qbankmesem5

7/24/2019 qbankmesem5

1/39

Question 1

A certain coaxial cable has characteristic impedance 50 ohms and velocity factor 0.63. Calculate the wave

velocity on this cable in cm/nanosecond; also the inductance and capacitance per unit lenth of the cable.

!he wave velocity in vacuum times the velocity factor is e"ual to the wave velocity on the cable.

!he wave velocity in vacuum is 30 cm per nanosecond# or 3 times $0%& metres per second. !hus

the wave velocity is 30'0.63 ( $&.)0 cm per nanosecond or in *+ units# $.&) times $0%& metres

per second. !he inductance per unit lenth , and capacitance per unit lenth C# in -/m or /m respectively#

are related to the velocity u in *+ units and the characteristic impedance o in ohms by therelationships o ( s"rt,/C1 and u ( $/s"rt,C1. !hus we see that , ( o/u and C ( $/o'u1.

2sin these we find , ( 50/$.&)'$0%&1 ( 64.55 n-/m and C ( $/50'$.&)'$0%&1 ( $05.& p/m.

,et us suppose that this cable feeds a load impedance 5 7$50 ohms. Calculate the complex reflection

coefficient amma# and express it both as real and imainary parts# and also as modulus and phase anle.

amma ( ,8o1/,o1 ( 57$508501/57$50501 ( $761/5761 ( honestly# by

calculator1 0.6$70.3)34 ( 0.&&'exp730.34de1

9etermine the normalised load impedance in this case and plot it on the *mith chart. :easure amma

from the *mith chart plot and show that it arees with the value you calculated.

ormalised load impedance ( ,/o ( $.573

Calculate the return loss in d


2/39

A certain 5 ohm coax cable feeds an antenna of assumed1 radiation impedance &0 7$5 ohms. Calculate

the return loss. Calculate the proportion of forward wave power which is radiated.

As in Hn $# amma ( &07$5851/&07$551 ( 57$51/$557$51 ( 0.04$70.0)& (

0.$0$5'exp766.04de1 so the return loss is 80'lo$00.$0$51 ( $).&d


3/39

A coaxial cable feeds a load 7$1o. !he line is lossless and has characteristic impedance o. 9efine#

and find values for# the followinM8

!he Goltae *tandin Fave atio G*F1 on the line.

!he G*F is a property of the standin wave pattern on the line. +t is a dimensionless number#

e"ual to the ratio of voltae at a standin wave pattern maximum to the voltae at a standin

wave pattern minimum# $/4 of a transmission line wavelenth away alon the line.

umerically# the G*F ( $=s=1/$8=s=1 where =s= is the modulus of the complex reflectioncoefficient or scatterin parameter somewhere alon the line. *trictly spea@in# the G*F is only

defined precisely in terms of the wave pattern on lossless lines. -owever# the definition in termsof =s= extends the concept of the G*F to lines havin attenuation or loss# and in this case the

value of the G*F depends on where it is measured alon the line.

+n this case# s ( ,8o1/,o1 ( $71/371 and =s= ( s"rt/$01 ( 0.44 so the G*F (

.6$&

!he return loss# in deci


4/39

!he input impedance if the line lenth is $.3 wavelenths. and o is 5 ohms.

!he input impedance is the ratio of line voltae to current at the input# or enerator end# of the

line.

+n our case in/o ( $s1/$8s1 where s is ta@en at the line input. Aain# =s= doesnIt vary alon the

line so it is still 0.44 at the line input. !he line lenth is $.3 wavelenths ( 3'0.5.3

wavelenths. Nach 0.5 wavelenth represents a chane in phase of the s parameter of 8360

derees and can be inored. !hus we are left with a phase shift input 8O load 8O input of8360'.3' derees ( 8$65.60 derees. At the input therefore the scatterin parameter has phase

6.58$65.60 ( 8$3).03 derees.

Convertin to real and imainary parts# at the input the scatterin parameter s ( 80.33870.)3

so in ( o0.663870.)31/$.3370.)31 with o(5 ohms. !he trusty calculator finds in

( 3$.)) 873.45 ohms

!he scatterin parameter at the input to the transmission line.

*ee the previous section

Question 4.

9efine the term Jscatterin matrixJ for a 8port microwave networ@. Nxplain carefully the output and

input variables related by the scatterin parameters. Nxplain why the scatterin parameters have

dimensionless complex character. Nxplain clearly the meanin of the term Jreference planeJ with respect

to which the scatterin parameters are defined and measured. Nxplain what difference in s parametersoccurs if the reference plane is moved further from the 8port. Nxplain why scatterin parameters are

usually iven in the form Jmodulus and phase anleJ.

:ost of the section above is dealt with in the notes on scatterin parameters .

+f the reference plane is moved the s parameters chane phase but not modulus. !hus it is best to

express them in this format.

Frite down the port scatterin matrix for a lossless transmission line of electrical lenth $/5 lambda#

ivin the scatterin parameters in both Jmodulus and phaseJ and then Jreal and imainaryJ

representations.

!he round trip distance is /5 wavelenth# or 360'/5 derees which is $44 derees. !he phase

retards as we move with the wave. Bne pass of the line is therefore 8 derees# and the line is

lossless. !hus s$(s$ ( $ anle 8 derees. s$$(s ( 0 because there are no inherent

reflections within the lenth of line. +n terms of real and imainary parts# s$(s$( 0.30)870.)5$

!his transmission line is connected to an antenna which has scatterin parameter 0.$ anle 0 derees.

9etermine the fraction of the forward wave power which is radiated. 9esin a matchin networ@ to be

placed at the input to the transmission line so that the enerator sees a perfect match at its terminals.

!he reflected wave power is 0.$'0.$ ( $L of the incident power. !hus 0.)) of the incident power

is radiated. ))L1.

Bne miht thin@ this is a ood enouh match. -owever# at the input of the transmission line the

reflection coefficient $8port s parameter1 is 0.$ anle 8$44 derees usin the results of the last

section. !his is 0. wavelenths towards the enerator from the antenna normalised impedance of

$0.$1/$80.$1 ( $. 70

which ives us an input impedance of read raphically1

0.&4 8 70.$

from the *:+!- plot 0. wavelenths towards the enerator

or we can CA,C2,A!N the reflection coefficient amma
http://personal.ee.surrey.ac.uk/Personal/D.Jefferies/sparam.htmlhttp://personal.ee.surrey.ac.uk/Personal/D.Jefferies/sparam.html


5/39

0.$ anle 8$44 derees1 ( 80.0&0) 8 70.05&&

which ives us an input impedance of calculated1

$ 8 0.0&0) 8 70.05&&1/$ 0.0&0) 70.05&&1

which evaluates to be

0.&44) 8 70.$004

which is satisfactorily in areement with the *:+!- plot.

Fe therefore need a sinle stub match to the impedance

0.&44) 8 70.$004

A stub of series impedance 870. placed 0.$&6 wavelenths further towards the enerator does

this. !he stub lenth for an impedance 870. is 0.46& wavelenths from a short.

Question 5

Calculate the rane of fre"uencies over which only the lowest order waveuide mode will propaate inrectanular waveuides havin the followin cross sectional dimensions.

0cm by $5cm cm by $.5cm mm by $.5mm

cm by $cm

cm by 0.&cm

cm by cm

cm by 4cm

!he cutoff wavelenth for the !N$0 mode is twice the lonest cross section dimension.

ememberin 30 ( fre">-?1 times wavelenth we have cutoff for the cm uides as 30/4 (

.5>-?. or the 0 cm uide it is 50:-? and for the mm uide it is 5 >-?.

!he next mode is either the !N0 mode at twice the cutoff fre"uency of the !N$0 mode# or the!N0$ mode if this is lower.

o !hus# 0cm by $5cm has 50:-? to $>-?

cm by $.5cm# .5>-? to $0>-?# and mm by $.5mm# 5>-? to $00>-?.

cm by $cm and cm by 0.&cm both have .5>-? to $5>-?.

cm by cm has two deenerate modes# !N$0 and !N0$ both with cutoff .5>-?. !here

is thus no band of fre"uency over which only one mode will propaate.cm by 4cm has lon dimension 4cm so the rane is 3.5>-? to .5>-?.

or the cm by $cm waveuide# determine the lowest fre"uency of the lowest mode which will

propaate. Calculate the uide roup and phase velocities# and the propaation anle alpha# for

fre"uencies $00:-? 300:-? and $>-? hiher than the cutoff fre"uency.

rom the eometry of !N$0 mode propaation# refer to the boo@ of your choice# if the free space

wavelenth is lambda# the uide width a# the propaation anle alpha# and the uide wavelenthlambda> we can write

o sinalpha1 ( lambda/a1

o cosalpha1 ( lambda/lambda>

o tanalpha1 ( lambda>/a1

!he cutoff of the cm by $cm uide is 500:-? so we want the values above at 600# &00# and

&500:-?.

