Understanding Trace-to-Trace Coupling - Understanding Signal Integrity - page 95

1.176E-10 − 5.306E-12 − 6.049E-13

(9.3b)

C =− 5.306E-12 1.180E-10 − 5.306E-12

− 6.049E-13 − 5.36E-12 1.176E-10

In this example, the self-inductance ( L 11 ) is 287.6 nH/m, the mutual induc-
tance from trace 1 to 2 ( L 12 ) is 35.64 nH/m, and the mutual inductance from trace
1 to 3 ( L 13 ) is 9.859 nH/m.
The total capacitance to ground on trace 1 ( C 11 ) is 117.6 pF/m when traces
2 and 3 are grounded. The mutual capacitance from trace 1 to 2 ( C 12 ) is − 5.306

pF/m; it is 0.6049 pF/m from trace 1 to 3 ( C 13 ). In the following sections we will

see the reasoning and benefits of showing the mutual capacitances as negative num-
bers. Here we notice that the total mutual capacitance is the sum of the two mutual
terms:

−

5.911 pF.
This makes intuitive sense. Because traces 1 and 2 are closer together than
traces 1 and 3, the mutual capacitance between traces 1 and 2 should be higher
than between traces 1 and 3, and since the self-capacitance includes all the mutual
capacitance, C 11 must be higher than the value of the mutual terms. This is further
explored in the Problems.
By examining these matrices, we notice that the mutual terms on either side
of the diagonal are the same. For instance, L 12 has the same value as L 21 (35.64
nH/m). This is reasonable because we would expect the coupling between traces
1 and 2 to be the same whether it is measured from traces 1 to 2, or from traces 2
to 1. SPICE-type simulators take advantage of this symmetry and require that only
half of the matrix be entered when creating transmission line models.

−

9.2.2 Why Do Some Field Solvers Report Capacitances with Negative
Numbers?
Field solvers commonly report the mutual capacitances as negative numbers, but
this is not a universal practice. The negative value for the mutual capacitance is
mathematically correct and is an artifact of the way in which charge is calculated
when the field solver computes the capacitance of a structure. Those interested in
these details are referred to [1-3].
Although the negative values are proper, their presence is a common source of
confusion because all of the hand calculations treat the mutual capacitance values
as positive numbers.
As shown in the Problems, calculations throughout this topic always take the
mutual capacitance as positive, even if the field solver reports them as negative.

9.3 How Does Switching Alter the Trace Inductance and Capacitance?

Figure 9.2 shows a trace flanked by neighboring traces. The aggressors switch at the
same time as the victim trace, and all of them have the same signal amplitude. This
implies that the voltage on each trace has the same rate of change. In practice load-
ing differences may make the signal rise times unequal, but to simplify the analysis
we will assume that they are the same.