Differential and Common Modes
SPICE doesn't inherently understand the differences between and impacts of differential and common modes. If a pair of edge-coupled traces turns a corner (Figure 7.4, below), the trace on the outside edge will travel farther than the one on the inside edge.

This causes a conversion of some of the differential energy into common-mode energy. This can be partially modeled by adding a section to the transmission line at the point in the model where the bend takes place, but this is only an approximation. In the real physical world, the subsequent trace will radiate energy much worse than did the section where the signals were matched. SPICE will not model this effect.

On the other hand, the procedure of adding a section of transmission line for the corner is valid in that the SPICE simulation will likely accurately model the increased crosstalk caused by the added common mode. So it is recommended that this procedure is used.

Recall that a short segment of transmission line often has a disproportionate impact on simulation time. Where segments are short, as this would be, it is much preferable to model the segment as an equivalent L C section. The equations to do this were presented earlier and will be again later. In this short segment, you can ignore loss.

With any instance where an edge-coupled differential pair is to make a bend, add a small segment of trace to the inside trace to account for the phase shift caused by the bend. It is true that in broadside-coupled traces this would not be necessary, but layer-to-layer registration is typically too poorly controlled to make broadside practical in commercial printed circuit boards.

Figure 7.4. Detail of a Corner

At lower frequencies, it was acceptable to equalize line lengths by matching the total length of the net to the required value. At microwave frequencies with differential signaling, this procedure is no longer adequate. A definition is needed to describe the requirements.

Define a feature as anything in the signal path that is not a simple, straight, isolated, plain old differential pair of traces. Look at Figure 7.5 below for examples of features in a typical layout. In traversing a link between two pieces of silicon, there will usually be numerous features, such as packages, corners, vias, perhaps a connector, maybe a passive component like a capacitor.

For purposes of this discussion, all these are features. The traces between features are defined as segments. So an interconnect consists of segments separated by features.

With these definitions, the following statement is made: Within a differential pair, trace lengths are to be adjusted or equalized to maintain precisely complementary phase alignment, also called balancing, of the signals on a segment-by-segment basis.

It is not adequate simply to match physical lengths at the end of the link; it is best done segment by segment. This ideal is not always practical in real layouts. Yet, even such a thing as a 90-degree bend in the traces followed a half- inch later by a complementary bend produces significant and easily measurable impact on the signal.

Figure 7.5. Examples of Features

The reason that this careful matching at the segment level is desirable is that now segment lengths are significant compared to wavelengths. The more significant portion of a wavelength means that these segments can become much better antennas radiating the common-mode signal.

Similarly, the higher frequency produces much better coupling of the common mode into the planar waveguides produced by the reference planes and results in both additional radiation and coupling to the resonant modes of that waveguide.

As will be detailed later, the common-mode portion of the signal produces crosstalk typically an order-of-magnitude worse than does the differential portion. The simplest way of viewing all this is that common mode is "bad" and differential mode is "good." Avoid making or transporting common mode.

Again, if a small section of transmission line is added to account for each corner, there is an impact on simulation time. Simulation goes much faster if you use an LC equivalent rather than an actual transmission-line equivalent.

The error will be small as long as the length of a segment that is represented by a single L-C segment is less than "one- tenth the wavelength of the highest frequency of interest in the circuit. This frequency will usually be about 1.5 times the data rate.

If the data rate is 2.5 gigabits per second, this frequency should be at least 3.75 gigahertz. Wavelength in FR4 at this frequency will be about 1.6 inches, so no segment greater than .16 inch in length should be modeled as a single L-C section for that data rate.

In most cases, a simple bend in a differential pair will not add this much trace, so there is no problem. The appropriate values for the elements of the L-C segment are easy to calculate. They derive simply from the two equations:

Note that stripline fields are totally immersed in the board material, so their effective dielectric constant equals the dielectric constant of the board material. This is not true with microstrip. The effective dielectric constant for microstrip will typically be somewhat less than the dielectric constant of the board because some of the field lines are in air.

That final equation presumes a non-magnetic material—usually a safe presumption in circuit boards. The trick is in selection of the units for the speed of light. The inductance and capacitance in the above formulae are always in terms of per unit length.

The unit length is established by the units used for the speed of light (c) in the above equation. Light travels at about 186,000 miles per second. If you use that value in these equations, the capacitance and inductance will be per mile. A more reasonable light-speed value might be 3*10^10 cm per second.

With a typical dielectric constant for FR4 of about four, the signal velocity would be about half the speed of light. In that case, for a specified characteristic impedance:

Recall the capacitance in these equations is capacitance per centimeter of trace. Substitute the capacitance back into the impedance equation to get the inductance. To get the inductance and capacitance values that are used in the SPICE model of the segment, multiply each by the length in centimeters of the desired segment.

It is usual to model such a transmission line segment either as two half-inductances with a capacitor to ground in the middle, or two half-capacitances with an inductor between. Three ways that a segment of transmission line can be implemented are shown in Figure 7.6 below.

Though the three act essentially identically at middle frequencies, they act very differently at very high frequencies. If you care whether the circuit presents an open or short at very high frequency, you choose which of the three on that basis. In a SPICE simulation, these models run a lot faster than a short transmission line segment runs.

Figure 7.6. Three Ways for an L-C Section

One final comment on this procedure. Although it is simple enough that it could be easily programmed into a spreadsheet, the procedure isn't always quite as simple as has been shown here.

In the case of stripline, the dielectric constant is that of the circuit board material, so the procedure is simple. It is simple because the signal velocity can be derived from the relative dielectric constant. Namely, the velocity equals the speed of light divided by the square root of the relative dielectric constant.

In the case of microstrip, it isn't that easy. There, the dielectric constant that you need to use is the effective dielectric constant made up partly of air and partly of board.

In this case, the method might not work exactly as shown because the effective dielectric constant might not be known. One solution would be to let the simulator calculate the velocity for you and use that calculation with the two fundamental equations. Or, you can find one of the analytic approximation formulae for the values of L and C.

The point of all this is that SPICE simulations often go faster if discreet equivalents are substituted where there are very short sections of transmission line, particularly in the case of lossy transmission lines. Calculating the values of capacitance and inductance that simultaneously yield the right impedance and velocity is easy to do.