rf&me lecture 6_microwave transmission lines

8/23/2019 Rf&Me Lecture 6_microwave Transmission Lines

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Microwave Transmission Lines

Engr. Ghulam Shabbir


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Resonant Transmission Lines

Standing Wave and Standing-Wave Ratio

A transmission line that is not properly terminated (i.e., a

line terminated in an impedance not equal to its

characteristic impedance) is called a resonant (unmatched)

line. A resonant circuit is one in which capacitive and inductive

reactances cancel each other. An example of such a circuit,

at a specific frequency, is a parallel or series LC circuit.

Antennas and transmission lines are resonant circuits at

many different frequencies.

A resonant line may be terminated in an open, in a short, in

some capacitive, inductive or resistive values other than the

characteristic impedance of the line.


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Resonant Transmission Lines


In any such case, the line cannot deliver full energy to a

load. Some of the energy is reflected back to the source

and forms standing waves on the line.

On a resonant line, some of the energy sent down the

line will be reflected back to the source, resulting instanding waves.

Every l/2 along a resonant line, high voltage and low

current points appear. Halfway between these points,

the opposite is true.


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Standing Wave


Incident Wave

The wave of voltage or current which travels from source to

load is called the incident wave.

If the load impedance is equal to the characteristic

impedance (Zo) then all energy provided to the load with thehelp of incident wave will be absorbed by the load.

Reflected Wave

If the impedance of load is not equal to the characteristics

impedance, then there will be the mismatching and some of

the energy will be reflected back.

Standing Wave

The resultant waveform which is produced along the

transmission line due to interaction of incident wave and

reflected wave is known as standing wave.


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When an AC signal is transmitted to a transmission line,some of the signal is reflected back toward the source.

This reflected wave come back towards the source but

the source cannot absorb it, due to which it stays in the

transmission line, and attenuates the transfer of power

from source to load because it is out of phase w.r.t

source.

Standing Wave



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Standing Wave


In a transmission line, the transmitter is feeding power(current and voltage) at a certain frequency or wavelength

into the line.

At the antenna or load end, some of the power is absorbed

by the load or radiated. The rest of the power is reflected back along the line towards

the transmitter.

All along the transmission line the forward and the reflected

current and voltages combine to give the total current and

voltage anywhere along the line.

As long as the load impedance, line length and frequency do

not change, a stable pattern of voltage and current peaks and

valleys will appear on the line.

That is called a "standing wave," since it doesn't change.


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Standing Wave

The general solutions of the transmissionline equation

consist of two waves, traveling in opposite directions with

unequal amplitude as shown:

so


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Standing Wave

With no loss in generality it can be assumed that V+e_ and

V-e are real. Then the voltage-wave equation can be

expressed as

This is called the equation of the voltage standing wave,

where

Which is called the standing wave pattern of the voltage or

the amplitude of the standing wave, and

which is called the phase pattern of the standing wave.


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Standing Wave

The maximum and minimum values of Eq. (3-3-3) can befound as usual by differentiating the equation with respect

to z and equating the result to zero.

By doing so and substituting the proper values ofz in the

equation, we find that1. The maximum amplitude is

2. The minimum amplitude is

and this occurs at z = (2n 1)/, where n = 0, 1,

2, ..


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Standing Wave3. The distance between any two successive maxima or

minima is one-half wavelength, since

Then

It is evident that there are no zeros in the minimum.

Similarly,


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Standing Wave

Standingwave pattern in a lossy line

The standingwave patterns of two oppositely travelingwaves with unequal amplitude in lossy or lossless line are

shown as


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Standing Wave

Voltage standingwave pattern in a lossless line


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Standing Wave

A further study reveals that

1. When V+ , V= 0, the standingwave pattern

becomes

2. When V+ = , V 0, the standingwave pattern

becomes

3. When the positive wave and the negative wave have equal

amplitudes (that is, |V+e| = |V-e

|) or the magnitude of

the reflection coefficient is unity, the standingwave

pattern with a zero phase is given by

which is called a purely standing wave


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Standing Wave

Similarly, the equation of a pure standing wave for the current is

Equations (3-3-12) and (3-3-13) show that the voltage and

current standing waves are 90 out of phase along the line. The points of zero current are called the current nodes.

