mutual coupling in antenna arrays

Hon Tat Hui Mutual Coupling in Antenna Arrays

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Mutual Coupling in Antenna Arrays

1 Introduction

The electromagnetic interaction between the antenna elements in an antenna array is called mutual coupling. By its nature, mutual coupling exhibits differently in transmitting and receiving antenna arrays and therefore has to be treated differently.The effect of mutual coupling is serious if the element spacing is small. It will affect the antenna array mainly in the following ways:


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1. change the array radiation pattern2. change the array manifold (the received element voltages)3. change the matching characteristic of the antenna

elements (change the input impedances) We will mainly study the first two effects in this chapter. We will consider the change of array radiation pattern in a transmitting antenna array while study the change of the array manifold in a receiving antenna array.


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2 Mutual Coupling in Transmitting Antenna Arrays

In a transmitting antenna array, the mutual coupling effect renders the pattern multiplication principle being inapplicable for obtaining the array radiation pattern. This is because it results in the element patterns being not all the same.We study an example of a two-element dipole array. We characterize the mutual coupling effect using the mutual impedance (the conventional mutual impedance).


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Consider two transmitting antennas (such as dipoles) as shown on next page. They are separated by a distance of d and the excitation voltage sources, Vs1 and Vs2, have a phase difference of but an equal magnitude. Hence if there is no mutual coupling effect, the excitation currents also differ by a phase difference of and have an equal magnitude. When the mutual coupling effect is taken into account, the two coupled antennas can be modelled as two equivalent circuits as shown on page 6.

2.1 Definition of the Mutual Impedance


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Far field observation point, r

d

y

xDipole 2Dipole 1

Dipoles are parallel to the zdirection

Two transmitting dipoles

Now because of the mutual coupling effect, there are two additional excitation sources (the controlled voltage sources) in the equivalent circuits. These controlled voltage sources are to model the coupled voltages induced by the currents on the other antennas (see next page).


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a

Zg1

Vs1 V12Z11

b

ab

cd

c

Zg2

Vs2 V21Z11

d

Excitationvoltage source I1 I2

Coupledvoltage

Antenna input impedance ZA

SourceinternalImpedance (Zg1 = Zg2 = ZL)

Vs1Zg2

Vs2

Zg1

Antenna 1 Antenna 2

I1 I2

Terminalcurrent d


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1 1

12

12

2

12

2 0, 0

mutual impedance with antenna 2 excited

coupled voltage across antenna 1's open-circuit terminalexcitation current at antenna 2's shorted terminal

s

oc

I V

ZVI

VI

ab

Voc12

(1)

cdI2


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2 2

21

21

1

21

1 0, 0

mutual impedance with antenna 1 excited

coupled voltage across antenna 2's open-circuit terminalexcitation current at antenna 1's shorted terminal

s

oc

I V

ZVI

VI

cd

Voc21 Note that for passive antennas, Z12 = Z21

(2)

abI1


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The mutual impedance can be either measured or theoretically calculated. For measurement, usually the two antennas in the array are treated as a two-port network and its s parameters, S11, S12, S21, and S22, are measured. The z parameters, Z11, Z12, Z21, and Z22, are then obtained from the s parameters as follows:

11 22 12 21 1211 0 12 0

11 22 12 21 11 22 12 21

11 22 12 212121 0 22 0

11 22 12 21 11 22 12 21

1 1 2 1 1 1 1

1 12 1 1 1 1

S S S S SZ Z Z ZS S S S S S S S

S S S SSZ Z Z ZS S S S S S S S

where Z0 is the system impedance. The Z12 and Z21 of the z parameters are then the mutual impedances.

(3)


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Alternatively, the mutual impedance can be measured directly using the method described in ref. [1]. A rigorous theoretical method for the calculation of the mutual impedance can be found in ref. [2].For an N-element antenna array, the mutual impedances can be obtained by considering two antennas at a time. The total mutual impedances of the array, Zij(i,j=1,2,,N) will then be the set of two-antenna mutual impedances for all possible pair of antennas in the array.


