UNIT IV PASSIVE FILTERS. Energy Transfer All the systems are designed to carryout the following jobs: 1.Energy generation. 2. Energy transportation. 3.

UNIT IV

PASSIVE FILTERS

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 = VoltageElectron flow = currentElectrons’ oscillation = WaveElectron transfer = LightElectron 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 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.

Electron - waves

Major Topics for discussioni) Circuit domain ( Filters )

ii) Semi Field domain (Transmission Line : Voltage-Current – 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 – WaveguideGuided 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 = 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

Quantum of energy E = h f; h =6.626x10-34 J-sQuantum 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 effect2.Transit time –3.Moving electron induce current4. Delay 5. Retardation-.Radiation6.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 EffectSkin 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 Wavelength

50 Hz 6,000 Km 3 KHz 100 Km 30 KHz 10 Km 300 KHz 1 Km 3 MHz 100 m 30 MHz 10 m 300 MHz 1 m 3 GHz 10 cm 30 GHz 1 cm 300 GHz 1 mm

V= V0 sin (0 )V= V0 sin (90)

V= V0 sin (180)

V= V0 sin (360)

Circuit domain :Dimension <<

C= f x = 300,000 km/secGiven f = 30 kHz ; = 10 kmHence circuit dimensions << = 10 km

Medium = Conducting medium. = Conductors in circuits.

Electrons = Energy PacketEnergy E = eV electron volts; W= V X I

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Circuit TheoryConnecting 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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0o 180o 360o

At 3 KHz No Phase variation across the Resistor

For f =3 KHz, = 10 Km

= 10 Km

R

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/secGiven f = 3000 MHz ; = 10 cmHence circuit dimensions = 10 cm

Dielectric medium – Free spaceWaves = E/H fieldes

Energy E = h.f joulesTotal 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.

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0o 180o 360o

Resistor

For f =30 GHz, = 1cm

0o 180o 360o

= 1cm

At 30GHz 360o Phase variation across the Resistor

Balanced transmission line opened out to form dipole radiator

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Transmission Line

Voltage Variation along the line

Reactive drop

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

fC

jXC 2

fLjX L 2

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.

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 inductanceRlead = Lead resistanceRDC = Dielectric leakageRAC =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 ResistanceCL =Lead capacitanceRcore =Core loss resistance

Phase Shift in Transmission Line

Space Effect

0o 180o 360o

Magnitude of

C = fMHz met = 300 For f = 3 KHz, = 100 KM

For f =3 GHz, = 10cm

For f =30 GHz, = 1cm

C = f x

0o 180o 360o

At 3 KHz No Phase variation across the Resistor

For f =3 KHz, = 10 Km

= 10 Km

R

Circuit TheoryConnecting wires introduces no drop and no delay. The wires between the componenets are of same potential. Shape and size of wires are ignored.

0o 180o 360o

Resistor

For f =30 GHz, = 1cm

0o 180o 360o

= 1cm

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 FilterResonant 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

i

iiin ZZZ

ZZZZZZZ

232

221111

)(

Input impedance at the 1,1 terminal iin ZZ 11

Likewise, the impedance looking into the 2,2 terminal is required to be iZ2

i

ii ZZZ

ZZZZZ

131

11322

)(

Upon solving for iiandZZ 21

32

133221311

))((

ZZ

ZZZZZZZZZ i

31

133221322

))((

ZZ

ZZZZZZZZZ i

32

133221

32

3211

211

ZZ

ZZZZZZ

ZZ

ZZZZ

ZZZ

sc

oc

scoci

scoci

ZZZ

ZZZ

222

111

2

1

2

1

V

V

I

I (1)

Then the voltage ratios and current ratios can be represented by

ii ZZ 21 o

o

i

i

I

V

I

V

2

2

1

1 then

If the image impedances are equal

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.

BI

I

AV

V

2

1

2

1

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 as a

nn

n

V

V

V

VX

V

VX

V

VX

V

V 11

4

3

3

2

2

1 .....

(2)

Which may also me stated as

43214321 .1.. AAAAAAAA

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

ncbancb eexxexee .........

Is an application in which addition is substituted for multiplication.

If the voltage ratios are defined as

etceV

Ve

V

Ve

V

V cba ;.......;;4

3

3

2

2

1

Eq. (2) becomes

ncba

n

eV

V ........1

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

(3)ndcbaV

V ..........ln

2

1

Thus it is common to define under conditions of equal impedance associated with input and output circuits.

