Name:

Exam in the course Antenna Engineering

2011-05-28
ANTENNA ENGINEERING (SSY100)
(E4) 2010/11 (Period IV)
Saturday 28 May 1400-1800 hours.
Teachers: Adjunct Prof Jan Carlsson, Prof Per-Simon Kildal, Associate Prof Jian Yang
Questions: Jan Carlsson, tel. 0703 665169
The exam consists of 2 parts. Part A is printed on colored paper and must be solved
without using the textbook. When you have delivered the colored text and the
solutions of Part A (latest 17:00), the textbook can be used for Part B of the exam.
You are allowed to use the following:
For Part A: Pocket calculator of your own choice
For Part B only: Mathematical tables including Beta
Pocket calculator of your own choice
Kildal’s compendium “Foundations of Antennas: A Unified
Approach for LOS and Multipath”
(The textbook can contain own notes and marks on its original printed pages. No
other notes are allowed.)

Tentamen består av 2 delar. Del A har tryckts på färgade papper och skall lösas utan att
använda läroboken. När du har inlämnat dom färgade arken med uppgifterna för del A
och dina svar på dessa uppgifter (senast 17:00), kan du ta fram läroboken för att lösa del
B.
Tillåtna hjälpmedel:
För del A: Valfri räknedosa
För del B: Matematiska Tabeller inkluderad Beta
Valfri räknedosa
Kildals lärobok “Foundations of Antennas: A Unified Approach
for LOS and Multipath”
(Boken kan innehålla egna notater skrivna på dom inbundna sidorna. Extra ark med
notater tillåts inte.)

Exam in the course Antenna Engineering, 2011-05-28 -1-

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Exam in the course Antenna Engineering, 2011-05-28 -2-

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PART A (must be delivered before textbook can be used)

1.0 On Labs (25p)

1.1. Some questions about Lab1. Be careful that you use the correct terminology from
the course literature.

1.1.1. Which environment is emulated by reverberation rhamber and which

environment is emulated by anechoic chamber? Explain.
1.1.2. Which quantities can we measure in reverberation chamber, and which
ones can we measure in anechoic chamber?
1.1.3. Explain the different steps involved when measuring in a reverberation
chamber.

1.2. In Lab 2, you have measured a standard gain horn and a microstrip slot array
antenna; see Fig. 1.1. The measured radiation patterns are shown in Fig. 1.2 and
Fig. 1.3.

1.2.1. Which of the two figures show radiation patterns for standard gain horn
and which one for microstrip slot array antenna? Explain briefly.
1.2.2. Please sketch (redraw) the patterns and mark which curve (blue or red in
the original graphs) is for E-plane and which one is for H-plane. Explain
briefly.
1.2.3. In Fig. 1.2, the radiation pattern in one plane (red curve) has much broader
beam width than the other, why?
1.2.4. In Fig.1.3, the radiation pattern in one plane (red curve) has lower first
sidelobe than the other, why?
1.2.5. What is the length of the slot in the slot array antenna in mm? The
operating frequency for the slot array is 9.5 GHz.
1.2.6. As you know, the slot array antenna has the maximum radiation direction
at 30o from the broadside. What is the maximum allowed spacing between
the slots to avoid grating lobes? If you do not remember the complete
formula, you can give the value for the full scan case. Explain.

1.3. In Lab 3, you could choose to realize one of several different patch antennas; see
Fig.1.4. Explain which polarization(s) each of these patch antenna shapes radiate,
i.e. horizontal linear, vertical linear or circular, provided the dimensions are
properly chosen.

Exam in the course Antenna Engineering, 2011-05-28 -3-

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Standard gain horn Microstrip slot array

antenna

Figure 1.1 Antennas in Lab2.

Figure 1.2 Measured radiation patterns on both E- and H-planes for antenna 1

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Fig. 1.3 M
Measured raadiation paatterns on b
both E- and
d H-planes for antenn
na 2

λg/22
λg/2

λg/2
/
λg/2

(aa) (b) (cc)

λg/2
λg/2

λg/2 λg/2

(d) (e)

F
Figure 1.4 different patch
p anten
nnas.

