Chapter 5b: Mutual Coupling inAntenna Arrays

Antennas and Propagation

Chapter 5bAntennas and Propagation Slide 2

Definition

IEEE Standard Definitions and Terms for AntennasDefines mutual coupling as follows:

2.244 mutual coupling effect (A) (on the radiation pattern of an array antenna). For array antennas, thechange in antenna pattern from the case when a particular feeding structure is attached to the array andmutual impedances among elements are ignored in deducing the excitation to the case when the same feedingstructure is attached to the array and mutual impedances among elements are included in deducing theexcitation.(B) (on input impedance of an array element). For array antennas, the change in input impedance of anarray element from the case when all other elements are present but open-circuited to the case when all otherelements are present and excited.

(A) ⇒ Radiation patterns are modified by nearby antennas.(B) ⇒ Input characteristics (like impedance) modified

In this lecture we will study this effect in detail.

Chapter 5bAntennas and Propagation Slide 3

Mutual Coupling Effect

From Standpoint of Radiation IntegralsWhat does placing an antenna nearby do?Change the boundary condition of the problemWill affect both radiation and terminal properties

Two scenariosDifferent function of

antennas

But, can be analyzedusing sameequations (a) Transmit Mode (b) Receive Mode

Chapter 5bAntennas and Propagation Slide 4

Transmit Mode

Intuitive Explanation of Modified RadiationDrive element mRadiated field intercepted by nCauses radiation from nTotal radiation is combination of m and nEffective radiation when we drive m

is thus changed

Input CharacteristicsPower radiated by n is intercepted

again by mChanges current flowing on mInput impedance at m is modified

Chapter 5bAntennas and Propagation Slide 5

Transmit Mode Analysis

Example 2-antenna systemLoad/drive port 1Goal: Find impedance/pattern

looking into port 2

Analysis

Clearly, the input characteristics at port 2 change due to what we connect at port 1!

Chapter 5bAntennas and Propagation Slide 6

Transmit Mode Analysis (2)

Radiation Pattern

Also, effective pattern of element 2 has changed due to load on antenna 1!

Pattern of kth antenna, other antennas “open circuit”

LetImportant note:

This is called an “embedded pattern”In contrast to “isolated pattern” ofsingle element

Chapter 5bAntennas and Propagation Slide 7

Receive Mode

Intuitive Explanation Plane wave impinges on element mCurrent flows on mPower reradiated (scattered!)Some reradiated power received by nThis causes radiation by element nSome power received by mCurrent on m is modified

Important PointEffective receive aperture of element m

depends on loading of element n(and vice versa!)

Chapter 5bAntennas and Propagation Slide 8

Mutual Coupling Effect

Severity Controlled byRadiation patterns of the two antennas

or distribution of near-fieldsSpacing of the antennaOrientation of antennasLoading

Ways to avoid couplingPlace elements far apartCareful design of antennas

Chapter 5bAntennas and Propagation Slide 9

Infinite Array

PurposeSimplifies analysisCan understand effect more intuitivelyAlso useful to analyze performance of

large arrays

AssumeRegular array of elements (uniform spacing)Identical elementsUniform phase shift driving / uniform plane-wave excitation

Leads toJust constant phase shift in signals from one element to next

Chapter 5bAntennas and Propagation Slide 10

Infinite Array: Transmit Mode

AnalysisVoltage on mnth element (element at mth row, nth column)

“Driving impedance” of mnth element

For uniform plane-wave excitaitonlinear phase shift across array of voc

because array is infinite, have linear phase shift of currents also!

Mutual impedance from pqth to mnth element.

Chapter 5bAntennas and Propagation Slide 11

Infinite Array: Transmit Mode (2)

ImplicationsDriving impedances of all the elements are identicalMakes analysis much simplerComplete system can be understood by driving 1 antenna

Chapter 5bAntennas and Propagation Slide 12

Infinite Array: Receive Mode

Infinite resistive sheet model

Reflection of sheet

Receive power is scan angle dependent.