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The following figure is an example which shows the conventional mutual impedance Z12 between two monopole antennas against their separation at a frequency of 2.4 GHz. The dimensions of the monopole antennas are: monopole length = 3 cm (0.24 at 2.4 GHz), radius of the monopole wires = 0.3 mm. Antenna 1 is open-circuited while antenna 2 is connected to a current source (with no terminal load).


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Using the mutual impedance, the coupled voltages V12and V21 can be expressed as follows:

12 12 2 21 21 1Z ZV I V I

I1 and I2 are the actual terminal currents at the antennas when there is mutual coupling effect. From the antenna equivalent circuits,

1 12 2 211 2 s s

L A L A

V V V VI IZ Z Z Z

2.2 Effect on the Array Radiation Pattern

(4)

(5)


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Is1 and Is2 are the terminal currents at the antennas when there is no mutual coupling effect.

1 21 2 s ss s

L A L A

V VI IZ Z Z Z

Our aim is to express I1 and I2 in terms of Is1 and Is2.

1 121

2 121

s

L A

sL A

V VIZ Z

I ZIZ Z

2 212

1 212

s

L A

sL A

V VIZ Z

I ZIZ Z

(6)

(7)


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From these two relations, we can find:

12 211 2 2 1

1 212 21 12 21

2 2

1 1

s s s sA L A L

A L A L

Z ZI I I IZ Z Z ZI I

Z Z Z ZZ Z Z Z

That is:

1 1 12 2 2 2 21 11 1 s s s sI I Z I I I Z ID D

(8)

(9)


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12 21

2

1212

2121

1A L

A L

A L

Z ZDZ ZZZ

Z ZZZ

Z Z

where

Now if we want to find the array pattern E on the horizontal plane (=/2) with mutual coupling effect, then E is just equal to the array factor.

cos1 2

1

1=AF jkdI I eI

EVector magnitude, not absolute value

(10)

(12)

(11)

(13)


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cos1 2

1

cos1 12 2 2 21 1

1

cos cos1 2 12 2 1 12 21

1

cos cos1 212

1 1

c1

1

1

1

1 with

1 with

1

jkd

jkds s s s

jkd jkds s s s

j jkd j jkd js s

s

j kds

I I eI

I Z I I Z I eI D

I I e Z I I e Z ZI D

I Ie e Z e e eI D II eI D

E

os cos12original pattern additional pattern

1 j kdjZ e e (14)


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We see that the array pattern now consists of two parts: the original array pattern plus an additional pattern:

which has a reverse current phase - and a modified amplitude with a multiplication of a complex number Z12ej. Note that all parameters in the above formula can be calculated except I1 which will be removed after normalization. Normalization of the above formula can only be done when its maximum value is known, for example by numerical calculation.

cos12 1

j kdjZ e e (15)


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Example 1Find the normalized array pattern |En| on the horizontal plane (=/2) of a two-monopole array with the following parameters with mutual coupling taken into account:

1 2 1 21, , 1, 1504, 4

js s s sI I e I I

d

12 21 21 8 21 9 47.3 22.3 50

A

L

Z Z . - j .Z jZ

d

2sI1sI

Absolute value


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1 2

1 2

1 2

1, 1

0 , 150 2.62 rad

js s

s s

s s

I I eI I

I I

Solution

12 2112 0 16 0 26

A L A L

Z ZZ . j .Z Z Z Z

24 2

kd

12 21

21 1.042 0.09A L

Z ZD jZ Z

As the required array pattern |En| is on the horizontal plane, it is equal to the normalized array factor |AFn|.


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cos cos112

1

2 cos2.62

1

2 cos2.62 2.62

2 cos

1

AF 1 1

0.95 0.08 1

0.16 0.26 1

0.94 0.37 1 1.14 0.40

j kd j kdjs

jj

jj j

j

I e Z e eI D

j e eI

j e e e

j j eI

E

The pattern of is shown on next page.

2 cos1 1.14 0.40 jf j e


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Normalization

The pattern of f attains the maximum value when = 180. When = 180,

180

2 cos

1801

1

0.94 0.37 1 1.14 0.40

1.83

jj j eI

I

E

Hence we normalize |E| by this factor (1.83/|I1|) to get:


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2 cos

1

1

2 cos

0.94 0.37 1 1.14 0.40

1.83

0.521 1.14 0.40

j

n

j

j j eI

I

j e

E

The polar plot of |En| is shown on next page.


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cosno mutual coupling effect

1 1 j jkdn e e E

The case when there is no mutual coupling is shown below for comparison.

where is a constant to makethe largest value of |AFn| equal to one ( = 1.73).


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2.3 Effect on the Array ManifoldThe total voltages (due to the excitation sources and the coupled voltages) Vtot1 and Vtot2 across the whole lengthsof the two antenna elements can be expressed as the summation of the excitation voltage Vs1 and Vs2 due to the excitation sources alone and the coupled voltages V12and V21 as follows:

1 1 12

2 2 21

tot s

tot s

V V VV V V

The total voltages Vtot1 and Vtot2 can be expressed in terms of the terminal currents I1 and I2.

(16)(17)


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1 1 11

2 2 22

tot L

tot L

V I Z ZV I Z Z

(18)(19)

From (16) & (17),

121 1 2

22

212 2 1

11

s tot totL

s tot totL

ZV V VZ Z

ZV V VZ Z

Putting (18) & (19) into (20) & (21), we have

(20)(21)

(22)

(23)

1 1 12 1 2 12

2 2 21 2 1 21

s tot tot

s tot tot

V V V V I ZV V V V I Z


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Hence when there is mutual coupling, the total excitation voltages for the two antennas, Vtot1 and Vtot2, are related to the excitation voltages without mutual coupling, Vs1 and Vs2, by the following matrix formula:

(24)

12

221 1

2 221

11

1

1

Ls tot

s tot

L

ZZ ZV V

V VZZ Z


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112

1 122

221

112 2

1 2

11 22

1

1

1

N

s totL L NN

N

L L NNs tot

N NsN totN

L L

ZZV VZ Z Z Z

ZZZ Z Z ZV V

Z ZV VZ Z Z Z

For an N-element transmitting antenna array, the uncoupled array manifold Vs1, Vs2, , and VsN (the uncoupled excitation voltages) is related to the coupled array manifold Vtot1, Vtot2, , and VtotN (contaminated with the coupled voltages) by the following formula:

(25)


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3 Mutual Coupling in Receiving Antenna ArraysIn a receiving antenna array, strictly speaking, we cannot use the conventional mutual impedances to measure the mutual coupling effect (see refs. [3]-[7]). This is because the excitation source for a receiving antenna array is at the far-field zone and not at the terminals of the antenna elements. Usually the current distributions on the antenna elements (which cause the mutual coupling) in a receiving antenna array are different from those on the antenna elements in a transmitting antenna array.Hence we need to re-define the mutual impedances in a receiving antenna array as follows:


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ZL1ab

cd

V1ZL2V2

Antenna 1 Antenna 2

I1 I2

d

Plane wave

The scenario for defining the receiving mutual impedance

3.1 Definition of the Receiving Mutual Impedance


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12

12

2

receving mutual impedance with a receiving current ( ) at antenna 2

coupled voltage across antenna 1's terminal loadreceiving current through antenna 2's terminal load

tZcurrent distribution

VI

1 12

V UI

ZL1U1

Antenna 1Plane wave

U1 = Received voltage across antenna 1s terminal load with antenna 2 removed (isolation voltage 1)

(26)


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21

21

1

receving mutual impedance with a receiving current ( ) at antenna 1

coupled voltage across antenna 2's terminal loadreceiving current through antenna 1's terminal load

tZcurrent distribution

VI

2 21

V UI

ZL2U2

Antenna 2Plane wave

U2 = Received voltage across antenna 2s terminal load with antenna 1 removed (isolation voltage 2)

(27)


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Note that in general, the definition of the receiving mutual impedance implies that the receiving mutual impedance is dependent on the direction of the plane wave, which is used as the excitation source. But for omni-directional antennas such as dipole and monopole antennas, the receiving mutual impedance is independent of the azimuth angle of the excitation plane wave.Furthermore, for a general antenna array, as the excitation source is not applied to the feeding ports of the antennas directly (the antennas being excited by an external plane wave source), reciprocity of the receiving mutual impedances does not hold and hence Zt

ij Ztji in general.


