Crosstalk Equations

The single line characteristic impedance of a balanced pair of coupled transmission lines is Z_o. When the odd mode impedance is Z_oo and the even mode impedance is Z_oe, the reverse voltage crosstalk coefficient for both strong and weak coupling is:

where Z_diff = 2Z_oo is the differential impedance.

Another equation for weak coupling can be obtained as follows.

Using (1.1) we get:

Equation (1.7) is the new equation. Its accuracy depends on the approximations used in (1.4) and (1.5).

The next section examines the effect of these approximations.

Approximations

The accuracy of the two approximations, (1.4) and (1.6), is examined for two configurations, a stripline and a surface microstrip.

In both cases the independent lines have a characteristic impedance of 50 ohms. In both cases the substrate is FR4 with a dielectric constant of 4.2.

a)  Stripline.  Height of substrate is 500 mm, track thickness is 35 mm and the rectangular track width is 176.567 mm. The tracks are midway between the ground planes. 

 Figure 1 shows the values of Z_oo, Z_oe, Z_o^" = Å(Z_oeZ_oo), and (Z_oe + Z_oo)/2 as the separation of the tracks is varied.

Fig 1.

From Fig. 1, it can be seen that for wide separations (ie. weak coupling), that Z_o = Z_o^" = Å( Z_oeZ_oo), eqn. (1.4) and Z_o = (Z_oe + Z_oo)/2, eqn. (1.6). However for small separations, where the coupling is stronger, Z_o^" < Z_o and (Z_oe + Z_oo)/2 < Z_o. The difference is noticeable when the separation < 250 mm or a separation/width ratio < 1.4.

Figure 2 shows the variation of the reverse voltage crosstalk, calculated using different equations, as the separation of the tracks is varied.

In Fig. 2, the curve XE is calculated using the exact equation, (1.4), the curve XOE is calculated using the approximate equation, (1.7), and the curve XDF is calculated using the approximate equation (1.2).

From the figure it can be seen that the crosstalk calculated from the equation containing the differential impedance, (1.2), is always greater than the exact value and becomes significantly in error for small separations. The crosstalk, calculated using the approximate equation containing Z_oe and Z_oo only, is always slightly less than the exact value: there is good agreement with the exact value even at small separations.

The reason for this agreement is readily explained by reference to (1.5). There are no approximations in the numerator. However, Z_o in the large bracket of the denominator is an approximation using (1.4). Thus this “value” of “Z_o” will decrease with respect to the original value of Z_o as the separation decreases. This value is divided by the sum, Z_oe + Z_oo which also decreases with respect to the original value of Z_o. Thus the ratio of these two terms remains approximately constant and hence the value of the crosstalk, calculated using (1.7), is almost the same, for all separations, as that calculated using the exact value (1.1)

Fig. 2.

b) Surface Microstrip. Height of substrate is 500 mm, track thickness is 35 mm and the rectangular track width is 953.136 mm.

 Figure 3 shows the values of the impedances etc., similar to those shown in Fig.1

Fig. 3

Figure 4, Shows the reverse voltage crosstalk similar to that shown in Fig.2.

The same comments used in figs. 1 and 2 apply to Figs. 3 and 4. except that the difference between Z₀" and Z₀ occurs for separations less than 750 mm or a separation/width ratio of 0.79. This ratio is less than that for the stripline used here.

Fig. 4

Figure 5 shows the comparison between the reverse voltage crosstalk for the surface microstrip (XE(M) curve) and stripline (XE(S) curve), for a common separation/track width ratio.

This figure shows that for large ratios, where the coupling is very weak, the crosstalk is less for the stripline: for smaller ratios the crosstalk is larger for the stripline. The crosstalk is equal for a ratio of 3.2.

Fig. 5

Conclusions

The calculations show that the reverse voltage crosstalk coefficient calculated using only the even mode impedance, Z_oe, and the odd mode impedance, Z_oo, through (1.7), is slightly less than, but also a very good approximation to, the exact value, (1.1) for most practical separations where crosstalk needs to be estimated rather than the impedance of coupled lines. The calculation using the differential impedance and characteristic impedance, (1.2) gives too large an estimate of the crosstalk for small separations.

Where both the even and odd mode impedances are available, e.g. as in the Si8000m, then (1.7) is recommended.

Note: Eqn (1.1) uses Zo of the two identical (but uncoupled) feed lines to transmit to the coupled lines. This value of Z₀ (call it Z_feed) can be quite different from the uncoupled impedance, Z_uncoup, of the coupled lines. For instance (1.1) was derived to examine the effect when a 50 Ohm differential CITS probe (for example) is used to measure a differential impedance of 75 Ohms. In this latter case Z_uncoup will be different from 50 Ohms.

Eqn 1.7 was derived from 1.1 assuming that Z_feed = Z_uncoup and the approximations given. In most cases of crosstalk (rather than "full" coupling) the assumption is that this is undesired and the "feed" line is now an uncoupled line. Under these circumstances both 1.1 and 1.7 give almost identical results. When Z_uncoup is not known then you can use 1.7. If Z_uncoup (= Z₀) is known then 1.1 can be used which assures accuracy in all circumstances: