Transcript Part 1

Part-1
EC 2305 (V sem)
Transmission Lines and Waveguides
17.7.13
Dr.N.Gunasekaran
Dean, ECE
Rajalakshmi Engineering College
Thandalam, Chennai- 602 105
[email protected]
Energy Transfer
All the systems are designed to carryout the
following jobs:
1.Energy generation.
2. Energy transportation.
3. Energy consumption.
Here we are concerned with energy transfer.
Electrons
Electron is part of everything on earth. Electrons
are the driving force for every activity on earth.
Electron is a energy packet, Source of energy,
capable of doing any work.
Electron accumulation = Voltage
Electron flow
= current
Electrons’ oscillation = Wave
Electron transfer
= Light
Electron emission
= Heat.
No mass ;
No inertia;
Highly mobile;
No wear and tear;
No splitting of
electron;
No shortage;
Excellent service
under wider
different conditions:
Vacuum, gas, solid;
Controlled by
Fields :
accelerated,
retarded, change
directions,
increase and
decrease of stream
of electrons;
instant reaction
due to zero inertia.
Energy = Electron - Wave
Energy is transferred from place to by two
means:
1. Current : Flow
conductors.
of
electrons
through
2. Wave : Wave propagation in space, using
guiding systems or unguided system (free
space).
In this subject, except free space energy
transfer, other means are discussed.
Electron
-
waves
Major Topics for discussion
i) Circuit domain ( Filters )
ii) Semi Field domain (Transmission Line : VoltageCurrent – Fields)
iii)More Field domain (Coaxial line)
iv)Field domain : TEM waves ( Parallel plate guiding)
v) Fully Field domain : TE-TM modes ( Waveguide )
Transmission Line – Waveguide
Guided communication
System
Frequency
Energy Flow
Circuits
LF, MF, HF Inside Conductor
Transmission Lines
VHF
Outside Cond.
Coaxial Lines
UHF
Outside Cond.
Waveguides
SHF
Outside Cond.
Optical Fiber
1015 Hz Inside Fiber
Energy
V = Voltage = Size of energy packet / electron.
I = Current = Number of energy packet flow / sec
Total energy flow / sec = V X I
System
Power Flow
Medium
Circuits
P=VxI
Conductor
Transmission Lines
P=ExH
Free space
Coaxial Lines
P=ExH
Free space
Waveguides
P=ExH
Free space
Optical Fiber
P=ExH
Glass
Quantum of energy E = h f; h =6.626x10-34 J-s
Quantum physics states the EM waves are
composed of packets of energy called photons.
At high frequencies each photon has more energy.
Photons of infrared, visible, and higher frequencies
have enough energy to affect the vibrational and
rotational states of molecules and electrons in the
orbits of atoms in the materials.
Photons at radio waves do not have enough energy
to affect the bound electrons in the materials.
System
Energy Flow
Circuits
Inside Conductor
Transmission Lines
TEM mode
Coaxial Lines
TEM mode
Waveguides
TE and TM modes
Optical Fiber
TE and TM modes
Problems at high frequency operation
1.Circuits radiates and accept radiation : Information
loss. Conductors become guides, current’s flow
becomes field flow
2.EMI-EMC problems: Aggressor – Victim problems
3.Links in circuit behave as distributed parameters.
4. Links become transmission Line: Z0 , ρ, .
5.Delay – Phase shift-Retardation.
6. Digital circuits involves high frequency problems.
7. High energy particle behaviour.
High Frequency Effects
1.Skin effect
2.Transit time –
3.Moving electron induce current
4. Delay
5. Retardation-.Radiation
6.Phase reversal of fields.
7.Displacement current.
8.Cavity
High Frequency effects
1.Fields inside the conductor is zero.
2.Energy radiates from the conductors.
3.Conductor no longer behaves as simple
conductor with R=0
4.Conductor offers R, L, G, C along its length.
5.Signal gets delayed or phase shifted.
Skin Effect
Skin effect makes the current flow simply a surface
phenomenon. No current that vary with time can
penetrate a perfect conducting medium. Iac = 0
The penetration of Electric field into the conducting
medium is zero because of induced voltage effect.
Thus inside the perfect conductor E = 0
The penetration of magnetic field into the conducting
medium is zero since current exists only at the
surface. H=0.
Circuits Radiate at high
frequency opearation
D →λ
Skin Effect
As frequency increases, current flow becomes a
surface phenomenon.
Conductor radiates at high frequencies
Circuit theory Model
OR
Lumped Model
(   100s Km ); ( D <<  )
 Is our scale
•
Frequency f
50 Hz
3 KHz
30 KHz
300 KHz
3 MHz
30 MHz
300 MHz
3 GHz
30 GHz
300 GHz
Wavelength 
6,000 Km
100 Km
10 Km
1 Km
100 m
10 m
1m
10 cm
1 cm
1 mm
V= V0 sin
(0 )
V= V0 sin (90)
V= V0 sin (360)
V= V0 sin (180)

