EE451_Lec8_ Solving the four fundamental EM equations transmission_lines.pdf

Contributions
Solving the four fundamental EM equations
Speed of light
Light is electromagnetic in nature
Other contributions to EE like mathematical forms of Krichoff’s laws, theory of colors etc

1. ةبيط ةعماج
EE 451 –Electromagnetic Waves
Lecture 8: Transmission Line Theory
Omar Siddiqui
Department of Electrical Engineering
College of Engineering
Taibah University
Madinah
Email:[email protected]

2. College of Engineering, Taibah UniversityEE451 Electromagnetics
Wave Propagation in different Media
Wave Propagation in Media
Guided Media Unguided Media
Propagation takes place by waves guided
by physically bounded path
Propagation takes place by electromagnetic fields
set up in a physically unrestricted medium
Examples: Transmission lines, optical fiberExamples: Free space, dielectric media
Different transmission lines

3. College of Engineering, Taibah UniversityEE451 Electromagnetics
Practical Examples of Wave Propagation in different media
Coaxial cables used in cell
towers/satellite systems
Microstrip lines in Planar circuits
Optical Fiber

4. Wavelength Comparison between low frequency and high frequency
-Voltage and current also travel in the form of waves but its not obvious when frequency is low
-For power lines, the traveling wave has a wavelength of 5000 km
-For a microwave transmission line operating at 1 GHz, the wavelength is 30 cm
-Compare the two transmission lines at a wavelength and a smaller distance of 100 cm
V
o
60 Hz
⇒�=
�
�
=5000��
Power transmission line (60 Hz)
MW transmission line (1 GHz)
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
5000km
V
o
-
+-
+
-V
o
V
o
10cm
V
o
-
+-
V
o
-
+
Voltage distribution over distance
V
o
10cm
V
o
-
+
-
V
o
-
+
V
o
1 GHz30cm
V
o
-
+-
+
-V
o
⇒�=
�
�
=30��

5. College of Engineering, Taibah UniversityEE451 Electromagnetics
Lumped element model of an unguided medium is a transmission line analog
Lossless LC Model)(zI
??????Δ�
??????(�)??????Δ�zˆ
መ�H
 H
 H
 zˆ
ො�
??????Δ�
??????Δ�
??????Δ�??????Δ�
Free Space Propagation
Ƹ�
The propagation in the unguided medium (free space) can be modeled by the lumped element
equivalent if following parameters are assumed:
??????=�,??????=??????
Where L and C are called the distributed inductance (H/m) and capacitance (F/m) of the lumped
model. The voltages and currents can be obtained by integrating:
dx
-
+
??????=−න
�
??????.��ො�
??????�=න
�
�.��
??????
ො�
Both transmission line and the unguided medium have same equivalent circuit model. Hence
the electric fields and magnetic fields equations can be converted to voltages and currents
�=�??????=????????????

6. Voltages and Currents
College of Engineering, Taibah UniversityEE451 Electromagnetics
Equivalent Voltage and Current equations in transmission line
Transmission line LC Model)(zI
??????Δ�
??????(�)??????Δ�zˆ
መ�H
 H
 H
 zˆ
�(�)=ො�
??????
??????0
+
??????
�
−��??????
??????(�)=ො�??????
??????0
+
�
−��??????
Fields in unguided medium space
�=??????�??????
??????=
??????
�0
+
??????
�0
+=
??????
??????
??????Δ�
??????Δ�
??????Δ�??????Δ�zj
eVzV
−+
=
0
)( zj
e
Z
V
zI
−
+
=
0
0
)(
�=??????????????????
�
0=
??????
0
+
�
0
+
=
??????
??????
Free Space Propagation
dx
-
+
??????
Wave impedance
Characteristic impedance

