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Electromagnetics:

Electromagnetic Field Theory

Smith Charts

Lecture Outline

•Construction of the Smith Chart

•Admittance and impedance

•Circuit theory•Determining VSWR and • Impedance transformation

• Impedance matching

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Slide 3

Construction of the Smith Chart

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Polar Plot of Reflection Coefficient 

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The Smith chart is based on a polar plot of the voltage reflection coefficient .  The outer boundary corresponds to || = 1.  The reflection coefficient in any passive system must be|| ≤ 1.

je

radius on Smith chart

angle measured CCW from right side of chart

Normalized Impedance z

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All impedances are normalized.  This is usually done with respect to the characteristic impedance of the transmission line Z0.

0

Zz

Z

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Normalized Reflection Coefficient 

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The reflection coefficient  can be written in terms of normalized impedances.

0

0 0 0

00

0 0

1

1

L

L L

LL L

ZZZ Z Z Z z

ZZZ Z zZ Z

Derivation of Smith Chart:Solve for Normalized Load Impedance zL

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Solve the previous equation for load impedance to get

1

1

1 1

1

1

1 1

1

1

L

L

L L

L L

L L

L

L

z

z

z z

z z

z z

z

z

1

1Lz

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Derivation of Smith Chart:Real and imaginary parts

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The normalized load impedance zL and reflection coefficient  can be written in terms of real and imaginary parts.

L L L r i z r jx j

L

r iL L

r i

r iL L

r i

1

11

1

1

1

z

jr jx

j

jr jx

j

Substituting these into the load impedance equation yields

Derivation of Smith Chart:Solve for rL and xL

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Solve the previous equation for rL and xL by setting the real and imaginary parts equal.

r iL L

r i

r i r i

r i r i

2r r r i i r i

2 2r i

2 2r i r i i r i i

2 2r i

2 2r i i

2 2r i

2 2r i i

2 22 2r i r i

1

1

1 1

1 1

1 1 1 1

1

1

1

1 2

1

1 2

1 1

jr jx

j

j j

j j

j j

j j j j

j

j

2 2r i

L 2 2r i

iL 2 2

r i

1

1

2

1

r

x

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Derivation of Smith Chart:Rearrange equation for rL

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We rearrange the equation for rL so that it has the form of a circle.

2 2

2 2

2 22 2

222 2

222 2

2 2 2 2

2 2

2 2

can be factored

1

1

11

11 0

12 1 0

2 1 0

2 1 1 1 0

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r iL

r i

r ir i

L

irr i

L L L

irr r i

L L L

L r L r r L i i L

L r L r L i L

L rr i

L

r

r

r r r

r r r

r r r r

r r r r

r

r

10

1L

L

r

r

2 2

2

2 2

2

2 22

2 2

2 2 22

2 2

2

22

10

1 1 1

1

1 1 1

1 1

1 1 1

1

1 1 1

1

1 1

L L Lr i

L L L

L L Lr i

L L L

L LL Lr i

L L L

L L Lr i

L L L

Lr i

L L

r r r

r r r

r r r

r r r

r rr r

r r r

r r r

r r r

r

r r

Derivation of Smith Chart:Rearrange equation for xL

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Rearrange the equation for xL so that it has the form of a circle.

iL 2 2

r i

2 2 ir i

L

2 2r i i

Lswap terms

can be factored

22

r i 2L L

2

1

21

21 0

1 11 0

x

x

x

x x

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Derivation of Smith Chart:Two families of circles

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Constant Resistance Circles

2 2

2Lr i

L L

1

1 1

r

r r

Constant Reactance Circles

2 2

2

r iL L

1 11

x x

These have centers at These have centers at

Lr i

L

01

r

r

r iL

11

x

Radii Radii

L

1

1 r L

1

x

Derivation of Smith Chart:Putting it all together

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Lines of constant resistance

We ignore what is outside the || = 1 circle.

We don’t draw the constant || circles.

This is the Smith chart!

+

Lines of constant inductive reactance

Lines of constant capacitive reactance

=

Superposition

+

Lines of constant reflection coefficient

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Alternate Way of Visualizing the Smith Chart

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Lines of constant reactanceLines of constant resistance Reactance Regions

L

C

opencircuit

shortcircuit

3D Smith Chart

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The 3D Smith Chart unifies passive and active circuit design.

2D 3D

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Summary of Smith Chart

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Impedanceand

Admittance

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Admittance Coordinates

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We could have derived the Smith chart in terms of admittance.

You can make an admittance Smithchart by rotating the standardSmith chart by 180.

Impedance/AdmittanceConversion

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The Smith chart is just a plot of complex numbers. These could be admittance as well as impedance.

