Transmission Lines Txt

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overview

At low frequencies, an electrical circuit is completely characterized by the electrical parameters likeresistance, inductance etc. and the physical size of the electrical components plays no role in the circuitanalysis. As the frequency increases however, the size of the components becomes important, that is tosay that, the space starts playing a role in the performance of the circuit. The voltage and currents exist

in the form of waves. Even a change in the length of a simple connecting wire may alter the behavior ofthe circuit. The circuit approach then has to be re-investigated with inclusion of the space into theanalysis. This approach is then called the transmission line approach.

One can then conveniently divide the subject of electromagnetics in two parts, the

static electromagnetics and the time varying electromagnetics. As will be clear

subsequently, the time varying electric and magnetic fields always constitute a wave

phenomenon called the electromagnetic wave which is the prime subject of discussion

of this book. The phenomenon of electromagnetism in totality is governed by the four

Maxwell's equations, which can be derived from the physical laws like the Gauss

Law, the Ampere's law and the Faraday's low of electromagnetic induction. The

electromagnetic theory is the generalization of the circuit theory, or the circuit theory

is rather a special case of the electromagnetic theory. Although every phenomena of

electricity and magnetism can be analyzed in the frame work of electromagnetic

theory, at low frequencies the circuit approach is adequate. As the frequency increases

the inadequacy of the circuit approach is felt and one is forced to follow the

electromagnetic field approach.

Objectives

At low frequencies, an electrical circuit is completely characterized by the electrical

parameters like resistance, inductance etc. and the physical size of the electricalcomponents plays no role in the circuit analysis. As the frequency increases however,

the size of the components becomes important, that is to say that, the space starts

playing a role in the performance of the circuit. The voltage and currents exist in the

form of waves. Even a change in the length of a simple connecting wire may alter the

behavior of the circuit. The circuit approach then has to be re-investigated with

inclusion of the space into the analysis. This approach is then called the transmission

line approach.


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Course Faculty :

Prof. R. K. Shevgaonkaremail: [email protected]

Click Here to Read more about the Faculty

The voltage and currents exist in the form of waves. Even a change in the length of a

simple connecting wire may alter the behavior of the circuit. The circuit approach then has tobe re-investigated with inclusion of the space into the analysis. This approach is then calledthe transmission line approach. Although the primary objective of a transmission line is tocarry electromagnetic energy efficiently from one location to other, they find wide applications

in high frequency circuit design. As the frequency increases, any discontinuity in the

circuit path leads to electromagnetic radiation. Also at high frequencies, the transit

time of the signals can not be ignored. In the era of high speed computers, where data

rates are approaching to few Gb/sec, the phenomena related to the electromagneticwaves, like the bit distortion, signal reflection, impedance matching play a vital role

in high speed communication networks. An antenna is a device which can launch

and receive electromagnetic waves efficiently. But for the large antennas, the

communication between an earth station and a satellite is practically impossible. The

communication which can be established with few watts of power, would need few

MW of power in the absence of proper antennas. However, antenna research is still

very active. With recent advances in mobile communication, design of compact,

efficient, multi-frequency antennas have received a new impetus in the last decade.

Objectives In this course you will learn the following What is a Transmission Line?

Various types of two conductor transmission lines and their special features.Balanced and unbalanced transmission line. Transit time effect on a transmission line

at high frequencies. Dominance of the reactive component over resistive component.

Concept of distributed elements and conditions under which the lumped element

circuit model is applicable. Approach to investigate transmission line characteristics

treating the line as collection of infinitesimal lumped section.

Various Types OfTransmission Line
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As the name suggests, the transmision line is a structure which can transport electrical energy from one point toanother.

At low frequencies, a transmission line consists of two linear conductors separated by a distance. When an electricalsource is applied between the two conductors, the line gets energized and the electrical energy flows along the

length of the conductors.

A two-conductor transmission line may appear in any of the forms shown in the figure

Co-axial cable

Consists of a solid conducting rod surrounded by the two conductors. This line has good isolation of the electricalenergy and therfore has low Electromagnetic Interference (EMI).

Parallel wire transmission line

Consists of two parallel conducting rods. In this case the electrical energy is distributed between and around the rods.Theoretically the electric and magnetic fields extend over infinite distance though their strength reduces as thedistance from the line. Obviously this line has higher EMI.

Microstrip lineConsists of a dielectric substrate having ground plane on one side and a thin metallic strip on the other side. Themajority of the fields are confined in the dielectric substrate between the strip and the ground plane. Some fringingfield exist above the substrate which decay rapidly as a function of height. This line is usually found in printed circuitboards at high frequencies.

Balanced and Un-balanced line

If the two conductors are symmetric around the ground, then the line is called the balanced line, otherwise the line isan un-balanced line. Transmission lines (a), (c) and (d) are un-balanced line, whereas the line (b) is a balanced line.

TransitTime Effect

It is important to note that No Signal can travel with infinite velocity. That is to say that if a voltage or current changes atsome location, its effect cannot be felt instantaneously at some other location. There is a finite delay between the 'cause' andthe effect. This is called the ' Transit Time' effect.

Consider the two-conductor line which is connected to a sinusoidal signal generator of frequency at one end and aload impedance at the other end. Due to the transit time effect the voltage applied at AA' will not appear instantaneously atBB'.

Let the signal travel with velocity along the line. Then the Transit time


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, = length of the line.

At some instant let the voltage at AA' be . Then will appear at BB' only after . However, during this time the

voltage at AA' changes to (say ) .

Important Observation

Even for ideal conductors i.e., no resistance, there is a voltage difference between AA' and BB'

When is transmit-time effect important?

Ideally the transit time effect should be included in analysis of all electrical circuits. However if the time period of the

signal is much larger than the transit time, we may ignore the effect of transmit time. That is, the transit timeeffect can be neglected if

Transit time effect becomes important when the length of the line becomes comparable to the wavelength. As the frequencyincreases, the wavelength reduces, and the transit time effect becomes more and more important.

Distributed CircuitElements

Due to transit time effect, the Kirchoff's laws cannot be applied to the circuit at a whole. However, if we take a small

sectionof the line such that its length is , the transit time effect would be negligible and consequently theKirchoff'slaws can be applied.


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A conductor carrying a current has magnetic field and consequently has flux linkage. The conductor thereforehasinductance.

Similarly the two conductors form a capacitance.

Due to transit time effect the whole line inductance or capacitance cannot be assumed to be located at a particular point inspace. The inductance and capacitance are distributed throughout the length of the line. These are therefore called the 'Distributed Parameters' of the line.

Distributed Circuit Elements

For non ideal conductors there is resistance along the length of the line. Also if themedium separating the conductors is non - ideal, there is leakage current through themedium which can be accounted for by placing equivalent conductance between theconductors.

In the presence of transit time effect, all the line parameters, the inductance, thecapacitance, the resistance, and the conductance are of distributed nature.

The distributed parameters can be defined per unit length of the line.