Circuit domain :Dimension << 
C= f x  = 300,000 km/sec
Given f = 30 kHz ;  = 10 km
Hence circuit dimensions <<  = 10 km
Medium = Conducting medium.
= Conductors in circuits.
Electrons = Energy Packet
Energy E = eV electron volts;
W= V X I
Circuit Theory
Connecting wires introduces no drop and no delay. The wires
between the components are of same potential. Shape and
size of wires are ignored.
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For f =3 KHz,
 = 10 Km
R
0o
180o
360o
 = 10 Km
At 3 KHz No Phase variation across the Resistor
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D <
;
D << 
• When circuit dimension is very small
compared to operating wavelength ( D <<  ) ,
circuit theory approximation can be made.
• No phase shift the signal undergoes by virtue
of distance travelled in a circuit.
• Circuit / circuit components/ devices/ links will
not radiate or radiation is very negligible.
Field domain : Dimension  
C= f x  = 300,000 km/sec
Given f = 3000 MHz ;  = 10 cm
Hence circuit dimensions   = 10 cm
Dielectric medium – Free space
Waves = E/H fieldes
Energy E = h.f joules
Total radiated power W =  EXH ds joules
Lumped circuit Model
• Electric circuits are modeled by means of lumped
elements and Kirchhoff’s law.
• The circuit elements R, L, C are given values in
those lumped circuit models, for example R=10
K, L = 10 H c= 10 pf.
• These models are physical elements and hence
the element values depend on the structure and
dimensions of the physical elements.
For f =30 GHz,
0o
 = 1cm
360o
180o
 = 1cm
Resistor
0o
180o
360o
At 30GHz 360o Phase variation across the Resistor
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Balanced transmission line opened out to
form dipole radiator
Reactive drop
Transmission Line
Voltage Variation along
the line
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Frequency dependent parasitic elements
At high frequency operation all ideal
components deviate from their ideal behavior
mainly due to parasitic capacitance and
parasitic inductance.
Any two conductors separated by some
dielectric will have capacitor between them.
Any conductor carrying current will have an
inductance.
Reactance XC and XL
Parasitic capacitance and parasitic inductance
create reactance that varies with frequency
j
XC 
2fC
X L  j 2fL
At DC, capacitance impedance is infinity; an open
circuit. The capacitive reactance decreases with
frequency. At DC an inductive impedance is zero; a
short circuit. The impedance of inductive reactance
increase with frequency.
Thus these real components behave different at high
frequency operation.
Cp =Parasitic capacitance due to leads of resistor,
parallel to R. At high frequency it shunts the resistor
reducing its value.
Llead = Due to resistor and material of resistor.
High value R are not recommended for high frequency
operation.
Caution: Minimize the lead size, Use surface mounted
device.
Llead = Lead inductance
Rlead = Lead resistance
RDC = Dielectric leakage
RAC =Dielectric Frictional loss due to polarization.
At high frequency operation, the component acts as
L. Large values of C are not useful at high frequency
operation.
RL =Lead Resistance
CL =Lead capacitance
Rcore =Core loss resistance
Phase Shift in Transmission Line
Space Effect
0o
180o
360o

