the smith chart & it's applications

THE SMITH CHARTTHE SMITH CHART

Part 3 Part 3 -- The Smith Chart and its ApplicationsThe Smith Chart and its Applications

L T iL T iLec. Topics Lec. Topics

8. Introduction to the Smith Chart 8. Introduction to the Smith Chart Principle of Operation Principle of Operation Construction of the Smith Chart Construction of the Smith Chart Key Points on the Smith ChartKey Points on the Smith ChartKey Points on the Smith Chart Key Points on the Smith Chart Using Smith Chart with Load and Line Combinations Using Smith Chart with Load and Line Combinations Smith Chart and General Transmission Lines Smith Chart and General Transmission Lines Eff t f V i ti i FEff t f V i ti i FEffect of Variation in Frequency Effect of Variation in Frequency

9. Smith Chart and VSWR 9. Smith Chart and VSWR Using the Smith Chart and VSWR to Find ZLUsing the Smith Chart and VSWR to Find ZLAdding Components Using a Smith Chart Adding Components Using a Smith Chart Matching with Smith Chart and Series ComponentsMatching with Smith Chart and Series ComponentsMatching with Smith Chart and Series Components Matching with Smith Chart and Series Components Admittance Using a Smith Chart Admittance Using a Smith Chart Single Stub Matching Single Stub Matching

THE SMITH CHARTTHE SMITH CHART Devised in 1944 by Devised in 1944 by Phillip H. Smith, Bell Labs, Phillip H. Smith, Bell Labs, U.S.A.U.S.A.

A graphical aid for A graphical aid for transmission line transmission line calculations.calculations.

Phillip Smith: 1905Phillip Smith: 1905--19871987 Used nowadays for Used nowadays for designing “matching” designing “matching”

Phillip Smith: 1905Phillip Smith: 1905 19871987

circuits and displaying RF circuits and displaying RF data.data.

OL

OL

ZZZZ

ll e 2)(

11

OL

OL

ZZ

ZZ

l

l

Ol eeZZ

2

2

)(

)(

11

1e1

ll

ZZLL

ll

LL

GeneratorGeneratorTransmission lineTransmission line LoadLoad

The Smith Chart basically enables you to The Smith Chart basically enables you to

GeneratorGenerator

y yy yconvert between convert between and Z graphically, either at and Z graphically, either at the load or at an arbitrary point down the line.the load or at an arbitrary point down the line.

1Simplified Smith ChartSimplified Smith ChartIn this simplifiedIn this simplified 2

3

0.5In this simplifiedIn this simplifiedversion of theversion of theSmith Chart Smith Chart

50.2most of themost of thegrid of linesgrid of linesh bh b 10

0 0 0.2 0.5 1 2 3 5 10

has been has been removed.removed.

0 2-5

-10For clarity, in For clarity, in these lectures these lectures thi ithi i -0.2

-3

this version this version will be used to will be used to illustrate theillustrate the

-0.5

-1

-2illustrate the illustrate the properties of the properties of the Smith Chart.Smith Chart.

is a complex quantity:is a complex quantity: ρρ = |= |ρρ|e|e jjθθ

SMITH CHART BASICSSMITH CHART BASICS is a complex quantity: is a complex quantity: ρρ | |ρρ|e|e jj

For all ZFor all ZLL, 0 , 0 < < ||ρρ| | < < 11 Circle of Circle of UNIT RADIUSUNIT RADIUSin comple n mber planein comple n mber planein complex number planein complex number plane

+jX+jX

R+jXR+jX

θθ||ρρ|| (1,0)(1,0)

R+jXR+jX

θθ||ρρ|| (1,0)(1,0)

RR00

jXjX--jXjX

Complex Impedance PlaneComplex Impedance Plane Complex Reflection Coefficient PlaneComplex Reflection Coefficient Plane

OL

ZZZZ

11

OL ZZOL ZZ

ρρ ZZLL

1

To avoid having to use a different Smith Chart for eveTo avoid having to use a different Smith Chart for evevalue of Zvalue of Zoo, the , the normalised impedance, z ,normalised impedance, z , is used: is used: oo pp

z = Zz = ZLL/ Z/ Zoo

The normalised impedance z can then be The normalised impedance z can then be “denormalised” to obtain the load impedance Z“denormalised” to obtain the load impedance ZLL by by multiplying by Zmultiplying by Zoo::