To determine admittance from impedance (or the other way around)…

1. Plot the impedance point on the Smith chart.2. Draw a circle centered on the Smith chart that passes through the point (i.e. constant VSWR).3. Draw a line from the impedance point, through the center, and to the other side of the circle.4. The intersection at the other side is the admittance.

impedance admittance

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Visualizing Impedance/Admittance Conversion

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Example #1 – Step 1Plot the impedance on the chart

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0.2 0.4z j

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Example #1 – Step 2Draw a constant VSWR circle

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0.2 0.4z j

Example #1 – Step 3Draw line through center of chart

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0.2 0.4z j

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Example #1 – Step 4Read off admittance

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0.2 0.4z j

1.0 2.0y j

Example #2 – Step 1Plot the impedance on the chart

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0.5 0.3z j

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Example #1 – Step 2Draw a constant VSWR circle

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0.5 0.3z j

Example #2 – Step 3Draw line through center of chart

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0.5 0.3z j

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Example #2 – Step 4Read off admittance

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1.0 2.0y j

0.5 0.3z j

DeterminingVSWR and 

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Determining VSWR

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1. Plot the normalized load impedance on the Smith chart.2. Draw a circuit centered on the Smith chart that intersections this point.3. The VSWR is read where the circle crosses the real axis on right side.

Example: 50 line connected to 75+j10 load impedance.

0

75 101.5 0.2

50LZ j

z jZ

impedance

VSWR = 1.55

1

VSWR

Example #1 –What is the VSWR?

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50

3.3157 nH

1.9894 pF

in

inin

0

0.4 0.

20 40

20 40

0 8

5

Z j

Z jz

Zj

VSWR 4.3

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Example #1 –What is the reflection coefficient?

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0.62

Impedance  Transformation

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Normalized Impedance Transformation Formula

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The impedance transformation formula was

0in 0

0

tan

tanL

L

Z jZZ Z

Z jZ

This can now be written in terms of the reflection coefficient .

00in 0 0

0 0

0 00 00 0

0 0 0

0.5 0.5cos sin

cos sin 0.5 0.5

j j j jLL

j j j jL Z

j jj j j jL LL L

j j j j jL L L L

Z e e Z e eZ jZZ Z Z

Z jZ Z e e Z e e

Z Z e Z Z eZ e Z e Z e Z eZ Z

Z e Z e Z e Z e Z Z e Z

0

0

20

0 0 20

0

11

11

j

jL

j jL

j jL

jL

Z e

Z Z e

Z Z e eZ Z

Z Z e e

Z Z e

Normalized the input impedance by dividing by Z0.

2

in 2

1

1

j

j

ez

e

Interpreting the Formula

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The normalized impedance transformation formula was

2

in 2

1

1

j

j

ez

e

Recognizing that  = ||ej, this equation can be written as

22

in 2 2

1 1

1 1

jj j

j j j

e e ez

e e e

Thus we see that traversing along the transmission line simply changes the phase of the reflection coefficient.

As we move away from the load and toward the source, we subtract phase from .  On the  Smith chart, we rotate clockwise (CW) around the constant VSWR circle by an amount 2l.  A complete rotation corresponds to /2.

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Impedance Transformationon the Smith chart

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1. Plot the normalized load impedance on the Smith chart.2. Move clockwise around the middle of the Smith chart as we move away from the

load (toward generator). One rotation is /2 in the transmission line.3. The final point is the input impedance of the line.

Example #2 – Impedance Transformation:Normalize the Parameters

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0 50 Z 50 25 LZ j

0.67

1 0.5 Lz j

0.67

in

1 0.5 tan 2 0.67tan1.299 0.485

1 tan 1 1 0.5 tan 2 0.67L

L

j jz jz j

jz j j

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Example #2 – Impedance Transformation: Plot load impedance

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0.67

1 0.5 Lz j

Example #2 – Impedance Transformation: Walk away from load 0.67

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0.67

1 0.5 Lz j

Since the Smith chart repeats every 0.5, traversing 0.67 is the same as traversing 0.17.

Here we start at 0.145 on the Smith chart.

We traverse around the chart to 0.145 + 0.17 = 0.315.

0.145

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Example #2 – Impedance Transformation: Determine input impedance

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0.67

1 0.5 Lz j

Reflection at the load will be the same regardless of the length of line.

Therefore the VSWR will the same.

The input impedance must lie on the same VSWR plane.

inZ

in 1.3 0.5z j

Example #2 – Impedance Transformation: Denormalize

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0.67

1 0.5 Lz j

To determine the actual input impedance, we denormalize.

inZ

in 0 in 50 1.3 0.5 65 25 Z Z z j j

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