NeI

I

V

V

2

1

2

1 (4)

The unit of “N” has been given the name nepers and defined as

Nnepers2

1

2

1 lnlnI

I

V

V (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,

NeP

P 2

2

1

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 FilterResonant 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

i

iiin ZZZ

ZZZZZZZ

232

221111

)(

Input impedance at the 1,1 terminal iin ZZ 11

Likewise, the impedance looking into the 2,2 terminal is required to be iZ2

i

ii ZZZ

ZZZZZ

131

11322

)(

Upon solving for iiandZZ 21

32

133221311

))((

ZZ

ZZZZZZZZZ i

31

133221322

))((

ZZ

ZZZZZZZZZ i

32

133221

32

3211

211

ZZ

ZZZZZZ

ZZ

ZZZZ

ZZZ

sc

oc

scoci

scoci

ZZZ

ZZZ

222

111

2

1

2

1

V

V

I

I (1)

Then the voltage ratios and current ratios can be represented by

ii ZZ 21 o

o

i

i

I

V

I

V

2

2

1

1 then

If the image impedances are equal

Performance of Unsymmetrical T & Networks

Dr.N.GunasekaranDean, ECE

Rajalakshmi Engineering CollegeThandalam, Chennai- 602 105

Part-2EC 2305 (V sem)Transmission Lines and Waveguides24.7.13

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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

currentOutput

current Input

ePerformanc

If the ratios of voltage to current at input and output of the network are equal then

2

1

2

1

V

V

I

I (1)

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

nn

n

V

V

V

VX

V

VX

V

VX

V

V 11

4

3

3

2

2

1 .....

(2)

Which may also me stated as

43214321 .1.. AAAAAAAA

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

ncbancb eeeee .........

is an application in which addition is substituted for multiplication.

If the voltage ratios are defined as

etceV

Ve

V

Ve

V

V cba ;.......;;4

3

3

2

2

1

Eq. (2) becomes

ncba

n

eV

V ........1

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

(3)ndcbaV

V ..........ln

2

1

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

NeI

I

V

V

2

1

2

1

(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

Nnepers2

1

2

1 lnlnI

I

V

V (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,

NeP

P 2

2

1

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 =

2

1

P

P log

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

2

1

P

P log 10 dB (6)

In case of equal impedance in input and output circuits,

2

1

2

1

V

V log 20

I

I log 20 dB (7)

Equating the values for the power ratios,

10102 dBNe 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 a logarithmically and not on a linear one.

Performance parameters of a “series of identical networks”.

1.Characteristic Impedance

2. Propagation constant

0Z

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 isCharacteristic impedance of

symmetrical networks

Symmetrical T section from L sections

For symmetrical network the series arms of T network are equal

21

21ZZZ

Symmetrical from L sections

22ZZZ ca

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 sectionThe parallel shunt arms will be combined

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, .

iZ1 iZ2

ii ZZ 21

oitii ZZZZ 21

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

0Z0Z

0i0R then Z ZIf ZZ

0ZZR 0ZZ i

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.

0Z

0Z

initR ZZZZ 0

Characteristic Impedance of Symmetrical T section network

021

01

211

2

)2(

2 ZZZ

ZZZZZ in

For T Network terminated in 0Z

When 01 ZZ in

21

212

0

022

2022140

4

1

0121

ZZZ

Z

zz

zzzzZ

Z

ZZZ

(9)

Characteristic Impedance for a symmetrical T section

2

12121

21

0 41(

4 Z

ZZZZZ

ZZ T

Characteristic impedance is that impedance, if it terminates a symmetrical network, its input impedance will also be

0Z

0Z

0Z is fully decided by the network’s intrinsic properties, such as physical dimensions and electrical properties of network.

(!0)

Characteristic Impedance section 0Z

202

021

202

021

1

222

2)22

(

ZZZZZ

Z

ZZZZZ

Z

Z in

When , for symmetrical

01 ZZ in

2

1

210

41 ZZZZ

Z

Characteristic Impedance

(11)

2021

21

21

211

1

21

1

4

2

22

2

Tscoc

scsc

ococ

ZZZZ

ZZ

ZZ

ZZZ

ZZ

ZZ

ZZ

20

21

122

0 4

4Z

ZZ

ZZZZ scc

scocZZZ 0

(12)

(13)

propagation constant

The magnitude ratio does not express the complete network performance , the phase angle between the currents being needed as well.

NeI

I

V

V

2

1

2

1

The use of exponential can be extended to include the phasor current ratio.

eI

I

2

1(14)

jWhere is a complex number defined by

Hence jee

I

I 2

1

If AI

I

2

1

eI

IA

2

1 je

(15)

With Z0 termination, it is also true,

eV

V

2

1

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 Z0

nI

I

I

I

I

I

I

I 1

4

3

3

2

2

1 ........ from which

neeee ........321

and taking the natural logarithm,

the ratio of currents is

n ..................4321

Thus the overall propagation constant is equal to the sum of the individual propagation constants.

