ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 269

0.025

0.020

0.015

0.010

0.005

0 10 20 30 40

Frequency, GHz

w 0

Figure 6-12 When a harmonic oscillator is being driven at a frequency much larger
than its natural or resonant frequency ( ω 0 <ω ), the losses vanish. This example plots the
loss generated from the Debye equation (6-28) with a pole at ω 0 = ω 2 i = 5 GHz.

the real and imaginary parts of the complex permittivity . Furthermore, if ω 0 ω ,

the loss tangent reduces to

1

ω
ω 0

ε ∞
ε ∞ + ε

tan | δ |≈ − (6-33)

which indicates an increase in the loss tangent with frequency. So for dielectrics
that can be described by the Debye equation, equation (6-32) shows that ε (and
subsequently ε r ) will decrease with frequency, and (6-33) shows that the loss
tangent will increase with frequency.
A question arises regarding the very high frequency response of the Debye
model. If the model is subjected to a very high frequency oscillatory forcing
function, where ω 0 <ω , there will be no time for the system to respond before
the forcing function has switched direction, so the losses will vanish as ω becomes
large, as is demonstrated by Figure 6-12, which has a pole at ω 0

=

5 GHz.

6.4.2 Mathematical Limits

For a transmission-line model to be physically consistent with the laws of
nature, certain mathematical limits must be obeyed. Presently, the vast majority
of engineers utilize models to design high-speed digital systems that employ
frequency-invariant values of the dielectric permittivity and loss tangent. This
assumption, although perfectly valid at low frequencies, induces amplitude
errors and phase miscalculations for digital data rates faster than 1 to 2 Gb/s
propagating on transmission lines because realistic dielectric materials have
frequency-dependent properties that must be modeled correctly. As data rates
increase, the spectral content of the digital pulse trains also increase and errors