7. College of Engineering, Taibah UniversityEE451 Electromagnetics
Full Solution of Voltages and Currents on a Transmission Line
Transmission line lossless LC Model
��=
??????
0
+
�
0
�
−��??????
−
??????
0
+
�
0
�
+��??????
�=??????????????????
�
0=
??????
0
+
�
0
+
=
??????
??????)(zI
??????Δ�
??????(�)??????Δ�
??????Δ�
??????Δ�??????Δ�
dx
-
+
Characteristic impedance
??????(�)=??????
0
+
�
−�??????
+??????
0
−
�
+�??????
??????(�)=??????
0
+
�
−��??????
+??????
0
−
�
+��??????
�=(??????+�????????????)(�+�????????????)�=�+��0=
Transmission line lossy RLC Model
??????Δ�??????Δ�
�Δ�??????Δ�??????(�)
-
+)(zI
��=
??????
0
+
�
0
�
−��??????
−
??????
0
+
�
0
�
+��??????
??????
0
−
�
+��????????????
0
+
�
−��?????? ??????
0
−
�
+��??????
??????
0
+
�
−��??????
Where R=resistance/m and G=conductance/mCjG
LjR
Z


+
+
=
0
Characteristic impedance
Two solutions of traveling voltages exist on a transmission line. The waves traveling in
forward direction and the reverse direction. Here we consider lossless and lossy models
Where is the phase constant. For lossless line
Where gis the propagation constant

8. Voltage and Current Equations for Lossy and Lossless Lines in time domain
Lossy ModelLossless Model
??????(�)=??????
0
+
�
−�??????
+??????
0
−
�
+�??????
??????(�)=??????
0
+
�
−��??????
+??????
0
−
�
+��??????
⇒??????��
�????????????
=??????
0
+
�
−��??????
�
�????????????
+??????
0
−
�
+��??????
�
�????????????
College of Engineering, Taibah UniversityEE451 Electromagnetics
�=?????????????????? �=(??????+�????????????)(�+�????????????)�=�+��
�=0
??????(�,�)=??????���??????(�)�
�????????????
Time domain:
⇒??????��
�????????????
=??????
0
+
�
�(????????????−�??????)
+??????
0
−
�
�(????????????+��??????)
⇒??????�,�=??????
0
+
���(??????�−��)+??????
0
−
(??????�+��)
??????(�,�)=??????���??????(�)�
�????????????
Time domain:
⇒??????��
�????????????
=??????
0
+
�
−��??????
�
�????????????
+??????
0
−
�
+��??????
�
�????????????
⇒??????��
�????????????
=??????
0
+
�
−�??????
�
�(????????????−�??????)
+??????
0
−
�
−�??????
�
�(????????????+��??????)
⇒??????�,�=??????
0
+
�
−�??????
���(??????�−��)+??????
0
−
�
−�??????
(??????�+��)
⇒??????��
�????????????
=??????
0
+
�
−�??????
�
−��??????
�
�????????????
+??????
0
−
�
−�??????
�
+��??????
�
�????????????

9. -The voltage equation ignoring the reflection on a transmission line is given by:
-The phasevelocitycanbecalculatedbyconsideringapointontheconstantphase
-Letusconsiderthe voltage lot for two different times.Notethattheblue dot has
moved from 0m to 0.1m in 0.333ns
-Velocity calculations:
Wavelength and Velocity
�(�,�)=??????
0
+
cos(??????�−��)
t = 0ns
t = 0.33ns
Taking derivative of both sides:⇒??????��−���=0
⇒�=
��
��
=
??????
�
⇒�=
??????
??????????????????
[??????�−�(�+�)]−[??????�−��]=−2??????
⇒−��=−2??????
⇒�=
2??????
??????/�
=
2??????�
2??????�
l
College of Engineering, Taibah UniversityEE451 Electromagnetics
Phase Velocity
??????�−��=��������Since the phase is constant:
Since dz/dt is the definition of velocity:
Since :�=?????????????????? ⇒�=
1
????????????
Wavelength
The angle between the two consecutive peaks of one of
the plots should be equal to 2p:
⇒�=
2??????
�
Also�=
??????
�
⇒�=
??????
�
⇒�=
�
�

10. What causes the reflections?)(zV )(zI
Z
Lzj
eV
−+
0 zj
eV
+−
0
??????(�)=??????
0
+
�
−��??????
+??????
0
−
�
+��??????�(�)=
??????
0
+
�
0
�
−��??????
−
??????
0
−
�
0
�
+��??????
Reflections are due to impedance mismatch (Z
L≠ Z
o)0
0
ZZ
ZZ
L
L
+
−
=
Conditions for no reflection0
ZZIf
L
=
⇒Γ=0zj
eVzV
−+
=
0
)( zj
e
Z
V
zI
−
+
=
0
0
)( Then
1.
2. (Theoretical)The line is infinite, so that the reflections do not come back
⇒??????�=??????
0
+
�
−��??????
+Γ??????
0
+( )
zjzj
ee
Z
V
zI
 +−
+
−=
0
0
)(
When Z
L≠ Z
o Reflection CoefficientΓ=
??????
0
−
??????
0
+
⇒??????
0
−
=Γ??????
0
+
⇒??????�=??????
0
+
(�
−��??????
+Γ�
+��??????
)
�
College of Engineering, Taibah UniversityEE451 Electromagnetics

11. Calculation of Reflection Coefficient ()zjzj
eVeVzV
 +−−+
+=
00)( zjzj
e
Z
V
e
Z
V
zI
 +
−
−
+
−=
0
0
0
0
)(
z = 0
I
L
??????(0)=??????
0
+
+??????
0
−
and�(0)=
??????
0
+
�
0
−
??????
0
−
�
0
⇒�
??????=
??????(0)
�(0)
⇒??????
0
+
(�
??????−�
0)=??????
0
−
(�
??????+�
0)
⇒
??????
0
−
??????
0
+
=
�
??????−�
0
�
??????+�
0
??????(�)
�(�)
-
+
V
L
�
0,�
??????
0
−
�
+��????????????
0
+
�
−��??????
Γ=
??????
0
−
??????
0
+The Reflection coefficient is defined as:
We know the load impedance. So we apply the boundary conditions at z =0, Z
in= Z
L
⇒�
??????=
??????
0
+
+??????
0
−
(??????
0
+
−??????
0
−
)/�
0
⇒�
????????????
0
+
−�
????????????
0
−
=�
0??????
0
+
+�
0??????
0
−
⇒Γ=
�
??????−�
0
�
??????+�
0
College of Engineering, Taibah UniversityEE451 Electromagnetics
Z
L
The voltage and current equations are given by:
�
��0
where

12. College of Engineering, Taibah University, MadinahEE453 MW Engineering
Demonstration of Traveling waves on an infinite TL
Lossy Model
Lossless Model)cos(),(
0 zteVtzv
z


−=
−+ )cos(),(
0 ztVtzv −=
+
Let f = 3 GHz l= 3x10
8
/3x10
9
= 0.1m
0 0.1 0.2 0.3 0.40=
= 2p/l= 2p/0.1= 62.8
0 0.1 0.2 0.3 0.4mNp/1=
Effect of Losses

13. Lossy Model)cos(),(
0 zteVtzv
z


−=
−+
Let f = 3 GHz l= 3x10
8
/3x10
9
= 0.1m
0 0.1 0.2 0.3 0.4
= 2p/l= 2p/0.1= 62.8
0 0.1 0.2 0.3 0.4mNp/2=
Lossy Model)cos(),(
0 zteVtzv
z


−=
−+ mNp/1=
Demonstration of Traveling waves on an infinite TL
Effect of Increase in attenuation ()
College of Engineering, Taibah UniversityEE451 Electromagnetics

