TOPICS IN HIGH-SPEED CHANNEL MODELING - Advanced Signal Integrity for High-Speed Digital Design - page 437

Since it is assumed that w h , the variation in the y -direction is assumed
constant. Therefore, the variation with y drops out of (3-48), and the general
solution is given by

(x,y) = C 1 + C 2 x (10-13)

The electric field is calculated using the electrostatic potential shown as discussed
in Section 2.4.1.

E =−∇ (x, y) =− a x v s
h

(10-14)

Therefore, the electric field propagating in the z -direction is

=− a x v s

E(x, y,z) = a x Ee − jβz

h e − jβz

(10-15)

where the propagation constant is as defined in Section 2.3.4:

2 πf √ µε = ω √ µε β = rad / m (10-16)

The magnetic field is calculated by dividing (10-15) by the intrinsic impedance
of the waveguide as described in Section 2.3.4.

µ
ε

η ≡ (10-17)

1
η

v s
h e − jβz

H(x,y,z) = a y

(10-18)

The propagation velocity is calculated from

1
√ µ r µ 0 ε r ε 0

c
√ µ r ε r

ν p = = m / s (2-52)

where µ r is almost always unity for practical digital designs.

REFERENCES

Hall, Stephen, Garrett Hall, and James McCall, 2000, High-Speed Digital System Design ,
Wiley-Interscience, New York.
Hall, Stephen, Steven G. Pytel, Paul G. Huray, Daniel Hua, Anusha Moonshiram,
Gary A. Brist, and Edin Sijercic, 2007, Multi-GHz causal transmission line mod-
eling using a 3-D hemispherical surface roughness approach, IEEE Transactions on
Microwave Theory and Techniques , vol. 55, no. 12, Dec.
Liang, Tao, Stephen Hall, Howard Heck, and Gary Brist, 2006, A practical method for
modeling PCB transmission lines with conductor surface roughness and wideband
dielectric properties, presented at IEEE MTT-S International, June, Microwave Sym-
posium Digest , pp. 1780-1783.