Pulsed EM Fields: Their Modeling and Potentialities by Martin - - PowerPoint PPT Presentation

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Brno University of Technology c 2014 SIX Research Centre

Title and affiliation Pulsed EM Fields: Their Modeling and Potentialities by Martin ˇ Stumpf Brno University of Technology SIX Research Centre Technick´ a 3082/12 • 616 00 Brno • The Czech Republic T: +420-5-4114-6539

• E: stumpf@feec.vutbr.cz

Presentation given at: Hamburg, Germany, 9 January 2015

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Motivation Presentation motivation

• ∀ PHYSICAL PHENOMENA ∈ SPACE – TIME
• R3 × R + CAUSALITY

Re(s) Im(s) s+

CAUSAL f(t)

• ANALYTIC ˆ

f(s) FREQUENCY DOMAIN Re(s) = 0

×

• THE CONCEPT OF ’FREQUENCY’ = MATHEMATICAL ARTIFACT
• ’FREQUENCY DOMAIN’ does away with

CAUSALITY UNIQUENESS

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Outline Presentation overview

• Generalized-Ray Theory (GRT) and its extensions
• EM Green’s functions in layered media
• Applications to bounded domains amd inclusion of relaxation phenomena
• Time-Domain RADAR Imaging
• Time-Domain Modeling of P/G Structures
• Analytical modeling of rectangular P/G structures
• Time-Domain Contour Integral Method (TD-CIM)
• EM radiation, (self-)reciprocity and mutual coupling
• Conclusions

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Generalized-Ray Theory

GENERALIZED-RAY THEORY

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GRT: Notation Conventions in notation

• SUBSCRIPT NOTATION
• Latin lower-case = {1, 2, 3}, Greek lower-case = {1, 2}
• position vector: x → xn (tensor of rank 1)
• spatial derivative: ∂/∂xn → ∂n (tensor of rank 1)
• Kronecker symbol: δm,n (symmetrical unit tensor of rank 2)
• Levi-Civita symbol: ei,j,k (completely antisymmetrical unit tensor of rank 3)
• SUMMATION CONVENTION
• δm,m = 3

m=1 δm,m = 3

• ek,l,mek,l,m = 3

k=1

3

l=1

3

m=1 ek,l,mek,l,m = 6

• ∇[∇ · v] → ∂i∂kvk = δi,j∂j∂kvk
• ∇ × [∇ × v] → ei,j,kek,l,m∂j∂lvm

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GRT: Problem description Problem definition

• ND domains described via ǫN = ǫN(x3) and µN = µN(x3),

for N = {1, ..., ND} with cN = (ǫNµN)−1/2 > 0

• the configuration is linear, instantaneously reacting,

time invariant, shift invariant in the horizontal direction

• the EM field is causally related to the action of

an impulsive source placed at x3 = x3;S

• a source starts to act at t = 0 and prior to this instant the EM fields vanish

throughout the configuration

• an arbitrary probe position
• no relaxation mechanism incorporated

x3

receiver

▽

{ǫND, µND} DND x3;ND {ǫS+1, µS+1} DS+1 x3;S+1 {ǫS, µS} DS x3;S source

×

{ǫ2, µ2} D2 x3;2 {ǫ1, µ1} D1

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GRT: EM field decomposition EM field decomposition† −ek,m,p∂mHp + ǫ∂tEk = −Jk ej,n,r∂nEr + µ∂tHj = −Kj for k = {1, 2, 3} and p = {1, 2, 3} : π = {1, 2} p = 3 for j = {1, 2, 3} and r = {1, 2, 3} : ρ = {1, 2} r = 3 −ek,µ,π∂µHπ − ek,µ,3∂µH3 − ek,3,π∂3Hπ + ǫ∂tEk = −Jk ej,ν,ρ∂νEρ + ej,ν,3∂νE3 + ej,3,ρ∂3Eρ + µ∂tHj = −Kj DECOMPOSED EM FIELD EQUATIONS

• for a point source: {Jk, Kj}(x, t) = {jk, kj}(t)δ(x1, x2, x3 − x3;S) ⇒

limx3↓x3;N Hπ − limx3↑x3;N Hπ = e3,π,κjκ(t)δ(x1, x2)δN,S limx3↓x3;N Eρ − limx3↑x3;N Eρ = e3,ι,ρkι(t)δ(x1, x2)δN,S INTERFACE CONDITIONS

†M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP, vol. 61, no. 5, pp. 2676–2687, May 2013.

