Prof. David R. JacksonDept. of ECE

Notes 2

ECE 5317-6351 Microwave Engineering

Fall 2011

Smith Charts

1

Recall,

0

0

20

20

2

2

00

0 0

1

1

1

1

1

1

1

1

L

L

L

L

V e

Ve

V V e e

VI e e

Z

V eZ

Z

ZZI e

Generalized reflection Coefficient: 2L e

Generalized Reflection Coefficient I(-l )

V(-l )+

l

ZL-

0,Z

2

2

2LL

I

L

j

R

e e

j

e

Lossless transmission line ( = 0)

2LjL e

0

0

LL

L

Z Z

Z Z

Generalized Reflection Coefficient (cont.)

Re 0

1

L

L

Z

For

0

0

0

0

L LL

L L

L L

L L

R jX Z

R jX Z

R Z jX

R Z jX

2 22 0

2 20

L LL

L L

R Z X

R Z X

Proof:

3

Complex Plane

2

2L

R I

jL

jL

j

e

e

Increasing l (toward generator)

l

Re

Im Decreasing l (toward load)

L

L

L

2L

Lossless line

4

0

1

1Z Z

0

1

1n

ZZ

Z

Define;n n n R IZ R jX j

Substitute into above expression for Zn(-l ):

Impedance (Z) Chart

1

1R I

Rn n

I

j

jR jX

Next, multiply both sides by the RHS denominator term and equate real and imaginary parts. Then solve the resulting equations for R and I in terms of Rn and Xn. This gives two equations. 5

1) Equation #1:2

2 1

1 1n

R In n

R

R R

Equation for a circle in the plane

,01

1

1

n

n

n

R

R

R

center

radius

Impedance (Z) Chart (cont.)

6

1

0nR 1nR

1nR 1 nR

R

I

2 2

2 1 11R I

n nX X

Equation for a circle in the plane:

11,

1n

n

X

X

center

radius

Impedance (Z) Chart (cont.)2) Equation #2:

7

1

0nX

1 nX

1nX 0 1nX

0 1nX

1nX

1nX

R

I

Impedance (Z) Chart (cont.)

Short-hand version

Rn = 1

Xn = 1

Xn = -1

8

Open ckt. (1

Imag. (reactive)impedance

Match pt. (0

Realimpedance

Short ckt. ( 1

Inductive (Xn > 0)

Capacitive (Xn < 0)

Impedance (Z) Chart (cont.)

planeRn = 1

Xn = 1

Xn = -1

9

Note:

0

11 1

1Y

Z Z

0

1

1Y

0

1

1n n nG jBY

YY

00

1Y

Z

Admittance (Y) Calculations

Define: 1

1nY

Same mathematical form as for Zn:

Conclusion: The same Smith chart can be used as an admittance calculator.

10 1

1nZ

Short ckt. ( 1

Imag. (reactive)admittance

Match pt. ( 0

Realadmittance

Open ckt. ( 1

Capacitive (Bn > 0)

Inductive (Bn < 0)

Admittance (Y) Calculations (cont.)

planeBn = 1

Bn = -1

Gn = 1

11

Impedance or Admittance (Z or Y) Calculations

The Smith chart can be used for either impedance or admittance calculations, as long as we are consistent.

12

As an alternative, we can continue to use the original plane, and add admittance curves to the chart.

1

1n n nG jBY

Admittance (Y) Chart

1

1n n nR jXZ

Compare with previous Smith chart derivation, which started with this equation:

If (Rn Xn) = (a, b) is some point on the Smith chart corresponding to = 0,Then (Gn Bn) = (a, b) corresponds to a point located at = - 0 (180o rotation).

Side note: A 180o rotation on a Smith chart makes a normalized impedance become its reciprocal. 13

Þ Rn = a circle, rotated 180o, becomes Gn = a circle. and Xn = b circle, rotated 180o, becomes Bn = b circle.

Open ckt.

Match pt.

Gn = 0

Short ckt.

Inductive (Bn < 0)

Capacitive (Bn > 0)

Gn = 1

Bn = +1

Bn = -1

Bn = 0

Admittance (Y) Chart (cont.)

plane

14

Short-hand version

Gn = 1

Bn = -1

Bn = 1

plane

15

Admittance (Y) Chart (cont.)

Impedance and Admittance (ZY) Chart

Short-hand version

Gn = 1 Rn = 1

Xn = 1

Xn = -1

Bn = -1

Bn = 1

plane

16

The SWR is given by the value of Rn on the positive real axis of the Smith chart.

Standing Wave Ratio

L

Proof:1

1L

L

SWR

2

2

2

2

1

1

1

1

1

1

L

L

n

jL

jL

j jL

j jL

Z

e

e

e e

e e

17

max 1

1L

nL

R

maxn nR R

At this link

http://www.sss-mag.com/topten5.html

Download the following .zip file

smith_v191.zip

Extract the following files

smith.exe smith.hlp smith.pdf

This is the application file

Electronic Smith Chart

18