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3.2 Measurement of the Receiving Mutual Impedance

The receiving mutual impedance can be either measured or theoretically calculated. For measurement, the antenna array has to be placed inside an anechoic chamber and its received power wave (the S12 parameter) is measured by a VNA machine. As an example, for a typical two-element monopole antenna array, the following procedure is used:1. Use the transmitting antenna in the anechoic chamber

as the plane wave source and place the monopole array at a fixed position (not rotating). (See the setup on next page.)


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The measurement of the receiving mutual impedances


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2. Measure S21 at monopole 1s terminal with monopole 2s terminal connected to a terminal load. Denote this as S21_1.

3. Measure S21 at monopole 2s terminal with monopole 1s terminal connected to a terminal load. Denote this as S21_2.

4. Measure S21 at monopole 1s terminal with monopole 2 removed (taken away from the array). Denote this as S21_1.

5. Measure S21 at monopole 2s terminal with monopole 1 removed (taken away from the array). Denote this as S21_2.


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Throughout the measurement, the relative positions of the two monopoles must not be changed with respect to the transmitting antenna as this change will alter the phase of S12. By definition,

21S

where is the square root of the power input to the transmitting antenna and is the square root of the power received by a monopole with V being the terminal voltage of the monopole and Z0 being the system impedance. Both and are complex values with magnitudes and phases.

0V Z

(28)


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Now convert the measured S21s into relative voltages as:

1

21_1 0

total received terminal voltage on monopole 1V

S Z

2

21_ 2 0

total received terminal voltage on monopole 2V

S Z

1

21_1 0

isolation voltage on monopole 1U

S Z

2

21_ 2 0

isolation voltage on monopole 2U

S Z

(29)

(30)

(31)

(32)


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Then the measured receiving mutual impedances are obtained as follows:

21_1 21_112 1 1 1 10

2 2 0 21_ 2t

S SV U V UZ ZI V Z S

21_ 2 21_ 221 2 2 2 20

1 1 0 21_1t

S SV U V UZ ZI V Z S

For an N-element antenna array, the mutual impedances can be measured by considering two antennas at a time using the above procedure and repeating it for all the possible pairs of elements in the array.

(33)

(34)


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0

25

20

15

10

5

0

-5

-10

-150.2 0.4 0.6 0.8 1.0

d

real

imaginary

The following figure is an example which shows the receiving mutual impedance Zt12 between two monopole antennas against their separation at a frequency of 2.4 GHz. The dimensions of the monopole antennas are: monopole length = 3 cm (0.24 at 2.4 GHz), radius of the monopole wires = 0.3 mm. Each monopole antenna is connected to a terminal load of ZL = 50 . The external plane wave used to excite the array comes from the horizontal direction with = 90and an arbitrary value of as the received currents on the two monopoles are independent of .

Antenna separation, d ()

Z

t

1

2

(

)


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3.3 Effect on the Array ManifoldUsing the receiving mutual impedances, the total received voltages V1 and V2 (contaminated with the coupled voltages) on the antenna elements can be expressed as the summation of the isolated terminal voltages U1 and U2 (without coupled voltages) and the coupled voltages V12 and V21 as follows:

12 12 21 1 12 1 2 1

21 21 12 2 21 2 1 2

t tL

t tL

VV U V U Z I U ZZVV U V U Z I U ZZ

(35)

(36)


