Lecture 9

The Smith Chart and Basic Impedance-Matching Concepts

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The Smith Chart: Γ plot in the Complex Plane• Smith’s chart is a graphical representation in the complex Γ plane of

the input impedance, the load impedance, and the reflection coefficient Γ of a loss-free TL

• it contains two families of curves (circles) in the complex Γ plane

• each circle corresponds to a fixed normalized resistance or reactance

Re r

Im i 1j

| | 1

0 1

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The Smith Chart: Normalized Impedance and Γ

0

0 0

1 where and1

=| | =

L L LL L L

L Lj

r i

Z Z z Zz r jxZ Z z Z

e j

11Lz

relation #1: normalized load impedance zL and reflection Γ

2 2

2 2

2 2

1(1 )

2(1 )

r iL

r i

iL

r i

r

x

2 22

2 22

11 1

1 1( 1)

Lr i

L L

r iL L

rr r

x x

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The Smith Chart: Resistance and Reactance Circles

2 22 1

1 1L

r iL L

rr r

2 22 1 1( 1)r i

L Lx x

let the abscissa be Γr and the ordinate be Γi (the complex Γ plane)

• resistance and reactance equations describe circles in the Γcomplex plane

• resistance circles have centers lying on the Γr axis (with Γi= 0, i.e., ordinate = 0)

• reactance circles have centers with abscissa coordinate = 1• a complex normalized impedance zL = rL + jxL is a point on

the Smith chart where the circle rL intersects the circle xL

resistance circles reactance circles

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The Smith Chart: Resistance Circles

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The Smith Chart: Reactance Circles

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The Smith Chart: Nomographs at the bottom of Smith’s chart (left side), nomograph is added to

read out with a ruler the following • (1st ruler) above: SWR, below: SWR in dB,• (2nd ruler) above: return loss in dB,

below: power reflection |Γ|2 (P)• (3rd ruler) above: reflection coefficient |Γ| (E or I)

perfect match

1020log | | 1020log SWR

21 | | 1T

21010log (1 | | )

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The Smith Chart: SWR Circles

a circle of radius |Γ| centered at Γ = 0 is the geometrical place for load impedances producing reflection of the same magnitude |Γ|

such a circle also corresponds to constant SWR

1 | |1 | |

SWR

SWR circle

0.4 0.7Lz j

3.87SWR

| | 0.59

The Smith Chart: Plotting Impedance and Reading Out Γ

0.5 1.0Lz j

0.5Lr

1Lx

| |

83

| | 0.62

What is ZL if Z0 = 50 Ω?0.135

R

getting |Γ| with a ruler:

2) measure 1) measure

3) | | / R

R

83

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The Smith Chart: Tracking Impedance Changes with L

20

1at generator: ,

1g j L

in gg

Z Z e

2

0 211

j L

in j LeZ Ze

relation #2: input impedance versus the TL length L

compare with1at load: 1Lz

2

211

j L

in j Leze

on the Smith chart, the point corresponding to zin is rotated by −2βL (decreasing angle, clockwise rotation) with respect to the point corresponding to zL along an SWR circle (toward generator)

one full circle on the Smith chart is 2βLmax = 2π, i.e., Lmax = λ/2; this reflects the π-periodicity of zin

(see L08, sl. 4)

r

i1j

0 1

0.5 0.5Lz j

1 1inz j

toward generator

toward load

/ 4L

11

The Smith Chart: Tracking Impedance Changes with L – 2

for Z0 = 50 Ω, the quarter-wavelength TL transforms a load of

25 25 LZ j to an input impedance of

50 50 inZ j

check and see whether

0 L inZ Z Z

For a frequency-independent load ZL, what would be the direction of the locus of Zin as frequency increases?

SWR circle

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The Smith Chart: Read Out Distance to Load• unknown distance to load

in terms of λ/nD D

• known load ZL

75 75 LZ j

• known Z0

0 50 Z

1.5 1.5Lz j A

• measured Zin

23 34 inZ j B

0.46 0.68inz j

toward generator 0.194AL

0.394BL

0.22n B AD L L n

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The Smith Chart: Reading Out SWR

, 1 1L Az j

, 2.6L Br

, 2.6L BSWR r

SWR circle

A

B

A BSWR SWR

,

,

11

L BB

L B

rr

,

1 | |1 | |

BB

B

B L B

SWR

SWR r

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The Admittance Smith Chart• normalized load admittance

11 1 1 1

1 1 1

j

L jLy ee

z

• normalized input admittance (at generator)2

12

11

j L

in in j Ley ze

• the relation between yin and yL is the same as that between zin and zL– one can get from load to generator (and vice versa) by following a circle clockwise (counter-clockwise)

