ELECTROMAGNETICS AND APPLICATIONS

Lecture 12Smith Chart & VSWR

Luca Daniel

Example: Open Circuit Load, ZL = :

In general: -j < Z(-D) < +j(can get ANY desired capacitance or inductance at a SINGLE )

capacitor Co

when z = 0, -/2,…

oL C1jZ j 1 j CD for kD 1

kD LC D

zz =-D 0

z0- 34

2

4

Im{Z}Open circuit

Short circuit

L oo

o L

Z jZ tan( kD)Z( D) Z

Z jZ tan( kD)

ojZ cotan(kD)

0 when z = -/4, -3/4,…

short circuit every half wavelength

open circuit every half wavelength

Examples of Z(z) transformations

Examples of Z(z) transformations (cont)

Example: half-wave and quarter-wave lines:

2tan =4

L oo

o L

Z jZ tan( kD)Z( D) Z

Z jZ tan( kD)

Zo

0 z

?

o

2

L

Z if D / 4

Z

L2Z if D , ,etc. tan 0

2 2

ZL

Example: quarter-wave with inductive load:

D

Zo

0 z/4

? LoZ( D) o o

2 2

L 0 0

Z Z 1Z j L j C

o

00 2

LC

Z

Alternative approach to finding Z(z)

V(z)

I(z)

ZL

zZ(z)

Zo,

0

+-

z = - D

(1) Normalize Load: LLn

0

ZZ

Z

(2) Compute reflection coef: LnL

Ln

Z 1Z 1

(3) “Rotate” reflection coef.j2k( D)

L( D) e (toward generator clockwise)

or “find” it on the plane (Smith Chart)

(4) Compute Normalized Line Impedance

no

Z( D) 1 ( D)Z ( D)Z 1 ( D)

or “read” it from Smith Chart

(5) “Un-Normalize” Line Impedance

0 nZ( D) Z Z ( D)

(D= /2 full rotation)

toward z=-D

= j

= -1

= -j

= 1

Re{Zn}=1Re{Zn} = 0

Im{Zn}=0resistor

Im{Zn} = -jcapacitor

= 0Zn = open

Im{Zn}=+jinductor

Re{Zn}=3

Im{}

Re{}

Zn = 0 short

Zn = 1Z =Z0

Voltage Standing Wave Ratio (VSWR)

V(z) ZL

z

Zo,

0

+-

toward z=-D

Im{}

Re{}

jkz jkzV(z) V e V e V 1 (z)

V(z)

z

maxV(z)

minV(z)

0-DL1 (z 0) 1

-1

1 (z)

1 (z)

1 (z)

L

L

1

1

max

min

V(z)VSWR

V(z)

V 1 (z)