BP1: Computational Electromagnetics Time Domain Modelling of Electromagnetic Field Propagation
via Wave Potentials
N. Georgieva Y. Rickard McMaster University McMaster University
Department of Electrical and Computer Engineering
McMaster University 1280 Main Street West
Hamilton, Ontario L8S 4K1, CANADA Tel: (905) 525 9140 / ext. 27141 Fax: (905) 523 4407 E-mail: [email protected], [email protected]
1
Generalised Theory of Vector Potentials in Time-Domain Electrodynamics Lossless inhomogeneous m∇ε ≠ 0, ∇µ ≠ 0, σ = 0, σm =
0=⋅∇
=⋅∇∂∂
−=×∇
+∂∂
=×∇
BD
tHE
JtEH
r
r
rr
rr
r
ρ
µ
ε
JAtE
tAE
AH
rrr
rr
rr
−
×∇×∇=
∂∂
⇔∇−∂∂
−=
×∇=
µε
Φ
µ
1
1⋅∇
⋅∇
×∇
×∇
H
E
r
r
∂
=
µ
edium 0:
0==
∂∂
=
−∂∂
−=
DB
tEH
Mt
HE
mr
r
rr
rr
r
ρ
ε
µ
Fr
×∇−1
MFt
HtF
rrr
r
−
×∇×∇=
∂
⇔∇−∂∂
−=
ε
Ψ
ε
1
2
Generalised Theory of Vector Potentials in Time-Domain Electrodynamics (cont’d)
JtAA
rr
rµµεµε
µµ =
∂∂
+Φ∂
∇+
×∇×∇ 2
21 Fr
µε
ε +
×∇×∇
1
⋅∇−=
∂Φ∂ At
r
µε11 =
∂Ψ∂t
Lorentz Gauge
0=∇µ and σ = 0: 0=∇ε a
JtAA
rr
rµµε −=
∂∂
−∇ 2
22 F
r−∇2
Homogeneous lossy medium, 0)( =∇ µε and σ ≠ 0
JtA
tAA
rrr
rµµσµε −=
∂∂
−∂∂
−∇ 2
22 F
rµε ∂−∇2
t∂Ψ∂µεA
tr
⋅−∇=Φ+∂Φ∂ µσµε
MtF
tr
r
εµεε =∂∂
+∂Ψ∂
∇ 2
2
⋅∇− F
r
εµ11
nd σm = 0:
MtF rr
εµε −=∂∂
2
2
, σm ≠ 0: r
rMtF
tF
mr
εµσ −=∂∂
−∂ 2
2
r
t∂
Fm ⋅−∇=Ψ+ µσ
3
EM Fields in Terms of Wave Potentials EM fields in terms of the magnetic vector potential A
r [3] 2τ
uAr
At
,tAE
AH
r
rr
rr
⋅∇−=∂∂
∇−∂∂
−=
×∇=
µεΦ
Φ
µ
1
1
Towards the new two-potential (two-scalar) model u r
1
2
21
1
2
2
1
12
τµ
τµ
ττµ
ττ
ττ
ττ
∂∂
−∂∂
=
∂∂
−∂∂
=
∂∂
−∂∂
=
u
u
u
Au
AH
uAAH
AAH
Fu E)uF(
tA
ττ
εr
r
−=×∇=∂∂ 1
1τ 2τArr
F
ME
10∂∂
−=∂∂
⇒=⋅∇t
Aµε
Φτ
r
10∂∂
−=∂∂
⇒=⋅∇t
Fµε
Ψτr
Fu Hut
FArr
=∇−∂∂
−=×∇ Ψµ τ1
1τ
2
1
2
2
1
1
τΦ
τΦ
Φ
ττ
ττ
∂∂
−∂∂
−=
∂∂
−∂∂
−=
∂∂
−∂∂
−=
tA
E
tA
E
utAE u
u
02
22 =
∂∂
−∇⇒tAA
uA u
uu µε
02
22 =
∂∂
−∇⇒tFF
uF u
uu µε
4
EM Fields in Terms of Wave Potentials (cont’d)
F;ufA A=rr
u
−∂∂
