Electromagnetics (ENGR 367) The Smith Chart: A Graphical Method for T-lines.

ElectromagneticsElectromagnetics(ENGR 367)(ENGR 367)

The Smith Chart:The Smith Chart:

A Graphical Method for T-linesA Graphical Method for T-lines

Advantages of Knowing Advantages of Knowing the Smith Chartthe Smith Chart

• Popularity: one of most widely used graphical Popularity: one of most widely used graphical methods for T-line analysis & designmethods for T-line analysis & design– as an aid in a pencil & paper approachas an aid in a pencil & paper approach– for display byfor display by measurement and CAD toolsmeasurement and CAD tools

• Labor-saving: facilitates complex number Labor-saving: facilitates complex number calculations without losing too much accuracycalculations without losing too much accuracy

• Serves as a valuable tool for visualizing key Serves as a valuable tool for visualizing key aspects of T-line systems as they depend on aspects of T-line systems as they depend on frequency or position along the linefrequency or position along the line

Requirements for Requirements for Understanding the Smith ChartUnderstanding the Smith Chart

• Have basic orientation to T-linesHave basic orientation to T-lines– Concepts (generator, load, matching, etc.)Concepts (generator, load, matching, etc.)

– Parameters (ZParameters (Z00, Z, Zinin, , , , , VSWR, Y=1/Z, etc.), VSWR, Y=1/Z, etc.)

• Become familiar with some unique Become familiar with some unique aspectsaspects– What curves on the graph representWhat curves on the graph represent– How to show Z or Y and move along the lineHow to show Z or Y and move along the line

• Practice using the Smith Chart to find Practice using the Smith Chart to find , , VSWR, ZVSWR, Zinin, and do T-line matching , and do T-line matching

Outline of Lecture on Smith Outline of Lecture on Smith ChartChart

• Basic layout: shows what curves Basic layout: shows what curves representrepresent

• Learn where to plot points: Learn where to plot points: complex impedance (or admittance)complex impedance (or admittance)

• Identify key reference pointsIdentify key reference points• Find out how to move along the lineFind out how to move along the line• Work examples of Smith Chart usageWork examples of Smith Chart usage• Solve impedance Matching ProblemsSolve impedance Matching Problems• Draw ConclusionsDraw Conclusions

Basic Layout of the Smith Basic Layout of the Smith ChartChart

• Within a circle of constant unit radius Within a circle of constant unit radius ((=1), graph explicitly shows a family =1), graph explicitly shows a family ofof– Circles of constant resistanceCircles of constant resistance– Curves of constant reactanceCurves of constant reactance

• A polar coordinate graph of A polar coordinate graph of = = eejj

exists w/r/to the center of the chart that exists w/r/to the center of the chart that may be evaluated for may be evaluated for 1 along with 1 along with VSWR VSWR 1 on scales beneath the chart 1 on scales beneath the chart

Plotting Points on the Smith Plotting Points on the Smith ChartChart

• Define Normalized ImpedanceDefine Normalized Impedance

• Reflection Coefficient in terms of zReflection Coefficient in terms of zLL

Note: this keeps values w/in range on S.C.Note: this keeps values w/in range on S.C.

L L L0 0

L L LZ R jXz r jx

Z Z

L 0 0 L

L 0 0 L

1/ 1

1/ 1

Z Z Z z

Z Z Z z

Plotting Points on the Smith Plotting Points on the Smith ChartChart

• Zero Reactance Line (Im{z} = x = 0):Zero Reactance Line (Im{z} = x = 0):

– Also Horizontal Center Line of Pure Resistance Re{z} Also Horizontal Center Line of Pure Resistance Re{z} = r= r

– Divides the Upper Half from the Lower HalfDivides the Upper Half from the Lower Half• Upper Half represents Im{z} > 0 (inductive reactance) Upper Half represents Im{z} > 0 (inductive reactance)

[or alternatively: Im{y} > 0 (capactive susceptance)] [or alternatively: Im{y} > 0 (capactive susceptance)]

• Lower Half represents Im{z} < 0 (capacitive reactance) Lower Half represents Im{z} < 0 (capacitive reactance) [or alternatively: Im{y} < 0 (inductive susceptance)] [or alternatively: Im{y} < 0 (inductive susceptance)]

Essential (Key) Reference Essential (Key) Reference PointsPoints

On The Smith ChartOn The Smith Chart1) Center (z = r = 1): matched condition1) Center (z = r = 1): matched condition

– Also represents Also represents = 0 and VSWR = 1 = 0 and VSWR = 1– Becomes the destination (objective) for Becomes the destination (objective) for

impedance matching problems (HOME !)impedance matching problems (HOME !)

