ec6503 transmission lines and waveguides by h.umma habiba, professor svce,chennai
TRANSCRIPT
EC6503 TRANSMISSION LINES AND WAVEGUIDES
BY
H.UMMA HABIBA, Professor
SVCE,Chennai.
Overview of syllabusOBJECTIVES
To introduce the various types of transmission lines and to discuss the losses associated.
To give thorough understanding about impedance transformation and matching.
To use the Smith chart in problem solving.
To impart knowledge on filter theories and waveguide theories
Overview of syllabus
OUTCOMESUpon completion of the course, students will be able to Discuss the propagation of signals through transmission lines.
Analyze signal propagation at Radio frequencies.
Explain radio propagation in guided systems.
Utilize cavity resonators
Overview of syllabus TEXT BOOK
1. John D Ryder, “Networks lines and fields”, Prentice Hall of India, New Delhi, 2005 REFERENCES1.William H Hayt and Jr John A Buck, “Engineering Electromagnetics” Tata Mc Graw-Hill Publishing Company Ltd, New Delhi, 2008
2.David K Cheng, “Field and Wave Electromagnetics”, Pearson Education Inc, Delhi, 2004
3.John D Kraus and Daniel A Fleisch, “Electromagnetics with Applications”, Mc Graw Hill Book Co,2005
4.GSN Raju, “Electromagnetic Field Theory and Transmission Lines”, Pearson Education, 2005
5.Bhag Singh Guru and HR Hiziroglu, “Electromagnetic Field Theory Fundamentals”, Vikas Publishing House, New Delhi, 2001.
6. N. Narayana Rao, “ Elements of Engineering Electromagnetics” 6
Overview of syllabus
UNIT I -TRANSMISSION LINE THEORY
General theory of Transmission lines - the transmission line - general solution - The infinite line - Wavelength, velocity of propagation - Waveform distortion - the distortion-less line - Loading and different methods of loading - Line not terminated in Z0 - Reflection coefficient - calculation of current, voltage, power delivered and efficiency of transmission - Input and transfer impedance - Open and short circuited lines - reflection factor and reflection loss.
Overview of syllabus
UNIT II -HIGH FREQUENCY TRANSMISSION LINES
Transmission line equations at radio frequencies - Line of Zero dissipation - Voltage and current on the dissipation-less line, Standing Waves, Nodes, Standing Wave Ratio - Input impedance of the dissipation-less line - Open and short circuited lines - Power and impedance measurement on lines - Reflection losses - Measurement of VSWR and wavelength.
Overview of syllabus
UNIT III-IMPEDANCE MATCHING IN HIGH FREQUENCY LINES
Impedance matching: Quarter wave transformer - Impedance matching by stubs - Single stub and double stub matching - Smith chart - Solutions of problems using Smith chart - Single and double stub matching using Smith chart.
Overview of syllabus
UNIT IV-PASSIVE FILTERS
Characteristic impedance of symmetrical networks - filter fundamentals, Design of filters: Constant K - Low Pass, High Pass, Band Pass, Band Elimination, m- derived sections - low pass, high pass composite filters.
Overview of syllabus
UNIT V -WAVE GUIDES AND CAVITY RESONATORS
General Wave behaviours along uniform Guiding structures, Transverse Electromagnetic waves, Transverse Magnetic waves, Transverse Electric waves, TM and TE waves between parallel plates, TM and TE waves in Rectangular wave guides, Bessel‟s differential equation and Bessel function, TM and TE waves in Circular wave guides, Rectangular and circular cavity Resonators.
UNIT I
TRANSMISSION LINE THEORY UNIFORM PLANE
WAVES
Transmission Line
Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss May or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz)
Properties
Coaxial cable (coax)Twin lead
(shown connected to a 4:1 impedance-transforming
balun)11
Transmission Line (cont.)
CAT 5 cable(twisted pair)
The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities.
12
Transmission Line (cont.)
Microstrip
h
w
r
r
w
Stripline
h
Transmission lines commonly met on printed-circuit boards
Coplanar strips
hr
w w
Coplanar waveguide (CPW)
hr
w
13
Transmission Line (cont.)
Transmission lines are commonly met on printed-circuit boards.
A microwave integrated circuit
Microstrip line
14
Fiber-Optic GuideProperties Uses a dielectric rod Can propagate a signal at any frequency (in theory) Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields
15
Fiber-Optic Guide (cont.)Two types of fiber-optic guides:
1) Single-mode fiber
2) Multi-mode fiber
Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength.
Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).
16
Fiber-Optic Guide (cont.)
http://en.wikipedia.org/wiki/Optical_fiber
Higher index core region
17
Waveguides
Has a single hollow metal pipe Can propagate a signal only at high frequency: > c
The width must be at least one-half of a wavelength
Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz)
Properties
http://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 18
Lumped circuits: resistors, capacitors,inductors
neglect time delays (phase)
account for propagation and time delays (phase change)
Transmission-Line Theory
Distributed circuit elements: transmission lines
We need transmission-line theory whenever the length of a line is significant compared with a wavelength.