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For an N-element receiving antenna array, the uncoupled array manifold U1, U2, , and UN (the isolation terminal voltages) can be calculated from the received array manifold V1, V2, , and VN (contaminated with the coupled voltages) using the following formula:

12 1

1 121 2

2 2

1 2

1

1

1

Nt t

L LN

t t

L L

N NN Nt t

L L

Z ZZ ZU V

Z ZU VZ Z

U VZ ZZ Z

(37)


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A seven-element monopole array is used to detect the directions of two incoming signals (plane waves) at 1 = 62.4 and 2 = 111.9, respectively. The received antenna terminal voltages, V1, V2, V3, V4, V5, V6, and V7 are analyzed by the direction-finding algorithm, MUSIC. The result is shown on next page. Three MUSIC power spectra are obtained with the uncoupled voltages determined by three different methods: (i) using the uncoupled voltages calculated by the receiving mutual impedances, (ii) using the uncoupled voltages calculated by the (conventional) mutual impedances, and (iii) using the coupled voltages directly. The average SNR of the received voltages is 39.1 dB.

Example 2


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0 20 40 60 80 100 120 140 160 180-50

-40

-30

-20

-10

0

P

o

w

e

r

(

d

B

)

Angle, (degree)

NC CMIM RMIM

Actual signal directions

The MUSIC power spectra for the estimation of two signal sources at 1 = 62.4and 2 = 111.9

(iii)

(i)

(ii)


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From this figure, it shows that the case of using the receiving mutual impedances (case (i)) for calculating the uncoupled voltages produces the most accurate direction finding results. As direction finding arrays are typical receiving arrays, this example tells that the antenna mutual coupling effect in a receiving antenna array has to be analyzed by using the receiving mutual impedances and not the (conventional) mutual impedances.(For more details on this example, please see ref. [8].)


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References:[1] R. C. Hansen, Micrwoave Scanning Antennas, volume II, p. 160, Academic

Press, 1966, p. 160.[2] E. C. Jordan, Electromagnetic Waves and Radiating Systems, Chapter 11,

Prentice-Hall Inc., Englewood Cliffs, N. J., 1968.[3] J. Daniel, Mutual coupling between antennas for emission or reception

application to passive and active dipole, IEEE Trans. Antennas Propagat., vol. 22, pp. 347-349, Mar. 1974.

[4] C. A. Balanis, Antenna Theory - Analysis and Design, John Wiley & Sons, New Jersey, 2005, pp. 478-481.

[5] Y. Wu, Z. Nie, On the improvement of the mutual coupling compensation in DOA estimation, Journal of Systems Engineering and Electronics, Vol. 19, No. 1, pp, 1-6, 2008.

[6] Y. Wu and Z. Nie, New mutual coupling compensation method and its application in DOA estimation, Front. Electr. Electron. Eng. China, vol. 4, no. 1, pp. 47-51, 2009.

[7] Hoi-Shun Lui, H. T. Hui, and M. S. Leong, "A note on the mutual coupling problems in transmitting and receiving antenna arrays, IEEE Antennas and Propagation Magazine, vol. 51, no. 5, pp. 171-176, 2009.


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Additional references:

[A1] C. A. Balanis, Antenna Theory, Analysis and Design, John Wiley & Sons, Inc., New Jersey, 2005.

[A2] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, Wiley, New York, 1998.

[A3] David K. Cheng, Field and Wave Electromagnetic, Addison-Wesley Pub. Co., New York, 1989.

[A4] John D. Kraus, Antennas, McGraw-Hill, New York, 1988.[A5] I. J. Gupta and A. A. Ksienski, Effect of mutual coupling on the performance of

adaptive arrays, IEEE Trans. Antennas Propagat., vol. 31, no. 5, pp. 785-791, 1983.

[8] Y. T. Yu, H. S. Lui, C. H. Niow, and H. T. Hui, Improved DOA estimations using the receiving mutual impedances for mutual coupling compensation: Anexperimental study, IEEE Transactions on Wireless Communications, vol. 10, no. 7, pp. 2228-2233, 2011.

mutual coupling in antenna arrays

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