Matching NetworksCCE 5220 RF and Microwave System Design Dr. Owen Casha B. Eng. (Hons.) Ph.D.12/01/2011 1

Maximum Power Transfer TheoremTo achieve maximum power transfer, one needs to match the load impedance to that of the source ZS = ZL* (Complex Conjugate) RS = RL and Xs = -XL12/01/2011

What should be done if ZS ZL* ?

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Matching NetworksMaximum 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 response12/01/2011 3

Passive Matching NetworksDiscrete Passive Networks (low gigahertz range) Microstrip lines Stub SectionsDiscrete Passive Network

Stub Section12/01/2011

Microstrip Line4

Two-Component Matching NetworksL-sections: capacitors / inductors Design:Analytical ApproachPrecise Suitable for Computer Synthesis

Smith ChartIntuitive Easier to verify FasterSmith Chart12/01/2011 5

Two-Component Matching Networks

Eight Possible Network Configurations

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Example 1: L-section Matching NetworkThe 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 ZT200 180 160 140 120 100 80 60 40 20 0 -20 -40 -60 -80 -100 1

Magnitude = 168

Phase = -26.6 deg

1.5

2 Frequency (GHz)

2.5

3

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The Smith ChartThe 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 ChartX=j constant arc inductive R=0.2 constant circle

capacitive X=-j constant arc

Origin

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Smith ChartThe 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.12/01/2011

WHY?

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

Normalise ZA and ZT* by 75ZA = 1 + j0.2 and ZT* = 2 - j

Draw constant R = 1 G = 0.4 S circle

circle and constant

Find intersection between R & G circles Determine inductance and capacitance value12/01/2011 12

Example 3: Design of general 2-component matching networksUsing 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 SelectionFor 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)12/01/2011 15

Frequency ResponseL-type matching networks consist of series and shunt combinations of capacitors and/or inductors. Classification:Low Pass High Pass Band Pass12/01/2011 16

Fundamental DefinitionsQ= fc f 2 f1

Quality factor (selectivity)

Low -3dB frequency

High -3dB frequency resonant frequency

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Example 4: Frequency ResponseDesign 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-3.5 -4 -4.5 Vout/Vs (dB) -5 -5.5 -6 -6.5 -7 -7.5 -80.5 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)12/01/2011 19

Matching Verification

Matching at 1 GHz

Vout1

Vin

R150R

C12.6pF

C22.65pF

L110nH

R280R

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Input Reflection Coefficient in1 0

Reflection Coefficient ( | | )

0.5

-50

Matching at 1 GHz

Z in Z s* in = Z in + Z s*0 0 0.5 1 1.5 Frequency (GHz) 2 2.5 -100 3

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Gain Vout / Vs dB

Nodal Quality Factor (Qn)QL = 1 / (2.2-0.402) = 0.56 QL/Qn = 0.46 ~ 0.5

Qn = 1.2 See smith chart

Nodal Q-factor

0.4 GHz

2.2 GHz

Qn QL 2Loaded Q-factor of matching network

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Design of a narrow-band matching networkDesign 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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Simulation3 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-factorDesigning 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 networks12/01/2011 25

T and Matching NetworksThe 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.12/01/2011

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Design of a T matching networkDesign 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 networkFor 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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ReferencesReinhold Ludwig and Pavel Bretchko: RF Circuit Design Theory and Applications, Chapter 8, Prentice Hall. ISBN 0-13-095323-7

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