ZZLL = z.Z= z.ZZZLL z.Z z.Zoo

In terms of its real and imaginary components, the In terms of its real and imaginary components, the normalised impedance can be written as:normalised impedance can be written as:normalised impedance can be written as:normalised impedance can be written as:

z = r + jxz = r + jx

CONSTRUCTION OF THE SMITH CHARTCONSTRUCTION OF THE SMITH CHART(pages 3(pages 3--5)5)

is for information onlyis for information onlyyy

IT IS NOT EXAMINABLEIT IS NOT EXAMINABLEIT IS NOT EXAMINABLEIT IS NOT EXAMINABLE

MAPPING BETWEEN IMPEDANCE PLANE MAPPING BETWEEN IMPEDANCE PLANE AND THE SMITH CHARTAND THE SMITH CHART

rr is the normalised resistance axisis the normalised resistance axisxx is the normalised reactance axisis the normalised reactance axis

Lines of constant normalised resistanceLines of constant normalised resistance

1

20.5ZZ oo

r = 0r = 0

Lines of constant normalised resistanceLines of constant normalised resistanceLines of constant normalised reactanceLines of constant normalised reactance

3

50.2x =

X/Z

x =

X/Z

0.40.4

0.20.210

0

-10

0 0.2 0.5 1 2 3 5 10 xx

rrr = R/Zr = R/Zoo

0.2 0.4 0.6 0.80.2 0.4 0.6 0.800

0 20 2

x = 0x = 0

-0.2

-2

-3

-5--0.20.2

--0.40.4-0.5

-1

-2

Complex Impedance PlaneComplex Impedance Plane Smith ChartSmith Chart

1

20.5xx3

mal

ised

mal

ised

tanc

e, x

tanc

e, x

5

10

0.2

norm

norm

reac

tre

act

0

-10

0 0.2 0.5 1 2 3 5 10

x x

r r normalised resistance, rnormalised resistance, r

-0.2-5

0 5 -2

-3

-0.5

-1

2

Example (page 2):Example (page 2):Step 1: Plot Step 1: Plot

p (p g )p (p g )

Measurements Measurements 0 7070 7074545ººon a slotted on a slotted

line with line with ZZ = 50= 50ΩΩ

0.7070.707

0.7070.7074545ºº

ZZoo = 50= 50ΩΩgive:give:||ρρ| = 0.707| = 0.707

4545ºº1100ºº||ρρ| 0.707| 0.707

and and θθ = 45= 45º. º. Find ZFind ZLL

Unit Circle inUnit Circle inUnit Circle in Unit Circle in Reflection Reflection

Coefficient PlaneCoefficient Plane

1

Example (page 2):Example (page 2):Step 2: Superimpose Smith Chart gridStep 2: Superimpose Smith Chart grid

2

3

0.5

p (p g )p (p g )

Measurements Measurements 0 7070 7074545ºº

50.2

on a slotted on a slotted line with line with ZZ = 50= 50ΩΩ

0.7070.707

0.7070.7074545ºº

10

0 0 0.2 0.5 1 2 3 5 10

x x

rr


4545ºº1100ºº

-10

r r ||ρρ| 0.707| 0.707and and θθ = 45= 45º. º. Find ZFind ZLL

-0.2

-3

-5

Unit Circle inUnit Circle in-0.5

1

-2

Unit Circle in Unit Circle in Reflection Reflection


1

Example (page 2):Example (page 2):Step 3: Read off normalised values for r and xStep 3: Read off normalised values for r and x

r = 1r = 1 x = 2x = 22

3

0.5

p (p g )p (p g )


r = 1r = 1, x = 2, x = 2

50.2


0.7070.707

0.7070.7074545ºº

10

0 0 0.2 0.5 1 2 3 5 10

x x

rr


4545ºº1100ºº

-10


-0.2

-3

-5


1

-2



1

Example (page 2):Example (page 2):Step 4: Calculate R and X by “denormalizing”: Step 4: Calculate R and X by “denormalizing”:

ZZLL = Z= Zoo(r + jx) = 50 + j100(r + jx) = 50 + j100ΩΩ2

3

0.5

p (p g )p (p g )


r=1r=1, x=2, x=2

50.2


0.7070.707

0.7070.7074545ºº

10

0 0 0.2 0.5 1 2 3 5 10

x x

rr


4545ºº1100ºº

-10


-0.2

-3

-5


1

-2



Open Circuit (Z =Open Circuit (Z = z = Z/Zz = Z/Z == ))KEY POINTS ON THE SMITH CHARTKEY POINTS ON THE SMITH CHART

