Physical bounds on small antennas as convexoptimization problems

Mats Gustafsson, Marius Cismasu, and Sven Nordebo

Department of Electrical and Information Technology

Lund University, Sweden

IEEE-APS, Chicago, 2012-07-09

Design of small antennas

R. H. Bhuiyan www.m0wwa.co.uk www.hyperexperience.com

I There are many advanced methods to design small antennas.

I Performance often in bandwidth, matching, and efficiency.

I How can new designs, geometries, and materials improveperformance?

I Here, what is the fundamental tradeoff between performanceand size?

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Tradeoff between performance and size

I Radiating (antenna) structure, V .

I Antenna volume, V1 ⊂ V .

I Current density J1 in V1.

I Radiated field, F (k), in direction kand polarization e.

Questions analyzed here, J1 for:

I maximum G(k, e)/Q.

I maximum G(k, e)/Q forD(k, e) ≥ D0 (superdirectivity).

I embedded antennas.

I also minimum Q for radiated fieldapproximately F (k) and effects ofOhmic losses.

VV1

y

x

z

k

e

^

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Background

I 1947 Wheeler: Bounds based on circuit models.I 1948 Chu: Bounds on Q and D/Q for spheres.I 1964 Collin & Rothchild: Closed form expressions of Q for arbitrary

spherical modes, see also Harrington, Collin, Fantes, Maclean, Gayi,Hansen, Hujanen, Sten, Thiele, Best, Yaghjian, Kildal... (most are based onChu’s approach using spherical modes.)

I 1999 Foltz & McLean, 2001 Sten, Koivisto, and Hujanen: Attempts forbounds in spheroidal volumes.

I 2006 Thal: Bounds on Q for small hollow spherical antennas.I 2007 Gustafsson, Sohl & Kristensson: Bounds on D/Q for arbitrary

geometries (and Q for small antennas).I 2010 Yaghjian & Stuart: Bounds on Q for dipole antennas in the limitka→ 0.

I 2011 Vandenbosch: Bounds on Q for small (non-magnetic) antennas in thelimit ka→ 0.

I 2011 Chalas, Sertel & Volakis: Bounds on Q using characteristic modes.I 2012 Gustafsson, Cismasu, & Jonsson: Optimal charge and current

distributions on antennas.I 2012 Bernland: Physical Limitations on the Scattering of High Order

Electromagnetic Vector Spherical Waves.I 2012 Gustafsson & Nordebo: Optimal antenna Q, superdirectivity, and

radiation patterns using convex optimization.

VV1

y

x

z

k

e

^

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

G/Q and D/Q

V

J(r)

ke ^

an

@V

Partial gain expressed in the partialradiation intensity P (k, e) and totalradiated Prad and dissipated power Ploss

G(k, e) = 4πP (k, e)

Prad + Ploss

Q-factor

Q =2ωW

Prad + Ploss=

2c0kW

Prad + Ploss,

where W = max{We,Wm} denotes themaximum of the stored electric and magnetic energies. The G/Qand (D/Q for lossless) quotient cancels Prad + Ploss

G(k, e)

Q=D(k, e)

Q=

2πP (k, e)

c0kW.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Stored EM energies from current densities J in V

V

J(r)

ke ^

an

@V

Use the expressions by Vandenbosch(2010) (and Geyi (2003) for small antennas).

Stored electric energy W(e)vac =

µ016πk2

w(e)

w(e) =

∫V

∫V∇1 · J1∇2 · J∗

2

cos(kR12)

R12

−k2

(k2J1·J∗

2−∇1·J1∇2·J∗2

)sin(kR12) dV1 dV2,

where J1 = J(r1),J2 = J(r2), R12 = |r1 − r2|. Stored

magnetic energy W(m)vac = µ0

16πk2w(m), where

w(m) =

∫V

∫Vk2J1 · J∗

2

cos(kR12)

R12

− k

2

(k2J1 · J∗

2 −∇1 · J1∇2 · J∗2

)sin(kR12) dV1 dV2.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Stored EM energies from current densities J in V II

Also the total radiated power Prad = η08πkprad with

prad =

∫V

∫V

(k2J1 · J∗

2 −∇1 · J1∇2 · J∗2

)sin(kR12)

R12dV1 dV2.

The normalized quantities w(e), w(m), and prad have dimensionsgiven by volume, m3, times the dimension of |J |2.

