Smith Chart & Matching Network

SmithChart & MatchingNetwork James Lu

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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NonmatchedImpedance(*z0) • ReflectionsleadtoZin variationswithline lengthandfrequency • Poweriswastedbecauseofreactivepower, whichcanalsodamageequipmentduring shortcircuit(forexample) • Onlypartialpowerisdeliveredtotheload. • SWR>1:therewillbevoltagemaximaon theline,voltagebreakdownathighpower levels • Noises(bouncesorechoes)

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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BenefitsofMatching(*=0) • Zin =ZO, independentoflinelength, andfrequency(overthebandwidth ofthematchingnetwork) • Maximumpowertransfertotheload isachieved • SWR=1:novoltagepeaksontheline • Nobounces(echoes)

JamesJ.Q.Lu

3 UQ

ECSE2100Fields&WavesI

LoadMatching What if the load cannot be made equal to Zo for some other reasons? Then, we need to build a matching network so that the source effectively sees a match load. Ps

M

Z0

ZL

* 0 Typically we only want to use lossless devices such as capacitors, inductors, transmission lines, in our matching network so that we do not dissipate any power in the network and deliver all the available power to the load.

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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NormalizedImpedance It will be easier if we normalize the load impedance to the characteristic impedance of the transmission line attached to the load. Z

z

Zo

z

r  jx

1 * 1 *

Since the impedance is a complex number, the reflection coefficient will be a complex number

*

r

u  jv

1  u 2  v2

1  u

2

JamesJ.Q.Lu

v

2v

x

1  u 2  v 2

2

ECSE2100Fields&WavesI

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SmithCharts The impedance as a function of reflection coefficient can be re-written in the form:

r

x

1  u 2  v2

1  u 2  v 2 2v

1  u

2

 v2

2

r · § 2 ¨u  ¸ v 1 r ¹ ©

u  1  §¨ v  1 ·¸ x¹ © 2

1

1  r 2 2

1 x2

These are equations for circles on the (u,v) plane

x  xo 2  ( y  yo ) 2

JamesJ.Q.Lu

ECSE2100Fields&WavesI

a2

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SmithChart– RealCircles Im^*` 1

*

1 Circle

0.5

r=0

r=1/3 r=1

1

0.5

0

Re^*`

r=2.5 0.5

1

0.5

1

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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SmithChart– ImaginaryCircles Im^*` 1

x=1/3

x=1

x=2.5

0.5

1

0.5

0

x=-1/3

0.5

0.5

x=-1

1

Re^*`

x=-2.5

1

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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SmithChart

Impedances, voltages, currents, etc. all repeat every half wavelength

zL zl

WT G

Inductive R SW

Pmin

L WT

S

*e  j 2 E l

1 * 1 *

j

4S

O

l

ro (at P max)

Open (z=f) *=1

Pmax

Capacitive

Purely imaginary impedances along the periphery

Purely real impedances along the horizontal centre line

JamesJ.Q.Lu

*l

ro *=0

y=1/(1+j) =0.5-j0.5

1  *l 1  *l

*e

z=1+j z=1

Short (z=0) *=1

z L 1 z L 1 1 * r L  jx L 1 *

* *L

ECSE2100Fields&WavesI

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SmithChartExample1 Given:

Zo

0.5‘45q

50:

WT G

*L

What is ZL? ZL

50:1.39  j1.35

L WT

69.5:  j 67.5:

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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SmithChartExample2 Given: ZL 15:  j25:

50:

Zo

WT G

What is *L? 15:  j25: 50: 0.3  j0.5

zL *L

0.6‘  123q

L WT

z1 = 2 + j z2 = 1.5 -j2 z3 = j4 z4 = 3 z5 = infinity z6 = 0 z7 = 1 z8 = 3.68 -j18 JamesJ.Q.Lu

ECSE2100Fields&WavesI

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SmithChartExample3 ZL

Zo

50:

W

6.78ns

50:  j50:

WT G

Given:

What is Zin at 50 MHz? 50:  j50: 50: 1.0  j1.0

*L *in l

0.445‘64q

*L e  j 2 El fOW

*L e  j 4S l / O 9

50 ˜ 10 ˜ 6.78 ˜ 10 O 6

*L e  j 2ZW 0.339O

L WT

zL

2ZW 244q

-in 180q *in 0.445‘180q Z in

50:0.38  j 0.0 19:

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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Admittance A matching network is going to be a combination of elements connected in series AND parallel. Z1 Impedance is well suited when working Z2 with series configurations. For example:

V

ZI

Z1  Z2

ZL

Impedance is NOT well suited when working with parallel configurations.

