or designed for electrical and
electronics engineers specializing
in radio frequency (RF)
engineering to assist them in
solving problems with
transmission lines and matching
circuits.
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The Smith Chart
X=j constant arc
R=0.2 constant circle
inductive
capacitive
Origin
X=-j constant arc
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Smith Chart
The addition of a reactance
connected in series with a
complex impedance results in
motion along a constantresistance circle.
A shunt connection produces
motion along a constantconductance circle.
Inductor – movement into the
upper half of the Smith Chart.
Capacitor – movement into
the lower half of the Smith
Chart.
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WHY?
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Example 2: L-Section Matching Network (Smith Chart)
Normalise ZA and ZT* by 75 Ω
ZA
= 1 + j0.2 and ZT* = 2 - j
Draw constant R = 1 Ω circle and constant
G = 0.4 S circle
Find intersection between R & G circles
Determine inductance and capacitance
value
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Example 3: Design of general 2-component matching networks
Using the smith chart, design all possible configurations
of discrete two element matching networks that match
the source impedance ZS = 50 + j25 Ω to the load ZL =
25 – j50 Ω. Assuming f = 2 GHz.
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Forbidden Regions (ZS = ZO = 50 Ω)
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Topology Selection
For any given load and input impedance
set there are at least two possible
configurations of the L-type networks that
achieve the required match.
Which network should one choose?
Availability
of components
DC
biasing
Stability
Frequency response / Q-Factor (Selectivity)
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Frequency Response
L-type matching networks
consist of series and shunt
combinations of capacitors
and/or inductors.
Classification:
Low
Pass
High Pass
Band Pass
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Fundamental Definitions
Q=
fc
f 2 − f1
Quality factor
(selectivity)
Low -3dB frequency
High -3dB frequency
resonant frequency
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Example 4: Frequency Response
Design two matching networks that transform a
complex load of resistance 80Ω and capacitance
2.65pF, into a 50Ω input impedance. (1 GHz)
Simulate their frequency response.
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Simulations
-3.5
-4
Vout/Vs (dB)
-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Frequency (GHz)
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Matching Verification
Matching at 1 GHz
Vout1
Vin
R1
50R
C1
C2
2.6pF
2.65pF
L1
10nH
R2
80R
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Input Reflection Coefficient Γin
0
0.5
Gain Vout / Vs dB
Reflection Coefficient ( | Γ| )
1
-50
Matching at 1 GHz
Z in − Z s*
Γin =
Z in + Z s*
0
0
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0.5
1
1.5
Frequency (GHz)
2
2.5
-100
3
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Nodal Quality Factor (Qn)
Qn = 1.2
See
smith chart
QL = 1 / (2.2-0.402) = 0.56
QL/Qn = 0.46 ~ 0.5
Nodal Q-factor
0.4 GHz
2.2 GHz
Qn
QL ≈
2
Loaded Q-factor of
matching network
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Design of a narrow-band matching network
Design two L-type networks that match a ZL = 25+j20 Ω
load impedance to a 50 Ω source at 1 GHz. Determine
the loaded quality factors of these networks from the
Smith Chart and compare them to the bandwidth
obtained from the frequency response.
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Simulation
3 dB
BW = 2 x (1.96-1) ~ 2 GHz
1.96 GHz
Qn = 1 (smith chart)
QL = 0.5
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Importance of Q-factor
Designing a broadband amplifier one uses
networks with low Q to increase the bandwidth
whilst for oscillator design it is desirable to
achieve high-Q networks to eliminate unwanted
harmonics in the output signal.
L-type matching networks provide no control
over the value of the nodal Q-factor.
One needs to introduce a third element in the
matching network:
T-matching
networks
π-matching networks
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T and π Matching Networks
The loaded quality factor of the matching
network can be estimated from the maximum
nodal Qn.
The addition of the 3rd element into the matching
network produces an additional node in the
circuit and allows the designer to control the
value of QL.
The following two examples illustrate the design
of T and π type matching networks with specified Qn
factor.
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Design of a T matching network
Design a T-type matching network that
transforms a load impedance ZL = 60-j30 Ω into
an input impedance of 10+j20 Ω and that has a
maximum nodal quality factor of 3.
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Design of a π-type matching network
For a broadband amplifier it is required to develop a πtype matching network that transforms a load impedance
of ZL = 10-j10 Ω into an impedance of Zin = 20+j40 Ω.
The design should involve the lowest possible nodal
quality factor, assuming that matching should be
achieved at a frequency of f = 2.4 GHz.
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References
Reinhold Ludwig and
Pavel Bretchko:
“RF Circuit Design –
Theory and
Applications”, Chapter
8, Prentice Hall.
ISBN 0-13-095323-7
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