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2.00 Far-ffield funcctions off differen

nt antennas (25p
p)
In Home
H assignnment 2, yoou were askeed to sketchh the co- andd cross- pollar radiation
n
pattterns in E- and
a H- planees for severral differentt basic antennnas. Howevver, in pracctice,
the question is opposite: you
y have som me specificaations on anntenna’s raddiation patteerns
m customerss, and you should
from s chooose proper aantennas to fulfill
f the sppecification
ns. For
eachh of the folllowing quesstions, the co-polar
c raddiation patterns in E- annd H-planess are
giveen. You shoould
a) Choosee a proper anntenna that can providee the requireed radiationn patterns. Write
W
the nam
me of the anttenna or inccremental soource;
b) Sketch a coordinatte system annd draw youur antenna inn it, and inddicate the
polarizaation.
c) Explainn which plannes are E-pllane and H--plane.

Notte that there may exist more

m than one
o solutionn for the sam
me radiationn pattern
requuirement.

2.1 Radiation ppatterns:

2 Radiation ppatterns:
2.2

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2.3 Radiation ppatterns:

2.4
4 Radiation ppatterns:

2.5 Radiation ppatterns:

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2.6
6 Radiation ppatterns:

2.7
7 Radiation ppatterns:

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PART B (You can use the textbook to solve this problem, but
only after PART A has been delivered)

3.0 Diversity antenna (25p)

We have access to two dipole antennas that we would like to use as a diversity antenna in
order to improve the performance of a radio unit that is used in an isotropic multipath
environment. In the figure below we show the CDF curves for the two antennas measured
separately (i.e. without the other antenna present) as well as the CDF when the two
antennas are placed close to each other, as they would be when used on the radio, and the
CDF when both antenna ports are selection combined. We also know that antenna 2 has
an efficiency of 0 dB.

0
10
Cumulative probability (CDF)

-1
10

Antenna 1, isolated
Antenna 2, isolated

Selection combined
-2 antenna 1 & 2 when
10
they are close to
each other

-3
10
-30 -25 -20 -15 -10 -5 0 5
Normalized power level (dB)

Figure 3.1. CDF:s for the two dipole antennas measured separately without the
other antenna present and when placed close to each other when
selection combination is used, respectively.

3.1 Why are the CDF curves for antennas 1 and 2 different and what does this
difference represents?

3.2 In the figure, sketch how the CDF curves change for the two ports (branches)
when the antennas are placed close to each other. Explain.

3.3 What is the gain of using the two antennas and selection combining if we compare
with using only antenna 1? Assume a CDF level of 1%.

Exam in the course Antenna Engineering, 2011-05-28 - 11 -

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3.4 What is the gain of using the two antennas and selection combining if we compare
with using an ideal reference antenna? Assume a CDF level of 1%.

3.5 Assume that we know that the efficiency for each branch decreases by 5 dB when
the antennas are closely spaced with the same spacing as for the diversity
combined curve. Determine then the absolute value of the complex correlation
coefficient between the two branches?

3.6 Now, we have got an offer from an antenna vendor to buy a diversity antenna
instead of using the two dipoles. The only information we have is that the antenna
is lossless and that it has the equivalent circuit shown in the figure below. The
values of the impedances in the figure are;
Z11 = Z 22 = 82.4 + j 32.1 ; Z12 = Z 21 = 76.1 − j 0.7 ohm .

What are the effective and apparent diversity gains of this antenna if selection
combining and a CDF-level of 1% are assumed?

I1 Z11 Z 22 I2

Port 1 Z12 I 2 +_ +_ Z 21 I1 Port 2

Figure 3.2. Equivalent circuit for diversity antenna.

3.7 If Port 1 is excited with a generator with 50 ohm internal impedance and Port 2 is
terminated with 50 ohm, how much power is dissipated in the 50 ohm load at Port
2 and how much power is radiated?

Exam in the course Antenna Engineering, 2011-05-28 - 12 -

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4.00 Dipole Array (225p)

Fig.. 4.1 shows an array anntenna consiisting of fouur halfwavee dipoles.

Figure 4.3 A 4-dipole lineear array anteenna

4.1 Draw the equivalent circuit

c of thhe array.

