Communications and Computers Engineering Department The Higher Institute of Engineering- El-Shorouk City

Electromagnetic Fields (ECE 271)

The Academic year 2022/2023- First Semester
Lecture (4)
Third year Electronics and Communications Engineering
Program 1
 Terminated Lossless Transmission Line
-The total voltage on the line is
− j z j z
V ( z ) = V1 e + V2 e
-The total current on the line is
V1 − j z V2 j z
I ( z) = e − e
ZO ZO
-The total voltage and current at the load are related by the load impedance
(Zr) at z=0 V (0) V1 + V2
Zr = = ZO
I (0) V1 − V2
Zr − Zo
V2 = V1
Zr + Zo
-The amplitude of the reflected voltage wave normalized by the amplitude
of the incident voltage wave is defined as the voltage reflection coefficient 
− j l
V2 Z r − Z o V2 e −2 j  l
 (0) = = &  (l ) = j l
=  (0)e & -1   (0)  1
V1 Z r + Z o V1 e 2
 Terminated Lossless Transmission Line
➢ The total voltage and current waves on the line can be written as
− j z j z
V ( z ) = V1 (e + e )
V1 − j  z j z
I ( z) = (e − e )
Zo
➢ It is seen that the voltage and current on the line consists of a
superposition of an incident and a reflected wave, such waves are
called Standing Waves.

➢ At  = 0 , there is no reflected wave (the load impedance Zr must be

equal to the characteristic impedance Zo). And the line is called flat.

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 Terminated Lossless Transmission Line
- When the load is mismatched,
− j z j z
V ( z ) = V1 (e + e )
2 j z
V ( z ) = V1 1 +  e
−2 j  l
at a distance z = -l , V ( −l ) = V1 1 +  e
j ( − 2  l )
V (−l ) = V1 1 +  e
where,  is the phase of the reflection coefficient.
the voltage magnitude oscillates with position along the line

Vmax = V1 (1 +  ) The distance between successive maxima (or minima) is l = .
2

Vmin = V1 (1 −  ) The distance between a maximum and minimum is l = .
4
− the voltage standing wave ratio (VSWR) is a measure of mismatch of the line and can be expressed as:
Vmax 1 + 
VSWR = =
Vmin 1 − 
- VSWR is real number, lies in the range of 1  VSWR   4
 Special Cases of Lossless terminated Lines
❑Case 1: Short Circuit case: Zr=0

Zr − Zo 0 − Zo
= = = −1
Zr + Zo 0 + Zo
1+  Voltage
VSWR = =
1− 

The voltage and current on the line can be obtained as:

− j z j z − j z j z
V ( z ) = V1 (e + e ) = V1 (e −e ) Current
V1 − j  z j z V − j z j z
I ( z) = (e −  e ) = 1 (e +e )
Zo Zo
2j j l − j l
V (−l ) = V1 (e − e ) = 2 jV1 sin(  l )
2j
Impedance
2 V1 j l − j l V
I (−l ) = (e + e ) = 2 1 cos(  l )
2 Zo Zo
input impedance: Z input = jZ o tan(  l ) 5
 Special Cases of Lossless terminated Lines
❑Case 2: Open Circuit case: Zr=∞
Zr − Zo 1 − Zo Zr
= = =1
Zr + Zo 1 + Zo Zr
1+ 
VSWR = =
1−  Voltage
The voltage and current on the line can be obtained as:
− j z j z − j z j z
V ( z ) = V1 (e + e ) = V1 (e +e )
V1 − j  z j z V1 − j  z j z
I ( z) = (e − e ) = (e −e ) Current
Zo Zo
2 j l − j l
V (−l ) = V1 (e + e ) = 2V1 cos(  l )
2
2 j V1 j l − j l V
I (−l ) = (e − e ) = 2 j 1 sin(  l )
2 j Zo Zo Impedance
input impedance: Z input = − jZ o cot(  l )
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Special Cases of Lossless terminated Lines
Z r + jZ o tan(  l )
Z input = Z o
Z o + jZ r tan(  l )

➢ For a transmission line, l=
2 , we get Zinput= Zr .

➢A half-wavelength line (or any multiple of 2 ) does not transform the

load impedance, regardless of its characteristic impedance.

n Zo2
➢For a transmission line with l=,we get Zinput .
, for n=1,3,5,... =
4 Zr
Such transmission lines are known as quarter-wave transmission lines and
can transform the load impedance in an inverse manner depending on its
characteristic impedance.
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Example:
Calculate the reflection coefficient of lossless T.L with characteristic
impedance Zo=50 Ω for the following load impedance
(a) Zr=25 Ω (b) Zr=75 Ω (c) Zr=30-j40 Ω

Solution:
Z r − Z o 25 − 50
(a)  = = = −0.33
Z r + Z o 25 + 50
Z r − Z o 75 − 50
(b)  = = = 0.2
Z r + Z o 75 + 50
Z r − Z o 30 − j 40 − 50 −20 − j 40
(c)  = = = = − j 0.5
Z r + Z o 30 − j 40 + 50 80 − j 40

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Example:
A 50-Ω lossless transmission line is to be matched to a resistive load
impedance with Zr=100Ω via a quarter-wave section as shown,
thereby eliminating reflections along the feedline. Find the
characteristic impedance of the quarter-wave transformer.

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Solution:
• To eliminate reflections at terminal AA’, the input impedance Zin looking
into the quarter-wave line should be equal to Z01 (the characteristic
impedance of the feedline). Thus, Zin = 50Ω .

Z 022
Z in =
Zr
Z 02 = 50  100 = 70.7
• Since the lines are lossless, all the incident power will end up getting
transferred into the load Zr .

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 Example:
A source with 50  source impedance drives a 50  transmission line
that is 1/8 of wavelength long, terminated in a load Zr = 50 – j25 .
Calculate:
(i) The reflection coefficient
(ii) VSWR
(iii)The input impedance seen by the source.

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SOLUTION:
(i) The reflection coefficient,
Zr − Z0
=
Zr + Z0

=
( 50 − j 25) − 50 = 0.242 − 76o
( 50 − j 25) + 50

(ii) VSWR 1+ 
VSWR = = 1.64
1− 

(iii) The input impedance seen by the source, Zin

2   
 = =  tan =1
4
 8 4 Zr +
Z in = Z 0
jZ 0 tan 
Z 0 + jZ r tan 
50 − j 25 + j 50
= 50
50 + j 50 + 25
= 30.8 − j 3.8
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Power Reflection Coefficient and Transmission
coefficient
➢Power reflection coefficient = 
2

➢Power Transmission coefficient = 1- 

2

Example:
If the power reflection coefficient is -20dB, find the voltage reflection
coefficient and Power transmission coefficient.
Solution: 10 log  = −20dB
2

20
−
 = 10 = 0.01
2
10

 = 0.1
Power Transmission coefficient = 1 −  = 0.99
2
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