Chapter 5bAntennas and Propagation Slide 13

Compensating Mutual Coupling

Practical Uses of Mutual Coupling?Reconfigurable antennas: modify pattern, matchingRFID / spatial modulation: send information by switching a load

Coupling normally detrimentalModifies radiation patterns of elementsComplicates analysis of arrayCan correlate signals

Compensation methods1. Matching-based methods2. Digital compensation

Chapter 5bAntennas and Propagation Slide 14

Matching-Based Methods

Method allows “perfect” decouplingDecoupling network:

Input reflection matrix is the Hermitian of the antenna reflectionMatrix extension of 1-port conjugate match

Design so Γ is 0 for reference impedance (Z0=ZL) Input characteristics / patterns of ports are independentProblem: Designing D.N. for large N!

Chapter 5bAntennas and Propagation Slide 15

Digital Compensation

Given: we know ZA and ZL

Measure v on array elementsCan use linear equations to get voc

For “minimum scattering” elements, voc of elementvery close to v on an isolated element (i.e. otherantennas not present)

Problems with Digital Compensation Requires detailed array calibration (ZA, ZL, embedded patterns)Signals corrupted before noise and quantization (info. loss)

Chapter 5bAntennas and Propagation Slide 16

Supergain / Superdirectivity

PhenomenonAllows high gain for small antennasMathematics shows it is possible

Not practical: why?High QHigh matching sensitivityLow efficiency (ohmic losses)

Chapter 5bAntennas and Propagation Slide 17

Supergain Analysis

ConsiderUniform linear arrayRadiation in azimuth (xy) plan (θ = π/2)Dipoles

Assume embedded patterns areapproximately same as isolated patterns(minimum scattering assumption)

Radiation pattern of array

ULA along x-axis

Chapter 5bAntennas and Propagation Slide 18

Supergain Analysis (2)

Radiation intensity in direction φ is

Radiation intensity of uniform radiator with same input power:

Chapter 5bAntennas and Propagation Slide 19

Supergain Analysis (3)

Goal: directivity in direction φ0 as large as possible

Problem:

or

Simplify by remove A from constraint.

Now, we have a new problem:

What is the solution?

Chapter 5bAntennas and Propagation Slide 20

From Linear Algebra

Optimization Problem

Solution

x = eigenvector corresponding to the largest eigenvalue of Arefer to this as the “dominant” eigenvector

For our Problem:

a′ = dominant eigenvector of G′(φ0)

Chapter 5bAntennas and Propagation Slide 21

Supergain Analysis (4)

Analyze N=2 elements: SimpleReveals main effect

For φ0=0 and N=2, eigenvectors are close to

(a) Odd mode (b) Even mode

For φ0=0, odd mode dominates

Let

Note: α is an arbitrary scale factor

Chapter 5bAntennas and Propagation Slide 22

Supergain Analysis (5)

For odd excitation radiated field is

Radiated power is

Radiation intensity of isotropic radiatoror U0 = Prad/(2π) = aHAa

Directivity becomes

Consider limit as kΔ→0

D0 = 2

Chapter 5bAntennas and Propagation Slide 23

Result of Supergain Analysis

Two ElementsOdd-mode excitationVanishing separationDirectivity D0=2For single antenna D0=1

MeaningCan put two dipoles as close together as we likeGet twice directivity of single dipoleIf we pack in N, get factor of NCan make a tiny antenna as directive as we likeContradiction?

Chapter 5bAntennas and Propagation Slide 24

Result of Supergain Analysis (2)

Consider Antenna Weights

Now, as separation diminishes

MeaningFor finite radiated power,Antenna weights become infinite!

Currents are infinite but opposite

Not practicalHigh ohmic lossesHigh sensitivity

For most analysesSet constraints to avoid supergainsolutions

J1 = +∞ J2 = -∞

Chapter 5bAntennas and Propagation Slide 25

Summary

Mutual Coupling in Antenna ArraysAntenna elements affect each otherNetwork characteristicsRadiation patterns

CompensationPossible through decoupling network

or digital (SP) calibration

Supergain effectCoupled dipoles can have higher gain than uncoupledMostly mathematical. Should be avoided in real designs.