2 What are the values for the phaase shifters when the maximum radiation direction
4.2 d
(main beaam directionn) is alongg the z-axiss? Here wee assume uuniform exccitations
(V1=V2=VV3=V4).

4.3 Write the expression for the scan n impedancce of dipolee 3 under thhe condition n of 4.2.
(Here you do not neeed to produ uce any num merical valuue. It is suffficient to write
w the
expressionn and explaiin the proceedure of youur derivatioon. All the pparameters you use
should be kknown or easy
e to obtaiin from form
mulas or figgures. Explaain which formulas
fo
and figuress in the com
mpendium you
y are usingg.)

4.4
4 Write the expressionn of the em mbedded raddiation field function of dipole 3 when
dipole 3 iss excited annd the rest of
o the dipolees are termiinated with loads (sam
me as the
source imppedances) Z0. NOTE that now V1=V2=V4=0. = (Here you do not need to
produce anny numericaal value. It is sufficiennt to write the
t expressiion and expplain the
procedure of your derrivation. Alll the parameters you usse should be known orr easy to
obtain froom formulaas or figurres. Explaiin which formulas
f annd figures in the
compendiuum you are using.)

4.5 Write the eexpression for

f the radiaation field function
f of the
t whole array.
a

4.6
6 Now we w want to steerr the main beam
b to 30o from the z--axis. What are now thee values
of the phasse shifters? Are there any
a grating lobes
l now? If yes, in w
which direction?

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Name:

Exam in the course Antenna Engineering

2011-05-28
ANTENNA ENGINEERING (SSY100)
(E4) 2010/11 (Period IV)
Saturday 28 May 1400-1800 hours.
Teachers: Adjunct Prof Jan Carlsson, Prof Per-Simon Kildal, Associate Prof Jian Yang
Questions: Jan Carlsson, tel. 0703 665169
The exam consists of 2 parts. Part A is printed on colored paper and must be solved
without using the textbook. When you have delivered the colored text and the
solutions of Part A (latest 17:00), the textbook can be used for Part B of the exam.
You are allowed to use the following:
For Part A: Pocket calculator of your own choice
For Part B only: Mathematical tables including Beta
Pocket calculator of your own choice
Kildal’s compendium “Foundations of Antennas: A Unified
Approach for LOS and Multipath”
(The textbook can contain own notes and marks on its original printed pages. No
other notes are allowed.)

Tentamen består av 2 delar. Del A har tryckts på färgade papper och skall lösas utan att
använda läroboken. När du har inlämnat dom färgade arken med uppgifterna för del A
och dina svar på dessa uppgifter (senast 17:00), kan du ta fram läroboken för att lösa del
B.
Tillåtna hjälpmedel:
För del A: Valfri räknedosa
För del B: Matematiska Tabeller inkluderad Beta
Valfri räknedosa
Kildals lärobok “Foundations of Antennas: A Unified Approach
for LOS and Multipath”
(Boken kan innehålla egna notater skrivna på dom inbundna sidorna. Extra ark med
notater tillåts inte.)

Exam in the course Antenna Engineering, 2011-05-28 -1-

Name:

Exam in the course Antenna Engineering, 2011-05-28 -2-

Name:

PART A (must be delivered before textbook can be used)

1.0 On Labs (25p)

1.1. Some questions about Lab1. Be careful that you use the correct terminology from
the course literature.

1.1.1. Which environment is emulated by reverberation chamber and which

environment is emulated by anechoic chamber? Explain. (2p)

Solution:
A reverberation chamber emulates the multipath environment (due to
multiple reflections…). An anechoic chamber emulates the free-space
environment (no refection…). (Any answers explaining the correct physics
will get full points.)

1.1.2. Which quantities can we measure in reverberation chamber, and which

ones can we measure in anechoic chamber? (2p)

Solution:
We can measure the diversity gain, efficiency, MIMO capacity, TIS,
TRP… in a reverberation chamber. (Mentioning three quantities is
enough.) We can radiation pattern (CO & XP), gain, … in an anechoic
chamber. (Mentioning one quantities is enough.)