19
Transmission Line
2 conductors
4 per-unit-length parameters:
C = capacitance/length [F/m]
L = inductance/length [H/m]
R = resistance/length [/m]
G = conductance/length [ /m or S/m]
z
20
21
Transmission Line (cont.)
z
,i z t
+ + + + + + +- - - - - - - - - -
,v z tx x xB
21
Rz Lz
Gz Cz
z
v(z+z,t)
+
-
v(z,t)
+
-
i(z,t) i(z+z,t)
22
( , )( , ) ( , ) ( , )
( , )( , ) ( , ) ( , )
i z tv z t v z z t i z t R z L z
tv z z t
i z t i z z t v z z t G z C zt
Transmission Line (cont.)
22
Rz Lz
Gz Cz
z
v(z+z,t)
+
-
v(z,t)
+
-
i(z,t) i(z+z,t)
23
Hence
( , ) ( , ) ( , )( , )
( , ) ( , ) ( , )( , )
v z z t v z t i z tRi z t L
z ti z z t i z t v z z t
Gv z z t Cz t
Now let z 0:
v iRi L
z ti v
Gv Cz t
“Telegrapher’sEquations”
TEM Transmission Line (cont.)
23
24
To combine these, take the derivative of the first one
with respect to z:
2
2
2
2
v i iR L
z z z t
i iR L
z t z
vR Gv C
t
v vL G C
t t
Switch the order of the derivatives.
TEM Transmission Line (cont.)
24
25
2 2
2 2( ) 0
v v vRG v RC LG LC
z t t
The same equation also holds for i.
Hence, we have:
2 2
2 2
v v v vR Gv C L G C
z t t t
TEM Transmission Line (cont.)
25
26
2
2
2( ) ( ) 0
d VRG V RC LG j V LC V
dz
2 2
2 2( ) 0
v v vRG v RC LG LC
z t t
TEM Transmission Line (cont.)
Time-Harmonic Waves:
26
27
Note that
= series impedance/length
2
2
2( )
d VRG V j RC LG V LC V
dz
2( ) ( )( )RG j RC LG LC R j L G j C
Z R j L
Y G j C
= parallel admittance/length
Then we can write:2
2( )
d VZY V
dz
TEM Transmission Line (cont.)
27
28
Let
Convention:
Solution:
2 ZY
( ) z zV z Ae Be
1/2
( )( )R j L G j C
principal square root
2
2
2( )
d VV
dzThen
TEM Transmission Line (cont.)
is called the "propagation constant."
/2jz z e
j
0, 0
attenuationcontant
phaseconstant
28
29
TEM Transmission Line (cont.)
0 0( ) z z j zV z V e V e e
Forward travelling wave (a wave traveling in the positive z direction):
0
0
0
( , ) Re
Re
cos
z j z j t
j z j z j t
z
v z t V e e e
V e e e e
V e t z
g0t
z0
zV e
2
g
2g
The wave “repeats” when:
Hence:
29
30
Phase Velocity
Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest.
0( , ) cos( )zv z t V e t z
z
vp (phase velocity)
30
31
Phase Velocity (cont.)
0
constant
t z
dz
dtdz
dt
Set
Hence pv
1/2
Im ( )( )p
vR j L G j C
In expanded form:
31
32
Characteristic Impedance Z0
0
( )
( )
V zZ
I z
0
0
( )
( )
z
z
V z V e
I z I e
so 00
0
VZ
I
+ V+(z)-
I+ (z)
z
A wave is traveling in the positive z direction.
(Z0 is a number, not a function of z.)
32
33
Use Telegrapher’s Equation:
v iRi L
z t
sodV
RI j LIdz
ZI
Hence0 0
z zV e ZI e
Characteristic Impedance Z0 (cont.)
33
34
From this we have:
Using
We have
1/2
00
0
V Z ZZ
I Y
Y G j C
1/2
0
R j LZ
G j C
Characteristic Impedance Z0 (cont.)
Z R j L
Note: The principal branch of the square root is chosen, so that Re (Z0) > 0. 34
35
00
0 0
j z j j z
z z
z j zV e e
V z V e V
V e e e
e
e
0
0 cos
c
, R
os
e j t
z
z
V e t
v z t V z
z
V z
e
e t
Note:
wave in +z direction
wave in -z direction
General Case (Waves in Both Directions)
35
36
Backward-Traveling Wave
0
( )
( )
V zZ
I z
0
( )
( )
V zZ
I z
so
+ V -(z)-
I - (z)
z
A wave is traveling in the negative z direction.
Note: The reference directions for voltage and current are the same as for the forward wave.
36
37
General Case
0 0
0 00
( )
1( )
z z
z z
V z V e V e
I z V e V eZ
A general superposition of forward and backward traveling waves:
Most general case:
Note: The reference directions for voltage and current are the same for forward and backward waves. 37
+ V (z)-
I (z)
z
38
1
2
12
0
0 0
0 0
0 0
z z
z z
V z V e V e
V VI z e e
Z
j R j L G j C
R j LZ
G j
Z
C
I(z)
V(z)+- z
2mg
[m/s]pv
guided wavelength g
phase velocity vp
Summary of Basic TL formulas
38
Lossless Case
0, 0R G
1/ 2
( )( )j R j L G j C
j LC
so 0
LC
1/2
0
R j LZ
G j C
0
LZ
C
1pv
LC
pv
(indep. of freq.)(real and indep. of freq.)39
40
Lossless Case (cont.)1
pvLC
In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that
LC
The speed of light in a dielectric medium is1
dc
Hence, we have that p dv c
The phase velocity does not depend on the frequency, and it is always the speed of light (in the material).
(proof given later)
40
41
0 0z zV z V e V e
Where do we assign z = 0?
The usual choice is at the load.