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GRT: Transform domains Transform-domain field representation† ˆ Ek(x, s) = ∞

t=0 exp(−st)Ek(x, t)dt

LAPLACE TRANSFORMATION

• Lerch’s sequence: {s ∈ R : s = s0 + nh, s0 > 0, h > 0, n = 0, 1, 2, ...}

⇒ transformation property: ∂t → s (for zero initial conditions) WAVE-SLOWNESS FIELD REPRESENTATION ˆ Ek(x, s) = s

2π

2

α1∈R dα1

• α2∈R exp(−isαµxµ) ˜

Ek(α1, α2, x3, s)dα2 ⇒ transformation property: ∂ν → ˜ ∂ν = −isαν

†M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP, vol. 61, no. 5, pp. 2676–2687, May 2013.

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GRT: Source-type field representations Source-type field representations† ∂2

3 ˜

Eπ − s2γ2 ˜ Eπ = sµ ˜ Jπ − ˜ ∂π ˜ ∂ν ˜ Jν/sǫ − ˜ ∂π∂3 ˜ J3/sǫ − eπ,ρ,3∂3 ˜ Kρ + eπ,ρ,3 ˜ ∂ρ ˜ K3 ∂2

3 ˜

Hρ − s2γ2 ˜ Hρ = sǫ ˜ Kρ − ˜ ∂ρ ˜ ∂π ˜ Kπ/sµ − ˜ ∂ρ∂3 ˜ K3/sµ + eρ,κ,3∂3 ˜ Jκ − eρ,κ,3 ˜ ∂κ ˜ J3 + TRANSFORM-DOMAIN INTERFACE CONDITIONS AND LINEARITY

• transform-domain representations for tangential EM field components

˜ Eπ(α1, α2, x3, s) = ˜ ∂π∂3ˆ j3(s) ˜ GJ⊥/sǫ ˜ Hρ(α1, α2, x3, s) = eρ,π,3 ˜ ∂πˆ j3(s) ˜ GJ⊥ VERTICAL ELECTRIC DIPOLE ˜ Eπ(α1, α2, x3, s) = −sµˆ jπ(s) ˜ GJ + ˜ ∂π ˜ ∂νˆ jν(s) ˜ GQ/sǫ ˜ Hρ(α1, α2, x3, s) = eρ,3,πˆ jπ(s)∂3 ˜ GJ − ˜ ∂νˆ jν(s)eρ,3,π ˜ ∂π∂3 ˜ GQ − ˜ GJ s2γ2 HORIZONTAL ELECTRIC DIPOLE

†M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP, vol. 61, no. 5, pp. 2676–2687, May 2013.

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GRT: Generalized-ray EM field expansion Generalized-ray EM field expansion (1)† GENERAL SOLUTION ˜ G = W +

N exp[−sγN(x3 − x3;N)] + W − N exp[−sγN(x3;N+1 − x3)]

• {W +

N, W − N } are transform-domain upgoing/downgoing wave amplitudes

SCATTERING DESCRIPTION W +

N = ¯

S+−

N W − N + ¯

S++

N W + N−1 + X+ N

W −

N−1 = ¯

S−−

N W − N + ¯

S−+

N W + N−1 + X− N−1

{ ¯ S+−

N , ¯

S−−

N } = {S+− N , S−− N } exp(−sγNdN)

{ ¯ S++

N , ¯

S−+

N } = {S++ N , S−+ N } exp(−sγN−1dN−1) x3;N DN DN−1 W −

N

W +

N

W +

N−1

W −

N−1

†M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP, vol. 61, no. 5, pp. 2676–2687, May 2013.