!he values of lambda respectively are 3.)44cm# 3.&46cm# and 3.5)4cm.


6/39

!he values of alpha are &0.0derees# 4.06derees# and 6$.)3derees.

!he values of lambda> are 4.4$cm# $4.0cm# and .50cm.

!he roup velocities are c cosalpha1 ( 0.4&4 0.&3)# and $.4$# all in units of $0%& metres per

second.

!he phase velocities are c/cosalpha1 ( $&.56# $0.)# and 6.36# all in units of $0%& metres per

second.

Question 6.

Calculate the scatterin matrix of a 5 cm lenth of waveuide of internal rectanular cross section 5cm

by .cm at a fre"uency of 4.0 >-?. -int# you will need to find out the uide wavelenth.

!he cutoff fre"uency is 3>-?. !he uide wavelenth is $$.33&)cm. !here are .04& uide

wavelenths in the 5cm lenth of uide.

!he residual phase shift after allowin whole cycles is 83.3 derees.

!he s parameters are s$$(s(0# s$(s$ ( $ anle 83.3 derees.

!he wave impedance of a waveuide mode is defined as the ratio of the transverse component of electric

field strenth to the transverse component of manetic field strenth. *how that the wave impedance

depends on the anle alpha and the uide wavelenth# and derive expressions for the dependencies.

or a !N mode# the electric field is wholly transverse and the same as it is in a free space wave.

!he manetic field is turned by cosalpha1 from bein transverse# so is smaller than the free space

wave value by cosalpha1. !hus the wave impedance is $0 pi/cosalpha1 or $0 pi

lambda>/lambda.

or a !: mode the electric field is rotates throuh alpha but the manetic field is entirely

transverse. !he wave impedance is thus $0 pi cosalpha1 or $0 pi lambda/lambda>.

Calculate the wave impedance for the conditions of the first part of this "uestion. -int# you need to @now

that in a free space wave the wave impedance is $0 pi ohms.

Applyin the results above# the impedance at 4>-? is 50 ohms to the nearest ohm.

Question 7

*how that a condition for stability of a port is that there is no reflection ain for connections loo@in in

to either port; and that such a circuit is unconditionally stable for any passive load impedance and passive

enerator source impedance.

-ere# the important point is that there must be no reflection ain at either port for any conditions

of passive load attached to the other port. +f this condition is not met# the return power is more

than the incident power; we have in effect made a Jreflection amplifierJ which can cause

oscillations in a reactive load formed by a lenth of transmission line attached to an open or short

circuit.

!he condition for a lenth of line to oscillate is that the round trip ain# accountin for the losses

of the line and the reflection coefficients at each end# be reater than one.

or a passive load the manitude of the reflection coefficient is always less than one. !herefore#

for there to be round trip ain we re"uire the other reflection to have manitude reater than one.+f it doesnIt# no oscillations will occur.

A certain 8port has s parameters s$$ ( 0.$ anle 830 derees# s$ ( 0.05 anle 860 derees# s$ ( 30

anle 860 derees# and s ( 0.$5 anles 80 derees. All these s parameters are ta@en at a spot fre"uency.

let us write these s parameters formally as

P0.$ 830Q P0.05 860Q

P30 860Q P0.$5 80Q

the definin e"uations are

b$ ( s$$a$ s$a


7/39

b ( s$a$ sa

9iscuss which values of load impedance on port 7ust ive rise to reflection ain on port $.

let us call the reflection coefficient at the load

on port amma.

!hen a ( amma b because the output b of the 8port

is reflected from the load to form a.

2sin the e"uation b ( s$a$ sa

then we find a$/amma 8 s1 ( s$a$

2sin the e"uation b$ ( s$$a$ s$a

then we find b$ ( a$Ds$$ s$s$/$/amma 8 s1E

Fe want b$ to have manitude less than a$

for there to be no reflection ain at port $.

!hus the complex number s$$ s$s$/$/amma 8s1

has to lie inside the unit circle.

Bne can substitute the values above; + estimate

that if the reflection coefficient is less than

about =amma= ( 0.55 the circuit will be stable.

!hat puts the output transmission line G*F as

bein less than about 3.5

Question 8.

Frite down the * matrices of the followin microwave components.... Rou may assume they have ideal

characteristics.

A 30 d< amplifierAn isolator with 0d< front to bac@ transmission ratio

A perfect 3 port circulator

A perfect 4 port dual directional coupler# with couplin strenth 830 d-? is 5.3 d


11/39

unnily enouh# in the actual exam in $))5 no one ot this bit riht. !hat demonstrated to me that

they hadnIt understood what the * matrix actually represents.

Question 11.

9erive formulas for the phase velocity and roup velocity of waves on a lossless rectanular waveuide

supportin a !N$0 mode. *how that the product of these velocities is c%. D35LE

!here were two acceptable ways of answerin this part of the "uestion# one involvin a raphical

construction showin the waveuide field patterns and anle of propaation# and the other usinthe definitions $1 that the phase velocity ( Danular fre"uencyE /Dpropaation constantE and 1

that the roup velocity ( dDanular fre"uencyE /dDpropaation constantE which is the derivativeof anular fre"uency with respect to the propaation constant1.

As this is a standard piece of boo@wor@ + refer you to U 9 Sraus Nlectromanetics or other

standard text of your choice.

>eometrically# the uide wavelenth is iven by lambda/cosalpha1 where alpha is the anle of

propaation with respect to the uide axis# so the uide phase velocity is c/cosalpha1 since f

lambda ( c. !he uide roup velocity is the component of velocity alon the uide axis and is c

cosalpha1. Clearly the product of velocities is c%.

Alebraically# one obtains the same results from the waveuide formula usin the relationships

iven above.

A certain cavity wavemeter has an accurate micrometer which ives a readin of the distance between themovin short and the fixed short. !he waveuide a standard K band F>)0 type1 has dimensions of

0.)00 inches by 0.400 inches. 9etermine the micrometer readins when the wavemeter absorbs at the

followin fre"uenciesM i1 &.5& >-?# ii1 $0.04 >-?# iii1 $$.3& >-?. D35LEFe start our solution by calculatin the three free space wavelenths for the problem# by dividin 30

by the fre"uency in >-?. !hat ives it in cm# and the waveuide dimensions are in inches so you can

convert one to the other# as lon as you state which units you are usin for the end result1. !he free space

wavelenths in cm are i1 3.4)65 ii1 .)&&0 iii1 .636. Fe @eep a few more sinificant fiures at this

stae of the calculation as roundin occurs last.

Fe now find the uide wavelenths from the waveuide formula you can "uote this from memory orderive it1. !he critical uide dimension is a(0.)00 inches ( .&6 cm so a ( 4.5 cm and a1% (

0.)03. !he waveuide formula ives the uide wavelenth as s"rtD$/P$/lambda%18 $/a1%QE. !he

respective uide wavelenths are i1 5.46 cm ii1 3.)4 cm iii1 3.66 cm.ow when half a uide wavelenth fits between the shorts the cavity wavemeter is resonant and

absorbs power from the waveuide to which it is coupled. !hat ives us a micrometer readin of half the

uide wavelenth in each case# always assumin that we can nelect perturbations of the field

distributions due to the couplin hole.

!he micrometer readins are therefore i1 .$ cm or $.0 inches# ii1 $.) cm or 0. inches# andiii1 $.6$ cm or 0.635 inches. +t is ood practice to ive the result to the same number of sinificant

fiures as the oriinal data in the "uestion.

9escribe how such a wavemeter can be used in con7unction with a 30d< dual directional coupler to

measure the fre"uency of a microwave test bench. *tatin your assumptions# estimate the achievableprecision of the measurement. D30LE.

!he directional coupler samples $/$000 of the forward wave power and feeds it up a side arm to a

detector. !he side arm is coupled by means of a small hole in the common wall to the cavitywavemeter. !he cavity wavemeter has a loaded H of about $000 to $0#000 dependin on how the

inside surfaces are coated silver platin is common1# on the surface rouhness# and on the

couplin strenth to the side arm waveuide. !his ives an achievable resolution in fre"uency of

between 0.0$L and 0.$L# or $ to $0 :-? at K band.

Question 12.