1.1.3. Explain the different steps involved when measuring in a reverberation

chamber. (3p)

Solution:
For an efficiency measurement, one need to first perform a reference
measurement with a reference antenna with unity efficiency to get the
average power level; then do an efficiency measurement with the antenna
under test (AUT) to get another power level. The difference of the two
power level (in dB) is the efficiency of the AUT. (Any correct
measurement procedure for any application, e.g. diversity, TRP…, will get
full points.)

1.2. In Lab 2, you have measured a standard gain horn and a microstrip slot array
antenna; see Fig. 1.1. The measured radiation patterns are shown in Fig. 1.2 and
Fig. 1.3.

1.2.1. Which of the two figures show radiation patterns for standard gain horn
and which one for microstrip slot array antenna? Explain briefly. (2p)

Solution:

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Fig.1.2 is for microstrip slot array antenna, Fig1.3 is for standard gain
horn. Because microstrip slot array antenna has very different beamwidth
for E- and H-planes.

1.2.2. Please sketch (redraw) the patterns and mark which curve (blue or red in
the original graphs) is for E-plane and which one is for H-plane. Explain
briefly. (2p)

Solution:
Blue curve in Fig.1.2 is for E-plane of microstrip slot array antenna, in
Fig1.3 is for E-plane for standard gain horn. Red curves are for H-planes.

1.2.3. In Fig. 1.2, the radiation pattern in one plane (red curve) has much broader
beam width than the other, why? (3p)

Solution:
Size in H-plane is much smaller than that in E-plane.

1.2.4. In Fig.1.3, the radiation pattern in one plane (red curve) has lower first
sidelobe than the other, why? (3p)

Solution:
Tapered field distribution in H-plane.

1.2.5. What is the length of the slot in the slot array antenna in mm? The
operating frequency for the slot array is 9.5 GHz. (2p)

Solution:

Half wavelength in free space. l   15.8mm
2

1.2.6. As you know, the slot array antenna has the maximum radiation direction
at 30o from the broadside. What is the maximum allowed spacing between
the slots to avoid grating lobes? If you do not remember the complete
formula, you can give the value for the full scan case. Explain. (2p)

Solution:

d  21mm
1  cos 

1.3. In Lab 3, you could choose to realize one of several different patch antennas; see
Fig.1.4. Explain which polarization(s) each of these patch antenna shapes radiate,
i.e. horizontal linear, vertical linear or circular, provided the dimensions are
properly chosen. (4p)

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Solution:
(a) Linear polar: vertical;
(b) Linear polar: horizontal;
(c) Dual-linear polar;
(d) Circular polar;
(e) Circular Polar;

Exam in the course Antenna Engineering, 2011-05-28 -5-

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Standard gain horn Microstrip slot array

antenna

Figure 1.1 Antennas in Lab2.

Blue
Red

Figure 1.2 Measured radiation patterns on both E- and H-planes for antenna 1

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Blue

Red

Fig. 1.3 Measured radiation patterns on both E- and H-planes for antenna 2

λg/2
λg/2

λg/2
λg/2

(a) (b) (c)

λg/2
λg/2

λg/2 λg/2

(d) (e)

Figure 1.4 different patch antennas.

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2.0 Far-field functions of different antennas (25p)

In Home assignment 2, you were asked to sketch the co- and cross- polar radiation
patterns in E- and H- planes for several different basic antennas. However, in practice,
the question is opposite: you have some specifications on antenna’s radiation patterns
from customers, and you should choose proper antennas to fulfill the specifications. For
each of the following questions, the co-polar radiation patterns in E- and H-planes are
given. You should
a) Choose a proper antenna that can provide the required radiation patterns. Write
the name of the antenna or incremental source; (1.5 p, 1 p for last 3 radiation
patterns)
b) Sketch a coordinate system and draw your antenna in it, and indicate the
polarization. (1.5 p, 1 p for last 3 radiation patterns)
c) Explain which planes are E-plane and H-plane. (1 p)

Note that there may exist more than one solution for the same radiation pattern
requirement.

2.1 Radiation patterns:

E-plane H-plane
Solution: (4p)
x-directed incremental electric current, dipole.