R = Resistance of both conductors together for unit length of the line (ohms/m)

L = Inductance (self and mutual) for both conductors together for unit length of the line(Henery/m)

C = Capacitance between two conductors for unit length of the line (Farad/m)G = Leakage conductance between two conductors for unit length of the line (Mho/m).

LumpedCircuitModel

A small section of the line of length has

Resistance =

Inductance =

Capacitance =

Conductance =


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The lumped circuit for a small section of the line can be any one of that shown below :

Note

For the analysis of the transmission to be valid at all frequencies, should be much less than at all

frequencies. In other words the analysis is to be carried out in the limit .

All above representations are equivalent in the limit .

Recap

In this course you have learnt the following

What is a Transmission Line?Various types of two conductor transmission lines and their special features.Balanced and unbalanced transmission line.Transit time effect on a transmission line at high frequencies.Dominance of the reactive component over resistive component.Concept of distributed elements and conditions under which the lumped element circuit model is applicable.Approach to investigate transmission line characteristics treating the line as collection of infinitesimal lumped section.

Chapter2

Objectives

In this course you will learn the following

Kirchoff's laws applied to an infinitesimal section of a line.Voltage and current equations for the transmission line exerted with time harmonic voltage and current.Solution of voltage and current equationsPropagation constant of a line and its relation to the line parameters per unit length and frequency.Physical interpretation of voltage and current solutions.

Existance of voltage and current waves on a transmission line.


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Voltage & current equations for small section of a line

Let us consider a small section of a transmission line of length . Let the voltage at theinput be V and current at the input be I.

Due to voltage drop in the series arm, the output voltage will be different from the input

voltage, say .

Similarly due to current through the capacitance and the conductance the output current

will be different from the input through the current, say . Then we can write

Now if the lumped circuit model should be valid for arbitrarily high frequency (i.e. arbitrarily small

), the analysis has to be carried out in the limit

-------- (2.1)

-------- (2.2)

Important

In general, the voltage and the current are not related through algebraic equations but are governedby differential equations.

Comment

The lines are essentially electromagnetic field problem. The simplified circuit analysis based ondistributed circuit elements and the lumped circuit model gives the operating equation(in terms ofthe terminal quantities) as a one-dimensional wave equation is a proof that the equivalent circuitmodel is correct.

Solution ofVoltage &

Currentequations of


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TransmissionLine

Differentiating eqn. 2.1 and substituting from eqn. 2.2 we get,

Similarly differentiating eqn. 2.2 and substituting from2.1 we get,

Let us define a parameter as

The physical significance of will be explained later.

However, is a parameter which depends upon the line parameters R, L, C and G and the frequenc.

is called the propagation constant of the line, and is in general a complex quantity.

Solution of Voltage & Current equations of Transmission Line

Both voltage and current are governed by the same second order differential equation i.e,

The time harmonic function is implicit in these equations. The general solution to the differentialequations with harmonic time function can be written as,

Where, are the arbitrary complex constants which are to be evaluated from theboundary conditions.

Since is in general a complex quantity let us write

(Where and are realquantities)

Substituting for , the voltage and current on any point of the line ' x ' at any instant, ' t 'can be written as
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Physical Interpretation of Voltage & Current SolutionsLet us now understand the phenomenon represented by the two terms of the voltage and current

soutions.Let us consider the voltage solution. Take the first term of the solution

Assuming that , the voltage due to this term at any point ' x ' on the line at anyinstant, ' t ' is

The first term therefore represents a voltage whose amplitude reduces exponentially with distance, 'x ' and whose phase is a combination of space, ' x ' and time, ' t '.

The voltage is composite function of space and time.

Temporal variation of Voltage and Current

For a given location on the line, is constant, and therefore voltage varies sinusoidally with time,

with amplitude and frequency ' '. The phase of the voltage is .

Note and observe the following : Voltage at two locationsThe amplitudes at two locations are not the same.

Due to differing phase differences, the voltages at two locations do not reach to the maximum at thesame instant.

Spatial Variation of Voltage & Current

On the other hand, for a given time, is constant, and therefore the voltage has

decaying Spatial Sinusoidal function with spatial frequency and phase . That is, if we

instantaneously look at the voltage along the line we see decaying sinusoidal function in the space(see figure)


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Recap


Kirchoff's laws applied to an infinitesimal section of a line.Voltage and current equations for the transmission line exerted with time harmonic voltage and current.Solution of voltage and current equationsPropagation constant of a line and its relation to the line parameters per unit length and frequency.Physical interpretation of voltage and current solutions.Existance of voltage and current waves on a transmission line.

Chapter3

Objectives


Demonstration of wave motion.Forward and backward travelling waves.

Interpretation of the propagation constant .Attenuation and phase constants and their units.Definition of wavelength and its relation to the phase constant.Characteristic impedance of the transmission line.Relation between voltage and current for forward and backward travelling waves.

Forward Travelling Wave

Combining now the space-time we get what is called the 'Wave Motion'. See Figure. The voltage pattern appears travellingfrom left to right.

The first term of the voltage solution represents a voltage travelling wave in direction (left to right),

and gives the amplitude of the wave at . We call this wave, the ' Forward Travelling Wave '

Backward Travelling Wave


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Similarly, the second term in the voltage solution gives a travelling wave

but travelling in negative' x' direction (right to left) as shown in Figure gives the

amplitude of the wave at . This wave we call the'Backward Travelling Wave' .

CurrentTravellingWaves

The first term of the current solution represents a current travelling wave

in direction (left to right), and gives the amplitude of the wave at . We call thiswave, the ' Forward Current Travelling Wave '

Similarly, the second term in the current solution gives a travelling wave but

travelling in negative ' x ' direction (right to left) as shown in Figure. gives the amplitude of

the at . This wave we call the'Backward Current Travelling Wave' .

Important Conclusion

The Voltage and the Current exist in the form of waves on a transmission line.

In general, we can say that in a circuit, any time varying voltage and/or current always existin the form of waves, although the wave nature may not be evoked at low frequencies wherethe transit time effects are negligible.

ComplexPropogationConstant

The propagation constant in general is complex

The wave amplitude varies as . That is denotes the exponential decay of the wave along its directio

propagation. therefore is called the 'Attenuation Constant' of the line . It has the unit Neper/m . ForNeper/m, the wave amplitude reduces to 1/e of its initial value over a distance of 1m.

Many times the attenuation of a wave is measured in terms of dB/m. therefore can be given in dB/m, where

1 Neper/m = 8.68 dB/m

Note

In voltage/current expressions, should always be in Neper/m. Therefore if is given in dB/m it shobe converted to Neper/m before it is used in the voltage/current equations.

The wave phase has two components


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Time phase

Space phase

The parameter gives the phase change per unit length and hence called the 'Phase Constant' of the line.units areRadian/m.