2) Right Extreme (z = r = 2) Right Extreme (z = r = ): open circuit): open circuit– Also Also = 1 = 100°° and VSWR = and VSWR = – An open circuited stub has ZAn open circuited stub has Zinin that begins here that begins here

Essential (Key) Reference Essential (Key) Reference PointsPoints

On The Smith ChartOn The Smith Chart3) Left Extreme (z = r = 0): short circuit3) Left Extreme (z = r = 0): short circuit

– Also Also = 1 = 1180180°° and VSWR = and VSWR = – A short circuited stub has ZA short circuited stub has Zinin that begins here that begins here

• Key Reference Points on the Smith ChartKey Reference Points on the Smith Chart* * Center Point: matched condition (HOME)Center Point: matched condition (HOME)

O O Right Extreme: open circuit terminationRight Extreme: open circuit termination

Left Extreme: short circuit terminationLeft Extreme: short circuit termination

Moving Toward the Generator Moving Toward the Generator (or Load) on the Smith Chart (or Load) on the Smith Chart

• Use Circular Band at Outer Edge of Chart: Use Circular Band at Outer Edge of Chart: has scales that indicate distance (in has scales that indicate distance (in ))

– To go toward the generator, move CW To go toward the generator, move CW – To go toward the load, move CCWTo go toward the load, move CCW

• Normalized Input Impedance on the T-lineNormalized Input Impedance on the T-line( 2 )2

in in2 ( 2 )

in

11 1( ) where (0) @ load position

1 1 1

1 and ( / 2) for 0,1,2,...

1 load and all oth

j lj l

Lj l j l

L

eez l z z

e e

z m z m

er wave impedances repeat every / 2 along the line @ a rate of 2 l

Examples of Smith Chart Examples of Smith Chart UsageUsage

• Example A (conditions same as in Example A (conditions same as in TLS3)TLS3)GivenGiven: Z: Z00 = Z = ZLL = 300 = 300 , , = = ll = 0.8 = 0.8 = =

288288°)°)

FindFind: Z: Zinin, , , VSWR, VSWR

SolutionSolution: take the following steps on the : take the following steps on the S.C.S.C.

LL

0

3001) normalize load impedance: 1

300

2) locate the load point: here @ center of chart!

Zz

Z


• SolutionSolution: (Example A continued): (Example A continued)

in L

3) 0 and 1 for the matched condition

(since radius = 0 on the polar plot w/r/to center)

4) on the 0 circle for a rotation of

0.8 ( 576 ) CW toward the generator

(or for any rotat

VSWR

z z

in in 0

ion angle here) since the line is matched!

5) Un-normalize: (1)(300 ) 300 Z z Z


• Example B (conditions same as in Example B (conditions same as in TLS3)TLS3)GivenGiven: Z: ZLL = R = RininRRinin = 300 = 300 300 300 = 150 = 150

and Zand Z0 0 = 300 = 300 , , = = ll = 0.8 = 0.8 = 288 = 288° as ° as before before


SolutionSolution: take these steps on the chart: take these steps on the chart

L0

L

150 1) normalize: 0.50

300

2) locate point: 0.50 (directly left of center )

LZz

Z

r


• SolutionSolution: (Example B continued): (Example B continued)

L

L

1 0.5 1 1/ 2 13)

1 0.5 1 3/ 2 3

(or by radius of circle from center on RFL COEFF scale below Smith Chart)

1 1 1/ 3 4 / 34) 2

1 1 1/ 3 2 / 3

(or by radius of circle from center on SWR scale below

z

z

VSWR

L

in

in

Smith Chart)

5) rotate from CW 0.8 (0.5 0.3) on constant 1/ 3 circle

to get 1.55 0.69 (to lower right of center)

6) Un-normalize: (1.55 0.69)(300 ) (466 206)

z

z j

Z j j


• Example C (conditions same as in Example C (conditions same as in TLS3)TLS3)GivenGiven: Z: ZLL = (120 - j60) = (120 - j60) , Z, Z0 0 = 300 = 300

and and = = ll = 0.8 = 0.8 = 288 = 288° as ° as before before


SolutionSolution: follow these steps on the chart: follow these steps on the chartL

L L

120 601) normalize: 0.40 0.20

3002) locate point: (to lower left of center)

on 0.40 circle and 0.20 curve

jz j

r x


• SolutionSolution: (Example C continued): (Example C continued)L

L

L

( 2 )in in

13) 0.45 153

1

(can verify on RFL COEFF scale or 0 line)

1 1.454) 2.6 (or by SWR scale)

1 0.55

5) rotate 2 2(288 ) 576 (360 216 )

to get 2.52 0.46j l

z

z

x

VSWR

l

e z j

=

in

(to lower right of center)