For parallel loads it is better to work with admittance.

I

1 Z1

YV

JamesJ.Q.Lu

Z2

Y1

Y2

Z1Z2 Z1  Z2

ZL

Y1

Z1

YL

Y1  Y2

ECSE2100Fields&WavesI

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NormalizedAdmittance y

Y Yo y

g

1  u 2  v 2  2v

1  u

2

v

1 * 1 * 2

1  u 2  v2

b

g  jb

YZo

2

§ g · ¨¨ u  ¸¸  v 2 1 g ¹ ©

u  1  §¨ v  1 ·¸ b¹ © 2

1

1  g 2 2

1 b2

These are equations for circles on the (u,v) plane JamesJ.Q.Lu

ECSE2100Fields&WavesI

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AdmittanceSmithChart Conductance Circles 1

Susceptance Circles

Im^*`

1

Im^*`

b=-1/3

b=-1 b=-2.5

0.5

g=2.5 1

g=f

g=1/3

g=1 0.5

0

g=0

0.5

Re^*`1

0.5

b=f 1

0.5

0

Re^*`

0.5

1

b=1/3

b=2.5 0.5

0.5

b=1 1

1

JamesJ.Q.Lu

ECSE2100Fields&WavesI

15

ImpedanceandAdmittanceSmithCharts • For a matching network that contains elements connected in series and parallel, we will need two types of Smith charts ¾ impedance Smith chart ¾ admittance Smith Chart

• The admittance Smith chart is the impedance Smith chart rotated 180 degrees. – We could use one Smith chart and flip the reflection coefficient vector 180 degrees when switching between a series configuration to a parallel configuration.

JamesJ.Q.Lu

ECSE2100Fields&WavesI

16

AdmittanceSmithChartExample1 Given:

y 1  j1 WT G

Find * z • Procedure:

Plot y

•Plot 1+j1 on chart •vector = 0.445‘64q

Flip 180 degrees

•Flip vector 180 degrees

z

L WT

* 0.445‘  116q

0.5  j 0.5

JamesJ.Q.Lu

Read *&z

ECSE2100Fields&WavesI

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AdmittanceSmithChartExample2 Given:

Find Y • Procedure:

WT G

* 0.5‘  45q Zo 50:

• Plot *

Plot *

• Flip vector by 180 degrees Read y

y 0.38  j0.36 Y

L WT

• Read coordinate

Flip 180 degrees

1 0.38  j 0.36 50:

7.6  j 7.2 ˜10 3 S JamesJ.Q.Lu

ECSE2100Fields&WavesI

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MatchingExample Ps

Z0

50:

M

100:

* 0 Match 100: load to a 50: system at 100MHz A 100: resistor in parallel would do the trick, but ½ of the power would be dissipated in the matching network. We want to use only lossless elements such as inductors and capacitors so we don’t dissipate any power in the matching network

JamesJ.Q.Lu

ECSE2100Fields&WavesI

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MatchingExample

ƒ We won’t get any closer by adding series impedance so we will need to add something in parallel.

WT G

ƒ We need to go from z=2+j0 to z=1+j0 on the Smith chart

L WT

ƒ We need to flip over to the admittance chart

Impedance Chart JamesJ.Q.Lu

ECSE2100Fields&WavesI

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MatchingExample

L WT

ƒ Before we add the admittance, add a mirror of the r=1 circle as a guide.

WT G

ƒ y=0.5+j0

Admittance Chart JamesJ.Q.Lu

ECSE2100Fields&WavesI

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MatchingExample ƒ Before we add the admittance, add a mirror of the r=1 circle as a guide

WT G

ƒ y=0.5+j0

L WT

ƒ Now add positive imaginary admittance.