• standard Smith chart gives resistance and reactance values

• admittance Smith chart is exactly the same as the impedance Smith chart but rotated by 180° [see eq. (*) ] in the complex Γ plane

( )

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Reading Out Normalized Conductance and Susceptance Values

00

1LL L

L

Yy Y ZY z

YL – load admittanceY0 – characteristic admittance

L L LY G jB

conductance susceptance

L L Ly g jb

• normalized admittance

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Conductance and Susceptance Circles in Admittance Smith Chart combined impedance and conductance Smith Charts

open circuit0

( )1

LL

YZ

short circuit

( 0)1

LL

YZ

conductance circles resistance circles

reactance circlessusceptance circles

positive (capacitive) susceptance

negative (inductive) susceptance

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Switching Between Impedance and Admittance on Smith Chart

• impedance values from a standard Smith chart can be easily converted to admittance by rotation along a circle by exactly 180°

• rotation by 180° on the impedance Smith chart corresponds to impedance transformation by a quarter-wavelength TL

24

1 1( / 4)11

11

j

in j

L

ez Le

z

1( / 4)in LL

z L yz

• in impedance Smith chart, the point diametrically opposite from an impedance point shows its respective “admittance” value

18

Switching Between Impedance and Admittance: Example

Check whether in this example the yL found from the Smith chart satisfies 1

LL

yz

r

i1j

0 1

0.5 0.5Lz j

1 1inz j

toward generator

toward load

/ 4L 1 1Ly j

same as

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Example: L-network matching

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Quarter-wave Transformer Revisited

from L08, sl. 18:20

/4in LL

ZZZ

for impedance match at the input terminals of the λ/4 TL, Zin = ZG*

0 G LZ Z Z⇒ ⇒

inZ

0 / 4L

0( , )ZLZ

GZ

GV

loss-free line

1G Lz z ⇒ G Lz y

TL must be designed to have this specific Z0

20

Quarter-wave Transformer Revisited – 2 the impedance match with the λ/4 transformer holds perfectly at

one frequency only, f0, where L = λ0/4

this impedance-match device is narrow-band0 0

00 0

tan( ) 2( ) , where tan( ) 4 2

Lin

L

Z jZ L fZ f Z LZ jZ L f

0

0

( )| ( ) |( )

in

in

Z f ZfZ f Z

0

100 50 70.71

L

G

ZZZ

perfect match

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Optimal Power Delivery: Review (Homework) at the generator’s terminals, a loaded TL is equivalently represented

by its input impedance Zin

GZ

inZinV

inI

GV

active (or average) power delivered to the loaded TL (this is also the power delivered to the load ZL if the line is loss-free)

22 21 1 1 1( ) Re | | Re | | Re

2 2 2in

in av in in in in GG in in

ZP V I V Y VZ Z Z

22 2

1( ) | |2 ( ) ( )

inin av G

in G in G

RP VR R X X

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optimal matching is achieved when maximum active power is delivered to the load Zin – what is this optimal value of Zin?

assume generator’s impedance ZG = RG + jXG is known and fixed

opt max ( )in

in in inZZ P Z

find the optimal Rin and Xin by obtaining the respective derivatives

2 2 20 ( ) 0inG in in G

in

P R R X XR

0 ( ) 0inin in G

in

P X X XX

maximum power is delivered to the load under conditions of conjugate match

opt opt opt and in G in G in GR R X X Z Z

Optimal Power Delivery: Review (Homework)

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the impedance Smith chart depicts a normalized load impedance as a point in the complex Γ plane

the load impedance is normalized with respect to the characteristic impedance Z0 of the TL

the admittance Smith chart depicts a normalized load admittance yLas a point in the complex Γ plane

the admittance Smith chart is rarely used because the impedance Smith chart can be readily used as an admittance chart as well – the sense of rotation with increasing TL length is the same

resistance/reactance impedance values are determined from the resistance/reactance circles

the input impedance zin of a TL loaded with a known zL is found by following the SWR circle, starting from zL, and completing an angle of 2βL in a clockwise direction (on chart: “toward generator”)

Summary

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if the input impedance zin of a TL is known but the load zL is not, zL is determined by starting from zin , following the SWR circle and completing an angle of 2βL in a counter-clockwise direction (on chart: “toward load”)

the normalized admittance yL of a given impedance zL is found by reading out the value of the point diametrically opposite to zL on the Smith chart

many more applications of the Smith chart will be shown during the tutorial

Summary – 2