Prof. David R. Jackson Dept. of ECE

ECE 5317-6351 Microwave Engineering

Fall 2019

1

Notes 5 Smith Charts

Adapted from notes by Prof. Jeffery T. Williams

Recall: ( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( )( ) ( ) ( )

2 0 0

20 0

0 0

2

0 02

1 1

1 1

11 1 1

z z z L

z z z L

z L

z L

V z V e e V e z

V VI z e e e z Z Z

V z zeZ z Z Z I z e z

γ γ γ

γ γ γ

γ

γ

+ − + + −

+ + − + −

+

+

= + Γ = + Γ

= − Γ = − Γ

 + Γ + Γ = = =   − Γ − Γ   

Generalized reflection Coefficient: ( ) 2 zLz e γ+Γ = Γ

Generalized Reflection Coefficient

2

0

0

L L

L

Z Z Z Z

− Γ =

+

0,Z β ( )V z

( )I z

0z =z

LZ +

−

( )

( ) ( )

2

2L

z L

j z L

R I

z e

e e

z j z

γ

φ γ

+

+

Γ = Γ

= Γ

= Γ + Γ

Lossless transmission line (α = 0)

( ) ( )2Lj zLz e φ β+Γ = Γ

Generalized Reflection Coefficient (cont.)

{ }Re 0 1L LZ ≥ ⇒ Γ ≤

( ) ( ) ( ) ( )

0

0

0

0

L L L

L L

L L

L L

R jX Z R jX Z

R Z jX R Z jX

+ − Γ =

+ +

− + =

+ +

( ) ( )

2 2 2 0

2 2 0

1L LL L L

R Z X R Z X

− + ⇒ Γ = ≤

+ +

Proof:

3

Different forms for Γ(z) Magnitude property of Γ(z)

Complex Γ Plane ( )

( ) ( ) ( )

( )

( )

( )

2

2

2

2

L

L

R I

j z L

j z L

j d L

j d L

z

z j z

e

e

e

e

β

φ β

φ β

β

+

+

−

−

Γ = Γ

= Γ + Γ

= Γ

= Γ

= Γ

= Γ

Increasing d (moving towards

generator)

4

ReΓ

Im Γ

Lφ

LΓ

LΓ

2L dφ β−

Γ

Lossless line

z d= − d = distance from load

1

Note: Going λ/2 on the line

corresponds to going all the way around the Smith chart.

2 dβ

Clockwise movement!

( ) ( )( )0 1 1

z Z z Z

z  + Γ

=  − Γ 

( ) ( ) ( )( )0 1 1n

Z z z Z z

Z z  + Γ

≡ =  − Γ  Define

n n nZ R jX= +

Hence we have:

Z Chart

( ) ( )

1 1

R I n n

R I

j R jX

j  + Γ + Γ

+ =  − Γ + Γ 

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

5

Note: The z dependence is being

suppressed here.

The Z chart is the “usual” Smith chart.

Start with

1) Equation #1: 2 2

2 1 1 1

n R I

n n

R R R

    Γ − + Γ =   + +   

Equation for a circle in the Γ plane

,0 1 1

1

n

n

n

R R

R

  =  + 

= +

Center

Radius

Z Chart (cont.)

6

1

ΓR

ΓI

( ) 2 2

2 1 11R I n nX X

    Γ − + Γ − =   

   

Equation for a circle in the Γ plane

11,

1 n

n

X

X

  =  

 

=

Center

Radius

2) Equation #2:

7

1

ΓR

ΓI

Z Chart (cont.)

Short-hand version

8

Γ plane Γ plane

Rn = 1

Xn = 1

Xn = -1

Z Chart (cont.)

9

Open ckt. (Γ=1)

Imag. (reactive) impedance

Match pt. (Γ=0)

Real impedance

Short ckt. (Γ= −1)

Inductive (Xn > 0)

Capacitive (Xn < 0)

Γ plane Rn = 1

Xn = 1

Xn = -1

Γ plane

Z Chart (cont.)