I(z)
V(z)+-
zZL
z = 0
Terminating impedance (load)
Ampl. of voltage wave propagating in negative z direction at z = 0.
Ampl. of voltage wave propagating in positive z direction at z = 0.
Terminated Transmission Line
Note: The length l measures distance from the load: z41
What if we know
@V V z and
0 0V V V e
z zV z V e V e
0V V e
0 0V V V e
Terminated Transmission Line (cont.)
0 0z zV z V e V e
Hence
Can we use z = - l as a reference plane?
I(z)
V(z)+-
zZL
z = 0
Terminating impedance (load)
42
( ) ( )z zV z V e V e
Terminated Transmission Line (cont.)
0 0z zV z V e V e
Compare:
Note: This is simply a change of reference plane, from z = 0 to z = -l.
I(z)
V(z)+-
zZL
z = 0
Terminating impedance (load)
43
0 0z zV z V e V e
What is V(-l )?
0 0V V e V e
0 0
0 0
V VI e e
Z Z
propagating forwards
propagating backwards
Terminated Transmission Line (cont.)
l distance away from load
The current at z = - l is then
I(z)
V(z)+-
zZL
z = 0
Terminating impedance (load)
44
20
0
1 L
VI e e
Z
200
00 0 1
VV eV V e ee
VV
Total volt. at distance l from the load
Ampl. of volt. wave prop. towards load, at the load position (z = 0).
Similarly,
Ampl. of volt. wave prop. away from load, at the load position (z = 0). 0
21 LV e e
L Load reflection coefficient
Terminated Transmission Line (cont.)I(-l )
V(-l )+
l
ZL-
0,Z
l Reflection coefficient at z = - l
45
20
2
2
0
0
2
0
1
1
1
1
L
L
L
L
V V e e
VI e e
Z
V eZ Z
I e
Input impedance seen “looking” towards load at z = -l .
Terminated Transmission Line (cont.)
I(-l )
V(-l )+
l
ZL-
0,Z
Z
46
At the load (l = 0):
0
10
1L
LL
Z Z Z
Thus,
20
00
20
0
1
1
L
L
L
L
Z Ze
Z ZZ Z
Z Ze
Z Z
Terminated Transmission Line (cont.)
0
0
LL
L
Z Z
Z Z
2
0 2
1
1L
L
eZ Z
e
Recall
47
Simplifying, we have
00
0
tanh
tanhL
L
Z ZZ Z
Z Z
Terminated Transmission Line (cont.)
202
0 0 00 0 2
2 0 00
0
0 00
0 0
00
0
1
1
cosh sinh
cosh sinh
L
L L L
L LL
L
L L
L L
L
L
Z Ze
Z Z Z Z Z Z eZ Z Z
Z Z Z Z eZ Ze
Z Z
Z Z e Z Z eZ
Z Z e Z Z e
Z ZZ
Z Z
Hence, we have
48
20
20
0
2
0 2
1
1
1
1
j jL
j jL
jL
jL
V V e e
VI e e
Z
eZ Z
e
Impedance is periodic with period g/2
2
/ 2
g
g
Terminated Lossless Transmission Line
j j
Note: tanh tanh tanj j
tan repeats when
00
0
tan
tanL
L
Z jZZ Z
Z jZ
49
For the remainder of our transmission line discussion we will assume that the transmission line is lossless.
20
20
0
2
0 2
00
0
1
1
1
1
tan
tan
j jL
j jL
jL
jL
L
L
V V e e
VI e e
Z
V eZ Z
I e
Z jZZ
Z jZ
0
0
2
LL
L
g
p
Z Z
Z Z
v
Terminated Lossless Transmission Line
I(-l )
V(-l )+
l
ZL-
0 ,Z
Z
50
Matched load: (ZL=Z0)
0
0
0LL
L
Z Z
Z Z
For any l
No reflection from the load
A
Matched LoadI(-l )
V(-l )+
l
ZL-
0 ,Z
Z
0Z Z
0
0
0
j
j
V V e
VI e
Z
51
Short circuit load: (ZL = 0)
0
0
0
01
0
tan
L
Z
Z
Z jZ
Always imaginary!Note:
B
2g
scZ jX
S.C. can become an O.C. with a g/4 trans. line
0 1/4 1/2 3/4g/
XSC
inductive
capacitive
Short-Circuit Load
l
0 ,Z
0 tanscX Z
52
Using Transmission Lines to Synthesize Loads
A microwave filter constructed from microstrip.
This is very useful is microwave engineering.
53
00
0
tan
tanL
inL
Z jZ dZ Z d Z
Z jZ d
inTH
in TH
ZV d V
Z Z
I(-l)
V(-l)+
l
ZL
-0Z
ZTH
VTH
d
Zin
+
-
ZTH
VTH
+ZinV(-d)
+
-
Example
Find the voltage at any point on the line.
54
Note: 021 j
LjV V e e
0
0
LL
L
Z Z
Z Z
20 1j d j d in
THin TH
LV dZ
Ze V
ZV e
2
2
1
1
jj din L
TH j dm TH L
Z eV V e
Z Z e
At l = d :
Hence
Example (cont.)