2.2 Radiation patterns:

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E-plane H-plane
Solution: (4p)
Two crossed incremental electric dipoles located in xy-plane and excited with 90
deg phase difference: Huygen source, eleven antenna, incremental electric current
located 0.25 wavelengths above infinite ground plane.

E-plane H-plane
2.3 Radiation patterns:

E-plane H-plane
Solution: (4p)
Waveguide slot antenna (assume infinite ground plane), PIFA:

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E-plane H-plane
2.4 Radiation patterns:

E-plane H-plane
Solution: (4p)
Edge-fed microstrip antenna on high permittivity substrate (top view):

E-plane H-plane
2.5 Radiation patterns:

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E-plane H-plane
Solution: (3p)
Long pyramidal horn antenna with quadratic aperture fed by TE10 rectangular
waveguide mode:

E-plane H-plane

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2.6 Radiation patterns:

E-plane H-plane

Solution: (3p)
x-directed incremental electric current located 0.25 wavelengths above infinite
ground plane, Huygen source:

z
0dB

0dB
y
x
-20dB -10dB

-20dB -10dB

a) Polarization:
b) E-plane:
H-plane:
-180° 0° 180° -180° 0° 180°
c) xz-plane d) yz-plane

OR
Long pyramidal horn antenna with quadratic aperture with soft surfaces on all
walls, conical horn corrugated.

2.7 Radiation patterns:

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E-plane H-plane
Solution: (3p)
Long pyramidal horn antenna with quadratic aperture with hard surfaces on all
walls. Small arrays is also a possible answer.

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PART B (You can use the textbook to solve this problem, but
only after PART A has been delivered)

3.0 Diversity antenna (25p)

We have access to two dipole antennas that we would like to use as a diversity antenna in
order to improve the performance of a radio unit that is used in an isotropic multipath
environment. In the figure below we show the CDF curves for the two antennas measured
separately (i.e. without the other antenna present) as well as the CDF when the two
antennas are placed close to each other, as they would be when used on the radio, and the
CDF when both antenna ports are selection combined. We also know that antenna 2 has
an efficiency of 0 dB.

0
10
Cumulative probability (CDF)

-1
10

Antenna 1, isolated
Antenna 2, isolated

Selection combined
-2 antenna 1 & 2 when
10
they are close to
each other

-3
10
-30 -25 -20 -15 -10 -5 0 5
Normalized power level (dB)

Figure 3.1. CDF:s for the two dipole antennas measured separately without the
other antenna present and when placed close to each other when
selection combination is used, respectively.

3.1 Why are the CDF curves for antennas 1 and 2 different and what does this
difference represents?

Solution: (2p)
The reason for the different CDF curves is that the two antennas have different
efficiencies. The horizontal distance between the two curves represents the
difference in efficiency, from the figure it can be seen to be 3 dB.

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3.2 In the figure, sketch how the CDF curves change for the two ports (branches)
when the antennas are placed close to each other. Explain.

Solution: (4p)
See figure.
- The curves for the two branches should be shifted to the left (2p)
- The two curves should be shifted equally much (2p)

0
10

Branch 2 (ant. 2)
Cumulative probability (CDF)

Branch 1 (ant. 1)
-1
10

Antenna 1, isolated
Antenna 2, isolated

Selection combined
-2 antenna 1 & 2 when
10
they are close to
each other

-3
10
-30 -25 -20 -15 -10 -5 0 5
Normalized power level (dB)

3.3 What is the gain of using the two antennas and selection combining if we compare
with using only antenna 1? Assume a CDF level of 1%.

Solution: (2p)
From the figure the diversity gain can be seen to be approx. 7 dB.

3.4 What is the gain of using the two antennas and selection combining if we compare
with using an ideal reference antenna? Assume a CDF level of 1%.

Solution: (3p)
- From the figure the diversity gain can be seen to be approx. 4 dB (2p)
- It should be realized that antenna 2 represents an ideal reference since it has 0
dB efficiency (1p)

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3.5 Assume that we know that the efficiency for each branch decreases by 5 dB when
the antennas are closely spaced with the same spacing as for the diversity
combined curve. Determine then the absolute value of the complex correlation
coefficient between the two branches?