Magnitude of 
C = f met = 300
MHz
For f = 3 KHz,
 = 100 KM
For f =3 GHz,
 = 10cm
For f =30 GHz,
 = 1cm
C=fx
For f =3 KHz,
 = 10 Km
R
0o
180o
360o
 = 10 Km
At 3 KHz No Phase variation across the Resistor
Circuit Theory
Connecting wires introduces no drop and no delay. The wires
between the componenets are of same potential. Shape and
size of wires are ignored.
For f =30 GHz,
0o
 = 1cm
360o
180o
 = 1cm
Resistor
0o
180o
360o
At 30GHz 360o Phase variation across the Resistor
Filters
Any complicated network with terminal
voltage and current indicated
A T network which may be made
equivalent to the network in the box (a)
A  network equivalent to (b) and (a).
The T section as derived from
unsymmetrical L-sections, showing notation
used in symmetrical network analysis
The  section as derived from
unsymmetrical L-sections, showing notation
used in symmetrical network analysis
Examples of Transmission Line
Transmission Line in communication carry
1)Telephone signals
2)Computer data in LAN
3)TV signals in cable TV network
4)Telegraph signals
5)Antenna to transmitter link
TRASMISSION LINE
• It is a set of Conductors used for transmitting
electrical signals.
• Every connection in an electrical circuit is a
transmission line.
• Eg: Coaxial line, Twisted-wire
•
Parallel wire pairs
•
Strip line , Microstrip
A succession of n networks in cascade.
Two types of transmission lines.
Basic Transmission Line.
A transmission line whose load impedance is
resistive and equal to the surge impedance
appears as an equal resistance to the
generator.
Infinite parallel plane transmission line.
Transmission line is low pass
filter
Any complicated network can be
reduced to T or  network
T and  Network
Resonant circuit and Filter
Resonant circuits select relatively narrow band
of frequencies and reject others.
Reactive networks, called filters, are designed
to pass desired band of frequencies while
totally suppressing other band of frequencies.
The performance of filter circuits can be
represented in terms of Input current to output
current ratios.
Image Impedance Non-Symmetry Network
Input impedance at the 1,1 terminalZ1in  Z1i 
Z1in
Z1 ( Z 2  Z 2 i )
 Z1i Z  Z1 
Z 2  Z 3  Z 2i
Likewise, the impedance looking into the 2,2
Z 2i to be
terminal is required
Z 3 ( Z1  Z1i )
Z 2i  Z 2 
Z1  Z 3  Z1i
Upon solving for Z1i andZ 2i
( Z1  Z 3 )( Z1Z 2  Z 2 Z 3  Z 3 Z1 )
Z1i 
Z 2  Z3
( Z 2  Z 3 )( Z1Z 2  Z 2 Z 3  Z 3 Z1 )
Z 2i 
Z1  Z 3
Z1oc  Z1  Z 2
Z1sc
Z 2 Z3
Z1 Z 2  Z 2 Z 3  Z 3 Z1
 Z1 

Z 2  Z3
Z 2  Z3
Z1i  Z1oc Z1sc
Z 2i  Z 2oc Z 2 sc
If the image impedances are equal
Z1i  Z 2i
then
V1i V2o

I1i I 2o
Then the voltage ratios and current ratios can
be represented by
I1
V1

I 2 V2
(1)
Performance of
Unsymmetrical
T &  Networks
Performance parameters of a Network
(Active or Passive)
1. Gain of Loss of signal due to the Network
in terms of Voltage or Current ratios.
V1
 A
V2
I1
 B
I2
2. Delay of phase shift of the signal due to
network.
Performance of a N networks in
cascade
If several networks are used in succession as in
fig., the overall performance may be
appreciated
V3a
Vn1 V1
V1 V2 as
V2
X
V3
X
V4
X .....
Vn

Vn
(2)
Which may also me stated as
A1 . A2 .1A3 . A4  A1 A2 A3 A4      
Both the processes employing multiplication of
magnitudes. In general the process of addition or
subtraction may be carried out with greater ease
than the process of multiplication and division. It is
therefore of interest to note that

e xe xe x.....e  e
b
c
n
a  b  c ....  n
Is an application in which
substituted for multiplication.
addition
is
If the voltage ratios are defined as
V1
a V2
b V3
c
e ;
e ;
 e ;.......etc
V2
V3
V4
Eq. (2) becomes
V1
a  b  c ........  n
e
Vn
If the natural logarithm (ln) of both sides is
taken, then
V1
ln
 a  b  c  d ..........  n
V2
(3)
Thus it is common to define under conditions
of equal impedance associated with input and
output circuits.
V1
I1
N