Now for a wave the distance over which the phase changes by is called the ' wavelength ' . Thereforephase change per unit length

Characteristic

Impedance ofTransmissionLine

Substituting the voltage and current solutions in the differential equations, and noting that the equations mustsatisfied by two waves indepentely we get,

We define a parameter called the 'Characteristic Impedance' of the line as

Note

The ratio of Forward Voltage and Current waves is always , and the ratio of the Backward Voltage and

Current waves is always .

The parameters and completely define the voltage and current behaviour on a transmission line. Thtwo parameters are related to R, L, G, and C, and the frequency of the signal. In transmission line analysis

knowledge of and is adequate and the explicit knowledge of R, L, G, C is rarely needed.

Recap


Demonstration of wave motion.Forward and backward travelling waves.

Interpretation of the propagation constant .Attenuation and phase constants and their units.Definition of wavelength and its relation to the phase constant.


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Characteristic impedance of the transmission line.Relation between voltage and current for forward and backward travelling waves.

Objectives


Formation of Voltage and Current standing waves on a transmission line.Partial and full standing waves.How backward wave is developed?What is voltage reflection coefficient?Relation of the voltage reflection coefficient to the load impedance.Impedance transformation on a transmission line.

The voltage and current on the line are superposition of the two waves travelling in the opposite directions.

Where is the distance measured from the load towards the generator

The result is a 'Standing Wave'. Ofcourse in general it is a partial standing wave since the amplitudes of the two travellwaves may not be equal.

Figure shows the voltage standing wave on the line. We may note how the nature of the wave changes from 'travelling'

'standing' when we vary and . (Try different values for )

When , , there is no backward wave and therefore the net wave is the 'Forward Travelling Wave'.

On the other hand when , the wave will be fully standing wave.

Origin ofBackward Wave

In our discussion, the generator is connected to the left end of the line. So a voltage travelling wave moving aw(the forward wave) from the generator is understandable. However, one would wonder about the origin ofbackward wave. There is no energy source at the right end of the line.

The only possibility then is, that the forward wave reaches the right end of the line and does not find corrconditions for transfering the full power to the load impedance. The part of the energy then gets reflected fromload which results into the 'Backward Wave'.

The strength of the backward wave then should be related to the load impedance with which the line is terminate

Since the forward wave carrys energy towards the load, we call this wave as the 'Incident Wave'. The backwwave which carrys reflected energy from the load is called the 'Reflected Wave'. We therefore have


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Voltage Reflection Co-efficient and its Relation to Load Impedance

As a measure of reflected energy we define a quantity called ' Voltage Reflection Coefficient ' as

Impedance seen at any distance from the load in terms of the ' Reflection Coefficient ' then is

Inverting the relation we get the reflection coefficient at any point on the line which is at a

distance from the load is

Now at , the impedance . Therefore the reflection coefficientat the load end of the line is

Interesting to Note

The transmission line provides a medium of impedance for the energy flow. Any

departure from creates an impedance step. This impedance step disrupts the smooth flowof energy and the part of the energy is reflected. Larger the impedance step more is thereflected energy and higher the reflection coefficient.

Impedanceat anyPoint onthe Line

Impedance at a distance from the load is

and

We therefore get,


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Rearranging terms and noting that

and , we get

Important

Impedance measured at line is not same as and is location dependent.

Impedance seen by the generator for a given load impedance varies a function of

the line length and consequently the power supplied by the generator becomes afunction of line length.Just changing the connecting wires the circuit performance will change.

General Impedance Transformation

The impedance at any point of line is a transformed version of the load impedance.

Infact there is nothing special about the load impedance. The impedance transformation can be between any two locationsthe line. It should be remembered however, that the sign convention for the distance on the line must be correctly taken.

If the length is measured towards the generator it is taken positive.

If the length is measured away from the generator, it is taken negative.

In the figure if we go from X to Y, ' is negative and if we go from Y to X, is positive.

If the impedance at is , its transformed version at will be given by

-------- (2.3)

Inverting the relation we get,

-------- (2.4)


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It can be noted that the above two expressions 2.3 and 2.4 are same expression with and interchanged and

replaced byConclusion

Expression 2.3 is the general impedance transformation relation which can be used for transforming impedance on

one location on the line to the other. If the impedance is transformed to a point towards the generator, is positiv

and if it is transformed to a point away from the generator, is negative.

Recap


Formation of Voltage and Current standing waves on a transmission line.Partial and full standing waves.How backward wave is developed?What is voltage reflection coefficient?Relation of the voltage reflection coefficient to the load impedance.

Impedance transformation on a transmission line.

Objectives


What is a loss-less transmission line?Variation of voltage and current on a loss less line.Standing waves on a loss-less line.Voltage standing wave ratio (VSWR) and its relation to the voltage reflection co-efficient.Importance of VSWR and its values for various impedances.

Concept of return-loss (RL). Return loss a measure of reflection on the line.

Analysis of Loss Less Transmission Line

In any electrical circuit the power loss is due to ohmic elements. A loss less transmission line therefore implies

and . For a loss less transmission line hence we get

Propagation constant :

That is, and .

The charateristic impedance

The reflection coefficient at any point on the line is

The voltage and current expressions become


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Let the reflection coefficient at the load end be written in the amplitude and phase form as

then we have

As we move towards the generator the phase becomes more negative and point P rotates clockwise on the dotted

circle. The radius of the circle is . Length of the vector OP gives the magnitude of the quantity

SpatialVariationofCurrent&Voltage

The previous equations indicate that the amplitudes of the voltage and current vary as a function of distance on theline.


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Wherever or even multiple of , the quantity in the brackets is maximum in the voltage

expression, and minimum in the current expression. That is wherever the voltage amplitude is maximum,the current amplitude is minimum.

Similarly wherever , the voltage is minimum and the current is maximum

Note

The voltage and current variation at every point on the line is only.

The distance between two adjacent voltage maxima (or minima) or two adjacent current maxima (or minima)corresponds to

The distance between adjacent voltage and current maxima or minima corresponds to

We then say that the voltage and current are in space quardrature, i.e, when voltage is maximum the current isminimum and vice versa.

Voltage Standing Wave Ratio

The maximum and minimum peak voltages measured on the line are


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Let us define a quantity called ' Voltage Standing Wave Ratio (VSWR) ' as

Substituting for and we get

The VSWR is a measure of the reflection on the line. Higher the value of VSWR, higher isi.e., higher is the reflection and is lesser the power transfer to the load.

Since , we get

VSWR of 1 corresponds to the . That is the best situation.

Ideally for a perfect match VSWR = 1. However, generally a is considered acceptablein all experimental works.

ReturnLoss &ReflectionCo-efficient

The return loss is defined as

Return loss (RL) = -20 log dBThe return loss indicates the factor by which the reflected signal is down compared to the incident signal.

For perfect match and the return loss is , whereas for the worst case of the return loss is

Higher the return loss better is the match.

