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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