0 2
1
1j din
TH j din TH L
ZV V e
Z Z e
55
Some algebra: 2
0 2
1
1
j dL
in j dL
eZ Z d Z
e
2
20 20
2 220
0 2
20
20 0
2
0
20 0
0
111
1 11
1
1
1
1
j dL
j dj dLL
j d j dj dL TH LL
THj dL
j dL
j dTH L TH
j dL
j dTH THL
TH
in
in TH
eZ
Z ee
Z e Z eeZ Z
e
Z e
Z Z e Z Z
eZ
Z
Z
Z Z
Z Z Ze
Z Z
Z
2
0
20 0
0
1
1
j dL
j dTH THL
TH
e
Z Z Z Ze
Z Z
Example (cont.)
56
2
02
0
1
1
jj d L
TH j dTH S L
Z eV V e
Z Z e
20
20
1
1
j din L
j din TH TH S L
Z Z e
Z Z Z Z e
where 0
0
THS
TH
Z Z
Z Z
Example (cont.)
Therefore, we have the following alternative form for the result:
Hence, we have
57
2
02
0
1
1
jj d L
TH j dTH S L
Z eV V e
Z Z e
Example (cont.)
I(-l)
V(-l)+
l
ZL
-0Z
ZTH
VTH
d
Zin
+
-
Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load).
58
2 2
2 2 2 20
0
1 j d j dL L S
j d j d j d j dTH L S L L S L S
TH
e eZ
V d V e e e eZ Z
Example (cont.)
ZL0Z
ZTH
VTH
d
+
-
Wave-bounce method (illustrated for l = d ):
59
Example (cont.)
22 2
22 2 20
0
1
1
j d j dL S L S
j d j d j dTH L L S L S
TH
e e
ZV d V e e e
Z Z
Geometric series:
2
0
11 , 1
1n
n
z z z zz
2 2
2 2 2 20
0
1 j d j dL L S
j d j d j d j dTH L S L L S L S
TH
e eZ
V d V e e e eZ Z
2j dL Sz e
60
Example (cont.)
or
2
0
202
1
1
1
1
j dL s
THj dTH
L j dL s
eZV d V
Z Ze
e
2
02
0
1
1
j dL
TH j dTH L s
Z eV d V
Z Z e
This agrees with the previous result (setting l = d ).
Note: This is a very tedious method – not recommended.
Hence
61
I(-l)
V(-l)+
l
ZL-
0 ,Z
At a distance l from the load:
*
*
2
0 2 2 * 2*0
1Re 1 1
1R
2
e2
L L
Ve e
Z
V I
e
P
2
20 2 4
0
11
2 L
VP e e
Z
If Z0 real (low-loss transmission line)
Time- Average Power Flow
20
20
0
1
1
L
L
V V e e
VI e e
Z
j
*2 * 2
*2 2
L L
L L
e e
e e
pure imaginary
Note:
62
Low-loss line
2
20 2 4
0
2 2
20 02 2* *0 0
11
2
1 1
2 2
L
L
VP d e e
Z
V Ve e
Z Z
power in forward wave power in backward wave
2
20
0
11
2 L
VP d
Z
Lossless line ( = 0)
Time- Average Power Flow
I(-l)
V(-l)+
l
ZL-
0 ,Z
63
00
0
tan
tanL T
in TT L
Z jZZ Z
Z jZ
2
4 4 2g g
g
00
Tin T
L
jZZ Z
jZ
0
20
0
0in in
T
L
Z Z
ZZ
Z
Quarter-Wave Transformer
20T
inL
ZZ
Z
so
1/2
0 0T LZ Z Z
Hence
This requires ZL to be real.
ZLZ0 Z0T
Zin
64
20 1 Lj j
LV V e e
20
20
1
1 L
j jL
jj jL
V V e e
V e e e
max 0
min 0
1
1
L
L
V V
V V
max
min
V
VVoltage Standing Wave Ratio VSWR
Voltage Standing Wave Ratio
I(-l )
V(-l )+
l
ZL-
0 ,Z
1
1L
L
VSWR
z
1+ L
1
1- L
0
( )V z
V
/ 2z 0z
65
Coaxial Cable
Here we present a “case study” of one particular transmission line, the coaxial cable.
a
b ,r
Find C, L, G, R
We will assume no variation in the z direction, and take a length of one meter in the z direction in order top calculate the per-unit-length parameters.
66
For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics.
Coaxial Cable (cont.)
-l0
l0
a
b
r0 0
0
ˆ ˆ2 2 r
E
Find C (capacitance / length)
Coaxial cable
h = 1 [m]
r
From Gauss’s law:
0
0
ln2
B
AB
A
b
ra
V V E dr
bE d
a
67
-l0
l0
a
b
r
Coaxial cable
h = 1 [m]
r
0
0
0
1
ln2 r
QC
V ba
Hence
We then have
0 F/m2
[ ]ln
rCba
Coaxial Cable (cont.)
68
ˆ2
IH
Find L (inductance / length)
From Ampere’s law:
Coaxial cable
h = 1 [m]
r
I
0ˆ
2 r
IB
(1)b
a
B d S
h
I
I z
center conductorMagnetic flux:
Coaxial Cable (cont.)
69
Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency.
Coaxial cable
h = 1 [m]
r
I
0
0
0
1
2
ln2
b
r
a
b
r
a
r
H d
Id
I b
a
0
1ln
2r
bL
I a
0 H/mln [ ]2
r bL
a
Hence
Coaxial Cable (cont.)
70
0 H/mln [ ]2
r bL
a
Observation:
0 F/m2
[ ]ln
rCba
0 0 r rLC
This result actually holds for any transmission line.