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GRT: Generalized-ray EM field expansion Generalized-ray EM field expansion (2) HORIZONTAL-SOURCE COUPLING W A+

N

= ¯ SA+−

N

W A−

N

+ ¯ SA++

N

W A+

N−1 + XA+ N

W A−

N−1 = ¯

SA−−

N

W A−

N

+ ¯ SA−+

N

W A+

N−1 + XA− N−1

W B+

N

= ¯ P B+−

N

W B−

N

+ ¯ P B++

N

W B+

N−1 + ¯

CB+−

N

W A−

N

+ ¯ CB++

N

W A+

N−1 + XB+ N

W B−

N−1 = ¯

P B−−

N

W B−

N

+ ¯ P B−+

N

W B+

N−1 + ¯

CB−−

N

W A−

N

+ ¯ CB−+

N

W A+

N−1 + XB− N−1

GENERALIZED-RAY (NEUMANN) EXPANSION W = M

m=0 Sm · X + SM+1 · W

⇒ CAGNIARD-deHOOP INVERSION

×

X+

3

X−

2

x3;4 x3;3 x3;2 PEC

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GRT: Generalized-ray constituent (1) Generalized-ray constituent (1) GENERIC FORM ˆ R(x, s) = s 2π 2 ˆ Q(s) ∞

α1=−∞

dα1 ∞

α2=−∞

Π(α1, α2) × exp

• −s
• iα1x1 + iα2x2 +

N

• K=1

γK(α1, α2)ZK

• dα2
• mapping M : {α1, α2} → {p, q} with ˆ

R(x, s) = s2 ˆ Q(s) ˆ K(x, s) PROPAGATION FACTOR

ˆ K(x, s) = 1 4π2i ∞

q=−∞

dq i∞

p=−i∞

¯ Π(p, q) × exp

• −s
• pr +

N

• K=1

¯ γK(p, q)ZK

• dp

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GRT: Generalized-ray constituent (2) Generalized-ray constituent (2) CAGNIARD-deHOOP PATH pr + N

K=1 ¯

γK(p, q)ZK = τ for {τ ∈ R; τ > 0} HEAD-WAVE PART

Re(p) Im(p)

p-plane

CBW CHW

BODY-WAVE PART

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GRT: Generalized-ray constituent (3) Body-wave part s-DOMAIN PROPAGATOR

ˆ KBW (x, s) = ∞

τ=T BW (0)

exp(−sτ)dτ 1 π2 QBW (τ)

q=0

Im

• ¯

Π

• pBW , q

∂pBW ∂τ

• pBW , q
• dq

⇓

LERCH’S UNIQUENESS THEOREM

⇓

t-DOMAIN PROPAGATOR

KBW (x, t) = 1 π2 QBW (t)

q=0

Im

• ¯

Π∂pBW ∂t

• dq

⇒

t-DOMAIN CONSTITUENT

RBW (x, t) =

• if t < T BW (0)

∂2

t Q(t) (t)

∗ KBW (x, t) if t > T BW (0)

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GRT: Generalized-ray constituent (4) Head-wave part s-DOMAIN PROPAGATOR

ˆ KHW (x, s) = T BW (0)

τ=T HW (0)

exp(−sτ)dτ 1 π2 QHW (τ)

q=0

Im

• ¯

Π

• pHW , q

∂pHW ∂τ

• pHW , q
• dq

+ T B

τ=T BW (0)

exp(−sτ)dτ 1 π2 QHW (τ)

q=QBW (τ)

Im

• ¯

Π

• pHW , q

∂pHW ∂τ

• pHW , q
• dq

⇓

LERCH’S UNIQUENESS THEOREM

⇓

t-DOMAIN PROPAGATOR

KHW (x, t) = 1 π2 QHW (t)

q={0,QBW (t)}

Im

• ¯

Π∂pHW ∂t

• dq

⇒

t-DOMAIN CONSTITUENT

RHW (x, t) =

• if t < T HW (0)