14. �(−�)
??????(�)=??????
�[�
−��??????
+Γ�
+��??????
]�(�)=
??????
�
�
0
[�
−��??????
−Γ�
+��??????
]
-The voltage is not constant throughout over the TL. The total voltage and current at any point
on the transmission line is the sum of the incident and reflected voltages and currents
Voltage, Current and Impedance Variation along the Transmission Line
Γ=
�
??????−�
0
�
??????+�
0
-Since voltages and current vary, the input impedance also varies as follows:
�
���=
??????�
��
=�
0
�
−��??????
+Γ�
+��??????
�
−��??????
−Γ�
+��??????
-So the load impedance on a transmission line depends on the point of observation. An open
circuited transmission line can look like a shortcircuit somewhere along the transmission line
-Usually the voltages, currents and impedance are calculated as functions of distance from
the load (opposite to z-axis).
-Put z = -l results in:
College of Engineering, Taibah UniversityEE451 Electromagnetics
Z
L
-
+
V
L
I
L
z = 0
z = -l
??????(−�)
�
��−�
??????(−�)=??????
�[�
+���
+Γ�
−���
]�(−�)=??????
��
�
−1
[�
+���
−Γ�
−���
]
-
+
�
��−�=�
0
�
+���
+Γ�
−���
�
+���
−Γ�
−���
l
�
0

15. Standing Waves
The incident and reflected voltages meet along the transmission line and form
standing waves just as standing waves in water
College of Engineering, Taibah UniversityEE451 Electromagnetics

16. Calculation of Voltage and Current along the Transmission Line
College of Engineering, Taibah UniversityEE451 Electromagnetics
??????(−�)=??????
�[�
+���
+Γ�
−���
]
�(−�)=
??????
�
�
0
[�
+���
−Γ�
−���
]
Z
L
V
o
I
L
z = 0
VzPattern
-
+
⇒??????(−�)=??????
��
+���
[1+Γ�
−2���
]
e
-2jl
fluctuates between +1 and -1
⇒??????
��??????=??????
�1+Γ and??????
���=??????
�1−Γ
⇒�(−�)=
??????
�
�
0
�
+���
[1−Γ�
−2���
]
⇒�
��??????=
??????
�
�
0
1+Γ⇒�
���=
??????
�
�
0
1−Γ
??????(−�)=??????
��
+���
1+Γ�
−2���
⇒�(−�)=
??????
�
�
0
�
+���
1−Γ�
−2���
⇒??????
��??????=??????
�1+
�
??????−�
0
�
??????+�
0
⇒??????
���=??????
�1−
�
??????−�
0
�
??????+�
0
⇒�
��??????=??????
�1+
�
??????−�
0
�
??????+�
0
⇒�
���=??????
�1−
�
??????−�
0
�
??????+�
0
Voltage V(z), z=-l Current I(z ), z=-l
-V
o
??????
�1+Γ
??????
�1−Γ
V
L
Z
LI
L
z = 0
I(z)Pattern
-
+
V
L
??????
�
�
0
1+Γ
??????
�
�
0
1−Γ
??????
�
�
0
−
??????
�
�
0
Voltage amplitude on a transmission line fluctuates between V
maxand V
minwhich is called standing
wave pattern
The amplitude is given by:
�
0 �
0

17. Standing wave voltage patterns on a Transmission Line with different loadings
Γ=
�
??????−�
�
�
??????−�
�
Voltage amplitude on a transmission line fluctuates between V
maxand V
minwhich is called standing
wave pattern.
We will assume Z
o= 50 W andV
o= 1 V ??????
��??????=??????
�1+Γ
1.33??????
0.67??????
−0.67??????
−1.33??????
Γ=0.33
??????
��??????=1.33????????????
���=0.67??????
??????
���=??????
�1−Γ
0
V
o
-V
o
Z
L= 50 W Γ=0
??????
��??????=1????????????
���=1??????
100W50W
-0.5l-l-1.5l-2l
Z
L= 100 W
50W50W
-0.5l-l-1.5l-2l
500W50W
Γ=0.82
??????
��??????=1.82????????????
���=0.18??????
Z
L= 500 W
0.18??????
1.82??????
-0.5l-l-1.5l-2l 00
−0.18??????
−1.82??????
20W50W
-0.5l-l-1.5l-2l 0
Z
L= 20 WΓ=−0.43
??????
��??????=1.43????????????
���=0.57??????
1.43??????
0.57??????
−0.57??????
−1.43??????
50W SC
Γ=−1??????
��??????=2????????????
��??????=0??????
2??????
0??????
−2??????
OC
Γ=+1 ??????
��??????=2????????????
��??????=0??????
-0.5l-l-1.5l-2l -0.5l-l-1.5l-2l
1??????
−1??????