e
V2
I2
(4)
The unit of “N” has been given the name
nepers and defined as
V1
I1
N
 ln
nepers  ln
V2
I2
(5)
Two voltages, or two currents, differ by one
neper when one of them is “e” times as large as
the other.
Obviously, ratios of input to output power may
also may also be expressed In this fashion. That
is,
P1
 e2 N
P2
The number of nepers represents a convenient
measure of power loss or power gain of a
network.
Losses or gains of successive
Transmission Line
1.It provided guided communication to distance
with reasonable minimum attenuation
2.It overcomes the parasitic effects of lumped
elements due to high frequency operation.
3. High frequency operation introduces
distributed parameter effect.
4.Due to high frequency operation, energy
carried by fields rather than voltage and
currents.
5. Operation remains outside conductors.
6. Radiation and phase shift (delay) play
important roles.
7. Radiation effects are much reduced or
prevented by special arrangements.
8. Treating Tr.Line as infinite infinitesimal
symmetrical networks, network theory analysis
is adopted.
Analysis of Transmission line ( N
networks in cascade) based on
basic symmetrical T and 
networks
Transmission line is low pass
filter
Any complicated network can be
reduced to T or  network
T and  Network
Resonant circuit and Filter
Resonant circuits select relatively narrow band
of frequencies and reject others.
Reactive networks, called filters, are designed
to pass desired band of frequencies while
totally suppressing other band of frequencies.
The performance of filter circuits can be
represented in terms of Input current to output
current ratios.
Image Impedance Non-Symmetry Network
Input impedance at the 1,1 terminalZ1in  Z1i 
Z1in
Z1 ( Z 2  Z 2 i )
 Z1i Z  Z1 
Z 2  Z 3  Z 2i
Likewise, the impedance looking into the 2,2
Z 2i to be
terminal is required
Z 3 ( Z1  Z1i )
Z 2i  Z 2 
Z1  Z 3  Z1i
Upon solving for Z1i andZ 2i
( Z1  Z 3 )( Z1Z 2  Z 2 Z 3  Z 3 Z1 )
Z1i 
Z 2  Z3
( Z 2  Z 3 )( Z1Z 2  Z 2 Z 3  Z 3 Z1 )
Z 2i 
Z1  Z 3
Z1oc  Z1  Z 2
Z1sc
Z 2 Z3
Z1 Z 2  Z 2 Z 3  Z 3 Z1
 Z1 

Z 2  Z3
Z 2  Z3
Z1i  Z1oc Z1sc
Z 2i  Z 2oc Z 2 sc
If the image impedances are equal
Z1i  Z 2i
then
V1i V2o

I1i I 2o
Then the voltage ratios and current ratios can
be represented by
I1
V1

I 2 V2
(1)
Performance of
Unsymmetrical
T &  Networks
Part-2
EC 2305 (V sem)
Transmission Lines and Waveguides
24.7.13
Dr.N.Gunasekaran
Dean, ECE
Rajalakshmi Engineering College
Thandalam, Chennai- 602 105
[email protected]
Filters
Filters -Resonant circuits
Resonant circuits will select relatively narrow
bands of frequencies and reject others.
Reactive networks are available that will freely
pass desired band of frequencies while almost
suppressing other bands of frequencies.
Such reactive networks are called filters.
.
Ideal Filter
An ideal filter will pass all frequencies in a given
band without (attenuation) reduction in
magnitude, and totally suppress all other
frequencies. Such an ideal performance is not
possible but can be approached with complex
design.
Filter circuits are widely used and vary in
complexity from relatively simple power supply
filter of a.c. operated radio receiver to complex
filter sets used to separate the various voice
channels in carrier frequency telephone
circuits.
Application of Filter circuit
Whenever alternating currents occupying
different frequency bands are to be
separated, filter circuits have an application.
Neper - Decibel
In filter circuits the performance Indicator is
Performance 
Input current
Output current
If the ratios of voltage to current at input and
output of the network are equal then
I1
V1

I2
V2
(1)
If several networks are used in cascade as
shown if figure the overall performance will
become
V3
Vn 1
V1
V2
V1
X
X
X .....

V2
V3
V4
Vn
Vn
(2)
Which may also me stated as
A1 . A2 .1A3 . A4  A1 A2 A3 A4      
Both the processes employing multiplication of
magnitudes. In general the process of addition or
subtraction may be carried out with greater ease
than the process of multiplication and division. It is
therefore of interest to note that

e  e  e  .....e  e
b
c
n
a  b  c ....  n
is an application in which
substituted for multiplication.
addition
is
If the voltage ratios are defined as
V1
a V2
b V3
c
e ;
e ;
 e ;.......etc
V2
V3
V4
Eq. (2) becomes
V1
a  b  c ........  n
e
Vn
If the natural logarithm (ln) of both sides is
taken, then
V1
ln
 a  b  c  d ..........  n
V2
(3)
Consequently if the ratio of each individual
network is given as “ n “ to an exponent, the
logarithm of the current or voltage ratios for all
the networks in series is very easily obtained as
the simple sum of the various exponents. It has
become common, for this reason, to define
V1
I1