The voltage nodes and current nodes are interlaced a

quarter wavelength apart.

The voltage and current may be expressed as real functions

of time and space:


15/48Pure standing waves of voltage and current


Standing Wave

The amplitudes of Eqs. (3-3-14) and (3-3-15) varysinusoidally with time; the voltage is a maximum at the

instant when the current is zero and vice versa.

Figure below shows the pure-standing-wave patterns of the

phasor of Eqs. (3-3-12) and (3-3-13) for an open-terminalline.


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StandingWave Ratio


The ratio of the maximum voltage to the minimum voltage

is a measure of the mismatch and is called the "Voltage

Standing Wave Ratio" or VSWR.

In the same way, the ratio of the maximum to minimumcurrent is called the "Current Standing Wave Ratio" or

ISWR, where the I stands for current.

It can be shown that the ISWR is the same as the VSWR,

but VSWR is normally easier to measure. Normally we just say SWR, implying VSWR.


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StandingWave Ratio

Standing waves result from the simultaneous presence of

waves traveling in opposite directions on a transmission line.

The ratio of the maximum of the standing-wave pattern to

the minimum is defined as the standing-wave ratio,designated by . That is,


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StandingWave Ratio

The standing-wave ratio results from the fact that the two

traveling-wave components of Eq. (3-3-1) add in phase at

some points and subtract at other points.

The distance between two successive maxima or minima is/.

The standing-wave ratio of a pure traveling wave is unity and

that of a pure standing wave is infinite.

It should be noted that since the standing-wave ratios of

voltage and current are identical, no distinctions are made

between VSWR and ISWR.

When the standing-wave ratio is unity, there is no reflected

wave and the line is called a flat line.


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StandingWave Ratio

The standing-wave ratio cannot be defined on a lossy linebecause the standing-wave pattern changes markedly from

one position to another.

On a lossless line the ratio remains fairly constant, and it

may be defined for some region. For a lossless line, the ratio stays the same throughout the

line.

Since the reflected wave is defined as the product of an

incident wave and its reflection coefficient, the standing

wave ratio is related to the reflection coefficient by

and vice versa


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StandingWave Ratio

This relation is very useful for determining the reflection coefficientfrom the standing-wave ratio, which is usually found from the Smith

chart.

The following curve shows the relationship between reflection

coefficient | | and standing-wave ratio .

As a result of Eq. (3-3-17), since | | 1, the standing-wave ratio

i , .

From Eq. (3-3-18) the magnitude of the reflection coefficient is never

greater than unity.


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Example: 3-3-1 : Standingwave ratio


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Why match?

Impedance Matching

Matching the impedance of a network to the

impedance of a transmission line has two principal

advantages.

First, all incident power is delivered to the network.

Second, the generator is usually designed to workinto an impedance close to common transmission

line impedances.

If it does so, the load impedance has no reactive part

which can pull the generator frequency, and the VSWRon the line is unity or close to unity so the line length is

immaterial and the line connecting the generator to the

load is non-resonant.


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Impedance matching is very desirable with radio frequency

(RF) transmission lines. Standing waves lead to increased losses and frequently

cause the transmitter to malfunction.

A line terminated in its characteristic impedance has a

standing-wave ratio of unity and transmits a given powerwithout reflection.

Also, transmission efficiency is optimum where there is no

reflected power.

A flat line is non-resonant ; that is, its input impedance

always remains at the same value Z0 when the frequency

changes.

Impedance Matching


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Matching a transmission line has a special meaning, one

differing from that used in circuit theory to indicate equalimpedance seen looking both directions from a given

terminal pair for maximum power transfer.