18. Time domain picture of the Standing wave voltage patterns on a Transmission Line
College of Engineering, Taibah UniversityEE451 Electromagnetics
�(−�)??????(�)=??????
�[�
−��??????
+Γ�
+��??????
]�(�)=
??????
�
�
0
[�
−��??????
−Γ�
+��??????
]
Γ=
�
??????−�
0
�
??????+�
0
??????(−�)
�
��−�
-
+
The voltage and current variation along the transmission line is given by:
Z
L
�
0
V
o
I
L
z = 0
-
+
-V
o
??????
�1+Γ
??????
�1−Γ
V
L
??????
��??????
??????
���
??????
��??????=
??????
���=
??????(�)�
+�????????????
=??????
�[�
−��??????
+Γ�
+��??????
]�
+�????????????
Time domain equation can be obtained by multiplying with e
jt
:
⇒??????(�)�
+�????????????
=??????
�[�
−��??????
−Γ�
−��??????
+Γ�
−��??????
+Γ�
+��??????
]�
+�????????????
⇒??????(�)�
+�????????????
=??????
�[1−Γ�
−��??????
+Γ(�
−��??????
+�
+��??????
)]�
+�????????????
⇒??????(�)�
+�????????????
=??????
�[1−Γ�
−��??????
+2�����]�
+�????????????
⇒??????�,�=??????
�1−Γcos??????�−��+2Γ??????
������cos??????�
⇒??????�,�=??????�[??????(�)�
+�????????????
]
Standing waveTraveling wave
So voltage on a transmission line is a combination of
standing and traveling wave
Example:
Γ=0.33
??????
��??????=1.33????????????
���=0.67??????
Z
L= 100 W
100W
50W
-0.5l-l-1.5l-2l z = 0

19. Time domain picture of the Standing wave voltage patterns for some special cases
College of Engineering, Taibah UniversityEE451 Electromagnetics
The time domain voltage variation along the transmission line is given by:
1. Matched line (=0):
⇒??????�,�=??????
�cos??????�−��
(Traveling wave)
??????�,�=??????
�1−Γcos??????�−��+2Γ??????
������cos??????�
Standing waveTraveling wave
Let us look at some important cases
-0.5l-l-1.5l-2l z = 0
50W
50W
2. Open circuited line (=1):
⇒??????�,�=2??????
������cos??????�
⇒??????�,�=2??????
�cos??????������+2??????
�sin??????������−2??????
������cos??????�
50W OC
-0.5l-l-1.5l-2l z = 0
3. Short circuited line (=1):
??????�,�=2??????
�cos??????�−��−2Γ�����cos??????�
⇒??????�,�=2??????
������sin??????�
Pure Traveling wave
Pure standing wave
Pure standing wave
SC

20. College of Engineering, Taibah University, MadinahEE451 Electromagnetics
Impedance distribution on Transmission Line
➢We have seen in the previous slide that the phasor voltage changes
along the transmission line
➢This implies that the impedance also changes
V(Z)
z=0
Z
L
-lzjzj
eVeVzV
 ++−+
+=
00
)( zjzj
e
Z
V
e
Z
V
zI
 +
+
−
+

−=
0
0
0
0
)(
Z
in0
0
ZZ
ZZ
L
L
+
−
=
Impedance Calculations
At a pointl from the load:)(
)(
lI
lV
Z
in
−
−
= ljlj
ljlj
eVeV
eVeV
Z


−++
−++
−
+
=
00
00
0 lj
L
Llj
lj
L
Llj
e
ZZ
ZZ
e
e
ZZ
ZZ
e
Z


−
−
+
−
−
+
−
+
=
0
0
0
0
0 lj
L
lj
L
lj
L
lj
L
eZZeZZ
eZZeZZ
Z


−
−
−−+
−++
=
)()(
)()(
00
00
0 )()(
)()(
0
0
0 ljljljlj
L
ljljljlj
L
eeZeeZ
eeZeeZ
Z