For acceptable value of VSWR = 2,


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The return loss should be higher than 9.54

Recap


What is a loss-less transmission line?Variation of voltage and current on a loss less line.Standing waves on a loss-less line.Voltage standing wave ratio (VSWR) and its relation to the voltage reflection co-efficient.Importance of VSWR and its values for various impedances.Concept of return-loss (RL). Return loss a measure of reflection on the line.

Objectives


Impedance transformation on a loss-less line.For a given load impedance, maximum and minimum impedance seen on the line.Locations where impedance is maximum and minimum.

Important impedance characteristics of loss-less transmission line.Concept of matched impedance.

Impedance Variation on Transmission Line

The impedance at any point on the loss-less transmission line is

Substituting and noting that

and , we get

Which can be written in terms of normalized impedances as

The expression can be used for transforming impedance on any point on the loss-less transmissionline to any other point.

Maximum and Minimum Impedanceseen on Transmission Line

The maximum impedance occurs where the voltage is maximum and current is minimum, and itsvalue is


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Noting that the quantity inside the square bracket is the VSWR, we get

The minimum impedance occurs at a location where the voltage is minimum and the current ismaximum,and its value is

The magnitude of the impedance at any point on the loss-less line is bounded by

and

Important Characteristics of aLoss Less Transmission Line

(1)We know line characteristics repeat every , we have

Proof :

Therefore,


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(2)Normalized impedance inverts every distance

Proof :

In this case,

Hence we get,

Note

It is the Normalized impedance which inverts

every distance and not the absoluteimpedance.

Important Characteristicsof a Loss Less

Transmission LineThe input impedance of a line of length, which is terminated with an

impedance is

For (open circuit), and for (short circuit),

Interesting

An open circuited cable connected to the output of a circuit may heavily load the circuit if

the length of the cable is , since at the output of the circuit the impedance appearedwill be short circuit.


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CautionIn circuit measurements we invariably make connections between a circuit and anoscilloscope. For high input impedance oscilloscope we assume that the testing iscarried out in almost open circuit conditions. However, at high frequencies whenthe length of the cable connecting the circuit and the oscilloscope becomes

comparable to , the circuit does not see the open circuit. For long cablethe circuit sees short circuit and therefore the measurements may go completelywrong.

Important Characteristics of a Loss Less Transmission Line(3)

For load impedance , the impedance at any point on the line is

Proof :

Important

If a line is terminated in the characteristics impedance , the impedance at every point

on the line is . That is the input impedance of the line is independent of the length ofthe line.

This is called the 'Matched Load' condition.

Golden Rule

All high frequency measurements should be carried out in the matched load conditions

(i.e. ) so that the cable lengths used in measurement setups do not play anyrole.

Recap


Impedance transformation on a loss-less line.For a given load impedance, maximum and minimum impedance seen on the line.Locations where impedance is maximum and minimum.Important impedance characteristics of loss-less transmission line.Concept of matched impedance.

Objectives In this course you will learn the following Power delivered to a complex load

connected to a generator through a section of a line. Complex power at any location

on the line. How to obtain the amplitude of the forward travelling wave?


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Powerdelivered tothe loadobtainedusing Circuit

concept

Consider a loss-less transmission line with characteristic impedance . Let the line be terminated in a

complex load impedance . Since the load impedance is not equal to the characteristicimpedance, there is reflection on the line, and the voltage and the current on the line can be given as

Since the reference point is at the load end, the power delivered to the load is

Since the difference of any complex number and its conjugate is in the purely imaginary part, is apurely imaginary quantity. Therefore the power delivered to the load is

Powerdeliveredto the loadusingWaveconcept

The power delivered to the load can also be calculated using a different approach and that is, the power given tothe load is the difference of the power carrried by the incident wave towards the load and the power carried awayby the reflected wave. Since the travelling waves always see the characteristic impedance, the incident and

reflected powers and respectively are,

We therefore get


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Complex Power atany point on theline

The complex power at any point on the line is

Substituting for voltage and current at location ,

Note(i)

The i.e., the power loss at any point on the line is same as that at the load.This makes sense because since the line is lossless, any loss of power is only in the loadimpedance.

(ii) The imaginary power which is related to the energy stored in the reactive fields is a function of

length. This is due to the fact that for mismatched lines we have loads , and hencethere is voltage and current variations on the line due to standing waves. The capacitive andinductive energies are different at different locations.

Evaluation of Arbitrary

Constant

For Impedance calculations the knowledge of is not needed. However for power calculation we need to know

We can obtain by transforming the load impedance to the generator end ofthe line and then applying lumpled circuit analysis.


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The transformed impedance at the generator end is

From circuit (b) the voltage and current at AA' are

From Fig(a) the voltage and the current at the generator end are

Equating the two voltages we get

Since the line is lossless, the power supplied to the transformed

impedance is same as that supplied to the load .

Recap



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Power delivered to a complex load connected to a generator through a section of a line.

Complex power at any location on the line.

How to obtain the amplitude of the forward travelling wave?

Objectives


Impedance transformation from the complex impedance plane to the complex reflection coefficitent plane.

Constant resistance and constant reactance circles on complex - plane.

Simth chart - Orthogonal impedance coordinate system on complex - plane.

Location of various impedances on the Smith chart.

The graphical representation given in the following mainly describes the impedance/admittance characteritics of atransmission line.

Complex Impedance (Z) & Reflection co-efficient ( ) planes

Let us define the normalized impedance

For passive loads

and

A passive load can be denoted by a point in the right half of the complex Z-plane as shown in Fig(a)

The complex Reflection Coefficient is


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The complex can be expressed in cartesian and polar form as

Since for passive loads , the reflection coefficient can be denoted by a point with the unity circle in the

complex - plane, as shown in Fig (b). 'R' denotes the magnitude of the reflection coefficient and denotes the phase of thereflection coefficient.

Since there is one-to-one mapping between to , the entire right half Z-plane is mapped on to the region within the unity

circle in the -plane.

Transformation from Z to

Let us transform the points from the - plane to - plane.

Separating real and imaginary parts, we get

--------- (2.5)

--------- (2.6)

Equations (2.5) and (2.6) are the equations of circles. Equation (2.5) represents constant resistance circles and equation (2.6)represent constant reactance circles.

ConstantResistanceCircles

The constant resistance circles have their centres at ( , 0) and radii ( ). Figure below shows the

constant resistance circles for different values of ranging between 0 and .


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We can note following things about the constant resistance circles.

(a)

The circles always have centres on the real -axis ( -axis).(b)

All circles pass through the point (1,0) in the complex plane.(c)

For , the centerof the circle lies at the origin of the plane and it shifts to the right as increases.(d)

As r increases the radius of the circle goes on reducing and for the radius approaches zero, i.e.,the circle reduces to a point.

(e)

The outermost circle with center (0,0) and radius unity, corresponds to or in other wordsrepresents purely reactive impedances.

(f)

The right most point on the unity circle, represents as well as .