12/39

>ive a brief description of the physical principles overnin the behaviour of non8reciprocal ferrite

microwave components. 9iscuss the practical limitations imposed by power handlin and fre"uency8

selective effects. D40LE

+n ferrite biased by a 9C manetic field# the spins of the unpaired electrons precess around the

direction of the bias field. !he strenth of the bias field overns the rate of precession# and it may

be tuned so that the precession fre"uency lies near the fre"uency of microwaves inside the ferrite

loaded device.


13/39

eturn power is never presented to a filter output# passin from port to 3 top1 to to 3 middle1

to to 3 bottom1 where it is absorbed in the termination.

*ince the filters are perfect reflectors of power in their stop bands# it is clear that no -TA output

can drive any other -TAIs output terminals.

Question 1 3

9efine the terms Jcharacteristic impedanceJ# Jvelocity factorJ# Jforward travellin waveJ# Jreflection

coefficientJ# and Jreturn lossJ# in the context of propaation on a transmission line. D5LE !he characteristic impedance is the ratio of voltae to current in a forward travellin wave# when

there is no reverse travellin wave. A transmission line presents the characteristic impedance to aenerator for times until the first reflection arrives at the enerator.

!he velocity factor eta is the ratio of the Jwave velocity or wave phase velocityJ to the Jspeed of

liht in vacuumJ.

!he forward travellin wave is any disturbance travellin from enerator to load.

!he reflection coefficient amma is the complex ratio of bac@ward wave amplitude to the forward

wave amplitude# considerin monochromatic waves waves of a sinle fre"uency1.

!he return loss is the amount in d< by which the reflected wave power is less than the forward

wave power. umerically# the return loss is 80 loD$0E=amma=1.A buildin is wired with Nthernet cable havin velocity factor 0.6. !here is a brea@ in this cable metres

from a time domain reflectometer !91 which emits rectanular pulses of rise and fall times 0.5

nanoseconds and duration 5 nanoseconds. Fhat is the time delay until the return of the first reflection

from this brea@ D35LE

!he pulse velocity is 0.6 times 30 cm per nanosecond# or $& cm per nanosecond.

!he round trip distance to the brea@ is times times $00 cm# or 4400 cm.

!he round trip time is 4400/$& ( 44 nanoseconds to the nearest nanosecond. !his is 0.44

microseconds.

A shunt open circuit spur of lenth $ metres is added at a distance $0 metres from the !9. !he rest of

the cable is undamaed and is correctly terminated. 9escribe "uantitatively the pulse se"uence seen on

the !9. D40LE

9raw a diaram with > the enerator# which has internal impedance o matched to the line so

that it doesnIt produce reflections# placed $0 metres from T the 7unction which feeds a shunt

matched load o and a side arm $ metres lon to H which is an open circuit.

:easurin the impedance at > in the steady state we of course see 7ust a matched load o. Bne

miht be fooled by this into thin@in that there would be no reflections from the system.... !his is

the point of the "uestion.

!he reflection coefficient at T is formed by the o of the matched arm in parallel with o for the

open spur. !hus amma ( 0.58$1/0.5$1 which is 8$/3.

!he reflection coefficient at H is similarly $.

!he round trip time >T> is 000/$& ( $$$ nanoseconds to the nearest nanosecond.

!he round trip time THT is 400/$& ( $33 nanoseconds.

!he transmission coefficient at the 7unction T in any direction is calculated as followsM power

reflected ( 8$/318$/31 ( $/) of the incident power. Tower transmitted ( $8$/) ( &/) of the

incident power. !his splits so that 4/) of the incident power is transmitted down each of theconnected arms#

!he transmitted amplitude on each arm is therefore /3 of the incident amplitude.

!he se"uence of pulses at the enerator end is therefore amplitude $ at time ?ero initial pulse1

followed by 8$/3 at $$$ nanoseconds and then a pulse which has travelled once up and down the

spur# havin amplitude /3 times $ times /3 ( 4/) at $$$$33 ( 44 nanoseconds. !he next pulse

has travelled twice up and down the spur# and has amplitude /3 times $ times 8$/3 times $ times

/3 or 84/ at time $$$ times $33 ( 3 nanoseconds. +f you can et this far you can wor@

out the rest for yourselves. !he next pulse has travelled three times up and down the spur...


14/39

Question 14

9escribe the construction and principles of operation of a waveuide dual directional coupler havin

couplin strenth 830 d< and fractional bandwidth about $0L for use at K band fre"uencies. D30LE

or a 30d< dual directional coupler# narrow band# a two hole waveuide coupler is sufficient.

9raw two waveuides with the broad face common# with two small x shaped holes on the centre

line of the common face and spaced a "uarter of a uide wavelenth apart.

!he couplin strenth depends on the hole si?e. !he forward coupled waves add in phase# ivina combined couplin strenth with coupled power $/$000 of the incident power in the other

waveuide. !he waveuide dimensions are about 0.)00 inches by 0.400 inches F>)01.A 830d< coupler is used to sample the forward and bac@ward wave amplitudes on a waveuide connected

to an antenna. !he detectors on the samplin arms consist of totally absorbin matched diode detectors

which have accurate s"uare law characteristics. 9etermine the ratio of output voltaes on the detectors if

the reflection causes a G*F of $.& on the main arm of the coupler. D5LE

!he G*F is $.& so the modulus of the reflection coefficient# =amma=# is $.&8$1/$.&$1 (

0.&/.& ( 0.&5 and the s"uare of this is 0.0&$6 so &.$6L of the power is reflected. !he detector

voltaes are proportional to the coupled wave powers# so the bac@ward wave detector voltae is

about &L of the forward wave detector voltae. !he ratio of voltaes is therefore $.5.

A mismatch is introduced at each of the detector diodes# ivin a reflection coefficient of manitude 0.$.

or the same conditions as part b1# estimate the ratio of output voltaes at the detectors. D5LE Fe are not told of the phase delays in the system. !he first assumption is that the phase is

arbitrary and we add powers. !he power at the bac@ward wave detector is therefore $/$.5 of the

forward wave power plus 0.$ times 0.$ of the power incident on the forward wave detector. ow

$/$.5 $/$00 ( $/$0.). Bf course there will be some reflected power on the forward wave

detector which comes from the bac@ward wave detector# but this is less than $/$000 of the

oriinal forward wave detector power and we nelect it.

+f you li@e you can do a phasor addition wherein the amplitudes rather than the powers are added.

!his is not necessary in this particular example.

Fe learn from this example that it is most important to have a ood match on the detectors#

particularly the one samplin the forward wave power.

Nxplain how the fractional bandwidth of the dual directional coupler may be increased to 50L or more.

D0LE!o increase the fractional bandwidth we add more couplin holes# havin the same total

couplin strenth as the two oriinal holes. !hey are spaced so that the resultant reverse wave

amplitude phasor is small; the contributions from each hole add in a straiht line for forward

wave couplin# but for reverse couplin they add in a polyonal manner until the resultant isminimised. !he bandwidth enhancement depends on the number of holes. or this application a

collection of 0 holes spread out over several uide wavelenths would wor@ nicely.

Question 15

9escribe the principal uses of components in microwave transmission systems which contain ferrites.

Nxplain what is the most important property of the components which is due to the presence of the ferrite.

D30LE

errites have the property that their wave propaation behaviour can depend on the direction of

power flow alon a transmission line into which they are incorporated. !his property is referredto as bein Jnon8reciprocalJ. !his is contrasted with most other non8manetic passivecomponents where transmission does not depend on the direction of power flow between two of

the ports of a networ@. errite devices share the property of non8reciprocity with active devices

such as transistor amplifiers.

!hey are used to JisolateJ a enerator from reflections caused by load mismatch. !hey can be

used in power combiners and in switchin applications. !hey ma@e excellent microwave absorber

for linin anechoic chambers; the manetic field due to microwaves is lare near a metal


15/39

reflectin surface# and manetic loss placed here is more effective at absorbin power than is

resistive dielectric loss; electric transverse fields are small close to a metal surface.

Circulators are used to separate forward waves from bac@ward waves in many applications. !hey

can be used in combination with a passive matched load to construct an e"uivalent 8port to an

isolator.

A ferrite isolator is imperfect# havin forward loss of $ d< and a forward/bac@ward transmission ratio of

$& d


16/39

!he reflected power is separated from the incident power usin a circulator. !he circulator

forward transmission is $3$3 etc# and the sinal source is connected to port $. !he >unn diode

is connected to port and a matched load or isolator driven load1 is connected to port 3. +t is

important that the source does not see reflection ain on its own transmission line.

iii1 :icrowave bandpass filter desin and construction.

ilters are usually desined to wor@ between @nown enerator and load impedances. :icrowave

filters are no exception and it is usual to ta@e the reference impedances of source and load to bethe transmission line characteristic impedance.

or the purposes of this answer it is assumed that it is @nown how to desin lumped component

bandpass filters wor@in between @nown specified impedance levels.

eactive components can be enerated# at least at a sinle fre"uency# by shorted lenths of

transmission line of @nown lenth.

A bandpass filter is usually constructed from reactive components only# as this @eeps its in8band

insertion loss low and its noise performance is ood.

Admittance to impedance transformation can be accomplished by means of "uarter wave sections

of transmission line. !hus we see that we can used shunt reactive stubs placed at intervals alon a

transmission line to ma@e a filter.

!he shunt elements can be physical lenths of line# or in waveuide they and be irises# or

apertures between coupled cavities.

:icrowave filters have a fre"uency response which# in principle# repeats indefinitely as the

fre"uency is raised and the line lenths become proressively loner by added amounts of half awavelenth. !hus they are by no means direct e"uivalents of their lumped constant templates.

!hey can be fabricated from coupled microstrip lines# dielectric resonator sections# waveuide

with irises# or acoustic delay lines. !hey are reciprocal and it is often assumed that they are

lossless.

or a lossless bandpass filter the sum of the reflected and transmitted powers must e"ual the

incident power. !hus filters can be constructed in reflection mode as well as in transmission

mode; in reflection mode the pass band becomes a stop band and vice versa.

>roup delay as well as amplitude response is usually an important desin parameter of the

microwave filter.

Question 17

9efine the terms JdirectivityJ# JainJ# JefficiencyJ# and Jeffective areaJ for an antenna array. Nxplain how

the boresiht ain of an antenna is related to its effective area and efficiency. D5LE

!he directivity of an antenna is the ratio of the power directed alon boresiht to the power which

would be radiated e"ually in all directions if the antenna was isotropic. 2sually the directivity is

assumed to be e"ual to the boresiht ain assumin no losses in the antenna.

!he ain of an antenna is usually specified in the direction of maximum radiation# the boresiht#

and is e"ual to the total power radiated in this direction assumin it was radiated isotropicallyuniformly in all directions1 divided by the input power to the antenna structure. !he ain is less

than the directivity because of antenna losses# spillae# bloc@ae# and so on.

!he efficiency of an antenna is the ain divided by the directivity. +t is a measure of what

proportion of the input power is usefully radiated.

!he effective area of an antenna is numerically e"ual to the ain times 4 pi/lambda%1# and is the

area of a perfect antenna which had that particular value of ain. !he effective area is usually in

the case of an aperture antenna1 somewhat smaller than the physical area.

A s"uare array of 4 by 4 elements consistin of lambda/ dipoles each havin a maximum ain 3d< is fed

by $6 sinals of ad7ustable relative phase and e"ual amplitudes. Assumin )0L efficiency# calculate the

boresiht ain when the sinals are all in phase. D50LE


17/39

,et us call the amplitude of the individual sinals drivin each element A. !hen the input power

to an element is =A=% modulus of A s"uared# or AA'1. !he efficiency is )0L# or 0.)# and the

s"uare root of this is 0.)5 which represents how much of the individual amplitude A is radiated.

!he ain of an individual element# in terms of power# is # or in terms of amplitude# is s"rt1.

!his is 3d


18/39

!he stored enery per unit lenth1 times the propaation velocity1 e"uals the forward wave

power flow1. !hus the stored enery per unit lenth ( =G==G=/o times wave velocity1.

,oo@in at the units# enery1/lenth1 times lenth/time1 ives us enery/time1 ( power

Dbecause Uoules/sec(wattsE

A 5 ohm cable# assumed lossless# feeds an antenna havin radiation resistance 307$01 ohms at the

sinal fre"uency. +f the forward wave power is $0 watts# calculate the return loss and the radiated power.

D5LE !he characteristic impedance o ( 5 ohms# and the load impedance , ( 307$0 ohms. !he

reflection coefficient is# at the load terminals# ,8o1/,o1 ( 8457$0 divided by $057$0which ives us 0.3&0570.0&0 whose modulus s"uared is 0.6460.

!he return loss in d< is therefore 8$0 lo$0 0.64601 which is $.&)& dive a "ualitative description of a method by

which the antenna may be matched to the cable. D5LE

ow the forward wave voltae modulus is iven by =G= and we @now from earlier parts of the

"uestion that =G==G=/o1 ( $0 watts with o ( 5 ohms. !hus =G= ( s"rtD$0 times times 5E( s"rtD$500E so the forward wave voltae si?e is s"rtD$500E ( 3&.3 volts. !he bac@ward wave

power flow is similarly 6.46 watts so the bac@ward wave voltae si?e is s"rtD6.46 times times

5E ( s"rtD)6)E ( 3$.$3 volts.

At a voltae standin wave maximum# the forward and bac@ward wave voltae phasors add in

phase# so the pea@ enerator voltae is the sum 3&.33$.$3 ( 6).&6 volts; to find the rms voltae

we divide by s"rt1 to find 4).)0 volts.

!he enerator may be matched by usin a sinle shorted stub which may be either series or shunt;

by a double or triple stub tuner# or with some power loss by an isolator# althouh in this case the

radiated power will be less than that supplied by the enerator. +n the case of a stub match# the

reflections from the stubs1 7ust cancel the reflection from the load.

Question 19

9escribe the s8parameter representation of the properties of a linear microwave 8port circuit# ivin thedefinin e"uations and also statin precisely which variable are related by the s8parameters. D5LE

!he s8parameters relate inward and outward wave complex amplitudes. !hey are measured at a

Jreference planeJ with respect to each port. !he reference planes are positions alon the

transmission lines feedin the ports; if moved# the phases of the s8parameters will alter. +t is usual

to define the wave amplitudes so that the s"uare moduli of the amplitudes represent the wave

power flows.

!he amplitudes are complex phasors and so the s8parameters are complex "uantities. +f we call

the input amplitudes a$ and a at ports $ and respectively# and the outward wave amplitudes b$and b similarly# the definin e"uations are....

b$ ( s$$ a$ s$ a

b ( s$ a$ s a

where the four complex s8parameters are s$$ s$ s$ s.

+n terms of the port $ rms voltae G and rms current +# and the port $ transmission line

characteristic impedance o# we have for the normalised wave amplitudes a$ and b$


19/39

a$ ( Go+1/o1

b$ ( G8o+1/o1

and similar relationships hold for the amplitudes a and b

*tate# ivin reasons for your choice# which microwave components are described by the followin

collections of s8parameters. D$5L for each of the three parts belowE

s$$(0 s$($exp8701

s$($exp8701 s(0

!his is a section of lossless transmission line# two wavelenths lon 0 derees of phase shift (

times 3601. All the incident power is transferred to the output; there is no reflection at all if the

output port is terminated.

s$$(0 s$(0.0 exp87401

s$($exp87601 s(0

!his is an isolator# havin no insertion loss in the forward direction# so that all the power in port $

comes out of port and there is no intrinsic reflection from the isolator. +n the reverse direction#

the isolator produces no intrinsic reflection# but attenuates the wave passed bac@ to the port $ by

80 lo$00.01 d< which is 3.$ d


20/39

rom the second of these e"uations we find# rearranin# that

b$8s amma1 ( s$ a$

and we substitute for b in the first e"uation so that

b$ ( s$$ a$ s$ s$ amma a$1/$ 8 s amma1

!he complex reflection coefficient at port $ is 7ust b$/a$

which is clearly

b$/a$1 ( s$$ s$ s$ amma1/$ 8 s amma1

+f this "uantity has modulus less than unity# there can be no

reflection ain# and the circuit arranement will be

unconditionally stable.

Question 20

9escribe the behaviours of an ideal isolator and an ideal 38port circulator. >ive examples of the

applications of these components. *tate which @ind of material# used in their construction# ives rise to

their uni"ue properties. D40LE

An ideal isolator transmits waves in one direction without any attenuation# but possibly some

phase shift; it absorbs completely waves in the reverse direction. everse waves see a matched

impedance at the output port of the isolator so at that port s is ?ero and so there is no reflection

of waves impinin on the output port 1 of the isolator. +deally the isolator wor@s over a

sinificant bandwidth and is essentially a non8resonant device.

