TOPICS IN HIGH-SPEED CHANNEL MODELING - Advanced Signal Integrity for High-Speed Digital Design - page 415

10.1.2 Generating a Tabular Dielectric Model

The real portion of the dielectric permittivity, as calculated with

ε
m 2

ln (ω 2 /ω)
ln ( 10 )

ε (ω) ≈ ε ∞ +

(6-30a)

− m 1

is used to determine the frequency-dependent capacitance C ( ω ) by dividing
C ( ω ref ) by the value of the dielectric permittivity calculated at the reference fre-
quency ε (ω ref ) and multiplying by the frequency-dependent value using equation
(10-1), where boldfold type indicates a matrix for the multiconductor case:

ε (ω)
ε (ω ref )

C(ω) = C(ω ref )

C (ω ref )ε (ω)ε − 1 (ω ref ) → C (ω) = (10-1)

To implement equation (10-1), the effective dielectric permittivity must be cal-
culated from the reference values. As detailed in equation (3-74), the effective
permittivity can be calculated by dividing the capacitance of each value in C
by the corresponding value in C air , which is calculated by setting the dielectric
permittivity to unity ( ε r = 1).

C ε r, eff
C ε r = 1

= ε r, eff (3-74)

As described in Chapter 3, the values in C air can be calculated directly from
the inductance using equation (3-46), assuming that the magnetic permeability is
unity ( µ r =

1), which is almost always the case because copper is usually the
metal of choice.

1
c 2 C ε r = 1

L = (3-46)

Equations (3-74) and (3-46) lead to equation (10-2), which is the value of the
dielectric permittivity at the reference frequency:

ε (ω ref ) = C (ω ref ) C − 1

air

= c 2 [ C (ω ref ) L (ω ref ) ] (10-2)

The frequency-dependent loss tangent is calculated as

ε
m 2 − m 1 − π/ 2

ε (ω) ≈ (6-30b)

ln ( 10 )