Note: ( ) ( ) ( ) ( )0

11 1 1

z Y z

Z z Z z  − Γ

= =  + Γ 

( )( ) ( )( )0

1 1

z Y

z  + −Γ

=   − −Γ 

( ) ( ) ( )( )( )( ) ( ) ( )0 1 1n n n

zY z Y z G z jB z

Y z  + −Γ

⇒ ≡ = = +  − −Γ 

( )0 01 /Y Z≡

Admittance Calculations with the Z Chart

Define:

( ) ( )z z′Γ ≡ − Γ ( ) 1

1n Y z

′+ Γ =  ′− Γ 

Same mathematical form as for Zn:

Conclusion: The same Smith chart can be

used as an admittance calculator.

10

( ) 1 1n

Z z + Γ =  − Γ 

11

Short ckt. (Γ′ =1)

Imag. (reactive) admittance

Match pt. (Γ′ =0)

Real admittance

Open ckt. (Γ′ = −1)

Capacitive (Bn > 0)

Inductive (Bn < 0)

Γ′ planeBn = 1

Bn = -1

Gn = 1

Γ′ plane

Admittance Calculations with the Z Chart (cont.)

Impedance or Admittance Calculations with the Z Chart

12

The Smith chart can be used for either impedance or

admittance calculations, as long as we are consistent.

The complex plane is either the Γ plane or the Γ′ plane.

Normalized impedance or admittance coordinates

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

( ) ( )( )( )( ) ( ) ( ) 1 1n n n

z Y z G z jB z

z  + −Γ

= = +  − −Γ 

Y Chart

( ) ( )( )( )( ) ( ) ( ) 1 1n n n

z Z z R z jX z

z  + Γ

= = +  − Γ 

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

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

Rn = 1 circle, rotated 180o, becomes Gn = 1 circle. Xn = 1 circle, rotated 180o, becomes Bn = 1 circle.

( ) ( ) ( ) ( )

n n

n n

R z G z

X z B z

→

→ ( ) ( )z zΓ → −Γ (rotation of 180o)

Examples:

Y Chart (cont.)

14

Open ckt.

Match pt.

Gn = 0

Short ckt.

Inductive (Bn < 0)

Capacitive (Bn > 0)

Gn = 1

Bn = +1

Bn = -1

Bn = 0

Γ planeΓ plane

The Y chart is the “mirror image” of the usual Smith chart.

Short-hand version

Γ plane

15

Gn = 1

Bn = -1

Bn = 1

Y Chart (cont.)

All Four Possibilities for Smith Charts

16

Z chart, used for admittance

Γ′ plane

Z chart, used for impedance

Γ plane

The first two are the most common.

The third is sometimes convenient.

The fourth is almost never used.

1 2

Γ′ plane

4 Y chart, used for impedance

Γ plane

3 Y chart, used for admittance

ZY Chart

17

This is convenient for doing matching problems that involve both series and shunt elements (done later).

Short-hand version

Γ plane

Gn = 1 Rn = 1

Xn = 1

Xn = -1

Bn = -1

Bn = 1

Inductive

Capacitive

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

the Smith chart (Rnmax).

Standing Wave Ratio

Proof: 1

SWR 1

L

L

+ Γ =

− Γ

( ) ( )

2

2

2

2

1 1

1 1 1 1

L

L

n

j z L

j z L

j j z L

j j z L

z Z

z

e e e e e e

β

β

φ β

φ β

+

+

+

+

+ Γ =

− Γ

+ Γ =

− Γ

+ Γ =

− Γ

18

max 12 0 1

L L n

L

z Rφ β + Γ

+ = ⇒ = − Γ

LΓ

max n nR R=

2 0L zφ β+ =2L zφ β+

Γ plane

( )zΓ

At this link:

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

Download the following zip file: smith_v191.zip

Extract the following files:

smith.exe mith.hlp smith.pdf

This is the application file

Electronic Smith Chart

19

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

0 50 100 50L

Z Z j

= Ω = + Ω

, 0

2 1LL n ZZ j Z

= = +

( )

/ 4 0.4 0.2

/ 4 20 10

g

n

g

d Z j

Z j

λ

λ

=