 eN
V2
I2
(4)
under condition of equal impedance
associated with input and output circuits
The unit of “N” has been given the name
nepers and defined as
nepers
N
V1
I1
 ln
 ln
V2
I2
(5)
Two voltages, or two currents, differ by one
neper when one of them is “e” times as large as
the other.
Obviously, ratios of input to output power may
also may also be expressed In this fashion. That
is,
P1
2N
e
P2
The number of nepers represents a convenient
measure of power loss or power gain of a
network.
Loses or gains of successive networks then
may be introduced by addition or subtraction of
their appropriate N values.
“ bel “ - “ decibel “
The telephone industry
popularized a similar unit
to the base 10, naming
Alexander Graham Bell
The “bel” is defined as
power ratio,
P1
number of bels = log
proposed and has
based on logarithm
the unit “ bel “ for
the logarithm of a
P2
It has been found that a unit, one-tenth as large,
is more convenient, and the smaller unit is called
the decibel, abbreviated “db” , defined as
P1
dB  10 log
P2
(6)
In case of equal impedance in input and output
circuits,
I1
V1
dB  20 log  20 log
I2
V2
(7)
Equating the values for the power ratios,
e  10
2N
dB
10
Taking logarithm on both sides
8.686 N = dB
Or 1 neper = 8.686 dB
Is obtained as the relation between nepers
and decibel.
The ears hear sound intensities on
logarithmically and not on a linear one.
a
Part-3
EC 2305 (V sem)
Transmission Lines and Waveguides
31.7.13
Dr.N.Gunasekaran
Dean, ECE
Rajalakshmi Engineering College
Thandalam, Chennai- 602 105
[email protected]
Performance parameters of a
“series of identical networks”.
1.Characteristic Impedance
Z0
2. Propagation constant

For efficient propagation, the network is to be
terminated by Z0
and the propagation
constant  should be imaginary.
We should also attempt to express these
two performance constants in terms of
network components Z1 and Z2 .
What is
Characteristic impedance of
symmetrical networks
Symmetrical T section from
L sections
For symmetrical network the series arms of T
network are equal
Z1  Z 2 
Z1
2
Symmetrical  from L sections
Z a  Z c  2Z 2
Both T and  networks can be considered as
built of unsymmetrical L half sections, connected
together
in one fashion for T
and oppositely for the  network.
A series connection of several T or  networks
leads to so-called “ladder networks”
which are indistinguishable one from the other
except for the end or terminating L half section
as shown.
Ladder Network made from T section
Ladder Network built from  section
The parallel shunt arms will be combined
For a symmetrical network:
Z1i  Z 2i
the image impedance Z1i and
Z 2 i are
equal to each other and the image impedance
is then called characteristic impedance or
iterative impedance,
.
Z1i  Z 2i  Z it  Z o
That is , if a symmetrical T network is
terminated in Z 0 , its input impedance will
also be Z 0 , or the impedance transformation
ration is unity.
If ZR  Z 0 then Z i  Z 0
Zi  Z0
Z R  Z0
The term iterative impedance
terminating impedance Z 0
the input impedance of a
networks in which case Z 0
input to each network.
is apparent if the
is considered as
chain of similar
is iterated at the
Z R  Z 0  Z it  Z in
Characteristic Impedance
of Symmetrical T section network
For T Network terminated in Z 0
Z
1
Z
(
 Z0 )
2
Z1
2
Z1in  
2 Z1  Z  Z
2
0
2
When Z1in  Z 0
Z0 
Z12
4
2
1
 z1 z 2  z 2 z0 
Z1
2  z 2  z0
Z
Z 
 Z1 Z 2
4
2
0
Z1 Z 0
(9)
2
Characteristic Impedance
for a symmetrical T section
Z 0T
Z12
Z1

 Z1Z 2  Z1Z 2 (1 
4
4Z 2
(!0)
Characteristic impedance
Z 0 is that
impedance, if it terminates a symmetrical
network, its input impedance will also be Z 0
Z0
is fully decided by the network’s intrinsic
properties, such as physical dimensions and
electrical properties of network.
Characteristic Impedance
 section Z 0
Z1in

2Z 2 Z 0 
 Z1  ( 2 Z  Z )  2 Z 2
2
0 


2Z 2 Z 0
Z1 
2Z 2  Z 0  2Z 2
When Z1in  Z 0
, for symmetrical 
Characteristic
Impedance
Z1Z 2