−= utfE AAr
( fH AA ×∇=
µ1r
The T
Governing equations f
τετµ
τµτε
F
A
f
f:u
+
∂∂
∂∂
∂∂
∂∂
11
11
11
11
ufF=1τ 2τ
ΨΦ ∇−∂∂
−=∇ ut
fH FFr
) ( )ufEu FF ×∇=
ε1r
Mu field The TEu field
or the wave potentials: inhomogeneous, lossy medium
µεσ
εµτετ
εεσ
µετµτ
uFmFF
F
uAAA
A
Mt
ftff
uuf
Jt
ftff
uuf
−=∂∂
−∂∂
−
∂∂
∂∂
−
∂∂
∂∂
−=∂∂
−∂∂
−
∂∂
∂∂
−
∂∂
∂∂
+
2
2
22
2
2
22
111
111
5
Sources and Boundary Conditions Transverse current sources, if present, have to be transformed
r r Dielectric interfaces
rr n
Fn
Fn
An
A
FFAA
)(n
)(n
)(n
)(n
)(n
)(n)(
n)(
n
∂∂
=∂∂
∂∂
=∂∂
==
2
2
1
1
2
2
1
1
2
2
1
121
1111εεεε
εε
nF
nF
nA
nA
FFAA)()()()(
)()()()(
∂∂
=∂
∂∂
∂=
∂∂
==2
2
1
1
21
2121
11 ττττ
ττττ
εε
rrrr
rr
tM
tJ
∂∂
−
=∂∂
τ
τ
r
Magner
=
=∂∂
F
nA
τ
τ
r
=∂∂
=
nF
A
τ
τ
r
r
Electri
µεµΨ
εµεΦ
τττ
τττ
Mfut
Jfut
F
A
r
+
∂∂
−∇=
∂∂
∇
+
∂∂
−∇=
∂∂
∇
11
11
[9]
)uJ(
)uM(
u
u
×∇=
×∇
ε
µ1
1
tic wall
0
00
00
=
=∂∂
n
n
F
nA
c wall
0
00
=∂∂
=
nF
A
n
n
6
Space-Time Finite-Difference Discretization in a Rectangular Mesh
Numerical Aspects of the Finite Difference Implementation – Advantages * The field is fully described only by the two wave potentials, fA and fF. * The wave potentials are decoupled except at discontinuities such as conducting edges, wedges and convex curvatures (depending on the boundary conditions). * The wave potentials are smoother functions of space in comparison with field quantities. * The (fA, fF) model provides CPU time improvement.
NUMBER OF FLOATING POINT OPERATIONS / CELL
FDTD TD-WP (fA, fF) multi 6 2sum 24 20total 30 22
zf tA
zf /ttF
2∆+
y
z x
2=∆∆
=tc
hq
7Examples/Verification Rectangular Waveguide Az ⇒ TMz modes Fz ⇒ TEz modes
Excitation: [5] )nsin()n(BHW)n(e ttt ⋅⋅= ω 2
⋅−
⋅+
⋅−= tttt n
Ncosan
Ncosan
Ncosaa)n(BHW 3222
3210πππ
a0 = 0.35875 a1 = 0.48829 a2 = 0.14128 a3 = 0 01168
time step frequency [GH Excitation pulse Spectrum of the ex
b
a yzx
FzAz
a = 3 cm b = 1.5 cm
.
