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Matching Networks

CCE 5220

RF and Microwave System Design

Dr. Owen Casha B. Eng. (Hons.) Ph.D.

212/01/2011

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

What should

be done if

ZS ZL*?

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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

Microstrip LineStub Section

Discrete

Passive Network

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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 . Design an

L-section matching network, such that maximum power

is delivered to the antenna whose input impedance is

ZA = 75 +j15 .

L = 6.12 nH

C = 0.73 pF

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Simulation: Input Impedance ZT

-100-80-60-40-20

020406080

100120140160180200

1 1.5 2 2.5 3

Frequency (GHz)

Magnitude = 168

Phase = -26.6 deg

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The Smith Chart

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

Origin

R=0.2 constant circleX=j constant arc

X=-j constant arc

inductive

capacitive

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Smith Chart

The addition of a reactance connected in series with a complex impedance results in motion along a constant-resistance circle.

A shunt connection produces motion along a constant-conductance circle.

Inductor movement into the upper half of the Smith Chart.

Capacitor movement into the lower half of the Smith Chart. WHY?

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Example 2: L-Section Matching Network (Smith Chart)

Normalise ZA and ZT* by 75

ZA = 1 + j0.2 and ZT* = 2 - j

Draw constant R = 1 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 to the load ZL =

25 j50 . Assuming f = 2 GHz.

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Forbidden Regions (ZS = ZO = 50 )

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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

resonant frequency

Low -3dB frequency High -3dB frequency

12ff

fQ c

=

Quality factor

(selectivity)

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Example 4: Frequency Response

Design two matching networks that transform a

complex load of resistance 80 and capacitance

2.65pF, into a 50 input impedance. (1 GHz)

Simulate their frequency response.

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Simulations

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

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)

Vout/Vs (dB)

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Matching Verification

Matching at 1 GHz

VinR1

50R

C1

2.6pF

L110nH

C2

2.65pF

R280R

Vout1

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0 0.5 1 1.5 2 2.5 30

0.5

1

Reflection C

oeff

icie

nt

( |

| )

Frequency (GHz)

0 0.5 1 1.5 2 2.5 3-100

-50

0

Gain

Vout /

Vs d

B

Input Reflection Coefficient in

*

*

sin

sinin

ZZ

ZZ

+

=

Matching at 1 GHz

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Nodal Quality Factor (Qn)

2

nL

QQ

Loaded Q-factor of

matching network

Nodal Q-factor

QL = 1 / (2.2-0.402) = 0.56

QL/Qn = 0.46 ~ 0.5

Qn = 1.2

See

smith chart

0.4 GHz 2.2 GHz

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Design of a narrow-band matching network

Design two L-type networks that match a ZL = 25+j20

load impedance to a 50 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

1.96 GHz

BW = 2 x (1.96-1) ~ 2 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

-matching networks

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T and 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 type matching networks with specified Qnfactor.

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Design of a T matching network

Design a T-type matching network that

transforms a load impedance ZL = 60-j30 into

an input impedance of 10+j20 and that has a

maximum nodal quality factor of 3.

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Design of a -type matching network

For a broadband amplifier it is required to develop a -

type matching network that transforms a load impedance

of ZL = 10-j10 into an impedance of Zin = 20+j40 .

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