ConstantReactanceCircles

The constant reactance circles have their centers at and radii . The centres for these circles lie on a

vertical line passing through point (1,0) in the -plane. The constant reactance circles are shown in figure below

for different values of


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Note again that only those portions of the circles are of significance which lie within the unity circle in the -plane. The curves shown dotted portion do not correspond to any passive load impedance.

We can note following things about the constant reactance circles:

(a)

These circles have their centers on a vertical line passing through point .(b)

For positive the center lies above the real -axis and for negative , the center lies below the real -axis.

(c)

For the center is at and radius is .This circle therefore represents a straight line.(d)

As the magnitude of the reactance increases the center moves towards the real -axis and it lies on the

real -axis at (1,0) for .(e)

As the magnitude of the reactance increases, the radius of the circle, ,decreases and it approaches

zero as .(f)

All circles pass through the point .(g)

The real -axis ( -axis) corresponds to and therefore represents real impedances, i.e., purelyresistive impedances.

(h)

The right most point on the unity circle, , corresponds to as well as .

The Smith

Chart The Smith chart is a graphical figure which is obtained by superposing the constant resistance and the constant

reactance circles within the unity circle in the complex -plane. Since we have mapped here the impedances to

the -plane, let us call this Smith chart the Impedance Smith chart.


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Generally the axes are not drawn on the Smith chart. However one should not forget that the Smith chart is a

figure which is drawn on the complex -plane with its center as origin.

The intersection of constant resistance and constant reactance circles uniquely defines a complex load

impedance on the -plane.

Let us identify some special points on the Smith Chart.

(a)

The left most point A on the smith chart corresponds to and therefore represents idealshort-circuit load.

(b)

The right most point B on the Smith chart corresponds to , and therefore represents idealopen circuit load.

(c)

The center of the Smith chart M , corresponds to and hence represents the matched load.(d) Line AB represents pure resistive loads and the outermost circle passing through A and B represents

pure reactive loads.

(e) The upper most point C represents a pure inductive load of unity reactance and the lower most point Drepresents a pure capacitive load of unity reactance.

(f) In general the upper half of the Impedance Smith Chart represents the complex inductive loads and thelower half represents the complex capacitive loads.

A ready made Smith Chart looks as in the following : Figure

Recap In this course you have learnt the following Impedance transformation from the

complex impedance plane to the complex reflection coefficitent plane. Constant

resistant and constant reactance circles on complex - plane. Simth chart -

Orthogonal impedance coordinate system on complex - plane. Location of various

impedance on the Smith chart.

Objectives


What is a constant VSWR circle on the - plane?Properties of constant VSWR circles.Calculations of load reflection coefficient.
http://popup%28%27smith.html%27%29/http://popup%28%27smith.html%27%29/


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Calculation of reflection co-efficient at a distance from the load.Calculation of transformed impedance at a distance from the load.VSWR on the line.Location of voltage maximum or minimum.Identifying the type of load.

ConstantVSWRCircles

The reflection coefficient at any location on the line is

Let be written in polar form as

Then we get,

Where,

and

As we move along the line towards the generator increases and consequently, becomes more negative.However, R remains constant.

The point moves on a circle with centre (0,0) and radiurs R in the complex - plane.

If we move towards the generator, is positive and the point moves clockwise on the circle. For movementaway from the generator, the point moves anti-clockwise on the circle.

For a circle, . Since VSWR , is constant for the circle.

That is, all points on the circle have same VSWR, . This circle therefore is called the CONSTANT VSWRCIRCLE.

Properties


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ofConstantVSWRCircles

The constant VSWR circles are shown below

We can make following observations about the constant VSWR circles:

All the circles have same center, the origin of the complex -plane.

The origin in the -plane represents or . As we move radially outwards the and hence

increases monotonically and for the outermost (unity) circle, and .

The origin corresponds to the condition , i.e., no reflection on the line. This point represents the best

matching of the load as there is no reflected power on the line. For the outer most circle , and we get the

worst impedance matching as the entire power is reflected on the transmission line. We can therefore make ageneral statement that closer is the point to the origin of the -plane, (i.e., the center of the Smith chart) better isthe impedance matching.

As we have defined earlier, the indicates a distance towards the generator. As becomes more positive,

decreases and the point moves clockwise on the constant VSWR circle. If we move away from the generator,becomes negative and then the point on the circle moves in the anticlockwise direction.

Smith chart isa very usefultool forsolving

transmissionline problems.A variety ofcalculationscan be carriedout using theSmith chartwithoutgetting intocomplexcomputations.

(A) Calculation of Load Reflection Co-efficient

Let us find the reflection coefficient for a load impedance . First normalize the impedance with the

characteristic impedance to get


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Identify the constant resistance and the constant reactance circles corresponding to and respectively.

Intersection of the two circles marks the load impedance on the Smith chart as point P (see Figure )

Measure the radial distance of P from the centre of the chart M. This is the magnitude of the load reflection

coefficient .

The angle which the radius vector MP makes with the -axis is the phase of the load reflection coefficient .

Note

The Smith chart should be placed in such a way that the most clusterred portion of the chart lies on the

right side. The horizontal line towards right then indicates the real - axis.

(B) Calculation of Reflection Co-efficient at a Distance from the Load

Let the transmission line be terminated in a load impedance . Let us find the reflection coefficient at a distance from theload using the Smith chart.

First, mark the load impedance as described in (A)

Draw the constant VSWR circle passing through P.

Rotate the radius vector MP by an angle in the clockwise direction to get to point Q (See Fig ). Radial distance MQ

gives the magnitude of the reflection coefficient, . Angle which the radius vector MQ makes with the -axis gives the

phase of , .


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(C)Calculation of Transformed Impedance at a Distance from the Load

Suppose we have to obtain the transformed impedance at a distance from the load. Carry out the steps in (a) and (b ) toget to point Q. Now instead of measuring reflection coefficient, identify the constant resistance and the constant reactance

circles passing through Q. They provide the transformed normalized resistance and reactance and respectively.

Multiplying by we get the transformed impedance .

The same procedure is used for transforming impedance from any point on the line to any other point.

However if the distance ` ' is away from the generator, it should be treated negative and hence the rotation of the radius

vector must be by in the anticlockwise direction.

(D) VSWR on the line


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As we have seen earlier if the load impedance is not equal to there is a standing wave on the transmission line. We

also know that the maximum impedance seen on the line . This means that the maximum normalized

impedance measured on the line is nothing but the VSWR, . The task of finding is then very simple.

Find the normalized impedance ( ) corresponding to point T. Then,

Mark the impedance on the Smith chart (P).

Draw the circle with center as the centre of the Smith chart (M) and radius PM.

Mark point T where the circle intersects the line ( axis) on the right hand side of the Smith chart.

Read corresponding to point T.

