NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 217

∇× E c + jωµ H c = 0 (5-31)

∇× H c =

J

σ E c = (5-32)

E c and H c denote the field values inside the conductor. If it is assumed

where

that n is the normal vector pointing outward from the conductor surface and

z is the normal coordinate inward into the conductor, the gradient operator is
∇≈− n(∂/∂z) , reducing Maxwell's equations to

H c = j n × ∂ E c /∂z
µω

(5-33)

E c =− n × ∂ H c /∂z
σ

(5-34)

These equations can be solved to yield the fields inside the conductor. The first
step is to take the partial derivative of (5-34):

∂ E c
∂z =−

H c
∂z 2

∂ 2 1

σ n ×

(5-35)

Next is to take the cross product of (5-33) with the unit vector,

∂ E c
∂z

j
µω n ×

n × H c = n ×

so the vector identity from Appendix A can be used to simplify the math,

a × (b × c) = (a · c)b − (a · b)c

∂ E c
∂z

n − (n · n) ∂ E c
∂z

j
µω

n × H c = n ·

n · ∂ E c /∂z = where 0 because when (5-35) is substituted, the form becomes pro-

n × ∂ 2

H c /∂z 2 , which is zero. Furthermore,

portional to n · n · n = 1, yielding

∂ E c
∂z

j
µω

n × H c =−

Next, equation (5-35) is substituted for ∂ E c /∂z , yielding

1
σ n × ∂z 2

H c

∂ 2 j

µω

n × H c =

∂z 2

n × H c

∂ 2 j

µωσ

=