Open Circuit (Z = Open Circuit (Z = , z = Z/Z, z = Z/Zoo = = ))

0,11 O

ZZZZ

1

20.5

z = r + jx = z = r + jx = ,

OZZ

3

50.2

= = 1100ºº = = 1100ºº

10

0

-10

0 0.2 0.5 1 2 3 5 10 1100ººxx

rr

-0.2

-2

-3

-5

-0.5

-1

-2

Reflection Coefficient PlaneReflection Coefficient Plane Smith ChartSmith Chart

Short Circuit (Z = 0, z = Short Circuit (Z = 0, z = 0)0)

ZZ

180,11 O

O

ZZZZ

1

2

3

0.5

5

10

0.2

xx0

-0.2 -5

-10

0 0.2 0.5 1 2 3 5 10 1100 rr

-0.5 -2

-3

= = 11180180ºº + j 0 + j0+ j 0 + j0

= = 11180180ºº

-1


z = r + jx = 0 + j0z = r + jx = 0 + j0

Matched Load (Z = ZMatched Load (Z = Zoo, z = , z = 1)1)

ZZ 00

O

O

ZZZZ

1

2

3

0.5

5

10

0.2

xx0

-0.2 -5

-10

0 0.2 0.5 1 2 3 5 10 1100 rr

-0.5 -2

-3

= = 00 + j 1 + j0+ j 1 + j0

= = 00

-1


z = r + jx = 1 + j0z = r + jx = 1 + j0

Resistive Load (Z = R +Resistive Load (Z = R + jj0 , z = 0 , z = r)r)R andR and ZZ (for a lossless line) are REAL(for a lossless line) are REAL

O

O

ZRZR

R and R and ZZoo (for a lossless line) are REAL, (for a lossless line) are REAL, therefore therefore ρρ is real.is real.

R >R > ZZ ρρ ++veve θθ = 0= 0ºº1

20.5

R > R > ZZoo ρρ ++veve, , θθ = 0= 0ººR < R < ZZoo ρρ --veve, , θθ = 180= 180ºº

3

50.2

10

0

-10

0 0.2 0.5 1 2 3 5 10 1100R = R = 00xx

rrR =R =

R < ZR < Zoo R = ZR = Zoo R > ZR > Zoo

-0.2

0 5 -2

-3

-5

R R

-0.5

-1

2


Purely Reactive Load (Z = 0 +Purely Reactive Load (Z = 0 + jjX , z = X , z = 0 +0 + jjx)x)

)( 21.1

j

O

O eZjXZjX

1

20.5

3

50.2z = jx (inductive)z = jx (inductive)

10

0

-10

0 0.2 0.5 1 2 3 5 10 1100z = jz = j00xx

rrz = jz = j

-0.2

0 5 -2

-3

-5

z jz j

z = jx (capacitive)z = jx (capacitive)-0.5

-1

2


General Case (Z = R +General Case (Z = R + jjXX))

j

O

O eZZZZ ||

1

20.5

3

50.2

||ρρ||10

0

-10

0 0.2 0.5 1 2 3 5 10 1100xx

rrθθ

-0.2

0 5 -2

-3

-5

-0.5

-1

2


1Pure ReactancePure ReactancePure Pure

2

3

0.5 ResistanceResistance

50.2

Matched LoadMatched Load10

0 0 0.2 0.5 1 2 3 5 10 xx

rr

0 2-5

-10Short Short CircuitCircuit

Open Open CircuitCircuit

-0.2

-3

-0.5

-1

-2

RATIO OF VRATIO OF V--/V/V++ AT AN ARBITRARY DISTANCE AT AN ARBITRARY DISTANCE llFROM THE LOAD ZFROM THE LOAD ZLL

Zo Zo

x = -l x = 0

ZL

If the line is losslessIf the line is lossless ll = j= jll , , hence:hence:

--ll ee--2 2 ll = = ee--jj ll eejjee--jj ll |e|ej(j(-- 22ll))

i e the magnitude of the reflection coefficient ati e the magnitude of the reflection coefficient ati.e. the magnitude of the reflection coefficient at i.e. the magnitude of the reflection coefficient at x =x = --ll is the same as at x = 0 but its phase changes is the same as at x = 0 but its phase changes fromfrom toto 22ll .. An additional phase change ofAn additional phase change offromfrom to to 22ll . . An additional phase change ofAn additional phase change of--22ll has been added by the introduction of the has been added by the introduction of the length of line,length of line, ll ..

