Digital Signal Processing Reference
In-Depth Information
outside the
x
-
y
plane. However, the relationship between the wavelength and
the structure sizes in practical systems allows us to assume that the waves are
propagating in TEM mode until very high frequencies. Measurements have con-
firmed that the TEM assumption remains valid to at least 50 GHz for typical
transmission-line structures used in contemporary digital designs. The validity of
the TEM assumption for transmission lines is discussed further in Chapter 3.
2.3.3 Time-Harmonic Fields
A simplification of Maxwell's equations can be made if the time variation is
assumed to be steady-state sinusoidal or time harmonic in nature. Although per-
fect sinusoidal waveforms are rarely encountered in digital design, the trapezoidal
digital pulses usually employed can be constructed from a series of sinusoidal
waveforms via the Fourier transform, making this general simplification partic-
ularly useful. Time-harmonic electromagnetic fields will be generated whenever
their charge and current sources also have densities that have a sinusoidal vari-
ation with time. Assuming that the sinusoidal sources are steady state permits
the assumption that both
E
and
B
also reach steady state and vary according to
cos
(ωt
+
θ
E
)
and cos
(ωt
+
θ
B
)
, where
ω
=
2
πf
and
θ
is the phase of either the
electric or the magnetic field.
Generally, a sinusoidal waveform can be represented as
cos
φ
+
j
sin
φ
=
e
jφ
(2-31)
so the sinusoidal form of a time-harmonic field will vary according to the complex
exponential factor
e
jωt
, which leads to a reduction of Maxwell's equations from
a function space and time to simply space:
E(x, y,z,t)
=
E(x,y,z)e
jωt
(2-32a)
B(x, y,z,t)
=
B(x,y,z)e
jωt
(2-32b)
Equations (2-32) allow Maxwell's equations to be rewritten as
∂ B(e
jωt
)
∂t
∇×
( Ee
jωt
)
+
=
0
∂( De
jωt
)
∂t
∇×
( He
jωt
)
=
Je
jωt
+
∇·
( De
jωt
)
=
ρe
jωt
∇·
( Be
jωt
)
=
0
The curl and the gradient affect only space-dependent functions, and the
e
jωt
is
operated on only by the partial time derivatives. Therefore, after canceling all
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