∂2

t Q(t) (t)

∗ KHW (x, t) if t > T HW (0)

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Extensions of GRT

EXTENSIONS OF GENERALIZED-RAY THEORY

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GRT in bounded regions (1) An application of the Cagniard-deHoop method for solving initial-value problems in bounded regions⋆ PROBLEM SOLUTION u(x, t) = uF(x, t) + uG(x, t) uF(x, t) is the fundamental (CdH) solution uG(x, t) is the secondary (image) solution

• based on the symmetry properties of uF(x, t) :

u(x1, x2, x3, t) =

• m

(±1)muF(x1|x1;S + 2mW1, x2, x3, t) ±

• m

(±1)muF(x1|2mW1 − x1;S, x2, x3, t)

along x1 = {0, W1}

⋆M. ˇ Stumpf,“An application of the Cagniard-De Hoop technique for solving initial-boundary value problems in bounded regions,” Q. J. Mech. Appl. Math., vol. 66, no. 2, pp. 185–197, Feb. 2013.

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GRT in bounded regions (2) Parallel-plate waveguide†

2 4 6 8 10

• 0.1

0.1 0.2 0.3 0.4 ct/h H[n]

2 (x1, x3, t)/H2;ref

x1/h = 1.5 x3/h = 0.6 w/h = 1

p0 p1 p2 p3 p4

1 2 3 4 5 0.2 0.4 0.6 0.8 1 c0t/h V0(t)/Vmax h/w = 1

c0tw/w = 0.9236 c0tw/w = 0.1847

2 4 6 8 10

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 ct/h H[n]

2 (x1, x3, t)/H2;ref

x1/h = 1.5 x3/h = 0.6 w/h = 1

n = 0 n = 1 n = 2 n = 3 n = 4

D {ǫ, µ}

PEC plane PEC plane

w h E1 = V0(t)/w x1 x3

O

× P

MODAL SOLUTION RAY SOLUTION

†M. ˇ Stumpf, Z. Raida, “Pulsed electromagnetic waves between parallel plates: The modal-expansion and generalized- ray approach,” IEEE AP Mag., 2015 (in print).

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GRT in bounded regions (3) Rectangular waveguide

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 ct/W i(t)/imax ctw/W = 0.5 ν = 2

D {ǫ, µ}

W

i(t)

x3 x2 x1

O

x1/W x3/W 0.5 1 1.5 2 0.5 1

E2(x1, x3, t) × 10−3 [V/m] −5.5 −2.8 −1.4 1.4 2.8 5.5

a

x1/W x3/W 0.5 1 1.5 2 0.5 1

E2(x1, x3, t) × 10−3 [V/m] −5.5 −2.8 −1.4 1.4 2.8 5.5

b

a : c0t/W = 0.50 b : c0t/W = 0.75

x1/W x3/W 0.5 1 1.5 2 0.5 1

E2(x1, x3, t) × 10−3 [V/m] −5.5 −2.8 −1.4 1.4 2.8 5.5

a

x1/W x3/W 0.5 1 1.5 2 0.5 1

E2(x1, x3, t) × 10−3 [V/m] −5.5 −2.8 −1.4 1.4 2.8 5.5

b

a : c0t/W = 1.0 b : c0t/W = 2.0

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CdH method for bounded regions (4) Stratified acoustic waveguide‡

x3;2 x3;3 x3;S x3;5 x3;ND D1 D2 D3 DS D5 DND {c2, ρ2} {c3, ρ3} {cS, ρS} {c5, ρ5}

W x1 x3 x2

O

source

×

(1) CAGNIARD-deHOOP METHOD

⇓

‘FUNDAMENTAL’ SOLUTION

⇓

(2) METHOD OF IMAGES

⇓

‘TOTAL’ SOLUTION

‡M. ˇ Stumpf, B. Nilsson, “Pulsed acoustic field radiation in a laterally bounded layered fluid,”

• J. Eng. Math., vol. 87, no. 1, pp. 99-109, Aug. 2014.