ZZ

ll

ZZLL

Transmission lineTransmission line LoadLoad1

2

3

0.5

GeneratorGenerator

5

10

0.2

xx||ρρ||

θθ

ρρ of Zof ZLL onlyonly

ll

z = r+jxz = r+jx

0

-0.2 -5

-10

0 0.2 0.5 1 2 3 5 10 1100 rr22ββll

ρρ of Zof ZLL plusplus

22ββll

f l d d lif l d d li

-0.5 -2

-3

ρρ of Zof ZLL plusplusline of length line of length ll

Point rotates clockwise by Point rotates clockwise by 22ββl l radians (2x360 radians (2x360 l/l/λλ degrees) degrees)

z of load and line z of load and line of length of length ll

-1


yy ββ (( g )g )at a constant radiusat a constant radius

ZZ

ll

“Clockwise towards generator”“Clockwise towards generator” ZZLL

Transmission lineTransmission line LoadLoad

Clockwise towards generatorClockwise towards generator

1

2

3

0.5

GeneratorGenerator

5

10

0.2

xx||ρρ||

θθ

ρρ of Zof ZLL onlyonly

ll

z = r+jxz = r+jx

0

-0.2 -5

-10

0 0.2 0.5 1 2 3 5 10 1100 rr22ββll

ρρ of Zof ZLL plusplus

22ββll

f l d d lif l d d li

-0.5 -2

-3

ρρ of Zof ZLL plusplusline of length line of length ll

z of load and line z of load and line of length of length ll

-1

Reflection Coefficient PlaneReflection Coefficient Plane Smith ChartSmith ChartPoint rotates clockwise by Point rotates clockwise by 22ββl l radians (2x360 radians (2x360 l/l/λλ degrees) degrees)

Tutorial C, Question 2 Tutorial C, Question 2 Solution by Smith Solution by Smith ChartChart

To find ZTo find Z ::To find ZTo find Zinin::1.1. zzLL = Z= ZLL/Z/Zoo= (100+j50)/75= (100+j50)/75

zzLL3. 23. 2ll = 2x2= 2x2ll//λλ2 22 2 ffll//4 Read z4 Read z((ll))= (100+j50)/75= (100+j50)/75

= 1.33+j0.67= 1.33+j0.672 Plot2 Plot onon

= 2x2= 2x2ffll/v/v= 1.26 radians= 1.26 radians= 72= 72ºº zz

7272ºº4. Read z4. Read z((ll))

from chart:from chart:zz((ll)) = 1.75= 1.75--j0.45j0.452. Plot z2. Plot zLL ononSmith ChartSmith Chart= 72= 72 zz((ll))

((ll)) jj

5. Denormalise to find Z5. Denormalise to find Zinin::ZZ = Z= Z x zx zZZoo = 75 = 75 ΩΩZZLL = 100 + j50 = 100 + j50 ΩΩ

ZZinin = Z= Zo o x zx z((ll))= 75(1.75= 75(1.75--j0.45)j0.45)= = 131131--jj34 34 ΩΩll = 2.2 m, f = 100 MHz= 2.2 m, f = 100 MHz

v = 2 x 10v = 2 x 1088 msms--11

jj(cf. 126(cf. 126--j36 j36 ΩΩ by exact by exact

calculation)calculation)

(l)(l) == ee--2 2 ll == ee--22ααllee--jjll

The Smith Chart & The Smith Chart & GeneralGeneral Transmission LinesTransmission Lines

== ++ jj(l)(l) ee ee ee== eejjee--22ααl l ee--jjll

22 ll j(j( 22l)l)

