Wavesand

Transmission Lines

Wang C. Ng

Traveling Waves

Load

Envelop of a Standing Wave

Waves in a transmission line

• Electrical energy is transmitted as waves in a transmission line.

• Waves travel from the generator to the load (incident wave).

• If the resistance of the load does not match the characteristic impedance of the transmission line, part of the energy will be reflected back toward the generator. This is called the reflected wave

Reflection coefficient

• The ratio of the amplitude of the incident

wave (v+ ) and the amplitude the reflective

wave (v-) is called the reflection coefficient:

v

v

Reflection coefficient

• The reflection coefficient can be determine from the load impedance and the characteristic impedance of the line:

0

0

ZZ

ZZ

L

L

Short-circuited Load

• ZL = 0

= -1

• v - = - v + at the load

• As a result, vL = v + + v - = 0

Load

Open-circuited Load

• ZL = = +1

• v - = v + at the load

• As a result, vL = v + + v - = 2 v +

Load

Resistive Load

• ZL = Z0

= 0

• v - = 0 at the load

• As a result, vL = v +

Traveling Waves

Load

Resistive Load

• ZL = 0.5 Z0

= - 1/3

• v - = -0.333 v + at the load

• As a result, vL = v + + v - = 0.667 v +

Composite Waves

Load

Resistive Load

• ZL = 2 Z0

= + 1/3

• v - = 0.333 v + at the load

• As a result, vL = v + + v - = 1.333 v +

Composite Waves

Load

Reactive Load (Inductive)

• ZL = j Z0

= + j1

• v - = v +90 at the load

• As a result, vL = v + + v - = (1 + j1) v +

= 1.414 v +45

Composite Waves

Load

Reactive Load (Capacitive)

• ZL = -j Z0

= - j1

• v - = v +-90 at the load

• As a result, vL = v + + v - = (1 - j1) v +

= 1.414 v +-45

Composite Waves

Load

Smith Chart

Transmission Line

Calculator

-j2

-j4

-j1

-j0.5

j0.5

j1

j4

j2

j0 0 0.5 1 2 4

ZL / Z0 = zL = 1 + j 2

0 0.5 1

0.7 45

= 0.5 + j 0.5

real

imaginary

||

-j2

-j 4

-j1

-j0.5

j0.5

j1

j4

j2

j 0 0 0.5 1 2 4

zL = 1 + j 2 0.7 45

||

||

re

im

-j2

-j4

-j1

-j0.5

j0.5

j1

j4

j2

j0 0 0.5 1 2 4

zL = 1 + j 2

0.7 45

45

0

135

90

180

225

270

315

zL 1 j 2

zL 1

zL 1

0.5 0.5i

0.707

arg 45deg

-j2

-j4

-j1

-j0.5

j0.5

j1

j4

j2

j0 0 0.5 1 2 4

zL = 0.5- j 0.5 0.45 -120

45

0

135

90

180

225

270

315

zL 0.5 j 0.5

zL 1

zL 1

0.2 0.4i

0.447

arg 116.565 deg

| |

0 0.5 1

-j2

-j4

-j1

-j0.5

j0.5

j1

j4

j2

j0

0 0.5 1 2 4

45

0

135

90

180

225

270

315

D C B E A

F

G

Re Im Mag AngleA 0 1 0 OpenB 2 0 1/3 0 Resistive = 2Z 0

C 1 0 0 0 Matched loadD 0.5 0 1/3 180° Resistive = Z 0/2E 0 0 1 180 Short F 0 1 1 90 Inductive = jZ 0

G 0 -1 1 -90° Capacitive = -jZ 0

zL Load