Solution: (3p)
The apparent diversity gain is given by equation 3.12 in the book, i.e.
Gapp  10 1   (note that it is Ok to use other, better, formulas for the apparent
2

diversity gain). Since the efficiency has decreased 5 dB due to that the antennas
are closely spaced as compared to when they are far apart the apparent diversity
gain is 5+4=9 dB (from d). Thus, we have Gapp  109 10  10 1      0.61
2

3.6 Now, we have got an offer from an antenna vendor to buy a diversity antenna
instead of using the two dipoles. The only information we have is that the antenna
is lossless and that it has the equivalent circuit shown in the figure below. The
values of the impedances in the figure are
Z11  Z22  82.4  j32.1 ; Z12  Z21  76.1  j 0.7 ohm .

What are the effective and apparent diversity gains of this antenna if selection
combining and a CDF-level of 1% are assumed?

I1 Z11 Z22 I2

Port 1 Z12 I 2 +_ +_ Z21I1 Port 2

Figure 3.2. Equivalent circuit for diversity antenna.

Solution: (6p)
The apparent diversity gain can be obtained from the knowledge of the
correlation coefficient which in turn, since the antenna is lossless, can be
calculated from the S-parameters as given by equation 3.10 in the book (note
that there is an error in the book, it should be a square root in the denominator,
it is however Ok if the expression in the book is used). Thus, the first thing we
have to do is to determine the S-parameters. For the given problem
Z11  Z22 and Z12  Z21 so that S11  S22 and S12  S21 .
S11 and S21 can be determined from the following circuit;

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50 I1 Z11 Z22 I2

V1 V2
U +
_ Z12 I 2 +_ +_ Z21I1 50
 
V 1 V 2

Since port 2 is terminated in 50 ohm V2  0 and we have the following relations;
V1 V2
S11  
, S 21  
, V2  V2
V1 V1
U U
We also know that V1   V1  V1 
2 2
We can now express the S-parameters in the total voltages as;
2V1 2V
S11  S22   1 , S12  S21  2
U U
The currents in the left and right loops can be determined as;
U  Z12 I 2  Z 21 I1
I1  , I2 
Z11  50 Z 22  50
Solving for the currents gives;
Z 22  50 Z 21
I1  U , I 2  U
 Z11  50  Z 22  50   Z12 Z 21  Z11  50  Z 22  50   Z12 Z 21
We can now calculate the port voltages as;
Z11  Z 22  50   Z12 Z 21
V1  Z11 I1  Z12 I 2  U
 Z11  50  Z 22  50   Z12 Z 21
50Z 21
V2  50 I 2  U
 Z11  50  Z 22  50   Z12 Z 21
The S-parameters are now given by;
Z11  Z 22  50   Z12 Z 21
S11  S22  2  1  0.10  j 0.42
 Z11  50  Z 22  50   Z12 Z 21
100Z 21
S12  S21   0.43  j 0.35
 11
Z  50  Z 22  50   Z12 Z 21
Now, when we have the S-parameters we can compute the absolute value of the
complex correlation coefficient by using equation 3.10 in the book (or rather the
correct version of the equation).
S11* S12  S21
*
S22
   0.41
  2
1  S11  S21
2
   2
1  S22  S12
2

The apparent diversity gain is given by equation 3.12 in the book;
Gapp  10 1    9.12  9.60 dB
2

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The effective diversity gain is defined as the apparent diversity gain multiplied
with the radiation efficiency, equation 3.11. We have by using equation 3.7;

Geff  erad Gapp  1  S11  S21
2 2
10 1   2
 4.61  6.64 dB

3.7 If Port 1 is excited with a generator with 50 ohm internal impedance and Port 2 is
terminated with 50 ohm, how much power is dissipated in the 50 ohm load at Port
2 and how much power is radiated?

Solution: (5p)
2
The power terminated in the 50 ohm load at Port 2 is given by; Pterm 2  Pavail S21
Where the available power is the power the generator can deliver in a matched
2
U
load, i.e. Pavail 
200 (where RMS-value for the voltage is assumed).
Thus,
2
0.43  j 0.35
2
S21
 From (f)  U  1.54  103 U
2 2 2 2
Pterm 2  Pavail S 21  U
200 200
Since the antenna is lossless the radiated power is given by;

Prad  Pavail  Prefl  Pterm 2  Pavail 1  S11  S 21
2 2

1  0.10  j0.42   2.52 10
2
U 2 2 3 2
  0.43  j 0.35 U
200

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4.0 Dipole Array (25p)

Fig. 4.1 shows an array antenna consisting of four halfwave dipoles.