(E)Location of Voltage Maximum or Minimum

At the location of voltage maximum the impedance is maximum ( ), and at the location of voltage minimum the

impedance is minimum ( ). Hence point T indicates the location of voltage maximum and point S indicates the locationof voltage minimum in figure below


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To find distance of these points from the load, measure the angle between the load point P and point T and between point P

and point S respectively in the clockwise direction from P to T.The angle PMT in the clockwise direction when divided by

gives the distance of the voltage maximum from the load, . Similarly one can obtain distance of voltage minimum,by measuring angle PMS in clockwise direction.

Alternatively one can make use of the fact that the voltage maximum and minimum are seperated by a distance of .

That is

Identifying the Type ofLoad

If the load is inductive, point P lies in the upper half of the Smith chart. Then while moving clockwise oconstant-VSWR circle, we first meet point T and then we meet poing S. In other words, for inductive lvoltage maximum is closer to the load point than the voltage minimum. Exactly opposite occurs for capa

loads i.e., the voltage minimum is closer to the load than the maximum.

We can therefore quickly identify the load looking at the standing wave pattern.

If the pattern is like the one given in Figure (a), that is the voltage drops towards the load, the loinductive.

Similarly if the pattern is like that in Figure (b), that is, the voltage rises towards load, the locapacitive.

If the voltage is maximum or minimum at the load, the load impedance is presistive Figure (c).

For the purely reactive loads the point P will lie on the outermost circle making andThe pattern for pure inductive loads will be like that in Figure (d) and that for the pure capacative loads

likethat in Figure (e).


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Note

For identifying the load, observe two things on the standing wave pattern

(1) Minimum Voltage

(2) Location of Voltage minimum/maximum

Recap


What is a constant VSWR circle on the - plane?Properties of constant VSWR circles.Calculations of load reflection coefficient.Calculation of reflection co-efficient at a distance from the load.Calculation of transformed impedance at a distance from the load.VSWR on the line.Location of voltage maximum or minimum.Identifying the type of load.

Objectives


Admittance Transformation on Transmission Line.

Admittance Smith Chart.

AdmittanceTransformatio


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n onTransmissionLine

For parallel connections of transmission lines, the analysis is simpler if we deal with admittances rather thanimpedances. We therefore develope Admittance transformation relations for a transmission line.

To start with, we define the characteristic admittance which is the reciprocal of the characteristic

impedance , i.e.,

Also .

The admittance at location therefore is

This relation is identical to that for the impedance transformation.

Admittance Smith Chart

Let us define the characteristic admittance as .

Normalization of every admittance is done with the characteristic admittance of the transmission line. An

admittance when normalized with is noted by

The reflection coefficient is


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That is

Now if we take normalized impedance equal to i.e., and , we get for which is out of phase with

respect to the for . That means, for same numerical values, if the normalized load is impedance we get some point P on

the plane and if the load is admittance we get point P' which is diagonally opposite to P on the -plane (see Figure). P' is

obtained by rotating P by around the origin of the plane.

This is true for every and and consequently all constant resistance and constant reactance circles when rotated

by around the origin of the -plane give corresponding constant conductance (constant- ) and constant susceptance

(constant- ) circles respectively.

Admittance SmithChart(contd.)

TheAdmittance Smith Chart therefore appears as in the following : Figure

The admittance Smith chart therefore is obtained by rotating the impedance Smith chart by and replacing

by and by . Since it is just a matter of rotation, there is no need to have separate Smith charts forimpedance and admittance.

Generally we keep the Smith chart fixed and rotate the co-ordinate axis of the complex - plane by if thechart is used for admittance calculation.

AdmittanceSmith Chart(contd.)
http://popup%28%27smith180.html%27%29/http://popup%28%27smith180.html%27%29/


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Following points should be kept in mind while making their use of the Smith chart fortransmission line calculations.

(1) While calculating phase of the reflection coefficient from the admittance Smith chart the phase

must be measured from the rotated -axis.

(2)

Although the and can be interchanged with and respectively and a point and

will have the same spatial location on the Smith chart for and , physical conditionscorresponding to the two will not be identical. Let us take some specific examples.

Upper half of the Smith chart with represents inductive loads where as represents capacitiveloads.

Point A in Figure (a) is as well as . But , represents short

circuit load hereas, , represents an open circuit load. The point A therefore represents theshort circuit in the impedance chart whereas it represents the open circuit in the admittance chart.

Similarly point B in Figure (a) represents the open circuit for the impedance chart but in admittance chartit represents the short circuit.

In Figure (b), point T corresponds to the voltage maximum if the chart is the impedance chart, and avoltage minimum if the chart is the admittance chart. The opposite is true for point S. Now since thevoltage maximum coincides with the current minimum and vice-versa, the point T in admittance Smithchart represents the location of the current maximum and point S represents location of the current

minimum. So we find that the voltage standing wave pattern and the impedance have the samerelationship as the current standing wave pattern and the admittance.

As we have seen, the reflection coefficients for same normalized impedance and admittance values

are out of phase. Therefore any normalized impedance can be converted to normalizedadmittance and vice-versa by taking a diagonally opposite point on the constant VSWR circle. In Figure(b), P' gives normalized admittance corresponding to the normalized impedance at P. We can thereforeswitch between admittance and impedance Smith charts freely without any additional computation.

Recap



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Admittance Transformation on Transmission Line.

Admittance Smith Chart.

Objectives


Various applications of transmission lines.

How to measure complex impedance at high frequencies where phase measurement is unreliable.

How and why to use sections of transmission line as reactive elements in the high frequency circuits.Use of Smith chart and to design transmission line sections for realizing reactive impedances.

Measurementof UnknownImpedance

The unknown impedance which is to be measured is connected at the end of the transmission line as shown in

Figure below. The transmission line is excited with a source of desired frequency . From the standing wave

pattern, three quantities, namely the maximum voltage , minimum voltage , and the distance of

the voltage minimum from the load is measured. The ratio of and gives the VSWR on the line.

We know that at point B on the transmission line where the voltage is minimum, the impedance is real and its


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value is . The impedance is nothing but the transformed value of the load impedance .

We can therefore obtain the unknown impedance by transforming back from point B to point A. Let

the distance of the voltage minimum from the load be . Since the transformation from B to A is away fromthe generator, the distance BA is negative. The unknown impedance therefore is

Substituting for , we get

Separating real and imaginary parts we get

Measurement ofUnknown

Impedance(Practical Consideration)

While practically implementing the above scheme one would also notice that invariably the location

of unknown impedance is not precisely defined. As a result the measurement of may havesome error which in turn will result into an error in the load impedance.

To overcome this problem the measurement is carried out in two steps. First, the standing wavepattern is obtained with the unknown load as explained above. Now replace the unknownimpedance by an ideal short-circuit and obtain the standing wave pattern again. The two standingwave patterns are shown as below

At the short circuit point (which is also the location of the unknown impedance) the voltage is


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zero. The voltage is also zero at points which are multiple of away from it. i.e., at point C, E etc.The points C, E etc represent impedance conditions identical to that at A, that is, the impedance atC or E is equal to the unknown impedance. The unknown impedance therefore can be obtained by

transforming impedance at B or D to point C. If impedance is transformed from D to C the

distance is negative, whereas if the transformation is made from B to C the distance ispositive. The unknown impedance therefore can be evaluated as

One can note here that . In the impedance calculation either of or canbe used. As long as the sign of the distance is taken correctly it does not matter which of the minimais taken for impedance transformation.