Z
1 1
4Z 2
Z 0
(11)
Z1
Z1oc  Z oc   Z 2
2
Z1Z 2
Z1
2
Z1sc  Z sc  
2 Z1  Z
2
2
Z12
Z oc Z sc 
 Z1Z 2  Z 02T
4
2
2
4Z Z1
2
Z 0c Z sc 
 Z 0
Z1  4Z 2
Z0  Zoc Z sc
(13)
(12)
propagation constant 
V1
I1

 eN
V2
I2
The magnitude ratio does not express the
complete network performance , the phase
angle between the currents being needed as
well.
The use of exponential can be extended to
include the phasor current ratio.
I1

e
I2
(14)
Where

is a complex number defined by
    j
Hence
If
(15)
I1

  j
e e
I2
I1
 A 
I2
I1

A
e
I2
 e
j
With Z0 termination, it is also true,
V1
 e
V2
The term 
has been given the name
propagation constant

= attenuation constant, it determines
the magnitude ratio between input and
output quantities.
= It is the attenuation produced in
passing the network.
Units of attenuation is nepers
= phase constant. It determines the phase
angle between input and output
quantities.
= the phase shift introduced by the
network.
= The delay undergone by the signal as it
passes through the network.
= If phase shift occurs, it indicates the
propagation of signal through the network.
The unit of phase shift is radians.
If a number of sections all having a common Z
the ratio of currents is
I1 I 2 I 3
I1
   ........ 
I 2 I3 I 4
In
1
2
3
from which
e  e  e  ........  e
n
and taking the natural logarithm,
 1   2   3   4 ..................   n
(16)
Thus the overall propagation constant is equal to
the sum of the individual propagation constants.
Z 0 and  of symmetrical networks
 and the introduction of
Use the definition of

e as the ratio of currents for a
Z 0 termination leads to useful results
The T network in figure is considered
equivalent to any connected symmetrical
network terminated in a Z 0 termination.
From the mesh equations the current ratio
can be shown as
I1

I2
Z1
2
 Z2  Z0
Z2
e

(30)
Where the characteristic impedance is given
2
as
Z
Z 02 
1
4
 Z1 Z 2
(32)
Eliminating Z 0
Z1
cosh   1 
2Z 2
(33)

Z1
sinh 
2
4Z 2
(36)
The propagation constant can be related to
network parameters by use of (10) for Z OT
In (30) as
Z1
e  1 2

2Z 2

Z1 2 Z1
(
) 
2Z 2
Z2
Taking the natural logarithm

Z1

  ln 1 


2Z 2


 Z1 
Z1 

 2Z 
 Z 
2 
2


2
For a network of pure reactance it is not difficult
to compute.
The input impedance of any T network
terminated in any impedance ZR , may be
written in terms of hyperbolic functions of .
Writing
Z in
2
12
Z
 Z11 
Z 22
It is reduced to
 Z R cosh   Z0 sinh  

Z in  Z 0 
 Z0 cosh   ZR sinh  
(39)
For short circuit, Z R = 0
Z SC  Z 0 tanh 
For a open circuit Z R  
Z lim Z 
Z0

tanh 
(40)
(41)
From these these two equations it can be
shown that
Z SC
tanh  
Z OC
(42)
Z 0  Z OC Z SC
Thus the propagation constant  and the
characteristic impedance Z0 can be evaluated
using measurable parameters
Z SC and Z OC
Filter fundamentals
Pass band – Stop band:
The propagation constant is
    j
For  = 0 or I1  I 2
There is no attenuation , only phase shift occurs.
It is pass band.
when    ve; I1  I 2 , attenuatio n occurs;
- Stop band
 Is conveniently studied by use of the
expression.

Z1
sinh 
2
4Z 2
It is assumed that the network contains only
pure reactance and thus Z1 4Z will be real
2
and either positive or negative, depending on
the type of reactance used for Z1 and Z2
Expanding the above expression


j
sinh  sinh ( 
)
2
2 2
 sinh

cos

 j cosh
2
2
It contains much information.

