Crosstalk between Two Pairs of Parallel Conductors

We realise that the self inductance of two identical inductors in parallel is half that of one inductor.
Thus, if the two pairs of lines are widely spaced so that M is negligible, then

The capacitance of two capacitors in parallel is double that of one capacitor. The characteristic impedance of two transmission lines in parallel is half that of one on its own. Thus,

are based around

, where L and Zo are the values for one isolated pair of wires AB^[1].

Inductance is defined as the amount of magnetic flux which threads a circuit when unit electric current flows through the circuit. Now if unit electric current flows in the Even Mode (Fig.62), that is, down lines A and P and back on lines B and Q, then only half of the current flows in an individual line A. This causes magnetic flux of quantity L/2 to thread circuit AA'B'B, and magnetic flux of quanitity M/2 to thread circuit PP'Q'Q. The other half current, flowing down PP' and back on QQ', causes magnetic flux of quantity M/2 to thread circuit AA'B'B. This argument shows that

where f_e is the function of the geometry of the cross section of the conductors (see The analogy between L, C and R). Similarly,

In Figure 62, AP=d and PQ=a.

ranges from L/2 for lines widely separated (becoming the L of two independent inductors in parallel) to L for lines which are very close such that M approaches L in value^[2]; coupling approaches unity. Generally, therefore,

Also from first principles, L, the flux generated in AB by current in AB results from integrating from r to a, while M, the flux generated in AB by current in PQ results from integrating from d to the diagonal formed by d and a.

From Figure 50 and Figure 51, and in the text, we see that the ratio of Maximum Fast Crosstalk to Signal, or FX/Signal, equals

As further justification for the last formula, let us again approach it via inductance.
Looking at Figure 62, we see that the self inductance of pair AB is (the magnetic flux linking between A and B caused by

₎ plus (the flux between A and B caused by

₎.
Each of these results from integrating from r, the radius of a wire, to d, the distance between A and B.
This gives us

.
Now the mutual inductance results from the magnetic flux which these same currents cause to link between P and Q.
In this case we integrate (twice) between the distance BQ (=d) and the diagonal BP

.
The result is

As shown previously (see The analogy between L, C and R), Z_o is analogous to L.
We can see why the formula for Maximum Fast Crosstalk results as calculated above.
It is merely the ratio of fluxes linked across from AB to PQ divided by the flux linked in AB by its own current.

Reference 15, page749 discusses the use of resistive paper to determine values for Zo (Fig.47),

.
Also, values for Graph 48 and Graph 49 can be determined directly by painting conductors onto resistive paper; putting a voltage between A and voltage plane, and measuring the resulting voltage drop between P and voltage plane^[4].

Graphs 47 , 48 and 49 were developed using resistive paper with the occasional spot check sending real pulses down real printed circuit boards (ref.15).

Here is an attempt to calculate FX. We try to make the situation in Graph 49 approximate to that in Figure 62 and Table 1(f), which we have calculated.

First we note that the major problem is for flux lines to exit from close to the conductor, where there is severe crowding. Therefore we assume that the ruling variable is the perimeter of the conductor. In Fig.49, this is 0.0228". However, the portion (0.0128") which faces air, although in no way inferior for magnetic flux, is less useful for electric flux by a factor of 4.5. Scale that section (arbitrarily) down by x3. Thus the effective (epoxy glass equivalent) perimeter is 0.0100 + 0.0128/3 = 0.0143". Now a circular conductor, as in Fig.62, with that circumference, has radius 0.0023". A reasonable value for distance between A and P is 0.015" to be equivalent to where d in Fig.49 is 0.010". A reasonable value for distance AB (=2h in Figs.44, 49), called a on p24 and in Fig. 62, is 0.015".

This agrees with the central point on the middle curve in Fig.49, where d=0.010", w=0.010" and FX/signal =slightly over 10%.

The important point to remember is that the logarithms of wildly different numbers are very similar. e.g.

10, 100, 100, are a very similar 1, 2, 3. Now in our exercises, only the

(also called

) of dimensions appear in calculations. So calculations based on crude approximations lead to remarkably accurate results.

^[1]As the separation between the pairs is reduced, the self inductance rises from L/2 towards L.

^[2]because all flux lines caused by current in one pair thread the other pair, so that M=L.

^[3]We are allowed to do this strange manoeuver,

taking the square root of top and bottom, because

log10,000/log100 = log100/log10! However, we

tend to lose the indication of the diagonal

(hypotenuse BP, Fig.62), in this mathematical

trickery, and it all starts to look like a function of

distances squared, which it is not. The physical

reality is better illustrated by the more awkward

formula

^[4]Actually, slice lines A and P horizontally through their middles, and paint B and Q, again sliced through their middles, at the bottom of a strip of resistive paper.