Transmission Lines and Waveguides

Transmission Lines and Waveguides
Dr.N.Gunasekaran Dean, ECE 1

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. 2

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. 3

No wear and tear; No splitting of electron;
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; 4

Controlled by Fields : accelerated, retarded, change directions, increase and decrease of stream of electrons; instant reaction due to zero inertia. 5

Energy = Electron - Wave
Energy is transferred from place to by two means: Current : Flow of electrons through conductors. 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. 6

Electron waves 7 7

Major Topics for discussion
Circuit domain ( Filters ) Semi Field domain (Transmission Line : Voltage-Current – Fields) More Field domain (Coaxial line) Field domain : TEM waves ( Parallel plate guiding) Fully Field domain : TE-TM modes ( Waveguide ) 8

Transmission Line – Waveguide
Guided communication 14

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 Hz Inside Fiber 40

Energy V = Voltage = Size of energy packet / electron.
I = Current = Number of energy packet flow / sec Total energy flow / sec = V X I 41

System Power Flow Medium Circuits P = V x I Conductor
Transmission Lines P = E x H Free space Coaxial Lines P = E x H Free space Waveguides P = E x H Free space Optical Fiber P = E x H Glass 42

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. 43

Circuits Inside Conductor Transmission Lines TEM mode
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 44

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. 45

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 46

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. 47 47

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. 48

Circuits Radiate at high frequency opearation
49

As frequency increases, current flow becomes a surface phenomenon.
Skin Effect As frequency increases, current flow becomes a surface phenomenon. 50 50

Conductor radiates at high frequencies
51

(   100s Km ); ( D <<  )
Circuit theory Model OR Lumped Model (   100s Km ); ( D <<  ) 54

 Is our scale 55

Frequency f Wavelength  50 Hz 6,000 Km
Frequency f Wavelength  50 Hz ,000 Km 3 KHz Km 30 KHz Km 300 KHz Km 3 MHz m 30 MHz m 300 MHz m 3 GHz cm 30 GHz cm 300 GHz mm 56

V= V0 sin (90) V= V0 sin (360) V= V0 sin (0 ) V= V0 sin (180)  57

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 58

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. 59

At 3 KHz No Phase variation across the Resistor
For f =3 KHz,  = 10 Km R 0o o o  = 10 Km At 3 KHz No Phase variation across the Resistor 61

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. 62

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 63

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. 64

For f =30 GHz,  = 1cm Resistor  = 1cm
0o o o  = 1cm Resistor 360o 0o 180o At 30GHz 360o Phase variation across the Resistor 65

Balanced transmission line opened out to form dipole radiator
66

Reactive drop Voltage Variation along the line Transmission Line
67

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. 68

Reactance XC and XL Parasitic capacitance and parasitic inductance create reactance that varies with frequency 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. 69

Llead = Due to resistor and material of resistor.
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. 70

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. 71

Rcore =Core loss resistance
RL =Lead Resistance CL =Lead capacitance Rcore =Core loss resistance 72

Phase Shift in Transmission Line
73

Space Effect  0o o o 74

Magnitude of  C = fMHz met = 300 For f = 3 KHz,  = 100 KM
For f =3 GHz,  = 10cm For f =30 GHz,  = 1cm 75

C = f x  76

At 3 KHz No Phase variation across the Resistor
For f =3 KHz,  = 10 Km R 0o o o  = 10 Km At 3 KHz No Phase variation across the Resistor 77

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. 78

For f =30 GHz,  = 1cm Resistor  = 1cm
0o o o  = 1cm Resistor 360o 0o 180o At 30GHz 360o Phase variation across the Resistor 79

Filters 80

Any complicated network with terminal voltage and current indicated
81

A T network which may be made equivalent to the network in the box (a)
82

A  network equivalent to (b) and (a).
83

The T section as derived from unsymmetrical L-sections, showing notation used in symmetrical network analysis 84

The  section as derived from unsymmetrical L-sections, showing notation used in symmetrical network analysis 85

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 86

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 87

A succession of n networks in cascade.
88

Two types of transmission lines.
89

Basic Transmission Line.
90

A transmission line whose load impedance is resistive and equal to the surge impedance appears as an equal resistance to the generator. 91