−−
−−
++−
−++
= ljZlZ
ljZlZ
Z
L
L


sincos
)sincos
0
0
0
+
+
= )
cos
sin
(cos
)
cos
sin
(cos
0
0
0
l
l
jZZl
l
l
jZZl
Z
L
L






+
+
= ljZZ
ljZZ
ZZ
L
L
in


tan
tan
0
0
0
+
+
=

21. College of Engineering, Taibah University, MadinahEE451
Range of Reflection Coefficient and SWR0
0
ZZ
ZZ
L
L
+
−
=
UnmatchedZ
L=Z
00= −
+
=
1
1
SWR 1=SWR
Total reflection
Z
L>>Z
0Z
L<<Z
01→ =SWR
Matched Z
LZ
010  SWR1
SWR meter
5103
∞
211.5
041125
% REF POWER
SWR
5103
∞
211.5
041125
% REF POWER
SWR
5103
∞
211.5
041125
% REF POWER
SWR
5103 ∞
211.5
041125
% REF POWER
SWR

22. College of Engineering, Taibah University, MadinahEE451
An Example: VSWR should be set to 1 by matching0
0
ZZ
ZZ
L
L
+
−
= −
+
=
1
1
SWR
1
Unmatchedzj
eV
−+
0 zj
eV
+−
0
Matching
Network
1zj
eV
−+
0
Matching
Network
Matched

23. College of Engineering, Taibah UniversityEE451 Electromagnetics
Open and Short Circuited Parameter Summary
Open circuit Voltage:Short circuit Voltage:zjzj
eVeVzV
 ++−+
+=
00
)(
For Z
L= 0W1−= 0
0
ZZ
ZZ
L
L
+
−
=
⇒??????&#3627408487;=??????
0
+
&#3627408466;
−&#3627408471;&#3627409149;??????
−??????
0
+
&#3627408466;
+&#3627408471;&#3627409149;??????
⇒??????&#3627408487;=−??????
0
+
(&#3627408466;
+&#3627408471;&#3627409149;??????
−&#3627408466;
−&#3627408471;&#3627409149;??????
)
⇒??????&#3627408487;=−&#3627408471;2??????
0
+
&#3627408480;&#3627408470;&#3627408475;&#3627409149;&#3627408487;
= ∞
⇒Γ=+1
For Z
Lzjzj
e
Z
V
e
Z
V
zI
 +
+
−
+

−=
0
0
0
0
)(
Short circuit Current:
⇒&#3627408444;&#3627408487;=
??????
0
+
&#3627408461;
0
&#3627408466;
−&#3627408471;&#3627409149;??????
+
??????
0
+
&#3627408461;
0
&#3627408466;
+&#3627408471;&#3627409149;??????1−=
⇒&#3627408444;&#3627408487;=
2??????
0
+
&#3627408461;0
&#3627408464;&#3627408476;&#3627408480;&#3627409149;z
Short circuit Impedance:ljZZ
ljZZ
ZZ
L
L
in