Coaxial Cable (cont.)
71
0 H/mln [ ]2
r bL
a
For a lossless cable:
0 F/m2
[ ]ln
rCba
0
LZ
C
0 0
1ln [ ]
2r
r
bZ
a
00
0
376.7303 [ ]
Coaxial Cable (cont.)
72
-l0
l0
a
b
0 0
0
ˆ ˆ2 2 r
E
Find G (conductance / length)
Coaxial cable
h = 1 [m]
From Gauss’s law:
0
0
ln2
B
AB
A
b
ra
V V E dr
bE d
a
Coaxial Cable (cont.)
73
-l0
l0
a
b
J E
We then have leakIG
V
0
0
(1) 2
2
22
leak a
a
r
I J a
a E
aa
0
0
0
0
22
ln2
r
r
aa
Gba
2[S/m]
lnG
ba
or
Coaxial Cable (cont.)
74
Observation:
F/m2
[ ]ln
Cba
G C
This result actually holds for any transmission line.
2[S/m]
lnG
ba
0 r
Coaxial Cable (cont.)
75
G C
To be more general:
tanG
C
tanG
C
Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies.
Coaxial Cable (cont.)
As just derived,
The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral.
76
This is the loss tangent that would arise from conductivity effects.
General expression for loss tangent:
c
c c
j
j j
j
tan c
c
Effective permittivity that accounts for conductivity
Loss due to molecular friction Loss due to conductivity
Coaxial Cable (cont.)
77
Find R (resistance / length)
Coaxial cable
h = 1 [m]
Coaxial Cable (cont.)
,b rb
a
b
,a ra
a bR R R
1
2a saR Ra
1
2b sbR Rb
1sa
a a
R
1
sbb b
R
0
2a
ra a
0
2b
rb b
Rs = surface resistance of metal
78
General Transmission Line Formulas
tanG
C
0 0 r rLC
0losslessL
ZC
characteristic impedance of line (neglecting loss)(1)
(2)
(3)
Equations (1) and (2) can be used to find L and C if we know the material properties and the characteristic impedance of the lossless line.
Equation (3) can be used to find G if we know the material loss tangent.
a bR R R
tanG
C
(4)
Equation (4) can be used to find R (discussed later).
,iC i a b contour of conductor,
2
2
1( )
i
i s sz
C
R R J l dlI
79
General Transmission Line Formulas (cont.)
tanG C
0losslessL Z
0/ losslessC Z
R R
Al four per-unit-length parameters can be found from 0 ,losslessZ R
80
Common Transmission Lines
0 0
1ln [ ]
2lossless r
r
bZ
a
Coax
Twin-lead
100 cosh [ ]
2lossless r
r
hZ
a
2
1 2
12
s
ha
R Ra h
a
1 1
2 2sa sbR R Ra b
a
b
,r r
h
,r r
a a
81
Common Transmission Lines (cont.)
Microstrip
0 0
1 00
0 1
eff effr reff effr r
fZ f Z
f
0
1200
0 / 1.393 0.667 ln / 1.444effr
Zw h w h
( / 1)w h
21 ln
t hw w
t
h
w
r
t
82
Common Transmission Lines (cont.)
Microstrip ( / 1)w h
h
w
r
t
2
1.5
(0)(0)
1 4
effr reff eff
r rfF
1 1 11 /0
2 2 4.6 /1 12 /
eff r r rr
t h
w hh w
2
0
4 1 0.5 1 0.868ln 1r
h wF
h
83
At high frequency, discontinuity effects can become important.
Limitations of Transmission-Line Theory
Bend
incident
reflected
transmitted
The simple TL model does not account for the bend.
ZTH
ZLZ0
+-
84
At high frequency, radiation effects can become important.
When will radiation occur?
We want energy to travel from the generator to the load, without radiating.
Limitations of Transmission-Line Theory (cont.)
ZTH
ZLZ0
+-
85
ra
bz
The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, or under any circumstances.
The fields are confined to the region between the two conductors.
Limitations of Transmission-Line Theory (cont.)
86
The twin lead is an open type of transmission line – the fields extend out to infinity.
The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair.”)
+ -
Limitations of Transmission-Line Theory (cont.)
Having fields that extend to infinity is not the same thing as having radiation, however.
87
The infinite twin lead will not radiate by itself, regardless of how far apart the lines are.
h
incident
reflected
The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency.
*1ˆRe E H 0
2t
S
P dS
S
+ -
Limitations of Transmission-Line Theory (cont.)
No attenuation on an infinite lossless line
88
A discontinuity on the twin lead will cause radiation to occur.
Note: Radiation effects increase as the frequency increases.
Limitations of Transmission-Line Theory (cont.)
h
Incident wavepipe
Obstacle
Reflected wave
Bend h
Incident wave
bend
Reflected wave 89
To reduce radiation effects of the twin lead at discontinuities:
h
1) Reduce the separation distance h (keep h << ).2) Twist the lines (twisted pair).
Limitations of Transmission-Line Theory (cont.)