== ++ jj -- propagation constantpropagation constant -- attenuation constantattenuation constant

phase constantphase constant

1

20.5

== |e|e--22ααlleej(j(--22l)l) -- phase constantphase constant

3

50.2

(0)(0) z = r+jxz = r+jx

10

0

-10

0 0.2 0.5 1 2 3 5 10 1100xx

rr((ll))z of load and linez of load and line

-0.2

0 5 -2

-3

-5z of load and line z of load and line

of length of length ll

-0.5

-1

2


The Smith Chart and Variation of FrequencyThe Smith Chart and Variation of Frequency(l)(l) == ee--jj22ll for a lossless linefor a lossless line(l)(l)

rotation angle is rotation angle is --22l l = = --2.(22.(2//λλ).).ll = = --2.2.(2(2.f/v)..f/v).ll== --44ffll/v or constant x f/v or constant x f

1

2

3

0.5

dd zzdcdc

= = --44ffll/v or constant x f /v or constant x f

5

10

0.2

xx

cc f1f1 zzf1f1

0

-0.2 -5

-10

0 0.2 0.5 1 2 3 5 10 1100 rrf2f2 zzf2f2

-0.5 -2

-3f3f3 zzf3f3

-1


frequency f3 > f2 > f1frequency f3 > f2 > f1

SummarySummary

TheThe Smith ChartSmith Chart is used as a graphical aid foris used as a graphical aid forThe The Smith ChartSmith Chart is used as a graphical aid for is used as a graphical aid for converting between a load impedance, Z, and a converting between a load impedance, Z, and a reflection coefficient,reflection coefficient, . (This can be done with . (This can be done with ((or without sections of line being present.)or without sections of line being present.)

lje 21

1

line)losslessa(forljO eeZjXRZ

21

1

O lZ

) (if 0

11

OL

OL

ZZZZ

ZZLL

To avoid having to use a different Smith Chart for To avoid having to use a different Smith Chart for every value of Zevery value of Zoo, the normalised impedance, z, is , the normalised impedance, z, is used:used:

z = Zz = ZLL/Z/Zoo(z = r + jx)(z = r + jx)

z can then be denormalised to obtain the load z can then be denormalised to obtain the load impedance Zimpedance ZLL, by multiplying by Z, by multiplying by Zoo::

ZZLL = z.Z= z.Zoo

The Smith Chart shows how the complex The Smith Chart shows how the complex ppimpedance plane maps on to the reflection impedance plane maps on to the reflection coefficient circle of unit radius.coefficient circle of unit radius.

The The circlescircles correspond to correspond to lines of constant lines of constant normalised resistance, r. normalised resistance, r.

The The arcsarcs correspond to correspond to lines of constant lines of constant normalised reactance, x.normalised reactance, x.

Adding a length, Adding a length, ll, of lossless line to a load, , of lossless line to a load, g g ,g g , , ,, ,ZZLL, corresponds on the Smith Chart to rotating , corresponds on the Smith Chart to rotating at constant radius from zat constant radius from zLL CLOCKWISECLOCKWISELLthrough an angle through an angle 22ll..

(“CLOCKWISE TOWARDS GENERATOR”)(“CLOCKWISE TOWARDS GENERATOR”)

If the line is not lossless, the radius If the line is not lossless, the radius decreases as we rotate around the centre.decreases as we rotate around the centre.

Increasing the signal frequency causes zIncreasing the signal frequency causes zLLg g q yg g q y L L to rotate clockwise around Smith Chart at to rotate clockwise around Smith Chart at constant radius.constant radius.

THE SMITH CHART AND VSWRTHE SMITH CHART AND VSWR The impedance of a load and line combination is The impedance of a load and line combination is

ii hha maximum, a maximum, zzmaxmax, when , when θθ -- 22ll (= (= ρρ)) = = 00

zzmaxmax = VSWR= VSWR Intersection of circle through load point, Intersection of circle through load point, zzLL, with , with i hti ht h d h lf f i t i i VSWR dh d h lf f i t i i VSWR drightright--hand half of resistance axis gives VSWR and hand half of resistance axis gives VSWR and

zzmaxmax.. Adding components in series with load:Adding components in series with load: Adding components in series with load:Adding components in series with load:

Adding inductors: move clockwise around Adding inductors: move clockwise around constant r circleconstant r circleconstant r circleconstant r circleAdding capacitors: move anticlockwise Adding capacitors: move anticlockwise around constant r circlearound constant r circlea ou d co sta t c c ea ou d co sta t c c eAdding resistors: move along arc of constant Adding resistors: move along arc of constant x towards rx towards r--axisaxis

the smith chart & it's applications

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