Transmission Line as a Circuit Element

At frequencies of hundreds and thousands of MHz where lumped elements are hard to realize, theuse of sections of transmission line as reactive elements may be more convenient.

The turns in the wire of the inductor have small distributed capacitors.

As the frequency increases, the capacitance begins to play a role in the response of the circuit andbeyond the resonant frequency, the capacitance predominates the response. That is the inductancecoil effectively behaves like a capacitor.

Similarly, for a capacitance, there exist lead inductance. As the frequency increases, the leadinductance starts dominating over the capacitance and beyond the resonant frequency of the LCcombination, the capacitor effectively behaves like an inductor.

So, it is clear that at high frequencies, realization of reactive element is not that simple.

On the other hand at high frequencies, the wavelength and the length of the transmission line sectionreduces and becomes more manageable.

Use of SmithChart for

calculatingand


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From the impedance relation we can see that if a line of length is terminated in a short circuit or opencircuit (shown in Figure below) the input impedance of the transmission line is purely reactive.

The input impedance of a loss-less line can be written as

Since the range of 'tan' and 'cot' functions is from to , any reactance can be realized by proper

choice of . Moreover, any reactance can be realized by either open or short circuit termination. This is avery useful feature because depending upon the transmission line structure, terminating one way may be

easier than other. For example, for a microstrip type line ( see in later section), realizing an open circuit iseasier as short circuit would require drilling a hole in the substrate.

Now if a reactance is to be realized in a high frequency circuit one can use a short circuited line of

length or an open circuited line of length given by

Smith chart can be used to find or as follows:

Choose suitable characteristics impedance of the line,

Normalize the reactance to be realized (X) by to give normalized reactance .

(a) Mark the reactance jx to be realized on the Smith chart to get point 'X' in Figure.

(b)

Move in anticlockwise direction from point X to the short circuit (SC) point on the Smith chart to get(see Figure below).

(c)

Move from X in the anticlockwise direction upto open circuit (OC) to get as indicated in Figure.

(d) Note here that instead of reactance if we had to realize a normalized susceptance b, the procedure isidentical except that SC and OC points are interchanged.


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Line length and theirEquivalent Reactants

The following figure shows the range of transmission line lengths and thecorresponding reactances which can be realized at the input terminals of the line.

Recap


Various applications of transmission lines.

How to measure complex impedance at high frequencies where phase measurement is unreliable.

How and why to use sections of transmission line as reactive elements in the high frequency circuits.

Use of Smith chart and to design transmission line sections for realizing reactive impedances.

Objectives



47/57

What is the resonant section of a transmission line?

Frequency response of a resonant section of a line.

Input impedance of a resonant section of a line.

Voltage and current on a resonant section of a line.

Transmision Lines asResonantCircuits

If the length of a short or open circuited line is exact multiple of , the imput impedance of the line is zero

or . Let us plot the input impedance as a function of frequency ` ', for a given length of transmission and agiven termination (short circuit or open circuit).

Figure shows the variation of reactance as a function of frequency for open and short circuited sections of a

transmission line. It is clear that around frequencies .., for which the length is an integer multilple

of , the impedance variation is identical to an L-C resonant circuit. In the vicinity of these frequencies the linecan be used as a LC - resonant circuit.

Frequency


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responseofResonantCircuit

The impedance characteristics of a series and a parallel resonant circuit are shown in the figure below

Comparing Figure (1) with Figure (2), one can observe that a short circuited line behaves like a parallel resonant

circuit around frequencies and , whereas around and its behaviour is like a series resonant circuit.

In general a short circuited section of a line is equivalent to a parallel resonant circuit.

Similarly, the line is equivalent to a series resonant circuit.

A converse is true for an open circuited section of a line i.e., if the length of the line is equal to odd multiples of

, the line behaves like a series resonant circuit, and if the length of the line is equal to even multiple of , the linebehaves like a parallel resonant circuit.

Input

Impedance ofResonantLine

Input impedance of a resonant lossless line is either . However, in practice, the lines have finite loss. Thisloss has to be included in the calculations while analysing the resonant lines. The complex propagation

constant has to be used in impedance calculations of a resonant line.

The input impedance of a short or open circuited line having propagation constant can be written as

Note that although has been taken complex for a low-loss transmission line, is almost real. Substituting


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for , we get

For a low-loss line, taking , we have . Also . Hence we get

Similarly for an open circuited line we get

For resonant lines, is integer multiples of i.e., is integer multiples of . If we take odd

multiples of , , and we get

On the other hand if we take even multiples of , , giving

Conclusion

A parallel resonant section of a line has an impedance and a series resonant section has an

impedance .One can cross-check the result with that of an ideal loss-less line. In the absence of any loss the parallel resonantcircuit shows infinite impedance and a series resonant circuit shows zero impedance at the resonance.

Voltage &Currenton aResonantSection ofa line

Consider a short circuited section of a line having length equal to odd multiples of . This line is equivalent to a

parallel resonant circuit. Let the line be applied with a voltage between its input terminals as shown inFigure(a).


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The voltage and current standing wave patterns on the line are shown in Figure(b,c).

The voltage is zero at the short-circuit-end of the line and is maximum at the input end of the line. similarly, thecurrent is maximum at the short-circuit end and minimum at the input end of the line.

The maximum value of the voltage on the line is and maximum value of current is . For a short-

circuited line the voltage and current on the line are given as

Recap

In this course you havel learnt the following

What is the resonant section of a transmission line?

Frequency response of a resonant section of a line.

Input impedance of a resonant section of a line.

Voltage and current on a resonant section of a line.

Objectives

In this course you will learn about the following

Quality factor of a resonant circuit.

Energy stored in a resonant section of a line.

Quality factor of a resonant line.

Transmission lines as step-up transformers.

Relation of the step-up ratio with the quality factor.


51/57

The Quality Factor of a ResonantCircuit

The quality factor of a resonant circuit is given as

where is the resonant frequency of the circuit.

'Q' of a

ResonantLine

The energy stored in a long section of the line is

U = Capacitive energy + Inductive energy

Since , we have . Hence the inductive and capacitive energies are equal, and thetotal energy is

The energy lost per second is the power loss in the line. At parallel resonance, the line effectively appears like

a resistance of value . The power loss in the line therefore is


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The quality factor of the line therefore is

Again noting that , and . We get,

Where is the phase constant at resonant frequency.

One can note here that is independant of the length of the line as long as the loss is small.

In practice, generally the lines have loss low enough to give a of few hundreds very easily.

Since the -bandwidth of a resonant circuit is , higher value of implies highly tuned or frequencyselective circuits.