An ideal 38port circulator transmits power from port $ to port # from port to port 3# and from

port 3 to port $ without attenuation. +ntrinsically it has no loss. +t is matched on each port; thusthere is no reflected wave from port $ if ports and 3 are terminated and not driven. !he same

remar@s apply to cyclic rotation of the port numbers.

+solators are used to prevent the pullin of low H oscillators such as >unn oscillators by resonant

loads whose couplin may be time8dependent. !hey are used to protect hih power enerators

and amplifiers from the effects of ross miss8match at the loads. !hey are used to reduce the

G*F on transmission line and the reverse transmission throuh amplifiers which miht

otherwise o unstable.

Circulators are used in power combiners# and to separate forward from bac@ward waves so that

the return loss of a load may be measured directly. !hey are used with antennas to separate the

received sinal from the transmitted sinal.


21/39

!he reflection coefficient from the 5 ohm load at the port reference planes on ports and 3 is

iven by amma ( 58501/5501 ( 5/$5 ( 0. ( $/5.

!he sinal travels without loss from port $ to port # with a phase shift of 860 derees en route. +t

is reflected at the load and reduced in amplitude by a factor of 5. +t has another 860 derees of

phase shift to et to port 3# where it is reduced in amplitude by another factor of 5. inally# it has

another 860 derees or phase shift to et to port $ aain. !he total reduction in amplitude is $/5

times $/5 and the total phase shift is 3 times 860 derees. !hus the output wave at port $ is 0.04anle 8$&0 derees with respect to the input wave; and this is the complex reflection coefficient.

Nxplain how a circulator may be used in place of an isolator# statin any advantaes of this arranement.D5LE

+f the circulator has a matched load placed on port 3# and the load of interest is attached on port

and the enerator on port $# any reflected power from the load on port is all absorbed in the

matched termination on port 3. +t is therefore a perfect isolator and has better isolation properties

than a 8port isolator. !he power is absorbed by a resistive load# which can be cooled by a fan or

water circulation# and the ferrite does not et hot. !his is a technoloical advantae as its

manetic properties are temperature dependent.

Question 21

>ive a 7ustification of the formula# stated below# for the fundamental !N$0 mode in a rectanular

waveuide of cross8sectional dimensions a cm1 by b cm1.

$ $ $

888 ( 888 8 888

lambda1% lambda1% a1%

*@etch the waveuide cross8section# indicatin clearly which dimension is a. *tate the e"uivalent formula

for the uide wavelenth of the !N$$ mode. D40LE

+n a waveuide# the electrical conductivity of the metal of the uide walls results in the parallel

components of electric field# and the perpendicular components of time8varyin manetic field#

vanishin at the walls.

+n a wave propaatin in free space in a sinle direction only a Jplane waveJ1 the manetic field#

electric field# and direction of travel# are all mutually at riht anles to each other. !hus to satisfythe boundary conditions in our rectanular uide it is necessary for there to be more than one

propaatin plane wave ivin rise to the field patterns.

+f we reard propaation as bein at an anle to the uide walls# there are three components of

propaation constant; namely# two transverse and one alon the uide. !he repetition distance orwavelenth associated with the propaation alon the uide is called the Juide wavelenthJ

lambda. !he wavelenth associated with the vector resultant of these three orthoonal

propaation constants is 7ust the free space wavelenth lambda.

+n order for the fields to vanish at the uide walls there must be standin wave patterns in the

transverse propaation. !he transverse propaation constants must have wavelenths or repetition

distances such that an interal number of half8wavelenths fit inside the lateral uide dimensions.

+f a is the lonest lateral dimension# then a must be the lonest transverse uide repetition

distance or wavelenth# and the reciprocal of this times pi is the associated transversepropaation constant.

Addin the components by Tythaoras theorem# and cancellin the factors of pi# we arrive at

the formula stated. +t is not necessary# for the fundamental mode# to have standin waves patterns

in both lateral directions at once# for we are allowed electric fields normal to uide walls# and

chanin manetic fields parallel to uide walls.

!he e"uivalent formula for the !N$$ mode is


22/39

$/lambda1% ( $/lambda1% 8 $/a1% 8 $/b1%

+f a ( 3 cm and b ( $. cm# calculate the cutoff fre"uencies of the !N$0# !N0$# and !N$$ modes#

assumin that the waveuide is filled with air. Calculate the uide wavelenth# the phase velocity# and the

roup velocity# for propaatin sinals havin a fre"uency $00:-? above the !N$0 mode cutoff

fre"uency. D35LE At the cutoff fre"uency lambda increases without limit and the waves bounce laterally across the

uide without ma@in proress alon it. 2sin the relation D$/lambda1E ( 0 we find the

followin free space uide wavelenth cutoffsM

!N$0 ( 6 cm

!N0$ ( .4 cm

!N$$ ( s"rt$/$/36 $/5.611 ( .& cm

usin fre"uency times wavelenth1 ( 3 times $0%$0 cm per sec#

the velocity of microwave radiation in air# we find cutoff

fre"uencies of 5>-?# $.5>-?# and $3.46>-? respectively.

!he sinal fre"uency for this problem is therefore 5.$>-? and the correspondin free space

wavelenth is 3/0.5$ ( 5.&&4 cm. 2sin the waveuide formula we find that lambda ( ).&5

cm. :ultiplyin this by the sinal fre"uency 5.$ >-? we find the uide phase velocity $.5

times $0%$$ cm per second# and usin the result iven in the followin part of the "uestion the

uide roup velocity is 5.)$ times $0%) cm per second.

>ive a derivation of the relationshipM uide phase velocity1xuide roup velocity1 ( velocity of liht in

waveuide medium1%. D5LE

>iven the anular fre"uency omea# and the uide propaation constant beta# the phase velocity

which is fre"uency times uide wavelenth is easily seen to be omea/beta. !he roup velocity is

domea1/dbeta1# the derivation of which is not needed here# and expressin the waveuide

formula iven in the earlier part of this "uestion in terms of omea and beta# havin translated thefree space wavelenth lambda into fre"uency by usin c ( fre"uency times free space

wavelenth# the result follows transparently on differentiation.

+t is also possible to use a eometrical construction showin the propaation directions in the

waveuide as bein anle alpha to the uide wall. +nspection shows that the phase velocity is

iven by c/cosalpha1 and the roup velocity by c times cosalpha1 from which the result also

follows transparently.

!here was a wide spread of "uality in the written answers. :ost people manaed to write a sinificant

amount on the "uestions they attempted. Huestion 4 was popular# bein larely boo@wor@. !here were not

many ood answers to the s8parameter "uestion particularly on the descriptions of components# and the

problem# althouh the boo@wor@ part of the "uestion was "uite well handled. Huestion $ caused

difficulties for some people who didnIt realise that what isnIt reflected from the antenna has to be

transmitted. :any people transposed the reflected and transmitted powers here. !he precision of somepeople in ivin definitions needed improvement.

:ost people seemed a lot happier with descriptive answers than with answers re"uirin a little thouht

and calculation. !here were some over8lon answers to "uestion 3 which must have ta@en time away from

attempts at other "uestions. +ncidentally# there are three sections of phase shift to et round a three port

circulator# not two.

inally# well done everyone in what is enerally accepted to be a difficult sub7ect. + promise microwaves

will become clearer to you all in the future as the ideas sin@ in deeper. emember you only had $3 wee@s

from your first introduction to the sub7ect until you were as@ed to write an exam. emember also the


23/39

ob7ective of an exam is not primarily to JradeJ people but to fix the ideas more firmly in memory by

eneratin a little stress.

Question 22

9efine the terms JGelocity factorJ# JCharacteristic +mpedanceJ# Jeturn ,ossJ# JComplex reflection

coefficientJ and JThase delayJ for waves travellin on a lossless transmission line. D5LE

A certain coaxial line is lossless# and has inductance , -enries/metre and capacitance C arads/metre.

>ive expressions for the characteristic impedance in ohms and the velocity factor dimensionless1 for this

line. Calculate the inductance and capacitance of a $ metre lenth of this line if it has impedance 50 ohms

and velocity factor 0.6. D5LE

Calculate the complex reflection coefficient and return loss d


24/39

load# where the real part conductance1 of the transformed normalised admittance is $ and the

residual normalised susceptance as 0.4&. A further *:+!- chart plot ives the lenth of a

shorted stub needed to ma@e a normalised susceptance of 80.4& as about 0.$& wavelenths D5LE

Question 23.