z]
citation pulse
8
Rectangular Waveguide (cont’d) dominant mode wavelength and wave impedance
2 5
theory TD-WP
0
5
1 0
1 5
2 0
5 6 7 8 9 1 0
λ g (c
m)
0
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
2 5 0 0
5 6 7 8 9 1 0
theory TD-WP
frequency (GHz)
Z w (Ω
)
9
Right-Angle Waveguide Bend
The input and the output waveguides are identical (a = 3 cm, b = 1.5 cm)
Fz
Az
z xAx Fz z x
Ax
Ex
A better choice of potentials (only one scalar potential function)
y y
z y
x
E x∆h
(mV
)
component of the incident field, t = 620•∆t
10
Right-Angle Waveguide Bend (cont’d)
Ex component of the field at the bend, t = 1000•∆t
E x∆h
(mV
)
z y
x
Reflected and transmitted Ex component of the field, t = 1290•∆t
E x∆h
(mV
)
z y
x
11
Right-Angle Waveguide Bend (cont’d)
HP HFSS TD-WP
1.2
1.1
1.0
|S11
| and
|S21
|
0.9
0.8
0.7
0.6
0.5
0.4
0.3 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 frequency (GHz)
12
Microstrip Line (infinitesimally thin strip) Structure The potential Az at the strip plane, t = 600 ∆t
13
z y
x
A z/∆
h (A
/cm
) z
y
x FzAz
εr
w
h
w = 0.6 mmh = 0.6 mm εr = 9.6
The Ex field component (half a step below the strip plane), t = 600 ∆t
E x (V
/cm
)
z y
x
The Hy field component (half a step below the strip plane), t = 600 ∆t
z y
x
Hy (
A/c
m)
14
Dispersion Characteristics Relative guide wavelength λ0/λg= rε
3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 0
λ0/λg
formula of Hammerstad & Jensen [6] formula of Pramanick & Bhartia [7] Katehi and Alexopoulos (MoM) [8]
TDWP
100908070605040302010
frequency (GHz)
15
Microstrip Line (cont’d) Characteristic impedance
0
100 90 80 70 60 50 40
20
formula of Hammerstad & Jensen [6] formula of Pramanick & Bhartia [7] Katehi and Alexopoulos (MoM) [8]
TDWP
100806040
Z c (Ω
)
frequency (GHz)
16
References 1. James Clerk Maxwell, A Treatise on Electricity and Magnetism, Dover Publications,
Inc., New York, 1954, vol.2 2. Roger F. Harington, Time Harmonic Electromagnetic Fields, McGraw-Hill Book
Company, Inc., New York, 1961 3. N. Georgieva and E. Yamashita, “Time-Domain Vector-Potential Analysis of
Transmission Line Problems”, IEEE Trans. On Microwave Theory and Techn., vol. 46, No 4, pp. 404-410, April 1998
4. Allen Taflove, Computational Electromagnetics – The Finite-Difference Time-Domain Method, Artech House, 1995
5. F.J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform”, Proc. IEEE, vol. 66, No 1, pp. 51-83, Jan 1978
6. D.M. Pozar, Antenna Design Using Personal Computers, 1985 7. E. Hammerstadt and O. Jensen, “Accurate Models for Microstrip Computer-Aided
Design”, IEEE MTT-S Int. Microwave Symp. Digest, pp. 407-409, 1980 8. P.B. Katehi and N.G. Alexopoulos, “Frequency-Dependent Characteristics of
Microstrip Discontinuities in Millimeter-Wave Integrated Circuits”, IEEE Trans. On Microwave Theory and Techn., vol. 33, No 10, pp. 1029-1035, Oct 1985
9. Robert E. Collin, Field Theory of Guided Waves, IEEE Press, 1991
17
18
Appendix: Boundary Conditions (cont’d) The absorbing boundaries: use Liao extrapolation scheme (3rd order) [4]
211112 fDfDfD −=3221
2111
fffDfffD
−=−=
121110 fDfDf)x,t(ff ++==
)xx,tt(ff)xx,tt(ff
)xx,tt(ff
∆∆∆∆
∆∆
3322
3
2
1
−−=−−=
−−=