CAT 5 cable(twisted pair)
90
91
Two conductorwire Coaxial line Shielded
Strip line
Dielectric
92
Two conductorwire Coaxial line Shielded
Strip line
Dielectric
93
Rectangular guide
Circular guide
Ridge guide
Common Hollow-pipe waveguides
94
STRIP LINE CONFIGURATIONS
W
SINGLE STRIP LINE COUPLED LINES
COUPLED STRIPSTOP & BOTTOM
COUPLED ROUND BARS
95
MICROSTRIP LINE CONFIGURATIONS
TWO COUPLED MICROSTRIPS SINGLE MICROSTRIP
TWO SUSPENDED SUBSTRATE LINES
SUSPENDED SUBSTRATE LINE
96
TRANSMISSION MEDIA
• TRANSVERSE ELECTROMAGNETIC (TEM):– COAXIAL LINES– MICROSTRIP LINES (Quasi TEM)– STRIP LINES AND SUSPENDED SUBSTRATE
• METALLIC WAVEGUIDES:– RECTANGULAR WAVEGUIDES–CIRCULAR WAVEGUIDES
• DIELECTRIC LOADED WAVEGUIDES
ANALYSIS OF WAVE PROPAGATION ON THESETRANSMISSION MEDIA THROUGH MAXWELL’SEQUATIONS
97
Auxiliary Relations:
tyPermeabili Relative
H/m 104 ; 5.
Constant Dielectric Relative
F/m 10854.8 ; 4.
Current Convection ; 3.
Current Conduction ;ty Conductivi
Law) s(Ohm' 2.
Velocity ; Charge
Newton .1
r
12o
12
HHB
EED
JvJ
J
EJ
vq
BvEqF
or
r
oor
98
Maxwell’s Equations in Large Scale Form
SdDt
SdJldH
SdMSdBt
ldE
SdB
dvSdD
SSl
SSl
S
SV
0
ENEE482 99
• Maxwell’s Equations for the Time - Harmonic Case
DjJHMBjE
BD
EEtEE
eEEejEEE
jEEa
jEEajEEazyxE
ezyxEtzyxE
xrxixixr
jtjxixr
tjxixrx
zizrz
yiyryxixrx
tj
tj
,
0,
)/(tan , )cos(
]Re[])Re[(
)(
)()(),,(
]),,(Re[),,,(
: then,variationse Assume
122
22
100
Boundary Conditions at a General Material Interface
s
s
s
sttnn
s
snn
stt
JHHn
MEEn
BBn
DDn
JHHBB
DD
MEE
)(ˆ
)(ˆ
0)(ˆ
)(ˆ
;
Density Charge Surface
21
21
21
21
2121
21
21
D1n
D2n
h
s
h
E1t
E2t
101
)(ˆ)(ˆ
)(ˆ)(ˆ
)(ˆ)(ˆ
)(ˆ)(ˆ
0 ;
0
0
21
21
21
21
2121
21
21
HnHn
EnEn
BnBn
DnDn
HHBB
DD
EE
ttnn
nn
tt
Fields at a Dielectric Interface
102
HnHJ
B
Dn
ts
n
ˆ
0Bn 0
ˆρD
0En oE
:ConductorPerfect aat ConditionsBoundary
sn
t
+ + +n
s
Js
Ht
103
0)(ˆ
)(ˆ
0)(ˆ
0)(ˆ
Hn
MEn
Bn
Dn
s
104
2/ ; 0
; 0
:medium free Sourcea For
)(
)(
22
2222
2
vkHkH
kEkE
EjJj
HjEEE
Wave Equation
105
zayaxar
kakakakAeE
kkk
zyxE
zyxiEkz
E
y
E
x
E
Ekz
E
y
E
x
EEkE
zyx
zzyyxxzjkyjkxjk
x
zy
x
iiii
zys
Let ,
k
variablesof separation Using),,,(for Solve
,,, 0
0
20
222x
202
2
2
2
2
2
202
2
2
2
2
220
2
106
space. free of admittance intrinsic theis
377 space free of impedance interensic theis
1
11
1
waveplane called issolution The
.kn propagatio ofdirection thelar toperpendicu is vector The
00 Since
,Similarly ,
0
0
00
0
0
0
0
00
00
00
0
0
00
Y
EnEnYEnEnk
eEkeEj
eEj
H
HjE
E
EkEeEE
CeEBeEAeE
rkjrkjrkj
rkj
rkjz
rkjy
rkjx
107
s
j
1)1(
2j)(1
j
2
1
2j)(1
j jj
s
108
The field amplitude decays exponentially from its surface According to e-u/
s where u is the normal distance into theConductor, s is the skin depth
Hn ,1
: Impedance surface The
EJ , 2
msmts
m
s
ZJZEj
Z
109
Parallel Polarization
z
x
Ei
Et
Er
3
1
2
n1
n3
n20
0
0000
202
101
Y,
,
, 120
110
k
EnYHeEE
EnYHeEE
rrrnjk
r
iirnjk
i
110
sin , cos
sin , cos
sin , cos
sin sin
cossin
cossin
cossin
, ,
,
331333
221222
111111
3121
333
222
111
3032010
00
3313
EEEE
EEEE
EEEE
n
aan
aan
aan
nnkknnknk
nkknYY
EnYHeEE
zx
zx
zx
zx
zx
zx
xxxx
ttrnjk
t
111
Under steady-state sinusoidal time-varying Conditions, the time-average energy stored in theElectric field is
V
e
VV
e
dVEEW
dVEEdVDEW
*
**
4
thenreal, andconstant is If
4
1
4
1Re
112
S
V
V
m
dSHEP
dVHH
dVBHW
*
*
*
Re2
1
:by given is S surface
closeda across smittedpower tran average timeThe
constant and real is if 4
4
1Re
:is field magnetic thein storedenergy average Time
113
dVMHJEdVDEHB
j
dVMHJEdVDEHBj
dSHEVdHE
EJJ
JEEDjHMBj
EHHEHE
V
s
V
s
VV
SV
s
)(2
1
442
)(2
1)(
2
2
1
2
1
)(
)()(
****
****
**
***
***
114
dSHEP
dVEEHHj
dVEEHHdVEE
dSHEdVMHJE
j
S
V
VV
S
S
V
Ss
*0
**
***
**
2
1
)(2
)(22
1
2
1)(
2
1-
ty conductivi and j- ,
:by zedcharacteri is medium theIf
115
volume.