Infinite parallel plane transmission line.
93

Transmission line is low pass filter
95 95

Any complicated network can be reduced to T or  network
96

T and  Network 97

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. 98

Image Impedance Non-Symmetry Network
Input impedance at the 1,1 terminal 99

Likewise, the impedance looking into the 2,2 terminal is required to be
Upon solving for 100

(1) If the image impedances are equal then
Then the voltage ratios and current ratios can be represented by (1) 102

Performance of Unsymmetrical
T &  Networks 103

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. 2. Delay of phase shift of the signal due to network. 104

Performance of a N networks in cascade
If several networks are used in succession as in fig., the overall performance may be appreciated as a (2) 105

Which may also me stated as
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 Is an application in which addition is substituted for multiplication. 106

(3) If the voltage ratios are defined as Eq. (2) becomes
If the natural logarithm (ln) of both sides is taken, then (3) 107

Thus it is common to define under conditions of equal impedance associated with input and output circuits. (4) The unit of “N” has been given the name nepers and defined as (5) nepers Two voltages, or two currents, differ by one neper when one of them is “e” times as large as the other. 108

Losses or gains of successive
Obviously, ratios of input to output power may also may also be expressed In this fashion. That is, The number of nepers represents a convenient measure of power loss or power gain of a network. Losses or gains of successive 109

3. High frequency operation introduces distributed parameter effect.
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. 110

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. 111

Analysis of Transmission line ( N networks in cascade) based on basic symmetrical T and  networks
112

Transmission line is low pass filter
113 113

Any complicated network can be reduced to T or  network
114

T and  Network 115

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. 116

Image Impedance Non-Symmetry Network
Input impedance at the 1,1 terminal 117

Likewise, the impedance looking into the 2,2 terminal is required to be
Upon solving for 118

(1) If the image impedances are equal then
Then the voltage ratios and current ratios can be represented by (1) 120

Performance of Unsymmetrical
T &  Networks 123

Transmission Lines and Waveguides 24.7.13
Part-2 EC 2305 (V sem) Transmission Lines and Waveguides Dr.N.Gunasekaran Dean, ECE Rajalakshmi Engineering College Thandalam, Chennai 124

Filters 125

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. . 126

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. 128

Application of Filter circuit
Whenever alternating currents occupying different frequency bands are to be separated, filter circuits have an application. 129

Neper - Decibel (1) In filter circuits the performance Indicator is
If the ratios of voltage to current at input and output of the network are equal then (1) 130

If several networks are used in cascade as shown if figure the overall performance will become
(2) 131

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 is an application in which addition is substituted for multiplication. 132

If the natural logarithm (ln) of both sides is taken, then (3) 133

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 (4) under condition of equal impedance associated with input and output circuits 134

(5) The unit of “N” has been given the name nepers and defined as
Two voltages, or two currents, differ by one neper when one of them is “e” times as large as the other. 135

Obviously, ratios of input to output power may also may also be expressed In this fashion. That is,
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. 136

“ bel “ - “ decibel “ The telephone industry proposed and has popularized a similar unit based on logarithm to the base 10, naming the unit “ bel “ for Alexander Graham Bell The “bel” is defined as the logarithm of a power ratio, number of bels = 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 137

(6) (7) In case of equal impedance in input and output circuits,
Equating the values for the power ratios, Taking logarithm on both sides 138

Is obtained as the relation between nepers and decibel.
8.686 N = dB Or 1 neper = dB Is obtained as the relation between nepers and decibel. The ears hear sound intensities on a logarithmically and not on a linear one. 139

Part-3 EC 2305 (V sem) Transmission Lines and Waveguides Dr.N.Gunasekaran Dean, ECE Rajalakshmi Engineering College Thandalam, Chennai 140

Performance parameters of a “series of identical networks”.
1.Characteristic Impedance 2. Propagation constant For efficient propagation, the network is to be terminated by Z0 and the propagation constant  should be imaginary. 141

We should also attempt to express these two performance constants in terms of network components Z1 and Z2 . 142