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Time-domain plasmonics (1) Pulsed EM field response of a plasmonic sheet∗

• Experimental setup

x1 x3 h h × sources probe

plasma frequency ωP collision frequency νC GOLD - Au SILVER - Ag 8.55 [eV] 0.0184 [eV] 9.60 [eV] 0.0228 [eV]

∗M. ˇ Stumpf, G. A. E. Vandenbosch, “Line-source excited impulsive EM field response of thin plasmonic metal films,” Photonics Nanostr.-Fundam. Appl., vol. 11, no. 3, pp. 253–260, Aug. 2013.

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Time-domain plasmonics (2) Pulsed EM field response of a plasmonic sheet on a dielectric support≀

• Problem configuration

x1 x3 d

line source

h

{σ(t), ǫ0, µ0} D0 {ǫ0, µ0} D1 {ǫ1, µ0} vacuum dielectric thin metallic film

⇒ NO TRUE SURFACE WAVES!

≀M. ˇ Stumpf, G. A. E. Vandenbosch, “Impulsive electromagnetic response of thin plasmonic metal sheets,” Radio Science, vol. 49, no. 8, pp. 689–697, August 2014.

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Buried antenna problem (1) Pulsed EM field transfer between subsurface antennas (1)

• Problem configuration

D0 {ǫ0, µ0} D1 {ǫ1, µ1, σ}

× O i3 i2 i1 × LR × LT

FARADAY’S LAW ⇒ OPEN-CIRCUIT VOLTAGE V G(t) ≃ −µ1∂t[Hi(xC, t) + Hr(xC, t)] · AR

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Buried antenna problem (2) Pulsed EM field transfer between subsurface antennas (2) (A) CAGNIARD-deHOOP METHOD

• LERCH’S UNIQUENESS THEOREM
• (B) GAVER-STEHFEST L−1

Re(s) Im(s)

s-plane

s0 s1 s2 · · ·

×

reflected-field time evolution

x1 (m) x3 (m) t/tw = 15 2 4 6 8 10 12 −6 −4 −2

V G;r/ max |V G;r|(x1, 8, x3, t) −1 −0.5 0.5 1

x1 (m) x3 (m) t/tw = 20 2 4 6 8 10 12 −6 −4 −2

V G;r/ max |V G;r|(x1, 8, x3, t) −1 −0.5 0.5 1

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TD Radar Imaging

TIME-DOMAIN RADAR IMAGING

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Time-domain RADAR imaging (1) Time-domain object’s shape reconstruction⊙ (1) SCATTERED FIELD FAR-FIELD BEHAVIOR

{As, Es, Hs} (|x|, ξ, t) = {As;∞, Es;∞, Hs;∞} (ξ, t − |x|/c0) 4π|x| ×

• 1 + O(|x|−1)
• as

|x| → ∞

with

Es;∞ = −µ0 [∂tAs;∞ − ξ(ξ · ∂tAs;∞)] Hs;∞ = c−1

0 ξ × ∂tAs;∞

“SLANT-STACK TRANSFORMATION”

As;∞(ξ, t) =

• ∂D

∂J s(x′, t + ξ · x′/c0)dA(x′) +

• D

J s(x′, t + ξ · x′/c0)dV (x′)

⊙M. ˇ Stumpf, “Radar imaging of impenetrable and penetrable targets from finite-duration pulsed signatures,” IEEE Trans. AP, vol. 62, no. 6, pp. 3035–3042, June 2014.