*tate the electromanetic boundary conditions that constrain electromanetic wave propaation at the

interface between air and a perfect conductor. 9efine what is meant by the term Jtransverse

electromanetic waveJ# and ive a diaram showin the directions of propaation# electric field# and

manetic field in such a wave. >ive another diaram showin the orientation of a stac@ of metal plates

spaced an arbitrary distance apart which may be introduced into this transverse electromanetic wave

without disturbin it. D30LE

>ive a "ualitative description of how the boundary conditions are satisfied for propaation in a

rectanular hollow metal waveuide of cross sectional dimensions A metres by < metres# with A O -? to

$40>-?# usin rectanular metal pipe filled with a substance of relative dielectric constant ) at these

fre"uencies. *uest suitable waveuide dimensions# and calculate the uide wavelenth at a fre"uency

of $5>-?. Comment on the probable sources of attenuation in such a waveuide# and estimate the

surface rouhness in microns1 of the internal uide walls that it would be possible to tolerate. D40LE

Outline solution 2.

!he electric fields meet the perfect conductor walls at riht anles. !here can be no component of

electric field parallel to the metal wall# close to the wall. !he time8varyin manetic fields areparallel to the metal walls but have no perpendicular8to8the8wall component. !he direction of

manetic field is at riht anles to the local surface current on the metal walls. +n a transverseelectromanetic wave# the electric field vector and manetic field vector are both at riht anles

to each other and to the direction of travel of the wave. A stac@ of metal sheets may be placed so

that the electric field is at riht anles to the surfaces of the plates# and the manetic field parallel

to the surfaces of the plates# without violatin the boundary conditions. D30LE

+n a rectanular pipe waveuide the electric fields can run normal to the A faces and the manetic

fields can lie parallel to the A faces thereby satisfyin the boundary conditions on the A faces.

*tandin waves can exist with a transverse component of propaation between the < faces# whichcan be rearded as short circuit ends of transmission line# without re"uirin the fields to be ?ero

everywhere within the uide. !he lowest mode cutoff fre"uency happens when a half wavelenth

of free space radiation 7ust fits into the lonest transverse dimension of the uide. +n the case here

that ives a free space wavelenth of cms which occurs at a fre"uency of $5>-?# which is

therefore the cutoff fre"uency of a $cm uide. D30LE +f the relative dielectric constant is )# the wave velocity in the material fillin the uide is slower

than the speed of liht in vacuum by a factor of s"rt)1 ( 3. +f we assume the cutoff fre"uency of

the re"uired uide is $00>-?# in free space the wavelenth would be 3 mm and in the dielectric

the wavelenth would be $ mm. !he uide cross section dimensions could therefore be $/ mm

by $/4 mm# or 500 by 50 microns. Attenuation would be provided by dielectric loss# and by the

resistance of the uide walls since the s@in depth would be small. At $5 >-? the free space

wavelenth in the dielectric would be $/$.5 ( 0.& mm# and usin the waveuide formula we

discover the uide wavelenth to be $/s"rtD$/0.&1% 8 $E ( $.33 mm. +t would be appropriate to


25/39

@eep the surface rouhness below about $/$00 of a wavelenth# so we would li@e a surface polish

to better than $0 microns. D40LE

Question 24.

9efine the term Jscatterin matrixJ and state carefully the "uantities related by this matrix. or a 8port

networ@ state how many elements there are in this matrix. 9efine the term JreciprocityJ and state the

maximum number of independent elements there can be for a reciprocal scatterin matrix used to describe

a 38port 7unction. D30LE

>ive a brief description of how a ferrite displacement isolator wor@s. A certain isolator has forward

insertion loss of 0.5d< and reverse insertion loss of $$.5d


26/39

reflectors for sinals in their stop bands. !hus the sinal from the bottom -TA proceeds via each

circulator# rflected from the other filters# to the antenna. Clearly no power from the lower -TAs

enters any of the hiher -TAs because it is prevented by the stopbands of the associated filters.

eturn power from the antenna is passed directly to the termination at the bottom of the tree.

D30LE

Question 25.

9efine the terms Jisotropic radiatorJ# Jboresiht directionJ# JdirectivityJ# JainJ# and JN8 plane radiation

patternJ in the context of antenna desin. Nxplain why it is impossible to construct an isotropic radiator in

practice. D30LE

A certain transmitter has a final amplifier that delivers $0 @F to an antenna with &5L efficiency. !he

antenna has boresiht directivity of $4dive a formula relatin the effective area of an antenna to its boresiht ain and to the wavelenth of the

radiated sinal. Nstimate the area of a horn aperture antenna re"uired to ive a ain of $4d-?.

+f such a dish were used to transmit the $0@F sinal described above# estimate the power flow in watts

per s"uare metre at a distance of $0m from this horn alon boresiht# and comment on the safety

implications. D30LE

Outline solution 4.

An isotropic radiator is one that radiates uniformly in all directions# both in a?imuth and

elevation. !he power density and field strenths radiated by an isotrope do not depend at all on

the direction of radiation. or a directional antenna# if there is a sinle direction in which the

radiated power is maximum# this is termed the boresiht direction. !he directivity of an antenna

is a dimensionless number representin the ratio of the radiated power density on boresiht to the

radiated power density at the same place in space which would exist if radiated by an ideal

isotrope havin the same total radiated power. Bften the directivity is iven in d


27/39

!he ain > and effective area A of an antenna are related by the formula > ( 4'pi'A/lambda1%#

where lambda is the wavelenth of the radiation. At $3 >-? lambda ( 3/$.3 cms ( .3$ cms and

we found above that the $4d


28/39

iven by $/s"rtepsilon1 where epsilon is the Jeffective relative dielectric constantJ of the

insulator formin the line spacer.

o eflection coefficientM A complex dimensionless number which describes# at a iven

point alon the transmission line called the Jreference planeJ# the ratio of return wave

amplitude to forward wave amplitude. !he phase of the reflection coefficient chanes as

the reference plane is moved alon the transmission line. Bn a lossless line# the

manitude of the reflection coefficient is the same everywhere alon the line. !here areimportant special cases of the reflection coefficient when the reference plane coincides

with the terminals of the load impedance# or the enerator terminals.

o G*FM !he Goltae *tandin Fave atio is a dimensionless number representin the

ratio of the maximum voltae of the standin waves on the transmission line# at a

reference plane where the forward and bac@ward waves are in phase# to the minimum

voltae at a reference plane lambda/4 away from the maximum1 where the forward and

bac@ward waves are in anti8phase. !he G*F is a sensitive measure of the mismatch on

the line# and is closely related to the manitude of the reflection coefficient.D5LE

Callin the forward wave complex amplitude G and the bac@ward wave complex amplitude G8#

the manitude =amma= of the reflection coefficient amma is iven by =G= / =G8= so the G*F is

iven byG*F ( =G= =G8=1/=G= 8 =G8=1 ( $ =amma=1/$ 8 =amma=1

At the load# the reflection coefficient is amma,# say# and at the enerator it is amma> (

amma, expP7 pi d/lambdaQ1 for a line of lenth d and waves of wavelenth lambda. 2sin

the relationships

amma, ( ,8o1/,o1 and amma> ( 8o1/o1

we can determine in terms of the "uantities ,# o# lambda# and d by alebra.

D30LE

!he wave velocity is 0.6'30 ( $& cm/nanosec. *o at $00:-? ( 0.$ >-?# the wavelenth lambda

( $&0 cm ($.& metres. *o the $0 metre cable is $0/$.& wavelenths lon. !otal cable loss# onepass# ( $0'0.4/$.& ( .d


29/39

[TM] modes# and show how these modes satisfy the boundary conditions in rectanular waveuide.

D30LE

b1

9etermine the cutoff fre"uencies of the !N$0# !N0$# !:$$# and !N$$ modes in a waveuide filled with a

lossless dielectric of relative permittivity epsilonr1 ( 6.5# for uide dimensions $.& cm by 0.& cm. *tatewhich of these modes are degenerate .

D30LE

c1

,ist any advantaes and disadvantaes of usin dielectric filled waveuide# compared to air8filled.

D0LE

d1

Nstimate the power handlin capacity of this uide for the !N$0 mode if the dielectric brea@down

strenth is $0% G/m

D0LE

Outline solution (2).

!he parallel component of N and the normal component of d-/dt must be ?ero ad7acent to a

perfect conductor. *o the N field always meets a conductor at riht anles# and a chanin - field

always lies parallel to a conductor surface ad7acent to the surface. +n a !N mode there is a

lonitudinal - field# and in a !: mode there is a lonitudinal N field. -ere# lonitudinal is ta@en

to mean in the direction of propaation alon the axis of the waveuide. A s@etch of mode

patterns is needed here# see the waveuide notes .