tan
tan
0
0
0
+
+
=
⇒&#3627408461;
&#3627408470;&#3627408475;=&#3627408461;
0
&#3627408471;&#3627408461;
0tan&#3627409149;&#3627408473;
&#3627408461;
0
⇒&#3627408461;
&#3627408470;&#3627408475;=&#3627408471;&#3627408461;
0tan&#3627409149;&#3627408473;
⇒Γ=
1−&#3627408461;
0/&#3627408461;
??????
1+&#3627408461;
0/&#3627408461;
??????
⇒??????&#3627408487;=??????
0
+
(&#3627408466;
+&#3627408471;&#3627409149;??????
+&#3627408466;
−&#3627408471;&#3627409149;??????
)
⇒??????&#3627408487;=2??????
0
+
&#3627408464;&#3627408476;&#3627408480;&#3627409149;&#3627408487;
&#3627408444;&#3627408487;=
??????
0
+
&#3627408461;
0
&#3627408466;
−&#3627408471;&#3627409149;??????
+
Γ??????
0
+
&#3627408461;
0
&#3627408466;
+&#3627408471;&#3627409149;??????
Open circuit Current:
⇒&#3627408444;&#3627408487;=
??????
0
+
&#3627408461;
0
&#3627408466;
−&#3627408471;&#3627409149;??????
−
??????
0
+
&#3627408461;
0
&#3627408466;
+&#3627408471;&#3627409149;??????
⇒Γ=+1
⇒&#3627408444;&#3627408487;=−
2&#3627408471;??????
0
+
&#3627408461;0
&#3627408480;&#3627408470;&#3627408475;&#3627409149;z
Open circuit Impedance:ljZZ
ljZZ
ZZ
L
L
in


tan
tan
0
0
0
+
+
=
⇒&#3627408461;
&#3627408470;&#3627408475;=&#3627408461;
0
1
&#3627408471;tan&#3627409149;&#3627408473;
⇒&#3627408461;
&#3627408470;&#3627408475;=−&#3627408471;&#3627408461;
0cot&#3627409149;&#3627408473;zjzj
eVeVzV
 ++−+
+=
00
)( 0
0
ZZ
ZZ
L
L
+
−
=
⇒&#3627408461;
&#3627408470;&#3627408475;=&#3627408461;
0
1+&#3627408471;&#3627408461;
0tan&#3627409149;&#3627408473;/&#3627408461;
??????
&#3627408461;
0/&#3627408461;
??????+&#3627408471;tan&#3627409149;&#3627408473;
Because of the importance of Short and Open circuit in Electromagnetics, we study them in detail

24. Special Case 1: Short Circuited Line, V, I, Z plots
z=0
-l
Z
in0=
LZ zVjzV sin2)(
0
+
−= z
Z
V
zI cos2)(
0
0
+
= ljZZ
in
tan
0
=
|V(z)|
l−3l/4−l/2−l/4
|I(z)|
|Z
in(z)|
➢Voltage is zero at z=0 (short)
➢Current is maximum at z = 0
➢Impedance is zero at z = 0
because Z
L= 0
➢Also at l = l/2, and lfrom the
load the transmission line appears
to be a short circuited line
➢At l = l/4, 3l/4, from the load the
transmission line appears to be an
open circuited line
0VV1
0
=
+
College of Engineering, Taibah UniversityEE451 Electromagnetics

25. Special Case 2: Open Circuited Line, V, I, Z plots
z=0
-l
Z
in=
LZ
|V(z)|
−l/4−l/2−3l/4l
|I(z)|
|Z
in(z)|
➢Voltage is max at z=0 (open)
➢Current is zero at z = 0 (open)
➢Impedance is infinite at z = 0
because Z
L= inf
➢At l= 0, l/2, and lfrom the load
the transmission line appears to be
a open circuited line
➢At l = l/4, 3l/4, from the load the
transmission line appears to be an
short circuited line
0VV1
0
=
+ zVzV cos2)(
0
+
= z
Z
V
jzI sin2)(
0
0
+
−= zjZzZ
in
cot)(
0
+=
College of Engineering, Taibah UniversityEE451 Electromagnetics

26. Transmission and Reflection on two Transmission Line Interface111,, 222,,
Z=0i
E
 i
H
 r
E
 r
H
 t
E
 t
H
 i
k
ˆ r
k
ˆ  zj
eV
1
0
−+ zj
eV
1
0
++
 zj
eV
2
0


