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: talia@mcmaster.ca, rickardy@sprint.ca

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

r

MtF

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

7

Examples/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

−−=−−=

−−=