!he relative dielectric constant is 6.5# so the wave velocity interior to the uide is c/s"rt6.51 (

$.$&N& metres per second. !he cutoff wavelenth for !N$0 is twice the broad uide dimension#

or 3.6 cm so the cutoff fre"uency for the !N$0 mode is 0.$$&/0.0361 >-? ( 3. >-?. !hecutoff wavelenth for the !N0$ mode is twice the narrow uide dimension# so the cutofffre"uency wor@s out to be .36>-? by a similar arument. !he !N$$ and !:$$ modes are

deenerate and share a cutoff fre"uency of &.05 >-?.

D30LE

Advantaes of dielectric8loaded waveuide are; increased brea@down field strenth and power

handlin capacity# and reduced si?e for a iven fre"uency of use. 9isadvantaes include

fabrication problems# dielectric loss leadin to attenuation# material dispersion as well as

eometric dispersion# and possible arcin at the wall8dielectric interfaces.

D0LE

Fithin the !N$08only propaation band# we assume the roup velocity is $.$&N&1/ metres per

second. !he enery density per unit lenth of the uide is of the order epsilon016.51N%1A

where A is the uide cross8section in s"uare metres# N is the maximum r.m.s. electric fieldstrenth in the wave# and epsilon0 ( &.&5N8$1 arads/metre is the permittivity of free space.

!his enery density is of the order of 0.4 Uoules per metre lenth of uide# so the maximum

power flow is 0.4'$.$&N&1/ ( 4 :eawatts.

D0LE
http://personal.ee.surrey.ac.uk/Personal/D.Jefferies/wguide.htmlhttp://personal.ee.surrey.ac.uk/Personal/D.Jefferies/wguide.htmlhttp://personal.ee.surrey.ac.uk/Personal/D.Jefferies/wguide.html


30/39

Question 27.

a1

9efine the termscattering matrixand ive examples of the scatterin matrix values s8parameters1 for

each of the followin devicesM8

a lambda/3 lenth of lossless coaxial cable

a perfect 38port circulator

a perfect 48port magic teein waveuide.

D40LE

b1

At its terminals# a certain antenna has s$$ ( 8 70.. 2sin the *:+!- chart# or otherwise# determine the

drivin point impedance at the input to a 50 ohm feeder of lenth $.6 lambda which is connected to this

antenna. Assume neliible loss in the feeder.

D40LE

c1

Nxplain the construction of a sinle short8circuit stub match to an antenna. +ndicate how the bandwidth of

the stub match miht be maximised.

D0LE


An n8port microwave circuit has n input waves and n output waves. Fe relate the incomin wave

amplitudes ai to the outoin wave amplitudes bi by the matrix e"uation bi ( si7 a7 where the si7

are the complex dimensionless1 scatterin parameters# which can be assembled into a matrix

called thescattering matrix . +n matrix notation we can write this relationship as < ( *A.o or a lambda/3 transmission line we have s$$(s ( 0. +t is a 8port networ@. !he other s

parameters are iven by s$(s$ ( expP87 pi/3Q; they have unity manitude andneative phase parts.

o or a perfect 38port circulator with the se"uence $8O8O38O$.... we have s$$(s(s33 ( 0

and forward transmission s$(s3(s$3 ( $ exp P7 8thetaQ with theta some phase anle

dependin on the construction. !he other s parameters s$#s3# s3$ are all ?ero as there is

no reverse propaation around the circulator.

o or a perfect 48port maic tee# the ports may be numbered $ and 41 straiht throuh#

-8plane arm# and 3 N8plane arm. !hen all the scatterin parameters have manitude

$/s"rt1# and the matrix may be written as $/s"rt1 times

P s$$(s(s33(s44(s$4(s4$(s3(s3 ( 0# s$(s3$(s$3(s$ (s43(s34($# s4(s4 (

8$Q

D40LE

!he manitude of s$$ is $/5 and the anle 8)0 derees. !he lenth of the line is round trip

distance1 '80.$1'360 derees we have subtracted 43 whole wavelenths from the round trip

phase shift1 so the input reflection coefficient amma has manitude $/5 and phase anle 8)08

$.41 derees# so in/o ( D$ ammaE/D$ 8 ammaE ( 0.6)6 7 0.$556 so in ( 34.& 7.&

ohms.

D40LE


31/39

or this part# see the stub matchin notes

D0LE

Question 28.

a1

Nxplain the term isotropic radiator. -ow do practical antennas differ from the isotropic ideal 9efine the

terms directivity andgain of an antenna and ive an expression for the efficiency of a front feed

Casserain reflector antenna.

D30LE

b1

A formula for the ain i dainM the same as directivity but allowin for antenna loss in the near field and the

radiatin structure. !he isotropic source now has the same total radiated power as that

acceptedby the antenna in "uestion from its feeder.

o !he efficiency of an antenna is the proportion of total accepted input power that is

ultimately radiated. umerically# the efficiency ( the ain1 / the directivity1 and it

doesnIt matter which directions are ta@en providin they are the same1 for establishin

the ain and the directivity.
http://personal.ee.surrey.ac.uk/Personal/D.Jefferies/stubs.htmlhttp://personal.ee.surrey.ac.uk/Personal/D.Jefferies/stubs.html


32/39

o *ources of loss in an antenna include absorbin ob7ects in the near field# resistive loss in

the current8bearin structures# and spillover from the feed to a reflector antenna.

D30LE

!he effective area1/the actual area1 ( the aperture efficiency1 and the aperture efficiency may

be about 08&5L for a typical Casserain reflector antenna. Assumin an aperture efficiency of

&0L# and usin the formula > ( 4 pi Ae1/lambda%1 for the ain of the antenna in terms of the

effective area Ae# for this case we find > ( $0%4&/$01 ( 630)6# and the wavelenth at $3 >-? is0.03/$.3 metres ( 0.03$ metres so that Ae ( .6& s"uare metres and the actual area is Ae/0.&1 (

3.35 s"uare metres.

D0LE

!he ain is approximately e"ual to the solid anle of the whole sphere 4 pi steradians1 divided

by the solid anle of the cone of radiation pi theta%1 whence theta ( 0.00&0 radians ( 0.456

derees is the cone semi8anle.

D0LE

*idelobes arise mathematically from the ourier transform of a uniformly illuminated aperture.

!hey may be rearded as the rins of a diffraction pattern formed by a uniformly illuminatedaperture havin the shape of the reflector. !hey may be reduced by usin aperture illumination

taper# where the fields are reduced at the edes of the reflector from their values at the centre. or

minimum sidelobes# the taper should approximate to a >aussian. !he sidelobes are only reducedby the trade8off of a wider main beam# so the antenna needs to be larer for a iven beamwidth.

D30LE

Question 29

a1

9efine the terms Jcharacteristic impedanceJ# Jwave velocityJ# Jvelocity factorJ# JdispersionJ# Jcomplex

reflection coefficientJ.

D$0LE

b1

>ive a formula that relates the velocity factor and characteristic impedance of a transmission line to theinductance and capacitance of a $ metre lenth of line.

D$0LE

c1

Calculate the inductance and capacitance of a $0 metre lenth of 300 ohm parallel wire ribbon cable

which has velocity factor 0.)

D0LE

d1

A transmission line 7unction is formed from four microstrip lines# all havin the same characteristic

impedance# and connected toether in the form of a cross. Tower is input to port $ and ports #3 and 4 are

all terminated in matched loads. Nvaluate the si?e of the reflection coefficient at port $. 9etermine how

many d< of sinal loss occurs between the in8port and one of the out8ports.

D40LE

e1

9escribe "ualitatively# ivin reasons# what happens to a rectanular pulse of current propaatin alon a


33/39

uniform but lossy dispersive microstrip transmission line.

D0LE


9efinitions.

Characteristic impedanceM e"ual to s"rt,/C1 for waves on a lossless transmission line havin

distributed inductance , -enries/metre and distributed capacitance C arads/metre. +t is the ratio

of voltae to current in a forward travellin wave on the line; it is e"ual to the input impedance of

a very lon lenth of line for times shorter than the time it ta@es the reflections to return from the

far end. +t is that impedance which# when used as a load# ives rise to no reflected wave# or a

perfect match.

Fave velocity. Bn a coaxial transmission line the waves propaate as !N: modes. !he wave

velocity is therefore either the phase velocity or the roup velocity; it is the speed and direction1with which the wave crests travel alon the coaxial cable# and it is also the speed with which the

enery and the modulations travel.