in the storedenergy reactive the times2 and )( volumein the
heat lost topower the,P surface he through tsmittedpower tran the
of sum the toequal is )(P sources by the deliveredpower The
)(2
)(2
442
2
1Im
)(2
1P
0
s
0
***
*s
P
WWjPPP
WW
dVEEHH
dSHE
dVMHJE
ems
em
VS
ss
V
s
losspower average Time
2
1)(
2***
dVEEdVEEHHPVV
116
C
L RI
V
networka of impedance theof n DefinitioGeneral
2
1)(2P
)(2P
)4
1
4
1(2
2
1
)(2
1
2
1
2
1
*
2
***
***
II
WWjZ
WWjC
IILIIjRII
C
jLjRIIZIIVI
em
em
117
22
22
22
22
2
0D
equation. HelmholtzousInhomogene
condition) (Lorentz or
Let ,
,
1
, 0
,Let
k
JAkA
jA
jAk
JjAkAAA
JjAJEjAH
AjE
AjEAjE
AjBjEAB
118
Solution For Vector Potential
(x,y,z)(x’,y’, z’) R
rr’
J
VdR
erJrA
rrzzyyxxR
VdR
erJrAzyxA
jkR
V
jkR
)(4
)(
)()()(
current alinfinitism anfor )(4
)(),,(
222
119
Rg
z
Ldz
Cdz
I(z,t)
V(z,t)V(z,t)+v/z
dz
I(z,t)+I/z dz
Lumped element circuit model for a transmission line
120
Impedance sticCharacteri:
C
L , ,
)()(),(
)()(),(
1
0),(),(
0),(),(
2
2
2
2
2
2
2
2
c
ccc
Z
ZZ
VI
Z
VI
v
ztfI
v
ztfItzI
v
ztfV
v
ztfVtzV
LCv
t
tzILC
z
tzI
t
tzVLC
z
tzV
121
LCC
L
YZ
VYIVYIeIeIzI
eVeVzV
vzV
vdz
zVd
zCVjz
zI
zLIjz
zV
tVtV
cc
cczjzj
zjzj
gg
, 1
, ,)(
)(
, 0)()(
)()(
)()(
cos)(
2
2
2
2
122
ZL
Zc
Z
To generator
1/
1/ ,
1
1
tcoefficien on Reflecti
)(1
L
cL
cL
c
L
L
L
L
cL
LL
L
ZZ
ZZ
Z
ZV
V
VVZZ
VIIII
VVVV
123
tan
tan
1
1
)2
(sin41
,
)1(2
1
)1)(1(Re2
1)Re(
2
1
2/122
22
*2*
Lc
cL
c
inin
jL
zjjzj
zjL
zj
Lc
LLcLL
jZZ
jZZ
Z
ZZ
S
lVV
eeVeeV
eVeVV
VY
VYIVP
124
Transmission Lines & Waveguides
Wave Propagation in the Positive z-Direction is Represented By:e-jz
,
,
)(
)()()(
,,
,,,,,,
,,
,,,,,,
zttztt
zttztt
zjztzzztzt
zjzt
zjztzt
zjz
zjt
zt
zjz
zjt
zt
ejehjh
ejhhje
ehhjeajeeaje
ehhjeeeajE
eyxheyxh
zyxHzyxHzyxH
eyxeeyxe
zyxEzyxEzyxE
125
Modes Classification:
1. Transverse Electromagnetic (TEM) Waves
0 zz HE
2. Transverse Electric (TE), or H Modes
0but , 0 zz HE
3. Transverse Magnetic (TM), or E Modes
0But , 0 zz EH
4. Hybrid Modes
0 , 0 zz EH
126
TEM WAVES
zjz
zjtt
zjt
zjtt
t
t
t
tt
ttt
eeaYehH
eyxeeE
yx
yxyxe
h
e
eh
ˆ
),(
0,
entialScalar Pot 0,,
eha , 0
hea , 0
0 , 0
0
2
t0tzt
t0tz
t
127
wavesTEMfor
0])([ , 0)(
, but , 0
:equation Helmholtzsatisfy must field The
direction z-or in then propagatio for wave
H
E
Impedance Wave , 1
0
20
2t
20
2
222t
20
2
0y
x
00
0
k
kEkE
ajEkE
H
E
ZY
tttt
tztt
x
y
128
TE WAVES
0
, 0
,
0
let , 0)(
0),()(
0
,
22
222222
222
22
ttztt
tzztztt
ttzztt
zczt
czt
zzt
ehjh
ejhajhah
heahje
hkh
kkhk
hkyxh
HkH
129
hx
y
h
tztzt
ztc
t
Zh
e
kZ
haZk
hae
hk
jh
y
x
0
000
2
h
e
Impedance Wave
; ˆˆ
130
Admittance Wave
ˆ
0
let , 0)(
0),()(
0
0
2
22
222222
222
22
Yk
Y
eaYh
ek
je
eke
kkeke
ekyxe
EkE
e
zet
ztc
t
zczt
czzt
zzt
TM WAVES
131
TEM TRANSMISSION LINES
Parallel -plate Two-wire Coaxial
ab
COAXIAL LINES
132
a b
0
jkz-00
jkz-r
0
0
021
2
2
2
e a )/ln(
and e a )/ln(
)/ln(
)/ln(
0at 0,at ln
0for 01
)(1
Y
abr
VYH
abr
VE
ba
brV
rarVCrCrr
rrr
133
)/ln(ˆRe
2