6) Un-normalize: (2.52 0.46)(300 ) (755 138) Z j j


• Example D (conditions same as in Example D (conditions same as in TLS3)TLS3)GivenGiven: Z: ZLL = - j300 = - j300 , Z, Z0 0 = 300 = 300

and and = = ll = 0.8 = 0.8 = 288 = 288° as ° as before before


SolutionSolution: take these steps: take these steps

3001) 1 (located at bottom center edge)

3001

2) 1 901

L

jz j

jj

j


• SolutionSolution: (Example D continued): (Example D continued)

L

in in

in

3) rotate 0.3 CW toward generator to get

and 1.96 (on upper right edge)

4) ( 1.96)(300 ) 589 , a purely

inductive reactance since 0 on this circle!

z j

Z j j

r

Impedance Matching Impedance Matching ProblemsProblems

• Methods of T-line Matching (Load-to-line)Methods of T-line Matching (Load-to-line)– Single stubSingle stub

•Series insertionSeries insertion

•Parallel insertionParallel insertion

– Double stub tuner: usually a manuf’d device Double stub tuner: usually a manuf’d device with movable shorts (act like trombone slides)with movable shorts (act like trombone slides)

– Quarter-wave transformer (QWT): fixed freq.Quarter-wave transformer (QWT): fixed freq.

– Tapered transformer (~Tapered transformer (~gg long): for pulses and long): for pulses and wideband applicationswideband applications

Matching Problem Matching Problem Using the Smith Chart: Using the Smith Chart:

Example 1Example 1• Single Stub Matching (P3-4-9, K&F, 5/e, p. Single Stub Matching (P3-4-9, K&F, 5/e, p.

149)149)GivenGiven: a uniform Z: a uniform Z00 = 100 = 100 T-line T-line

terminated by a load Zterminated by a load ZLL = (150 + j50) = (150 + j50)

FindFind: a) min. dist. d: a) min. dist. d11 from load to parallel shorted stub from load to parallel shorted stub b) min. length d b) min. length d22 of the shorted stub to match of the shorted stub to match

SolutionSolution: first examine the schematic diagram: first examine the schematic diagram

Single Stub MatchingSingle Stub Matching

• SolutionSolution: take the following steps on S.C.: take the following steps on S.C.

L

L L

2 ( / 4)

in 2 ( / 4)

Lin L

150 501) Normalize the load: 1.5 0.50

1002) Convert to admittance: 1/ 0.60 0.20

1 1Note: ( / 4)

1 11 1 1

1 (0)

going halfway ( /4) around

j j

j j

jz j

y z j

e ez l

e e

yz z

the smith chart converts

any impedance to its reciprocal (admittance)!


• SolutionSolution: (continued): (continued)

L

1

B B

3) Find the intersection between 0.27

( 1.78) circle and the 1 circle

Note that two intersection points exist, but one

minimizes the length . In terms of admittance

1.0 0.58

VSWR g

d

y j g j

B

1 B A' (0.647 0.454) 0.193

b

d z z



in B 0

ssss

inss C B

4) Find the minimum length of a shorted stub, that when

inserted in parallel at B, produces a match ( ).

1 (exists at the right extreme)

Desire: 0 0.58

Z Z R

yz

y y jb j

C

2 C ss

2

This occurs at the pos. on the S.C.: 0.416

(0.416 0.250) 0.166

5) Adding the shorted stub of length in parallel at

B completes our path on the S. C. from B to C (HOME!).

z

d z z

d


• Design of the Single Stub Match has been Design of the Single Stub Match has been completed since location and length of completed since location and length of the shorted stub have been specified.the shorted stub have been specified.

• For further considerationFor further consideration– What if an open circuited stub had been What if an open circuited stub had been

used?used?– Actual physical location and length of the Actual physical location and length of the

stub depends on frequency and phase stub depends on frequency and phase velocity since the wavelength velocity since the wavelength = v = vpp/f. /f.

Matching Problem Matching Problem Using the Smith Chart: Using the Smith Chart:

Example 2Example 2• QWT Matching (P3-4-10, K&F, 5/e, p. 151)QWT Matching (P3-4-10, K&F, 5/e, p. 151)

GivenGiven: a 100 : a 100 T-line terminated by a Z T-line terminated by a ZLL = (300 + j200) = (300 + j200) loadload

FindFind: a) min. length d: a) min. length d11 to transform Z to transform ZLL to a pure resistance to a pure resistance

b) impedance of the QWT required for a matchb) impedance of the QWT required for a match

c) VSWR on each section of line c) VSWR on each section of line

SolutionSolution: first examine the schematic diagram: first examine the schematic diagram

QWT MatchingQWT Matching

• SolutionSolution: take the following steps on S.C.: take the following steps on S.C.

L

B B

B

300 2001) Normalize load: 3 2

1002) Rotate CW on the 0.63 ( 4.4) circle to

intersect with the horizontal ( 0) midline where

4.4 0 (4.4)(100 ) 440

11 Note:

1 1

jz j

VSWR

x

z j z

z

1 B A

4.4 here!

