ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 284

L

∆ z

R

∆ z

C ∆ z

G ∆ z

(a)

1 2 N s

L ∆ z L ∆ z L ∆ z

R ∆ z R ∆ z R ∆ z

C
∆

z

C ∆ z C ∆ z

(b)

Figure 6-24 (a) Model for a differential element of a transmission line; (b) full model.

For this equivalent circuit, equation (6-42) calculates the shunt admittance by
adding a conductance term G :

Y shunt = G + jωC (6-42)

The formula for G can be derived by

ε = ε − j σ dielectric
ω

= ε − jε (6-15)

ε
ε

tan | δ |= (6-16b)

which show clearly that the dielectric losses will be proportional to ε and tan | δ | ,

and therefore G .
Furthermore, we know from Section 6.4.1 that the real and imaginary parts
of the dielectric permittivity must be related. Consequently, there must also be a
relationship between the conductance G and the capacitance C . If the dielectric
losses are treated as an equivalent conductivity, we can say that the dielectric
carries a current of

J

= σ dielectric E [equation (2-7)]. Equation (3-1) says that the

voltage between the signal conductor and the reference plane is v =− a

E · dl

and that the total current is calculated from (2-20) as i = S

J · ds . Therefore, in

circuit terms, the conductance G can be written

S

σ dielectric S

J · E ·

d s d s

i
v =

G =

l =

(6-43a)

− a − a

E · E · l

d d