(16)

and of symmetrical networks

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

e0Z

0Z

eZ

ZZZ

I

I

2

021

2

1 2

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

0Z

Where the characteristic impedance is given as

21

212

0 4ZZ

ZZ

(30)

(32)

2

1

2Z

Z1 cosh (33)

2

1

42 sinh

Z

Z

(36)

Eliminating 0Z

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

OTZ

2

12

2

1

2

1 )2

(2

21Z

Z

Z

Z

Z

Ze

2

1

2

2

1

2

1

221ln

Z

Z

Z

Z

Z

Z

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

22

212

11 Z

ZZZ in

sinh Z coshZ

sinh Z cosh

R 0

00

Rin

ZZZ

For short circuit, = 0 RZ

tanh 0ZZSC For a open circuit RZ

tanh0

Zlim

ZZ

It is reduced to

(39)

(40)

(41)

SCOCZZZ 0

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

OCSC ZandZ

From these these two equations it can be shown that

OC

SC

Z

Z tanh

(42)

Filter fundamentals Pass band – Stop band:

The propagation constant is

jFor = 0 or

There is no attenuation , only phase shift occurs.It is pass band.

21 II

band Stop-

occurs;n attenuatio ,I ve; when 2 1 I

Is conveniently studied by use of the expression.

2

1

42 sinh

Z

Z

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

2

14Z

Z

21 Zand Z

)22

( sinh2

sinh j

2sin

2 cosh

2 cos

2 sinh

j

It contains much information.

thenreactances typesame the Zand ZIf 21 are

real. and positive is ratio or the 04 2

1 Z

Z

This condition implies a stop or attenuation band of frequencies.

The attenuation will be given by

2

11

4sinh2

Z

Z

thenreactances typeopposite Zand ZIf 21 are

imaginary. is radical or the 04 2

1 Z

Z

This results in the following conclusion for pass band.

04

12

1 Z

Z

The phase angle in this pass band will be given by

2

11

4sin2

Z

Z

14Z

Zwhen

2

1

Another condition for stop band is given as follows:

band. Stop 04 2

1 Z

Z

band stop 14Z

Z

2

1

band pass 04

12

1 Z

Z

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

0or Z 04 1

2

1 Z

Z

212

1 4or Z 14

ZZ

Z (48)

.reactances of typesopposite are Z& Zwhere 21

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.

21 Zand Z

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 of a reactance network are unlike reactance arms, then

21 Zand Z

221 kZZ

where k is a constant independent of frequency. Networks or filter circuits for which this relation holds good are called constant-k filters.

CjLjZ 21 Zand

221 kC

L RZZ (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 : 211 4Z- Z to0 Z

cff to0f

stopband cff

LCcf 1

Cf

fj

2 sinh

Variation of and with frequency for the low pass filter

then, 04Z

Z1- that so 1

f

f

2

1

c

For

)(2sin 0, , 1 1-

cc f

f

f

f

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

2

1C

OT f

f

C

LZ

2

1C

KOT f

fRZ

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

Variation of with frequency for low pass filter. k

OT

R

Z

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 m-derived filter

Variation of over the pass band for T and networks

0Z

(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 as

)4

1(2

121 Z

ZZZZOT

There are n such terminated section.

rs II , = sending and receiving end currents

then

n

r

s eI

I

2

12

2

1

2

1

2

12

2

1

2

1

)2

(2

1ln

)2

(2

21

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Ze

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.

= Propagation constant for one section

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 lengthShunt constants:Y + jCmhos/unit length

Thus one T section, representing an incremental length x of the line has a series impedance Zx ohms and a shunt admittance Yx 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

)4

1(0

xxYZ

xY

xZZ

)4

1(2

0

xZY

Y

ZZ

(4)

Allowing x to approach zero in the limit the value ofZ0 for the line of distributed constant is obtained as Y

ZZ 0 Ohms

(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 + …

= ZYx + ( 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 Z0 = Z / Y Ohms Propagation Constant γ = ZY

as x 0

Characteristic or surge impedanceSince 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 impedanceThe 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.

C

L

Cj

LjZ

0

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.

C

L

Cj

LjZ

0

502500)102200(

)105.5(12

6

x

x

Characteristic impedance line

32.4842.381038.62100

38.6210010

38.6238.5210110100

11010010

11010010

2

23

1

12

11

x

ZR

ZRRZ

x

ZR

ZRRZ

RRZ

S

S

S

S

62.4263.321032.48100

32.48100104

xZ

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

37137

370010

37100

371001010

X

ZR

xZRRZZ

LS

LS

Transmission Line

Transmission line is a critical link in any communication system.

Transmission lines behaves as follows:a)Connecting linkb) R – L – C componentsc)Resonant circuitd)Reactance impedancee) Impedance Transformer