CROSSTALK - Advanced Signal Integrity for High-Speed Digital Design - page 150

Equations (4-14) and (4-15) show that the effective capacitance of the system as
seen by our input signals is changed by the mutual capacitance and is a function
of the switching pattern. In particular, the even-mode capacitance is decreased
relative to the total capacitance, by an amount equal to the mutual capacitance.
Correspondingly, the odd-mode capacitance is increased by the mutual capaci-
tance, giving us the relationship C even <C total <C odd , where the total capacitance
is C total + C M .

In developing a general expression for the capacitance, we write the equation
for the current flow in the capacitive elements as a function of capacitances and
input voltages in matrix form:
i 1
i 2

= C 0

C g + dv 1 /dt

dv 2 /dt

C M − C M

= (4-16)

− C M C g + C M

If we add a third line to our system, as shown in Figure 4-6, we can extend
equation (4-16):

i 1
i 2
i 3

C 11 − C 12 − C 13 dv 1 /dt

dv 2 /dt
dv 3 /dt

=

− C 21 C 22 − C 23 (4-17)

− C 31 − C 32 C 33

In equation (4-17), the diagonal elements C 11 , C 22 , and C 33 represent the total
capacitances on lines 1, 2, and 3, respectively. The total capacitance is the sum
of the capacitance to ground (e.g., C 1 g for line 1) plus the mutual capacitances
between lines. The mutual capacitances are represented by C ij , where i and j
correspond to the lines coupled by the mutual capacitance. In other words,

C ij

C i = C i +

j = i

As is the case with the inductance matrix, the capacitance matrix is symmetric
since the capacitance does not depend on the polarity of the electric field.
We can generalize equation (4-17) to handle n capacitively coupled lines,

i 1
i 2
.
i n

C 11 − C 12 ··· − C 1 n dv 1 /dt

dv 2 /dt
.
dv n /dt

=

− C 21 C 22 − C 2 n

(4-18)

. .

. . .

− C n 1 − C n 2 ··· C nn

where the C ii are the total capacitances and the C ij are the mutual capacitances,
with C ij = C ji . Remember that total capacitance of line i is equal to sum of the
capacitance to ground for line i and the mutual capacitances between line i and
all of the other lines in the system.
A feature of the capacitance matrix that immediately stands out is that the
off-diagonal entries are negative. Although this may seem counterintuitive, it is