The transmission line sections therefore act as excellent frequency selective circuits at high frequencies.

Voltage orCurrentStep-up

TransformerLet us take a resonant transmission line of length . The line is open circuited at one end and short circuitedat other as shown in the figure.

Let us say there is a voltage source which induces a voltage in the line at some point 'X'. This induced voltage

will send two voltage waves and with equal amplitudes. Consider now one of the waves, say . This

wave travels upto point B to encounter a short circuit. Since the reflection coefficient for a short circuit is , thewave gets fully reflected with a phase reversal. The wave after travelling a distance BA reachs to the open

circuited end of the line and again gets fully reflected but with no phase reversal as for the open circuit.

After one round trip therefore when the wave reaches point X, its amplitude is same as its original value but

its phase is changed by , due to reflection at point B and due to propagation of a round trip distance

of . This wave therefore adds up with the induced voltage in phase and the added up wave travels on the

transmission line. The process is regenerative and the amplitude of the voltage wave goes on increasing.


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Exactly identical thing happens with the other wave .

Since the two waves travelling in the opposite directions identically grow in amplitude, the result is a continiouslygrowing standing wave on the line with appropriate voltage maximum at A and voltage minimum at B.

If the coupling of voltage is sustained, and the line is loss-less, there is no limit on the voltage and current, and

the voltage and current eventually would grow to .

However, if the line has a loss (no matter how small), then of course the voltage and current stabilize at somefinite values. As the voltage/current increases the ohmic loss also increases and when the power lost in the linejust equals the power supplied by the coupling source, the voltage/current stabilizes. It should be noted howeverthat the maximum stabilized voltage or current on the line could be much higher than the coupling voltage orcurrent.

This suggests a possibility of using a resonant transmission line as a step-up voltage or current transformer.

VoltageAmplificationon a

ResonantLine

As an illustrative example let us take a resonant section of a line, and instead of putting a short circuitput an ideal voltage source at point B as shown Figure (a). The open circuit at point A appears as almost

short (for a low loss line) at point B. The impedance seen by the voltage source is and a

current flows in terminal B. Since point B is a voltage minimum and current maximum, the source

current is equal to the maximum current on the line . The maximum voltage on the line then

is and it appears at point A. We therefore have


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Since for a low-loss line, we get . That is, the voltage at the open-circuited end of the

resonant line is much higher compared to the excitation voltage . That is, a section of a line can be usedfor stepping up a voltage.

The Voltage step-up ratio is

Taking (where is an odd integer), and substituting , the voltage step-up ratio can bewritten as

Since is typically few hundreds for a low-loss transmission line, a voltage amplification of few hundredsmay occur in a resonant line.

Recap


Quality factor of a resonant circuit.

Energy stored in a resonant section of a line.

Quality factor of a resonant line.

Transmission lines as step-up transformers.

Relation of the step-up ratio with the quality factor.

Objectives

In this course you will learn about the following

Impedance matching techniques.

Quater wavelength transformer matching its advantages and limitations.

Single stub matching technique and its special features.

Impedance Matching

Impedance matching is one of the important aspects of high frequency circuit analysis. To avoid reflections andfor maximum power transfer the circuits have to be impedance matched.

Transmission line sections can be used for the purpose of impedance matching.


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There are various impedance matching techniques which are discussed in the following :

Quarter Wavelength Transformer

This technique is generally used for matching a resistive load to a transmission line (a), for matching two resistiveloads(b),or for matching two transmission lines with unequal characteristic impedances (c) (see Figure). All cases

are identical in principle as all require matching between two purely resistive impedances.

Principle

Introduce a section of a transmission line(called transformer) between two resistances to be matched, such thatthe transformed impedances perfectly match at either end of the transformer section. That is, in Figure (a) say,

the impedance seen towards right at A should be , and impedance seen towards left at B should be R. So

when seen from transmission line side it appears to be terminated in , and when seen from load resistanceside it appears to be connected to a conjugately matched load R. Similar is true for Figure (b,c).

For the transformer we have two parameters to control, characteristic impedance of the transformer section, andthe length of the transformer section.

Let us assume that the characteristic impedance of the transformer section is . For length, thetransformer inverts the normalized impedance. Therefore the impedance seen at A towards right in Figure (a)would be

For matching at A, should be equal to , i.e.

Conclusion

Two resistive impedances can be matched by a section of a transmission line which is quarter-wavelength longand has characteristic impedance equal to the geometric mean of the two resistances.

The quarter wavelength transfer is commonly used at the junction of two transmission lines of unequalcharacteristic impedances.

Drawback

This technique needs special line of characteristics impedance for every pair of resistances to be matched.


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Single-Stub Matching Technique

A stub is a short-circuited section of a transmission line connected in parallel to the main transmission line. A stub ofappropriate length is placed at some distance from the load such that the impedance seen beyond the stub is equal to thecharacteristic impedance.

Suppose we have a load impedance connected to a transmission line with characteristic impedance (Figure a). Theobjective here is that no reflection should be seen by the generator. In other words, even if there are standing waves in the

vicinity of the load , the standing waves must vanish beyond certain distance from the load.

Conceptually this can be achieved by adding a stub to the main line such that the reflected wave from the short-circuit endof the stub and the reflected wave from the load on the main line completely cancel each other at point B to give no netreflected wave beyond point B towards the generator.

We make use of Smith chart for this purpose

Since we have a parallel connection of transmission lines, it is more convenient to solve the problem using admittancesrather than impedances. To convert the impedance into admittance also we make use of the Smith chart and avoid anyanalytical calculation.

Now onwards treat the Smith chart as the admittance chart

Matching Procedure

First mark the load admittance on the admittance smith chart (A).

Plot the constant circle on the smith chart .Move on the constant circle till you intersect the constant

circle this point of intersection corresponds to point (B). The distance traversed on the constant circle is .

This is the location of placing the stub on the transmission line from the load end .


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Find constant suseptance circle.

Find mirror image of the circle to get circle.

Mark on the outer most circle (D).

From (D) move circular clockwise upto s.c point (E) to get the stub length .

Advantage

The single-stub matching technique is superior to the quarter wavelength transformer as it makes use of only onetype of transmission line for the main line as well as the stub. This technique also in principle is capable ofmatching any complex load to the characteristic impedance/admittance. The single stub matching technique isquite popular in matching fixed impedances at microwave frequencies.

Drawback

The single stub matching technique although has overcome the drawback of the quarter wavelength transformertechnique, it still is not suitable for matching variable impedances. A change in load impedance results in achange in the length as well as the location of the stub. Even if changing length of a stub is a simpler task,

changing the location of a stub may not be easy in certain transmission line configurations. For example, if thetransmission line is a co-axial cable, the connection of a stub would need drilling of a hole in the outer conductor.

Recap

In this course you have learntthe following

Impedance matching techniques.

Quater wavelength transformer matching its advantages and limitations.

Single stub matching technique and its special features.

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