Figure 4.3 A 4-dipole linear array antenna

4.1 Draw the equivalent circuit of the array.

Solution (5p)

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4.2 What are the values for the phase shifters when the maximum radiation direction
(main beam direction) is along the z-axis? Here we assume uniform excitations
(V1=V2=V3=V4).

Solution (4p)
Φ1= Φ2= Φ3 =Φ4=0

4.3 Write the expression for the scan impedance of dipole 3 under the condition of 4.2.
(Here you do not need to produce any numerical value. It is sufficient to write the
expression and explain the procedure of your derivation. All the parameters you use
should be known or easy to obtain from formulas or figures. Explain which formulas
and figures in the compendium you are using.)

Solution (4p)
We define the followings
V1  V   I1   Z11  Z 0 Z12 Z13 Z14 
V  V  I   Z Z 22  Z 0 Z 23 Z 24 
Vex   2     , I   2 , Z   21

V3  V   I3   Z 31 Z 32 Z 33  Z 0 Z 34 
       
V4  V  I4   Z 41 Z 42 Z 43 Z 44  Z 0 
Where all Zij  Z ji can be found by using Fig. 10.9 in Per-Simon’s book and Vex is known.
So we have
Vex  ZI I  Z1Vex
Now I is solved. Then,
V
Z 3 scan   Z 0
I3
4.4 Write the expression of the embedded radiation field function of dipole 3 when
dipole 3 is excited and the rest of the dipoles are terminated with loads (same as the
source impedances) Z0. NOTE that now V1=V2=V4=0. (Here you do not need to
produce any numerical value. It is sufficient to write the expression and explain the
procedure of your derivation. All the parameters you use should be known or easy to
obtain from formulas or figures. Explain which formulas and figures in the
compendium you are using.)

Solution (4p)
We have now
V1   0   I1   Z11  Z 0 Z12 Z13 Z14 
V   0  I   Z Z 22  Z 0 Z 23 Z 24 

Vex   2
   
, I   2
, Z  21

V3  V   I3   Z 31 Z 32 Z 33  Z 0 Z 34 
       
V4   0  I4   Z 41 Z 42 Z 43 Z 44  Z 0 
Where all Zij  Z ji can be found by using Fig. 10.9 in Per-Simon’s book and Vex is
known. So we have
I  Z1Vex

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Now I is solved. Then, from (5.11) in Per-Simon’s book, we have the radiation
function of a halfwave dipole at the origin of the coordinate system as G dx  ,   with
a current of I0. Note that now ˆl  xˆ so (5.11) should be modified a bit. Then, the
embedded radiation function for dipole 3 is

I1 jkr1 r I I I
G d 3 _ embedded  ,    G dx  ,   e  G dx  ,   2 e jkr2 r  G dx  ,   3 e jkr3 r  G dx  ,   4 e jkr4 r
I0 I0 I0 I0

4.5 Write the expression for the radiation field function of the whole array.

Solution (4p)
Almost the same as the above expression but we should use I instead of I .
I1 jkr1 r I I I
GWholeArray  ,    G dx  ,   e  G dx  ,   2 e jkr2 r  G dx  ,   3 e jkr3 r  G dx  ,  4 e jkr4 r
I0 I0 I0 I0

4.6 Now we want to steer the main beam to 30o from the z-axis. What are now the values
of the phase shifters? Are there any grating lobes now? If yes, in which direction?

Solution (4p)
From (10.22), we have

cos  0  
kd a
Now 0  60o , kda  1.4 so    cos 0 kda  0.7
So phase shifters are 0, 0.7 , 1.4 , 2.1 , respectively.
Now there is a grating lobe, at
 
1  cos 1 cos 60o    cos
1
 0.5  1.43  cos 1  0.9286   158.2o
 da 

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