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Time-domain RADAR imaging (2) Time-domain object’s shape reconstruction⊙ (2) IMPENETRABLE (PEC) TARGET PHYSICAL OPTICS

∂J s PO ≃ 2 ν × Hi

• n

∂DI ⊂ ∂D

⇒ BACK-SCATTERED & CO-POLARIZED FAR-FIELD

Es;∞(−β, t)

PO

≃ ∂te(t)

(t)

∗

• x′∈∂DI δ(ξ − c0t/2) β · ν(x′)dA(x′)

IF e(t) = RAMP PULSE

Es;∞

ramp(−β, t)|T −1

w PO

≃ − α (2/c0Tw) S(c0t/2)

∂D ν S(ξ) ξ ξA ξS ξB

s h a d

• w

× O i3 i2 i1

D

′ {ǫ0, µ0}

β

⊙M. ˇ Stumpf, “Radar imaging of impenetrable and penetrable targets from finite-duration pulsed signatures,” IEEE Trans. AP, vol. 62, no. 6, pp. 3035–3042, June 2014.

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Time-domain RADAR imaging (3) Time-domain object’s shape reconstruction⊙ (3) PENETRABLE (DIELECTRIC) TARGET BORN APPROXIMATION

J s

B

≃ (ǫ − ǫ0)∂tEi in D

⇒ BACK-SCATTERED & CO-POLARIZED FAR-FIELD

Es;∞(−β, t)

B

≃ − [(ǫr − 1)/2c0]∂2

t e(t) (t)

∗ ξB

ξ=ξA

δ(ξ − c0t/2)S(ξ)dξ

IF e(t) = RAMP PULSE

Es;∞

ramp(−β, t)|T −1

w B

≃ − α [(ǫr − 1)/2c0Tw] S(c0t/2)

∂D ν S(ξ) ξ ξA ξS ξB

s h a d

• w

× O i3 i2 i1

D

′ {ǫ0, µ0}

β

⊙M. ˇ Stumpf, “Radar imaging of impenetrable and penetrable targets from finite-duration pulsed signatures,” IEEE Trans. AP, vol. 62, no. 6, pp. 3035–3042, June 2014.

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Time-domain RADAR imaging (4) Time-domain object’s shape reconstruction⊙ (4)

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 t/Tw T(t|Tw)

⇒

1 2 3 4 −0.1 −0.05 0.05 0.1 0.15 t − |x|/c0 [ns] α · ES;t

∞ (−β, t)

[V]

Tw = 2.40 [ns] α = i1 β = i3

⇐

0.1 0.05 0.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35

x1 [m] x2 [m] x3 [m]

≃

0.05 0.1 0.15 0.05 0.1 0.15 0.1 0.2 0.3 0.4

x1 [m] x2 [m] x3 [m]

⊙M. ˇ Stumpf, “Radar imaging of impenetrable and penetrable targets from finite-duration pulsed signatures,” IEEE Trans. AP, vol. 62, no. 6, pp. 3035–3042, June 2014.

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Modeling of P/G Plates

TIME-DOMAIN MODELING OF P/G STRUCTURES

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Time-domain analysis of P/G structures (1) Time-domain analysis of P/G structures

• Problem configuration & motivation

× O i3 i2 i1

ν Ω ∂Ω

• building block of multilayered printed-circuit boards (PCBs)
• populated by switching digital devices (ICs)

⇒ Signal Integrity & Power Integrity issues (SI/PI) ⇒ radiated ElectroMagnetic Interference (EMI)

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Time-domain analysis of P/G structures (2) Rectangular P/G structures with relaxation△

• Problem configuration

× O i3 i2 i1

Ω L W d {κ(t), µ0}

△M. ˇ Stumpf, “Time-Domain analysis of rectangular power-ground structures with relaxation,” IEEE Trans. EMC, vol. 56, no. 5, pp. 1095–1102, October 2014.