Characteristic impedance of symmetrical networks
What is Characteristic impedance of symmetrical networks 143

Symmetrical T section from L sections
For symmetrical network the series arms of T network are equal 144

Symmetrical  from L sections
145

and oppositely for the  network.
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. 146

Ladder Network made from T section
147

Ladder Network built from  section
The parallel shunt arms will be combined 148 148

For a symmetrical network:
the image impedance and are equal to each other and the image impedance is then called characteristic impedance or iterative impedance, 149

That is , if a symmetrical T network is terminated in , its input impedance will also be , or the impedance transformation ration is unity. 150

The term iterative impedance is apparent if the terminating impedance is considered as the input impedance of a chain of similar networks in which case is iterated at the input to each network. 151

Characteristic Impedance of Symmetrical T section network
152

For T Network terminated in
(9) When 153

Characteristic Impedance for a symmetrical T section
(!0) Characteristic impedance is that impedance, if it terminates a symmetrical network, its input impedance will also be is fully decided by the network’s intrinsic properties, such as physical dimensions and electrical properties of network. 154 154

Characteristic Impedance
 section 155

When , for symmetrical  Characteristic Impedance (11) 156

(12) (13) 157

propagation constant 
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. (14) 158

Where is a complex number defined by
(15) Hence If 159

With Z0 termination, it is also true,
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 160

= the phase shift introduced by the network.
= 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. 161

If a number of sections all having a common Z0
the ratio of currents is from which and taking the natural logarithm, (16) Thus the overall propagation constant is equal to the sum of the individual propagation constants. 162

and  of symmetrical networks
Use the definition of and the introduction of as the ratio of currents for a termination leads to useful results 163

Where the characteristic impedance is given as
The T network in figure is considered equivalent to any connected symmetrical network terminated in a termination. From the mesh equations the current ratio can be shown as (30) Where the characteristic impedance is given as (32) 164

Eliminating (33) (36) The propagation constant can be related to network parameters by use of (10) for In (30) as 165

Taking the natural logarithm
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 166

It is reduced to (39) For short circuit, = 0 (40) For a open circuit
(41) 167

From these these two equations it can be shown that
(42) Thus the propagation constant  and the characteristic impedance Z0 can be evaluated using measurable parameters 168

Filter fundamentals Pass band – Stop band:
The propagation constant is For  = 0 or There is no attenuation , only phase shift occurs. It is pass band. 169

 Is conveniently studied by use of the expression.
It is assumed that the network contains only pure reactance and thus will be real and either positive or negative, depending on the type of reactance used for Expanding the above expression 170

It contains much information.
This condition implies a stop or attenuation band of frequencies. 171

The attenuation will be given by
This results in the following conclusion for pass band. 172

The phase angle in this pass band will be given by
Another condition for stop band is given as follows: 173

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 (48) 175

Since may have number of combinations, as L and C elements, or as parallel and series combinations, a variety of types of performance are possible. 176

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 177

If of a reactance network are unlike reactance arms, then
where k is a constant independent of frequency. Networks or filter circuits for which this relation holds good are called constant-k filters. (51) 178

(b) reactance curves demonstrating that (a) is a low pass section or has pass band between Z1 = 0 and Z1 = - 4 Z2 179

Low pass filter Pass band : 180

Variation of  and  with frequency for the low pass filter
181

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. 182

Characteristic Impedance of T filter
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 183

Variation of with frequency for low pass filter.
184

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 185

m-derived T section (a) Derivation of a low pass section having a sharp cut-off section (b) reactance curves for (a) 186

m-derived low pass filter
187

Variation of attenuation for the prototype amd m-derived sections and the composite result of two in series. 188

Variation of phase shift  for m-derived filter
189

Variation of over the pass band for T and  networks
190

(a) m-derived T section; (b)  section formed by rearranging of (a); © circuit of (b) split into L sections. 191

Variation of Z1 of the L section over the pass band plotted for various m valus
192

Cascaded T sections = Transmission Line
196

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. 197

Lumped constants in a circuit
198

Transmission Line Theory
Transmission Line = N sections symmetrical T networks with matched termination 199