2
sin

2
If Z1 and Z2 are the same type reactances then
Z1
 0 or the ratio is positive and real.
4Z 2
This condition implies a stop or attenuation band
of frequencies.
The attenuation will be given by
  2 sinh
1
Z1
4Z 2
If Z1 and Z2 are opposite type reactances then
Z1
 0 or the radical is imaginary.
4Z 2
This results in the following conclusion for
pass band.
Z
1 
1
4Z 2
0
The phase angle in this pass band will be
given by
  2 sin
1
Z1
4Z 2
Another condition for stop band is given as
follows:
Z1
when
 1
4Z 2
Z1
0
4Z 2
Stop band.
Z1
1 
0
4Z 2
pass band
Z1
 1
4Z2
stop band
Cut-off frequency
The frequency at which the network changes
from pass band to stop band, or vice versa,
are called cut-off frequencies.
These frequencies occur when
Z1
 0 or Z1  0
4Z 2
Z1
 1 or Z1  4Z 2
4Z 2
(48)
where Z1 & Z2 are opposite types of reactances .
Since Z1 and Z2
may have number of
combinations, as L and C elements, or as
parallel and series combinations, a variety of
types of performance are possible.
Constant k- type low pass filter
(a) Low pass filter section; (b) reactance curves
demonstrating that (a) is a low pass section or has pass
band between Z1 = 0 and Z1 = - 4 Z2
If Z1 and Z2 of a reactance network are unlike
reactance arms, then
Z1Z 2  k
2
where k is a constant independent of
frequency. Networks or filter circuits for which
this relation holds good are called constant-k
filters.

j
Z1  jL and Z2 
Z1Z 2 
L
C
R
2
k
C
(51)
(b) reactance curves demonstrating that (a) is a
low pass section or has pass band between Z1 =
0 and Z1 = - 4 Z2
Low pass filter
Pass band :
Z1  0 to Z1  - 4Z2
f  0 to f  f c
f  f c  stopband
fc 

1
 LC
f
sinh  j
2
fC
Variation of  and  with frequency for the low
pass filter
f
Z1
For  1 so that - 1 
 0 , then
fc
4Z2
f
 1,
fc
  0,
f
  2sin ( )
fc
-1
Phase shift is zero at zero frequency and
increases gradually through the pass band,
reaching  at cut-off frequency and remaining
same at  at higher frequencies.
Characteristic Impedance of T filter
Z OT
2

 f  
L
1    

C   fC  


Z OT  RK
  f 2 
1    
  f C  
ZOT varies throughout the pass band, reaching a
value of zero at cut-off, then becomes imaginary
in the attenuation band, rising to infinity
reactance at infinite frequency
Z OT
Variation of R
k
filter.
with frequency for low pass
Constant k high pass filter
(a) High pass filter; (b) reactance curves demonstrating
that (a) is a high pass filter or pass band between Z1 = 0
and Z1 = - 4Z2
m-derived T section
(a) Derivation of a low pass section having a
sharp cut-off section (b) reactance curves
for (a)
m-derived low pass filter
Variation of attenuation for the prototype
amd m-derived sections and the
composite result of two in series.
Variation of phase shift  for mderived filter
Variation of Z 0 over the pass band for T
and  networks
(a) m-derived T section; (b)  section formed by
rearranging of (a); © circuit of (b) split into L sections.
Variation of Z1 of the L section over the
pass band plotted for various m valus
Cascaded T sections =
Transmission Line
Circuit Model/Lumped constant
Model Approach
• Normal circuit consists of Lumped elements
such as R, L, C and devices.
• The interconnecting links are treated as good
conductors maintaining same potential over
the interconnecting links. Effectively links
behaves as short between components and
devices.
• Circuits obey voltage loop equation and
current node equation.
Lumped constants in a
circuit
Transmission Line
Theory
Transmission Line = N sections symmetrical
T networks with matched termination
If the final section is terminated in its
characteristic impedance, the input impedance
at the first section is Z0. Since each section is
terminated by the input impedance of the
following section and the last section is
terminated by its Z0. , all sections are so
terminated.
Characteristic impedance
of T section is known
Z1
)
asZ OT  Z1Z 2 (1 
4Z 2
There are n such terminated
section.
I s , I r = sending and receiving end currents
Is
n
e
Ir
the
n
 = Propagation constant for
one section
Z1
Z1 2 Z1
e  1 2
 (
) 
2Z 2
2Z 2
Z2