Dielectric relaxation functions κ(t) diffusive type

κ(t) = ǫ0[ǫrδ(t) + (σ/ǫ0)H(t)]

Debije type

κ(t) = ǫ0{ǫ∞δ(t) + [(ǫr − ǫ∞)/τr] exp(−t/τr)H(t)}

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Time-domain analysis of P/G structures (3) Time-Domain Contour Integral Method (TD-CIM)⋄ (1)

• Reciprocity-based problem formulation

CONFIGURATION

× O i3 i2 i1

ν Ω ∂Ω

RECIPROCITY

Domain Ω ⊂ R2 Actual Field (A) Testing Field (B) Field State {E3, H1, H2} {EB

3 , HB 1 , HB 2 }

Material State {κ(t), µ0δ(t)} {κ(t), µ0δ(t)} Source State J3 ∂JB

3

⋄M. ˇ Stumpf, “The time-domain contour integral method - an approach to the analysis of double-plane circuits,” IEEE Trans. EMC, vol. 56, no. 2, pp. 367–374, April 2014.

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Time-domain analysis of P/G structures (4) Time-Domain Contour Integral Method (TD-CIM)⋄ (2) Global interaction quantity of the time convolution-type

1 2

• x∈∂Ω

E3(x, t)

(t)

∗ ∂JB

3 (x|xS, t)dl(x) −

• x∈∂Ω

E3(x, t)

(t)

∗ ν(x) · ∂J B(x|xS, t)dl(x) =

• x∈Ω

EB

3 (x|xS, t) (t)

∗ J3(x, t)dA(x) −

• x∈∂Ω

EB

3 (x|xS, t) (t)

∗ ν(x) · ∂J(x, t)dl(x)

EB

3 (x|xS, t) = −µ0∂t

• xT ∈∂Ω

G∞[r(x|xT), t]

(t)

∗ ∂JB

3 (xT|xS, t)dl(xT)

∂JB

κ (x|xS, t) = −∂κ

• xT ∈∂Ω

G∞[r(x|xT), t]

(t)

∗ ∂JB

3 (xT|xS, t)dl(xT)

⋄M. ˇ Stumpf, “The time-domain contour integral method - an approach to the analysis of double-plane circuits,” IEEE Trans. EMC, vol. 56, no. 2, pp. 367–374, April 2014.

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EM radiation of a P/G structure Pulsed EM radiation of a P/G structure⊲

× ×

× O i3 i2 i1 x ξ

D∞ {ǫ0, µ0} νT ΩT ∂ΩT

‘Slant-stack’ transformation

∂−1

t ET ∞(ξ, t) ≃ − c−1 0 ξ ×

• x∈∂ΩT VT(x, t + ξ · x/c0)τ T(x)dl(x)

⊲M. ˇ Stumpf, “Pulsed EM field radiation, mutual coupling, and reciprocity of thin planar antennas,” IEEE Trans. AP, vol. 62, no. 8, pp. 3943–3950, Aug. 2014.

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(Self-)Reciprocity of a P/G structure (Self-)Reciprocity of a P/G structure⊲

transmitting state

× ×

× O i3 i2 i1 ξ = −β

D∞ {ǫ0, µ0} ∂ΩR receiving state

× O i3 i2 i1 −β

D∞ {ǫ0, µ0} ∂ΩR

β ‘Time-derivative transmission/reception relation’ α · ET

∞(−β, t) ≃ VR(xS, t) provided that eI(t) = µ0∂tIT (t)

⊲M. ˇ Stumpf, “Pulsed EM field radiation, mutual coupling, and reciprocity of thin planar antennas,” IEEE Trans. AP, vol. 62, no. 8, pp. 3943–3950, Aug. 2014.

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Mutual EM coupling between P/G structures Mutual EM coupling between P/G structures⊲

transmitter DT ΩT ∂ΩT

× ×

receiver DR ΩR ∂ΩR D∞ {ǫ0, µ0}

× O i3 i2 i1

coupling path

Induced pulsed voltage

VR(xS, t)

(t)

∗ IB(t) ≃ −

• x∈∂ΩR VB(x|xS, t)

(t)

∗ τ R(x) · HI(x, t)dl(x)

⊲M. ˇ Stumpf, “Time-domain mutual coupling between power-ground structures,” in Proc. 2014 IEEE EMC Symp., Raleigh, NC, USA, Aug. 2014, pp. 240–243.

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The END Conclusions

There’s plenty of potentialities in employing the pulse shape! Thank you for your attention!