I Introduced by Vandenbosch in Reactive energies, impedance,and Q factor of radiating structures, IEEE-TAP 2010.

I In the limit ka→ 0 by Geyi in Physical limitations of antenna,IEEE-TAP 2003.

I Validation for wire antennas in Hazdra etal, Radiation Q-factorsof thin-wire dipole arrangements, IEEE-AWPL 2011.

I Some issues with ’negative stored energy’ for large structures inGustafsson etal, IEEE-TAP 2012.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

G/Q in the current density J

V

J(r)

ke ^

an

@V

The partial radiation intensity P (k, e)in direction k and for the polarization e is

P (k, e) =η0k

2

32π2

∣∣∣∣∫Ve∗ · J(r)ejkk·r dV

∣∣∣∣2We have the G/Q quotient

G(k, e)

Q= k3

∣∣∣∫V e∗ · J(r)ejkk·r dV∣∣∣2

max{w(e)(J), w(m)(J)}

≤ maxJ

k3

∣∣∣∫V e∗ · J(r)ejkk·r dV∣∣∣2

max{w(e)(J), w(m)(J)}Solve the optimization problem. Closed form solutions in the limitka→ 0 and convex optimization for larger (but small) structures.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Convex optimization

minimize f0(x)

subject to fi(x) ≤ 0, i = 1, ..., N1

Ax = b

convex

not convex

f(y)

f(x)

f(®x+¯y)

®f(x)+¯f(y)

where fi(x) are convex, i.e., fi(αx+ βy) ≤ αfi(x) + βfi(y) forα, β ∈ R, α+ β = 1, α, β ≥ 0.

Solved with efficient standard algorithms. No risk of getting trappedin a local minimum. A problem is ’solved’ if formulated as a convexoptimization problem.

Can be used in many convex formulations for antenna performanceexpressed in the current density J , e.g.,

I Radiated field F (k) = −k × k ×∫V J(r)ejkk·r dV is affine.

I Radiated power, stored electric and magnetic energies, andOhmic losses are positive semi-definite quadratic forms in J .

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Currents for maximal G/Q

Determine a current density J(r) in the volume V that maximizes thepartial-gain Q-factor quotient G(k, e)/Q.

I Partial radiation intensity P (k, e)

G(k, e)

Q=

2πP (k, e)

c0kmax{We,Wm}.

I Scale J and reformulate P = 1 asRe{e∗ · F } = 1.

I Convex optimization problem.

minimize max{JHWeJ,JHWmJ}

subject to Re{FHJ} = 1

V

J

y

x

z

k

e

^

Determines a current density J(r) in the volume V with minimal storedEM energy and unit partial radiation intensity in {k, e}.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Maximal G(k, x)/Q for planar rectangles

Solution of the convex optimizationproblem

min. max{JHWeJ,JHWmJ}

s.t. Re{FHJ} = 1

for current densities confined toplanar rectangles with side lengths `xand `y = {0.01, 0.1, 0.2, 0.5}`x.

Note `x/λ = k`x/(2π), giving`x = λ/2→ k`x = π → ka ≥ π/2.

0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

G(k,x)/(Qk a )^^ 33

` /¸x

` y

` xJ(r)

x

y z

^

^ ^

` =0.5` y x

` =0.2` y x

y =0.1` x

` =0.01` y x

k=y^

k=z^

a

+j

1.5

1.75

2

2.25

2.5

0.1 0.2 0.3 0.4 0.5

D(k,x)^^

` /¸x

` =0.5` y x

` =0.2` y xy =0.1` x

` =0.01` y x

1 6 16

1 6 16

1 6 16

k=y^k=z^

1 6 16

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

D/Q (or G/Q) bounds

I Similar to the forward scattering bounds for TM.

I Can design ’optimal’ electric dipole mode (TM) antennas.

I TE modes and TE+TM are not well understood.

I Typical matlab code using CVX

cvx_begin

variable J(n) complex;

dual variables We Wm

maximize(real(F’*J))

We: quad_form(J,Ze) <= 1;

Wm: quad_form(J,Zm) <= 1;

cvx_end

We now reformulate the complex optimization problem to analyzesuperdirectivity, antennas with a prescribed radiation pattern, losses,and antennas embedded in a PEC structure.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Superdirectivity

I A superdirective antenna has adirectivity that is much larger than fora typical reference antenna.

