Department of Electrical Engineering

School of Electrical Engineering and Computer Science (SEECS)

National University of Sciences & Technology (NUST)

Fall 2015

EE342 Microwave Engineering

Instructor : Dr. M. Umar Khanumar.khan@seecs.edu.pk

Transmission Line Equation

© Umar, 2015.

Previous Lecture Solution of Transmission line equation

Propagation constant, characteristic impedance

Lossless transmission line

This time Distortionless line

Input Impedance, SWR, Power

EE342 Microwave Engineering 2

© Umar, 2015.

Transmission-Line Equation

Bridges the gap between circuit theory and field analysis

Important for analysis of microwave circuits and devices

EE342 Microwave Engineering 3

© Umar, 2015.

Transmission-Line Equation

By applying KVL and KCL, we get two second order differential equations𝜕2𝑉𝑠𝜕𝑧2

− 𝛾2𝑉𝑠 = 0,𝜕2𝐼𝑠𝜕𝑧2

− 𝛾2𝐼𝑠 = 0

Where

𝛾 = 𝛼 + 𝑗𝛽 = (𝑅 + 𝑗𝜔𝐿)(𝐺 + 𝑗𝜔𝐶) is the propagation constant

EE342 Microwave Engineering 4

© Umar, 2015.

Transmission-Line Equation

The solution of the differential equation gives𝑉 𝑧 = 𝑉𝑜

+𝑒−𝛾𝑧 + 𝑉𝑜−𝑒𝛾𝑧

and

I 𝑧 = 𝐼𝑜+𝑒−𝛾𝑧 + 𝐼𝑜

−𝑒𝛾𝑧

The characteristic impedance is the ratio of positively traveling voltage wave to the current wave at any point on the line

𝑍𝑜 =𝑅 + 𝑗𝜔𝐿

𝐺 + 𝑗𝜔𝐶

EE342 Microwave Engineering 5

© Umar, 2015.

Lossless Line

For a lossless line, R = G = 0

Under such conditions, attenuation constant is zero

The characteristic impedance is real

𝛼 = 0 , 𝛽 = 𝜔 𝐿𝐶

𝑍𝑜 =𝐿

𝐶

EE342 Microwave Engineering 6

© Umar, 2015.

Distortionless Line

The line whose attenuation constant is not a function of frequency and phase constant is a linear function of frequency

The conditions for distortionless transmission line is:𝑅

𝐿=𝐺

𝐶 For such line, the propagation constant is :

𝛼 = 𝑅𝐺 , 𝛽 = 𝜔 𝐿𝐶

The characteristic impedance is :

𝑍𝑜 =𝐿

𝐶

EE342 Microwave Engineering 7

© Umar, 2015.

Summary

EE342 Microwave Engineering 8

© Umar, 2015.

Example

A transmission line operating at 500 MHz has 𝑍𝑜 = 80Ω, α =

0.04𝑁𝑝

𝑚, 𝛽 = 1.5

𝑟𝑎𝑑

𝑚. Find the line parameters R, L, C, G.

EE342 Microwave Engineering 9

© Umar, 2015.

Input Impedance, Standing Wave Ratio, Power Consider a transmission line connected to a load. The line extend

from z=0 at the generator to z= l at the load.

EE342 Microwave Engineering 10

© Umar, 2015.

Input Impedance, Standing Wave Ratio, Power The voltage and current waves on the line are :

𝑉 𝑧 = 𝑉𝑜+𝑒−𝛾𝑧 + 𝑉𝑜

−𝑒𝛾𝑧

I 𝑧 =𝑉𝑜+

𝑍𝑜𝑒−𝛾𝑧 −

𝑉𝑜−

𝑍𝑜𝑒𝛾𝑧

The conditions at the input and output are:𝑉 𝑧 = 0 = 𝑉𝑜 , 𝐼 𝑧 = 0 = 𝐼𝑜

𝑉 𝑧 = 𝑙 = 𝑉𝐿 , 𝐼 𝑧 = 𝑙 = 𝐼𝐿

EE342 Microwave Engineering 11

© Umar, 2015.

Input Impedance, Standing Wave Ratio, Power Input impedance at any point on the line is :

𝑍𝑖𝑛 =𝑉(𝑧)

𝐼(𝑧)

At the generator, the equation is :

𝑍𝑖𝑛 =𝑍𝑜(𝑉𝑜

+ + 𝑉𝑜−)

𝑉𝑜+ − 𝑉𝑜

−

Substituting for the load end and simplification

𝑍𝑖𝑛 = 𝑍𝑜𝑍𝐿 + 𝑍𝑜tanh(𝛾𝑙)

𝑍𝑜 + 𝑍𝐿tanh(𝛾𝑙)

EE342 Microwave Engineering 12

© Umar, 2015.

Input Impedance, Standing Wave Ratio, Power For a lossless line, the input impedance is :

𝑍𝑖𝑛 = 𝑍𝑜𝑍𝐿 + 𝑗𝑍𝑜tan(𝛽𝑙)

𝑍𝑜 + 𝑗𝑍𝐿tan(𝛽𝑙)

The voltage reflection coefficient (Γ) is the ratio of voltage reflection wave to the incident wave. At the load, it is :

Γ𝐿 =𝑉𝑜−𝑒𝛾𝑙

𝑉𝑜+𝑒−𝛾𝑙

EE342 Microwave Engineering 13

© Umar, 2015.

Input Impedance, Standing Wave Ratio, Power The voltage reflection coefficient (Γ) at any point on the line is given

as:

Γ(𝑧) =𝑉𝑜−

𝑉𝑜+ 𝑒

2𝛾𝑧

The standing wave ratio (SWR) is defined as:

𝑆𝑊𝑅 =1 + Γ𝐿1 − Γ𝐿

EE342 Microwave Engineering 14

© Umar, 2015.

Input Impedance, Standing Wave Ratio, Power The average input power at a distance ‘l’ from the load is

𝑃𝑎𝑣𝑔 =1

2𝑅𝑒 𝑉(𝑙)𝐼∗(𝑙)

Solving the above equation, we get power in terms of reflection coefficient at the load.

𝑃𝑎𝑣𝑔 =𝑉𝑜+ 2

2𝑍𝑜1 − Γ𝐿

EE342 Microwave Engineering 15

© Umar, 2015.

Special Case

Shorted Line If line is shorted , 𝑍𝐿 = 0

Open-Circuit Line If line is shorted , 𝑍𝐿 = ∞

Matched Line If line is shorted , 𝑍𝐿 = 𝑍0

EE342 Microwave Engineering 16