In circuit theory, maximum power transfer requires the load

impedance to be equal to the complex conjugate of the

generator. This condition is sometimes referred to as

conjugate match.

In transmission-line problems matching means simply

terminating the line in its characteristic impedance.

A common application of RF transmission line is the one inwhich there is a feeder connection between a transmitter

and antenna.

Usually the input impedance to the antenna itself is not

equal to the characteristic impedance of the line.

Impedance Matching


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The output impedance of the transmitter may not be equal

to the Z0 of the line. Matching devices are necessary to flatten the line.

A complete matched transmission - line system is shown as

under:

Impedance Matching

Matched transmission - line system


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For a low-loss or lossless transmission line at radio

frequency, the characteristic impedance Z0 of the line is

resistive.

At every point the impedances looking in opposite

directions are conjugate.

If Z0 is real, it is its own conjugate. Matching can be tried first on the load side to flatten the

line; then adjustment may be made on the transmitter side

to provide maximum power transfer.

At audio frequencies an iron-cored transmitter is almostuniversally used as an impedance-matching device.

Occasionally an iron-cored transmitter is also used at radio

frequencies.

Impedance Matching


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In a practical transmission-line system, the transmitter is

ordinarily matched to the coaxial cable for maximum power

transfer.

Because of the variable loads, however, an impedance-

matching technique is often required at the load side.

Since the matching problems involve parallel connectionson the transmission line, it is necessary to work out the

problems with admittances rather than impedance to

admittance by a rotation of 180.

Impedance Matching


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In a communication network , it is often desirable to adjust

its elements, if possible, so that the power delivered to the

load is maximum.

In order to derive the conditions under which maximum

power transfer occurs, consider a load impedance,

ZL=RL+jXL=|ZL|ejL, connected to a generator of internalimpedance, ZG=RG+jXG=|ZG|e

j G, generating a voltage VG.

The power, PL, delivered to the load is

PL =

+ 2 RL

=|

|

(+)+ +

RL

Impedance Matching


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In microwave and radio-frequency engineering, a stub is a length

of transmission line or waveguide that is connected at one endonly.

The free end of the stub is either left open-circuit or (especially in

the case of waveguides) short-circuited.

Neglecting transmission line losses, the input impedance of the

stub is purely reactive; either capacitive or inductive, depending on

the electrical length of the stub, and on whether it is open or short

circuit.

Stubs may thus be considered to be frequency-dependent

capacitors and frequency-dependent inductors.

Because stubs take on reactive properties as a function of their

electrical length, stubs are most common in UHF or

microwave circuits where the line lengths are more manageable.

Stubs are commonly used in antenna impedance matching circuits

and frequency selective filters.

Impedance MatchingStub


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Stubs are shorted or open circuit lengths of transmission

line intended to produce a pure reactance at the

attachment point, for the line frequency of interest.

Any value of reactance can be made, as the stub length is

varied from zero to half a wavelength. Look at the SMITH chart and find the outer circle where

the modulus of the reflection coefficient is one.

On this circle are the SHORT and OPEN points, and all

values of positive and negative reactance.

The resistance is zero everywhere.

To generate a specified reactance, start at a short circuit

(or maybe an open) and follow around towards the

generator until the desired reactance is obtained.

Cut the stub number of wavelengths long.

Impedance Matching

Why stub?


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It is important to keep the total stub length as short as

possible, if wider bandwidths are required.

Every time you add a half wavelength to the stub length

the reactance of the stub comes back to the same value.

It is good design practice to make stubs in the range 0 to

0.5 wavelengths long.

However, this may require an impractically short stub, so

one can make the stub just a little over 0.5 wavelengths.

Impedance Matching

Why stub?


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If it is allowed to use either short or open stubs, the total

stub length can always be kept in the range 0 - 0.25

wavelengths.

A length of transmission line of 0.25 wavelengths takes

half way round the SMITH chart and transforms an open

into a short, or vice versa.

On microstrip it is usually easier to leave stubs open

circuit, for constructional reasons.