Admittance Chart JamesJ.Q.Lu

ECSE2100Fields&WavesI

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MatchingExample ƒ Before we add the admittance, add a mirror of the r=1 circle as a guide

WT G

ƒ y=0.5+j0

ƒ Now add positive imaginary admittance jb = j0.5

j0.5 50:

j0.5

j2S100MHz C

L WT

jb

C 16pF 16pF

100:

JamesJ.Q.Lu

Admittance Chart ECSE2100Fields&WavesI

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MatchingExample ƒ Flip to the impedance Smith Chart

WT G

ƒ We will now add series impedance

L WT

ƒ We land at on the r=1 circle at x=-1, i.e. z = 1 – j1

Impedance Chart JamesJ.Q.Lu

ECSE2100Fields&WavesI

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ƒ Add positive imaginary admittance to get to z=1+j0

jx

 j1.0 50:

WT G

MatchingExample

j1.0 j2S100MHz L

L 80nH

L WT

80nH

16pF

100:

JamesJ.Q.Lu

ECSE2100Fields&WavesI

Impedance Chart 25

MatchingExample

WT G

ƒ This solution would have also worked

L WT

32pF

160nH

JamesJ.Q.Lu

100:

ECSE2100Fields&WavesI

Impedance Chart 26

MatchingBandwidth 0

80 nH 5

16 pF

100 :

WT G

Reflection Coefficient (dB)

10

150 MHz

15 20 25 30 35

50

60

70

80

90 100 110 Frequency (MHz)

120

130

140

150

L WT

40

50 MHz

Because the inductor and capacitor impedances change with frequency, the match works over a narrow frequency range JamesJ.Q.Lu

Impedance Chart

ECSE2100Fields&WavesI

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SingleStubTuner yl1

l1

yl2

Z0

Goal: *

Z0

0

zin=1 (yin=1)

ZL

Z0

l2

Open or Short

yin = yl1 + yl2 = 1 = (gl1 + jbl1) + jbl2

1  *l 1  *l

yl

1 zl

*l

*e  2 E l

gl1 = 1 (real-part condition) Stub length l = W up=fOW bl1 = -bl2 (imaginary-part condition) Phase shift: TJ=2El=2(2S/O)l=4S(l/O) (2 Degrees of freedom) Open: Zin = -jZocotEl, or zin = -jcotEl Short: Zin = jZotanEl, or zin = jtanEl JamesJ.Q.Lu

ECSE2100Fields&WavesI

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SingleStubTuner Match 100: load to a 50: system at 100MHz using two transmission lines connected in parallel WT G

ƒ Flip to Admittance chart ƒ y=0.5+j0

L WT

ƒ Adding length to Cable 1 rotates the reflection coefficient clockwise to g=1. l1 = 0.152O

ƒ y1l=1+j0.72

JamesJ.Q.Lu

Admittance Chart ECSE2100Fields&WavesI

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L WT

ƒ The stub has to add a normalized admittance of -0.72 to bring the trajectory to the center of the Smith Chart

WT G

SingleStubTuner

Admittance Chart JamesJ.Q.Lu

ECSE2100Fields&WavesI

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SingleStubTuner

L WT

ƒ By adding enough cable to the short stub, the admittance of the stub will reach to -0.72

WT G

ƒ An short stub of zero length has an admittance=jf

l2 = (0.401-0.25)O = 0.151O

Admittance Chart JamesJ.Q.Lu

ECSE2100Fields&WavesI

31

SingleStubTuner l1 = 0.152O

0

L WT

l2 = 0.151O

100:

WT G

*

Admittance Chart JamesJ.Q.Lu

ECSE2100Fields&WavesI

32

SingleStubTuner l1 = 0.347O

100:

0 WT G

*

l2 = 0.097O

L WT

This solution would have worked as well.

An open stub of zero length has an admittance=j0

Admittance Chart

JamesJ.Q.Lu

ECSE2100Fields&WavesI

33

SingleStubTunerMatchingBandwidth *

l1 = 0.347O

0

100: WT G

l2 = 0.097O 0 5

15 20

50 MHz

25

L WT

Reflection Coefficient (dB)

10

30 35 40

150 MHz 50

60

70

80

90 100 110 Frequency (MHz)

120

130

140

15

Because the cable phase changes linearly with frequency, the match works over a narrow frequency range JamesJ.Q.Lu

ECSE2100Fields&WavesI

Impedance Chart 34

Summary • Impedance matching is necessary to: – reduce VSWR – obtain maximum power transfer

• Lump reactive elements and a single stub can be used. • A quarter-wave line can also be used to transform resistance values, and act as an impedance inverter. • These matching network types are narrow-band: they are designed to operate at a single frequency only. JamesJ.Q.Lu

ECSE2100Fields&WavesI

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