1
e)/ln(
2e
)/ln(I
eˆ)/ln(
ˆˆ
200
2
0
*
jkz-2
0
00
jkz-00
jkz-00
ab
VYrdrdaHEP
ab
VYad
aba
VY
aaba
VYHaHnJ
z
b
a
zrs
• THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0
Ohms ln2
1
00
0
a
b
YI
VZ c
Zc OF COAXIAL LINE AS A FUNCTION OF b/a
134
r Zo=X
b/a
1
10
1000
20
40
60
80
100
120
140
160
180
200
220
240
260
135
EJ
YY
kkEaYHeE
kk
kkjjkjjjk
j
jYYjkk
rr
rzjkz
t
r
r
r
r
rr
r
rr
rrr
rrrr
ty conductivi the toequivalent is
, ˆ ,
, 2
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losses small For )(
)( and )(
0
0
00
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2/10
0
2/10
2/10
136
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*
**
0
0
*
0
**
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Re2
1
2Re
2
1 ,
1
:lossconductor the todue losspower The
222 ,
2
2- ,
22
1
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2121
YYab
ab
ab
YR
SdHEP
dHHR
dJJZPj
Z
kYY
dSEEY
P
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PePP
dSEEdSJJP
m
s
SS
sm
s
SS
sms
m
r
r
r
d
S
z
SS
Qc OF COAXIAL LINE AS A FUNCTION OF Zo
137
Q-C
op
per
of
Coaxia
l Lin
e
2000
2200
2400
2600
2800
3000
3200
34000 10
20
30
40
50
60
70
80
90
10
0
11
0
12
0
13
0
14
0
15
0
16
0
17
0
18
0
r Zc
GHz
c
fb
Q
138
w
y
x
d
jkzww
s
jkzd
yjkz
jkzjkz
t
t
ed
wVdxzHydxzJI
edyEVed
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VyeyxeyxE
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d
yVyx
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0
00
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00
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2
ˆ)ˆ(ˆ
V ,ˆˆ),(
ˆ),(),(
ˆ),( , ),(
Vd)(x,
0,(x,0) By Ay)(x,
dy0
w,x0 0),(
TEM Modes
139
zj-
zj-
22
22
2
e cos),,(
e sin),,(
sin),(
)(
,....3,2,1,0 , , 0B
d 0,yat 0),(
cossin ),(
0),(
yd
nA
k
jzyxH
yd
nAzyxE
yd
nAyxe
d
nK
nndk
yxe
ykBykAyxe
yxeky
nc
x
nz
nz
c
z
ccz
zc
140
0nfor 2
)Re(
0nfor 4
)Re(2
1ˆ
2
1
2 ,
Z
:is modes TM theof impedance waveThe
22
0E ,e cos),,(
2
2
2
2
*
0 00 0
*0
g
TM
xzj-
nc
nc
x
w
x
d
y y
w
x
d
y
p
x
y
cc
ync
y
Ak
d
Ak
d
dydxHEdydxzHEP
v
kH
E
d
kf
Hyd
nA
k
jzyxE
141
0nfor Np/m 22
22
2
lossconductor toduen Attenuatio
2
2
222
0
0c
d
kR
d
R
Ak
wRdxJ
RP
P
P
ssc
nc
sw
x ss
142
zj-
zj-
22
22
2
e sin),,(
e cos),,(
cos),(
)(
,....3,2,1 , , 0A
d 0,yat 0),(
cossin ),(
0),(
yd
nB
k
jzyxE
yd
nBzyxH
yd
nByxh
d
nk
nndk
yxe
ykBykAyxh
yxhky
nc
x
nz
nz
c
x
ccz
zc
143
Np/m 2
0nFor )Re(4
2
1ˆ
2
1
2 ,
Z
:is modes TM theof impedance waveThe
22
0E ,e sin),,(
2
c
2
2
*
0 00 0
*0
g
TE
yzj-
dk
Rk
Bk
dw
dydxHEdydxzHEP
v
k
H
E
d
kf
Hyd
nB
k
jzyxH
sc
nc
y
w
x
d
y x
w
x
d
y
p
y
x
cc
xnc
y
144
COUPLED LINES EVEN & ODDMODES OF EXCITATIONS
AXIS OF EVEN SYMMETRY AXIS OF ODD SYMMETRY
P.M.C. P.E.C.
EVEN MODE ELECTRICFIELD DISTRUBUTION
ODD MODE ELECTRIC FIELD DISTRIBUTION
eZ0 oZ0=EVEN MODE CHAR. IMPEDANCE
=ODD MODE CHAR. IMPEDANCE
Equal currents are flowing in the two lines
Equal &opposite currents areflowing in the two lines