Z1
Z1 2 Z1 
  ln 1 
 (
)  
2Z 2
Z2 
 2Z 2
A uniform transmission can be viewed as an
infinite section symmetrical T networks. Each
section will contributes proportionate to its share
,R, L, G, C per unit length. Thus lumped method
analysis can be extended to Transmission line
Certain the analysis developed for lumped
constants can be extended to distributed
components well.
The constants of an incremental length x of a
line are indicated.
Series constants:
R + j L
ohms/unit length
Shunt constants:
Y + jC
mhos/unit length
Thus one T section, representing an incremental
length x of the line has a series impedance Zx
ohms and a shunt admittance Yx mhos. The
characteristic impedance of all the incremental
sections are alike since the section are alike.
Thus the characteristic impedance of any small
section is that of the line as a whole.
Thus eqn. (1) gives the characteristic of the line
with distributed constant for one section is given
as
Zx
ZxYx
Z0 
(1 
)
Yx
4
Z
ZYx
Z0 
(1 
)
Y
4
2
(4)
Allowing x to approach zero in the limit the value
of
Z0 for the line
Z of distributed constant is obtained
Ohms
as Z 0  Y
(5)
Z and Y are in terms unit length of the line. The
ration Z/Y in independent of the length units
chosen.
Propagation Constant
Under Z0 termination
I1/ I2 = eγ
γ = Propagation constant
α + jβ
I1/ I2 = ( Z1/2 + Z2 + Z0 ) / Z2 = eγ
= 1 + Z1/ 2Z2 + Z0/ Z2
I1/ I2 = 1 + Z1/ 2Z2 + √ Z1/Z2 ( 1 + Z1 / 4Z2 )
Propagation Constant

Z1 / Z2 ( 1 + Z1 / 4 Z2 ) = Z / Z2 [ 1 + ½ (Z1 /4Z2)
– 1/8 (Z1 / 4Z2)2 + ……..]
e
= 1+ Z1 / Z2 + ½ (Z1 / Z2 )2 + 1/8 (Z1 / Z2 )3
– 1/128 (Z1 / Z2 )5
+ ……
Applying to incremental length x
e x = 1 +  ZY x + ½ (ZY)2 x 2 + 1/8 ((ZY)23 x3
– 1/128 (ZY)5 x5 + …
6.6)
Series expansion is done
e x
e x = 1 +  x + x 2 x2 / 2! + 3 x3 / 3! +
…
(6.7)
Equating the expansions and canceling unity terms
x + 2 x2 / 2 + 3 x3 / 6 + …
= ZYx + ( ZY)2 x2 )/2+ ( ZY)3 x3) / 8 +
…
Divide x
+ 2 x2 / 2 + 3 x3 / 6 …
= ZY + (ZY)2 x / 2 + (ZY)3 x2 / 8 + …
as x
0
γ = ZY
(8)
Characteristic Impedance
γ = ZY
Propagation Constant
as x
Z0 =  Z / Y Ohms
0
Characteristic or surge impedance
Since there no energy is coming back to the
source , there is no reactive effect.
Consequently the impedance of the line is pure
resistance.
This inherent line impedance is called the
characteristic impedance or surge impedance
of the line.
The characteristic impedance is determined by
the inductance and capacitance per unit
length .
These quantities are in turn depending upon the
size of the line conductors and spacing
Dimension of line decides line impedance
The closer the two conductors of the line and
greater their diameter, the higher the capacitance
and lower the inductance.
A line with large conductors closely spaced will
have low impedance.
A line with small conductors and widely spaced
will have relative large impedance.
The characteristic impedance of typical lines
ranges from a low of about 50 ohms in the coaxial
line type to a high of somewhat more than 600
ohms for a open wire type.
Z0 
jL
L

jC
C
Thus at high frequencies the characteristic
impedance Z0 of the transmission line
approaches a constant and is independent of
frequency.
Z0 depends only on L and C
Z0 is purely resistive in nature and absorb all
the power incident on it.
jL
L
(5.5x106 )
Z0 


 2500  50
12
jC
C
(2200 x10 )
Characteristic
line
impedance
Z1  R  R1  10  100  110
RS Z1
100 x110
Z2  R 
 10 
 10  52.38  62.38
RS  Z1
100  110
RS Z 2
100 x62.38
Z3  R 
 10 
 10  38.42  48.32
RS  Z 2
100  62.38
100 x 48.32
Z 4  10 
 10  32.63  42.62
100  48.32
With additional section added the input
impedance is decreasing further till it
reaches its characteristic impedance of
37. For a single section with
termination of 37 
RS xZL
100 X 37
3700
Z 0  Z1  R 
 10 
 10 
 37
RS  Z L
100  37
137
Transmission Line
Transmission line is a critical link in any
communication system.
Transmission lines behaves as follows:
a) Connecting link
b) R – L – C components
c)Resonant circuit
d)Reactance impedance
e) Impedance Transformer