Characteristic impedance of T section is known as
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 as There are n such terminated section. = sending and receiving end currents 200

 = Propagation constant for one section then
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 too. 201

The constants of an incremental length x of a line are indicated.
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 202

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 203

Allowing x to approach zero in the limit the value of
(4) Allowing x to approach zero in the limit the value of Z0 for the line of distributed constant is obtained as Ohms (5) Z and Y are in terms unit length of the line. The ration Z/Y in independent of the length units chosen. 204

I1/ I2 = eγ γ = 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 = Z1/ 2Z2 + √ Z1/Z2 ( 1 + Z1 / 4Z2 ) 205

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 + … ) 206

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 207

= ZY + (ZY)2 x / 2 + (ZY)3 x2 / 8 + … as x 0 γ = ZY (8)
+ 2 x2 / 2 + 3 x3 / 6 … = ZY + (ZY)2 x / 2 + (ZY)3 x2 / 8 + … as x γ = ZY (8) Characteristic Impedance Z0 =  Z / Y Ohms Propagation Constant γ = ZY 208

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 between the conductors. 209

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. 210

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. 211

Characteristic impedance line
212

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  213

Transmission Line Transmission line is a critical link in any communication system. Transmission lines behaves as follows: Connecting link b) R – L – C components c)Resonant circuit d)Reactance impedance e) Impedance Transformer 214

Part-2 EC 2305 (V sem) Transmission Lines and Waveguides Dr.N.Gunasekaran Dean, ECE Rajalakshmi Engineering College Thandalam, Chennai

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 of the circuit is expressed in terms of ratio of input –current to output-current magnitude. If the ratios of voltage to current at input and output of the network are equal then (1)

If several networks are used in cascade as shown if figure the overall performance will become
(2)

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 is an application in which addition is substituted for multiplication.

If the natural logarithm (ln) of both sides is taken, then (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 (4) under condition of equal impedance associated with input and output circuits

(5) The unit of “N” has been given the name nepers and defined as
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,
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 proposed and has popularized a similar unit based on logarithm to the base 10, naming the unit “ bel “ for Alexander Graham Bell The “bel” is defined as the logarithm of a power ratio, number of bels = 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

(6) (7) In case of equal impedance in input and output circuits,
Equating the values for the power ratios, Taking logarithm on both sides

8.686 N = dB Or 1 neper = dB Is obtained as the relation between nepers and decibel. The ears hear sound intensities on a logarithmically and not on a linear one.

Symmetrical T section Network
For symmetrical network the series arms of T network are equal

Symmetrical  Network

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.

The parallel shunt arms will be combined 236

For a symmetrical network the image impedance
and are equal to each other and the image impedance is then called characteristic impedance or iterative impedance, That is , if a symmetrical T network is terminated in , its input impedance will also be , or the impedance transformation ration is unity. The term iterative impedance is apparent if the terminating impedance is considered as the input impedance of a chain of similar networks in which case is iterated at the input to each network.

When

Characteristic impedance is that impedance, if it terminates a symmetrical network, its input impedance will also be is fully decided by the network’s intrinsic properties, such as physical dimensions and electrical properties of network. 240

 section

When , for symmetrical  Characteristic Impedance

Under the assumption of equal input and output impedances, which may be , for a symmetrical network, the current ratio. 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 if it be defined that under the condition of 244

Hence If

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 indicate the propagation of signal through the network. The unit of phase shift is radians.

the ratio of currents is from which and taking the natural logarithm, Thus the overall propagation constant is equal to the sum of the individual propagation constants.

Physical properties of symmetrical networks
Use the definition of and the introduction of as the ratio of current for a termination leads to useful results

The T network in figure is considered equivalent to any connected symmetrical network terminated in a termination. From the mesh equations the current ratio can be shown as

Where the characteristic impedance is given as
The propagation constant  can be related to the network parameters as follows:

For short circuit, = 0 For a open circuit

Thus the propagation constant  and the characteristic impedance Z0 can be evaluated using measurable parameters

The propagation constant is For  = 0 or There is no attenuation , only phase shift occurs. It is pass band.

is conveniently studied by use of the expression.
It can be proved from this, the pass band condition is as follows: where the two reactance are opposite type. The phase shift in pass band is given by :

Since may have number of combinations, as L and C elements, or as parallel and series combinations, a variety of types of performance are possible.