On coax line or parallel wire line, a short circuit stub hasless radiation from the ends: it is difficult to make a

perfect non-radiating open circuit as there are always

some end effects on the line.

Impedance Matching

Short or open stubs?


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Single stub-matching

ZL

open-stub

Zo

d

x

ZL

short-stub

Zo

d

x

Short-stub matching

Open-stub matching

Parallel configuration


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Single stub-matching

Short-stub matching

Open-stub matching

Serial configurationZL

short-stub

Zo

d

Zo

ZL

open-

stub

Zo

d

Zo

i

i


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Find out the load impedance ZL and transmission line

characteristic impedance Zo.

Calculate the normalized impedance z=(ZL/Zo).

Plot it on the SMITH chart. You are told the frequency and the velocity factor of the

line. Calculate the wavelength in meters. (or cm).

Follow the circle of constant radius on the SMITH chart

towards the generator until the locus crosses the r=1 circle.

Measure the number of wavelengths along the perimeterof the SMITH chart between the z point originally plotted,

and the r=1 circle intersection.

This tells you how far from the load to place your stub.

Impedance Matching

Stubs design procedure


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Read off from the r=1 intersection the reactance x' value.

Starting from a short (or open) follow the r=0 circle

around the outside of the SMITH chart until you come to

a point of reactance -x'.

Measure the number of wavelengths; this represents

from short/open end towards the generator. Cut your

stub this long.

The stub is placed in series with one of the transmission

line conductors. In coax this may be difficult to do technically.

One therefore often resorts to shunt stub matching,

where the stub and the original transmission line are

connected in parallel.

Impedance Matching



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It is easier to work in admittances. We notice that the SMITH chart can be used as an

admittance chart merely by rotating it through 180

degrees.

Normalized resistance becomes normalized conductance;

normalized reactance becomes normalized susceptance.

Admittances in parallel add:

short circuit point has infinite admittance

open circuit point zero admittance

The design procedure is the same as for series stubs.

Impedance Matching



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If you look at the SMITH chart you will find a circle of

constant real impedance r=1 which goes through the open

circuit point and the center of the chart.

If you plot any arbitrary impedance on the SMITH chart and

follow round at constant radius towards the generator, youmust cross the r=1 circle somewhere.

This transformation at constant radius represents motion

along the transmission line towards the generator.

One complete circuit of the SMITH chart represents a travelof one half wavelength towards the generator.

At this intersection point your generalized arbitrary load

impedance r + jx has transformed to 1 + jx', so at least the

real part of the impedance equals the characteristic

impedance of the line.

Impedance Matching

Single-stub Matching


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Note x' is different from x in general.

At this point you cut the line and add a pure reactance -jx'.

The total impedance looking into the sum of the line

impedance and -jx' is therefore 1 + jx' -jx' = 1 and the line is

matched.

Impedance Matching



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Although single-lumped inductors or capacitors can match

the transmission line, it is more common to use the

susceptive properties of short-circuits sections of

transmission lines. Short-circuited sections are preferable to open-circuited

ones because a good short circuit is easier to obtain than a

good open circuit.

For a lossless line with Yg = Y0 , maximum power transfer

requires Y11 = Y0 , where Y11 is the total admittance of theline and stub looking to the right at point 1-1.

The stub must be located at that point on the line where

the real part of the admittance , looking towards the load,

is Y0 .

Impedance Matching


Impedance Matching


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In a normalized unit y11 must be in the form

Impedance MatchingSingle-stub Matching

Single-stub Matching for Example 3-6-1


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Example 3-6-1: Single-stub Matching

Solution


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Graphic solution for Example 3-6-1


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Double-stub matching

The advantage of this technique is the position of stubs

( d and x) are fixed. The matching are done by changing

the length of stubs. The disadvantage of this technique

is not all impedances can be matched.

ZL

Zo

d x

S1

S2

Open orshort stubs

rf&me lecture 6_microwave transmission lines

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