Gelocity factorM !his is a dimensionless number# less than unity# which is the ratio of the speed ofwaves on the coaxial line to the speed of liht in vacuum# 3N& metres per second. !ypically the

velocity factor for coax lies between 0.55 and 0.&; it is closely iven by $/s"rtepsilon1 where

epsilon is the Jeffective relative dielectric constantJ of the insulator formin the line spacer.

9ispersionM this is a technical term for the case where the phase velocity of waves depends on

fre"uency or wavelenth# and a superposition ourier1 of waves JdispersesJ in distance as it

travels alon the line.

Complex reflection coefficientM A complex dimensionless number which describes# at a iven

point alon the transmission line called the Jreference planeJ# the ratio of return wave complex8

amplitude to forward wave complex8amplitude. !he phase of the reflection coefficient chanes as

the reference plane is moved alon the transmission line. Bn a lossless line# the manitude of the

reflection coefficient is the same everywhere alon the line. !here are important special cases of

the reflection coefficient when the reference plane coincides with the terminals of the loadimpedance# or the enerator terminals.

D$0LE

Callin the velocity factor eta and the velocity of liht c ; with the capacitance of the cable C

arads per metre and the inductance of the cable , -enries per metre we have

o ( characteristic impedance ( s"rt ,/C1

velocity of waves on the line ( eta c1 ( $/s"rt,C1

so , ( o/eta c1 and C ( $/o eta c1 by simple alebra.

D$0LE

+f o ( 300 ohms# eta ( 0.) and c ( 3N& then for $0 metres of cable the inductance , is $$.$micro8-enries and the capacitance C is $3 p.

D0LE

At the microstrip 7unction# the three outoin terminated transmission lines each present a shunt

impedance o. !here are thus three o impedances in parallel at the end of the input line# and

they form a load impedance , ( $/31o. !he reflection coefficient at the 7unction is therefore

D$/3 8 $E/D$/3 $E or 8$/. !he amount of power reflected is therefore D8$/ED8$/E or $/4# and so

3/4 of the incident power is transmitted.


34/39

power# and the loss is 6dive a description of the formalism of a scatterin matrix representation of a two8port microwave circuit#

statin clearly what each of the scatterin parameters represents physically.

D30LE

b1

A certain microwave component has the followin s8parameters

s$$ ( 0.$ anle 830 derees

s ( 0.3 anle 860 derees

s$ ( &.4 anle 8$0 derees

s$ ( 0.05 anle 8)0 derees

*tate what @ind of component is represented by these s8parameters.

D$0LE

c1

!he component is supplied on port $ with $0mF power level while port is connected to a matched load.

9etermine the power output on port .D$0LE

d1

!his component is embedded in a 50 ohm transmission system. 9etermine the input impedance at port $

if port is

i1 terminated by an impedance of 50 ohms

ii1 short circuited

D30LE

e1Frite down the s8matrix of a perfect microwave 38port circulator $OO3O$ with $0 derees phase delay

between successive ports.

D0LE


35/39


!he *8matrix lin@s incomin complex wave amplitudes a to outoin complex wave amplitudes

b . !he incomin power on port i is ai1ai1' ( the s"uared modulus of ai# and similarly for the b

amplitudes. +n matrix notation

b$ = s$$ s$ = a$

( = =

b = s$ s = a

or b ( * a

or fully

b$ ( s$$ a$ s$ a

b ( s$ a$ s a

ow# if port $ is driven and port is terminated then a ( 0 and b$ ( s$$ a$# and b ( s$ a$.

so =s$$=% is the proportion of input power reflected# and =s$=% is the proportion of input power

transmitted. !he phase anles of s$$ and s$ represent the phase shifts on reflection and

transmission# respectively.

*imilarly for port $ terminated and port driven.

D30LE

!he component is an amplifier as the manitude of s$ is larer than unity and the reverse

transmission s$ is small.

D$0LE

or $ milliwatt of input power# the output power delivered to the matched load on port is

&.4'&.4 milliwatts# from the property above. !his is 0.56 milliwatts. !hus for $0mF of input

power# the power delivered to the load will be 0. watts approximately.D$0LE

ow o is 50 ohms. +f port is terminated then b$ ( s$$ a$ and the reflection coefficient at port

$ is s$$. !herefore in/o ( $s$$1/$8s$$1 so in ( 5). 8 7 6.0 ohms.

D$5LE

+f there is a short circuit on port # the reflection coefficient from the short is 8$ and so a ( 8 b is

the wave amplitude returned to port from the reflection at the short.

!hus b$ ( s$$ a$ 8 s$ b

and b ( s$ a$ 8 s b

where we have substituted 8b1 for a in the e"uations listed in the first part of the answer.


36/39

*olvin these e"uations we obtain b$ in terms of a$ to et the reflection coefficientgamma at the

input port# which is iven by

amma ( Ds$$ 8 s$ s$1/P$sQE

and so# as before#

in ( oD$ammaE/D$8ammaE ( $$3 8 746 ohms on puttin in the numbers.

D$5LE

or the three port circulator we have s$$ ( s ( s33 ( 0 as it it intrinsically matched on each

port# and we also have s$ ( s3 ( s3$ ( 0 as there is no reverse cyclic transmission# and we are

left with s$ ( s3 ( s$3 ( manitude $ anle 8$0 derees.

D0LE

Question 31.

a1

*tate the boundary conditions for electromanetic fields at the interface between air and a perfect

conductor.

D$0LE

b1

A rectanular waveuide has cross section 4.& cm wide face1 by .4 cm narrow face1. 9etermine the

cutoff fre"uencies for each of the followin modesM !N$0 !N0$ !N0 !:$$

D0LE

c1Fhich two of the modes above are deenerate >ive reasons.

D$0LE

d1

9etermine the attenuation coefficient in d


37/39

always lies parallel to a conductor surface ad7acent to the surface.

D$0LE

or the Jm#nJ mode whether !N or !:1 we recall the waveuide formula

$/lambdacutoff1% ( m/a1% n/b1%

and we have fre"uencycutoff1 times lambdacutoff ( 3N& metres/sec

*o we obtain

cutoff fre"uencies !N$0 ( 3.$5>-?# !N0$(!N0 ( 6.5>-?# !:$$ ( 6.)&&>-?.

D0LE

!he !N0$ and !N0 modes are deenerate; they have the same cutoff fre"uencies and the uide

wavelenths for each mode will be the same at any iven fre"uency.

D$0LE

Fe now use the remembered1 formula

$/lambdauide1% ( $/lambda1% 8 $/lambdacutoff1%

and lambda ' fre"uency ( 3N& metres per second

to find $/lambdauide1% ( f% 8 fc%1/3N&1%.

as fc is reater than f# lambdauide is imainary and the attenuation coefficient alpha is iven by

alpha ( pi1/modulus of lambdauide1

Aain# puttin in the numbers we find the attenuation alpha ( 4.& epers per metre ( $5

d-? as a sensible sinle8mode !N$01 rane.

D30LE

Question 32.

a1

9istinuish between the termsgainand directivity of an antenna.

D$0LE


38/39

b1

Nxplain why a dipole antenna necessarily has a maximum directivity which is reater than unity.

D$0LE

c1

An array antenna consists of two vertically orientated half8wave dipoles placed with feeds a distance onewavelenth apart and orientated alon the line 7oinin their feed points. +n both a?imuth and elevation

planes# s@etch radiation patterns of i1 the dipoles considered as isolated elements# ii1 the array pattern of

two isotropes placed on the dipole centres# and iii1 the total radiation pattern of the antenna.

D50LE

d1

Nxplain what is meant by the term Jpattern multiplicationJ.

D$0LE

e1

9etermine the boresiht directivity in d


39/39

Fhen we combine the a?imuth patterns by pattern multiplication# the total radiation pattern in the

a?imuth plane remains circular omnidirectional1. -owever# the elevation pattern multiplication

puts an additional null alon the rod axes into the array pattern# so there are now a total of 6 lobes

and 6 nulls as we rotate around in elevation.

*@etches would help and et mar@s here.

D50LE

Tattern multiplication may be summarised alorithmically as

o choose a direction a?imuth# elevation1

o determine element directivity in this direction

o determine array factor directivity in this direction

o plot the product of these directivities in this direction1 on a 3d plot

o repeat for all a?imuth elevation1 anles at reasonable intervals to derive the total antenna

radiation pattern.

D$0LE !he array factor ain is there are two elements1 and the element factor ain is 7ust that of a

sinle dipole. !herefore the total antenna ain is 3d

qbankmesem5

Documents

Transcript of qbankmesem5