(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

(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)

Variation of attenuation for the prototype amd m-derived sections and the composite result of two in series.

Part-3 EC 2305 (V sem) Transmission Lines and Waveguides Dr.N.Gunasekaran Dean, ECE Rajalakshmi Engineering College Thandalam, Chennai

Performance parameters of a “series of identical networks”.
1.Characteristic Impedance 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

Symmetrical  from L sections

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.

The parallel shunt arms will be combined 276

For a symmetrical network:
the image impedance and are equal to each other and the image impedance is then called characteristic impedance or iterative impedance,

That is , if a symmetrical T network is terminated in , its input impedance will also be , or the impedance transformation ration is unity.

The term iterative impedance is apparent if the terminating impedance is considered as the input impedance of a chain of similar networks in which case is iterated at the input to each network.

(9) When

(!0) Characteristic impedance is that impedance, if it terminates a symmetrical network, its input impedance will also be is fully decided by the network’s intrinsic properties, such as physical dimensions and electrical properties of network. 282

 section

When , for symmetrical  Characteristic Impedance (11)

(12) (13)

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. (14)

(15) Hence If

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.

the ratio of currents is from which and taking the natural logarithm, (16) Thus the overall propagation constant is equal to the sum of the individual propagation constants.

and  of symmetrical networks
Use the definition of and the introduction of as the ratio of currents for a termination leads to useful results

The T network in figure is considered equivalent to any connected symmetrical network terminated in a termination. From the mesh equations the current ratio can be shown as (30) Where the characteristic impedance is given as (32)

Eliminating (33) (36) The propagation constant can be related to network parameters by use of (10) for In (30) as

Taking the natural logarithm
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

It is reduced to (39) For short circuit, = 0 (40) For a open circuit (41)

(42) Thus the propagation constant  and the characteristic impedance Z0 can be evaluated using measurable parameters

The propagation constant is For  = 0 or There is no attenuation , only phase shift occurs. It is pass band.

 Is conveniently studied by use of the expression.
It is assumed that the network contains only pure reactance and thus will be real and either positive or negative, depending on the type of reactance used for Expanding the above expression

It contains much information.
This condition implies a stop or attenuation band of frequencies.

The attenuation will be given by
This results in the following conclusion for pass band.

The phase angle in this pass band will be given by
Another condition for stop band is given as follows:

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 (48)

Since may have number of combinations, as L and C elements, or as parallel and series combinations, a variety of types of performance are possible.

(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 of a reactance network are unlike reactance arms, then
where k is a constant independent of frequency. Networks or filter circuits for which this relation holds good are called constant-k filters. (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 :

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
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

(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)

Variation of attenuation for the prototype amd m-derived sections and the composite result of two in series.

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

Characteristic impedance of T section is known as
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 as There are n such terminated section. = sending and receiving end currents

 = Propagation constant for one section
then 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 too.

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

Allowing x to approach zero in the limit the value of
(4) Allowing x to approach zero in the limit the value of Z0 for the line of distributed constant is obtained as Ohms (5) Z and Y are in terms unit length of the line. The ration Z/Y in independent of the length units chosen.

I1/ I2 = eγ γ = 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 = 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 + … )

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

= ZY + (ZY)2 x / 2 + (ZY)3 x2 / 8 + … as x 0 γ = ZY (8)
+ 2 x2 / 2 + 3 x3 / 6 … = ZY + (ZY)2 x / 2 + (ZY)3 x2 / 8 + … as x γ = ZY (8) Characteristic Impedance Z0 =  Z / Y Ohms Propagation Constant γ = ZY

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 between the conductors.

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.

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.

Characteristic impedance line

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 

Transmission Line Transmission line is a critical link in any communication system. Transmission lines behaves as follows: Connecting link b) R – L – C components c)Resonant circuit d)Reactance impedance e) Impedance Transformer

Transmission Lines and Waveguides

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