ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 24

electrical properties, such as crosstalk, reflections, standing waves, and differ-
ent modes of propagation in multiconductor systems (e.g., a bus). We begin by
manipulating Faraday's and Ampere's laws using some useful vector identities:

∂ B
∂t =

∇× E + 0 ( Faraday's law ) (2-1)

∂ D
∂t

∇× H = J + ( Ampere's law ) (2-2)

Taking the curl of (2-1) produces

∂ B
∂t

∇× ( ∇× E) =−∇×

µ r µ 0 H [from (2-8)], the equation above can be written in terms of
the electric field by substituting (2-2) into the right-hand part:

B

Since =

∂ B
∂t =−

∂( ∇× µ H)
∂t

∂ D
∂t

=− µ ∂
∂t

∇× ( ∇× E) =−∇×

J +

where µ = µ r µ 0 .
If it is assumed that the region of wave propagation is source-free, the current
density

J is zero. Combining equations (2-6) and (2-9) yields the relation D =

ε r ε 0 E = ε E and allows the equation to be expressed only in terms of

E :

E
∂t 2
The formula can be simplified further by using the following vector identity (see
Appendix A):

∇× ( ∇× E) =− µε ∂ 2

∇× ( ∇× E) =∇ ( ∇· E) −∇

2

E

Since we have assumed a source-free medium, the charge density is zero ( ρ = 0),

∇· E =

Gauss's law reduces to 0, yielding equation (2-27), which is known as

the wave equation for the electric field :

E
∂t 2

E − µε ∂ 2

2

∇ = 0 (2-27)

Using the identical technique, the wave equation for the magnetic field can be
derived:

H
∂t 2

H − εµ ∂ 2

2

∇ = 0 (2-28)

Note that equations (2-27) and (2-28) are similar except that the order of mul-
tiplication of µε is reversed. The order of multiplication for this derivation was
preserved because it will become important when using matrices to calculate the
solution for waves propagating on multiple transmission lines in Chapter 4.