Find length: (0.250 0.224) 0.026

VSWR

d z z



BB'

3) Find ( / 4) for a match with 100 line:

( 4) (440 )(100 ) 210

440 then re-normalize on the QWT to get 2.09

( / 4) 210

4) Rotate CW on the 0.36 ( 2.09) circle

halfway ( 4)

Z λ

Z λ/

zz

z

VSWR

λ/

C C

0 0

D

around the chart to input of the QWT

where 0.48 0 and (0.48)(210 ) 100.8

5) Now re-normalize on the 100 Ω line to get

1.0 at D (we have arrived at HOME for a match!)

z j Z

Z R

z VSWR


• Actual physical length of QWT depends on Actual physical length of QWT depends on operating frequency and phase velocity since operating frequency and phase velocity since =v=vpp/f/f

• For a microstrip QWT, Z(For a microstrip QWT, Z(/4) is determined by /4) is determined by its width a substrate of known its width a substrate of known rr and thickness and thickness (d), but must be within manufacturable limits(d), but must be within manufacturable limits

• As for Single Stub matching, QWT method As for Single Stub matching, QWT method works well for fixed frequency applications but works well for fixed frequency applications but its quality deteriorates for wideband its quality deteriorates for wideband applicationsapplications

Matching with Double Stub Matching with Double Stub TunerTuner

• Tuner design possible on the Smith ChartTuner design possible on the Smith Chart

• Tuner has one more degree of freedom Tuner has one more degree of freedom than the Single Stub, so more involvedthan the Single Stub, so more involved

• In practice, tuning for match usually done In practice, tuning for match usually done experimentally with a pre-fabricated deviceexperimentally with a pre-fabricated device

Variations of QWT Method for Variations of QWT Method for Wideband, Pulse & Digital Data Wideband, Pulse & Digital Data

Tx Tx DescriptiDescripti

on of on of MethodMethod

ElementsElements Real Real EstateEstate

(vs. QWT)(vs. QWT)

BandwidtBandwidth (vs. h (vs. QWT)QWT)

Shorter Shorter than than /4 /4 lengthslengths

2 2 /16 /16 transformerstransformers

Less (+)Less (+) Less (-)Less (-)

Lumped Lumped ElementsElements

LC sectionLC section Less (+)Less (+) Less (-)Less (-)

MultipleMultiple /4 /4

sectionssections

Match to Match to inter-inter-

mediate mediate imp’s.imp’s.

More (-)More (-) More (+)More (+)

Tapered Tapered sectionsection

1 continuous 1 continuous piecepiece

More (--)More (--) More (++)More (++)

Reference: Kraus & Fleisch, 5/e, pp. 155-166 (Bandwidth, Pulses & Transients).

ConclusionsConclusions

• The Smitch Chart serves as a graphical The Smitch Chart serves as a graphical method of T-line analysis and design to method of T-line analysis and design to help visualize aspects of the systemhelp visualize aspects of the system

• The Smith Chart represents complex The Smith Chart represents complex normalized impedances (or normalized impedances (or admittances) on curves within a polar admittances) on curves within a polar plot of plot of = 1 = 1

ConclusionsConclusions

• Moving toward the generator (or load) Moving toward the generator (or load) on the Smith Chart corresponds to on the Smith Chart corresponds to rotating on the constant rotating on the constant circle CW (or circle CW (or CCW)CCW)

• The Smith Chart may be used to find The Smith Chart may be used to find , , VSWR and ZVSWR and Zinin any position on a T-line any position on a T-line ahead of a complex load impedanceahead of a complex load impedance

ConclusionsConclusions• Fixed frequency T-line impedance Fixed frequency T-line impedance

matching methods whose design may matching methods whose design may be illustrated on the Smith Chart includebe illustrated on the Smith Chart include– Single & Double Stub TunerSingle & Double Stub Tuner– Quarter-wave Transformer (QWT)Quarter-wave Transformer (QWT)

• For wideband applications, a tapered For wideband applications, a tapered transformer provides more bandwidth at transformer provides more bandwidth at the expense of more space to the expense of more space to implementimplement

References and Other References and Other ResourcesResources

• Hayt & Buck, Hayt & Buck, Engineering Engineering ElectromagneticsElectromagnetics, 7/e, McGraw Hill: , 7/e, McGraw Hill: New York, 2006.New York, 2006.

• Kraus & Fleisch, Kraus & Fleisch, Electromagnetics Electromagnetics with Applicationswith Applications, 5/e, McGraw Hill: , 5/e, McGraw Hill: New York, 1999.New York, 1999.

Electromagnetics (ENGR 367) The Smith Chart: A Graphical Method for T-lines.

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