I Often low efficiency (low gain) andnarrow bandwidth.

I There is an interest in smallsuperdirective antennas, e.g., Bestetal. 2008 and Arceo & Balanis 2011, Best, etal., An Impedance-Matched

2-Element Superdirective Array,IEEE-TAP, 2008

Here, we add the constraint D ≥ D0 to the convex optimizationproblem for G/Q to determine the minimum Q for superdirectivelossless antennas. We can also add constraints on the losses.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Superdirectivity

Add the constraintPrad ≤ 4πD−1

0 the get theconvex optimization problem

min. max{JHWeJ,JHWmJ}

s.t. Re{FHJ} = 1

JHPJ ≤ k3D−10

Example for current densitiesconfined to planar rectangleswith side lengths `x and`y = 0.5`x.

Q

` /¸x

0.1 0.2 0.3 0.4 0.5

100

10

1

D(z,x)¸1.8^^

D(z,x)¸1.7^^

D(y,x)¸3^^

D(y,x)¸3.2^^

k=y^

k=z^

0.35 0.4 0.45 0.510

102

103

104

105

106

` /¸x

Q

1 6 16 30 48

1 6 16 30 48

1 6 16 30 48

` x` /2x

J(r)

xyz

^^ ^

k=y^

k=z^

k=x^

e=x^

e=x^

e=y^

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Superdirectivity with D ≥ D0 = 10

0.35 0.4 0.45 0.510

102

103

104

105

106

` /¸x

Q

1 6 16 30 48

1 6 16 30 48

1 6 16 30 48

` x` /2x

J(r)

xyz

^^ ^

k=y^

k=z^

k=x^

e=x^

e=x^

e=y^

Note, it gives a bound on Q as D is known.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Optimal performance for embedded antennas

I It is common with antennas embeddedin metallic structures.

I The induced currents radiate but theyare not arbitrary.

I Linear map from the antenna regionadds a (convex) constraint.

I Here, we assume that the surroundingstructure is PEC and add a constraintto account for the induced currents onthe surrounding structure in the G/Qformulation.

VV1

y

x

z

k

e

^

b)

V2

V1

J (r)1

J (r)2

PEC

a)

V1

J (r)1

V2

J (r)2

`y

`x

» `y

» `x

»`x

`y

x

y

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Center fed strip dipole

0.1 0.2 0.3 0.4 0.5

0.06

0.065

0.07

0.075

0.08

»=0.1

»=0.2

»=0.4

»=0.6»=0.8

»=1

V2

0.01`x

`x»`x

V1

` /¸x

G(z,x)/(Qk a )3 3^ ^

V2PEC

Almost independent of the feed width at the resonance just below`x = 0.5λ.

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Embedded antennas in a planar rectangle

1

10

100

1000

` /¸x

Q

0.1 0.2 0.3 0.4 0.5

x

y z

^

^ ^

` /2 x

` xa

V2

V1

PEC

»` x

»` /2 x

»=0.5

»=0.25

»=0.5

»=1

»=0.25

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden

Conclusions

I Closed form solution for small antennas.I Optimal current distributions. Spherical

dipole, capped dipole, and folded sphericalhelix. More in IF46, Small Antennas:Designs and Applications on Thursday.

I Convex optimization to determine boundsand optimal currents for larger structures:

I D/Q and G/Q.I Q for superdirective antennas.I Embedded antennas in PEC structures.I Q for antennas with prescribed far fields.

See also Physical bounds and optimal currents onantennas IEEE TAP, 60, 6, pp. 2672-2681, 2012and Antenna currents for optimal Q, superdirectiv-ity, and radiation patterns using convex optimiza-tion (www.eit.lth.se/staff/mats.gustafsson)

−3

−2.5

−2

−1.5

−1

−0.5

0

0 45 90 135 1800

0.2

0.4

0.6

0.8

1

45 90 135 180

−1

−0.5

0

0.5

1

a)

b)

c)

J /Jµ0µ

± ± ± ± ±µ

45 90 135 180± ± ± ±µJ /Jµ0µ

J/Jµ0

Á

µ

± ± ± ±

VV1

y

x

z

k

e

^

Mats Gustafsson, Department of Electrical and Information Technology, Lund University, Sweden