−+ 01
Z 02
Z
College of Engineering, Taibah UniversityEE451 Electromagnetics

27. Special Case: A lossless line terminated in another lossless line
z=0
Z
01 Z
02
 (reflected) (transmitted Wave)
 (incident)
z<0zjzj
eVeVzV
11
01011
)(
 ++−+
+= zj
eVzV
2
022
)(
−+
= zjzj
e
Z
V
e
Z
V
zI
11
01
01
01
01
1
)(
 +
+
−
+
−= zj
e
Z
V
zI
2
02
02
2
)(
−
+
=
At z = 0, the voltages and currents should be continuous, i.e. V
1=V
2and I
1= I
2+++
=+
020101
VVV 02
02
01
01
01
01
Z
V
Z
V
Z
V
+++
=−
Solving the two equations
Z>0
(infinite)01
02
01
01
01
01
Z
V
Z
V
Z
V
+++
=+ 







+=
+
+
0201
02
01
01 112
ZZ
V
Z
V 







+=
+
+
0201
02
01
01 112
ZZ
V
Z
V 






+
=
+
+
0201
0201
02
01
01
2
ZZ
ZZ
V
Z
V 0102
02
01
02 2
ZZ
Z
V
V
+
=
+
+ =
 is called the transmission coefficient0
0
ZZ
ZZ
L
L
+
−
= 0102
0102
ZZ
ZZ
+
−
=
College of Engineering, Taibah UniversityEE451 Electromagnetics

28. zjzj
eVeVzV
 ++−+
+=
00
)( Z
LW=50
0
Z GHzf3= = Z
02W=100
Case 5 (another TL)002
002
ZZ
ZZ
+
−
= 33.0= 002
022
ZZ
Z
+
= 33.1=
Z
01=50W Z
02=100W
z=0zj
eVzV
2
012
)(


−+
= zjzj
eVeVzV
11
01011
)(
 ++−+
+= W=50
01
Z W=100
02
Z −
+
=
1
1
SWR 98.1=SWR
Special Case: A lossless line terminated in another lossless line
College of Engineering, Taibah UniversityEE451 Electromagnetics

29. Impedance Transformationzjzj
eVeVzV
 ++−+
+=
00
)( zjzj
e
Z
V
e
Z
V
zI
 +
+
−
+

−=
0
0
0
0
)( 0
0
ZZ
ZZ
L
L
+
−
= ljZZ
ljZZ
ZZ
L
L
in


tan
tan
0
0
0
+
+
=
After calculations (previous slides)
Input impedance is periodic and changes over the line. This is called
Impedance transformation
e.g. At l = -ll
l
p
l
l
p
2
tan
2
tan
0
0
0
L
L
in
jZZ
jZZ
ZZ
+
+
= ljZZ
ljZZ
Z
L
L
l
p
l
p
2
tan
2
tan
0
0
0
+
+
= Lin
ZZ=
lLin
ZZ=
At l = -l/44
2
tan
4
2
tan
0
0
0
l
l
p
l
l
p
L
L
in
jZZ
jZZ
ZZ
+
+
= L
L
in
jZ
Z
jZ
Z
ZZ
+
+
=
2/tan
2/tan
0
0
0
p
p L
in
Z
Z
Z
2
0
=
l/4Lin
ZZZ /
2
0

0
0
College of Engineering, Taibah UniversityEE451 Electromagnetics

30. Quarter Wave Impedance Transformer
Z
0
Matching a load Z
Lto a line
Z
o
Z
0
Z
1
Z
L0
0
ZZ
ZZ
L
L
+
−
=
Z
L
l/4
Let us insert a quarter wave
section of another TLL
in
Z
Z
Z
2
1

For matchingin
ZZ=
0 L
Z
Z
Z
2
1
0
= LZZZ
01=
So the load can be matched to Z
oby
a l/4 section of impedance Z
1
College of Engineering, Taibah UniversityEE451 Electromagnetics

31. Reading from the book
College of Engineering, Taibah UniversityEE451 Electromagnetics
✓11.2 TRANSMISSION LINE PARAMETERS
✓11.3 TRANSMISSION LINE EQUATIONS
✓11.4 INPUT IMPEDANCE, STANDING WAVE RATIO, AND POWER
Sadiku, Elements of Electromagnetics 7
th
edition: Chapter 11 (553-572)