Differential Pairs and the Cost of Asymmetry

A differential pair carries information in the difference between two conductors. The receiver subtracts one from the other, so anything present equally on both is rejected. That rejection is the entire reason the pair exists: it makes the link immune to supply noise, ground bounce, and most coupled interference. The rejection is never perfect, and the quantity that degrades it most directly is the difference in the two conductors’ electrical lengths.

Two modes, two impedances, two velocities

Any pair of coupled conductors supports two orthogonal modes. In the odd mode the two conductors carry equal and opposite currents; in the even mode they carry equal currents in the same direction. A differential signal excites the odd mode. Any component common to both conductors excites the even mode.

The modes have different impedances because the field distributions differ. In the odd mode the conductors swing in opposite directions, so the mutual capacitance between them is charged to twice the voltage of either conductor and carries displacement current straight from one to the other; each conductor then behaves as though its capacitance to the reference were its own plus twice the mutual capacitance, and the impedance falls. In the even mode both conductors move together, no displacement current crosses the mutual capacitance, and each conductor sees only its own capacitance to the reference — a smaller capacitance, and therefore a higher impedance.

They also propagate at different velocities, and the reason is subtle. In an inhomogeneous medium — a microstrip pair with dielectric below and air above — the odd and even modes have different field distributions, and therefore different fractions of their energy in the dielectric. The mode with more field in air travels faster. The result is modal dispersion: a signal containing both modes separates into two arrivals, and the separation grows with length.

A stripline pair in a homogeneous dielectric has equal modal velocities, which is one of the reasons high-speed serial links are routed on inner layers. Even there, the modes differ in impedance, so a termination that matches one mode mismatches the other.

What skew does to a differential signal

Consider a pair driven with a clean differential step, but with one conductor longer than the other by an amount that delays its arrival by $\Delta t$. At the receiver, the differential voltage is the difference of the two arrivals. During the interval between them, one conductor has transitioned and the other has not, so the difference is at an intermediate level rather than at the full swing.

The effect on the eye is direct: the transition is spread over $\Delta t$, the eye closes horizontally by that amount, and the amplitude at the sampling point is reduced. If the skew is a large fraction of the unit interval, the eye closes completely.

The same skew also creates a common-mode component. While one conductor has transitioned and the other has not, the average of the two sits half a differential step above its quiescent level; once both have transitioned it returns. When the skew is comparable to the edge, the common-mode voltage therefore carries a pulse of roughly half the differential amplitude lasting $\Delta t$. That pulse is what radiates from the pair and what couples into adjacent structures.

The conversion is quantified by the ratio of the common-mode pulse to the differential step, which for a simple delay difference is roughly $\pi f \Delta t$ in the small-skew limit, where $f$ is the frequency of interest. At 5 GHz, 10 ps of skew produces a common-mode component of about 15% of the differential amplitude — large enough to matter for emissions even though the differential eye still looks acceptable.

Deriving a length tolerance

The useful way to set a length-matching rule is to start from the receiver’s skew budget and work backwards, rather than to apply a generic rule from a layout guide.

A receiver’s datasheet typically specifies a maximum intra-pair skew, or the interface standard does. A common figure for a multi-gigabit serial link is a few picoseconds; slower interfaces tolerate tens of picoseconds. That budget covers the package, the connector, and the board, so the board’s share is a fraction of it.

Propagation delay on an inner layer is about 6.5 ps/mm, so a 1 mm mismatch is 6.5 ps. A board allocation of 5 ps corresponds to about 0.8 mm of length difference. On an outer layer, at roughly 6.0 ps/mm, the same budget allows about 0.8 mm as well — the difference between layers matters less than the mismatch within a pair.

PS_PER_MM_STRIPLINE = 6.5
PS_PER_MM_MICROSTRIP = 6.0

def allowed_mismatch_mm(skew_budget_ps, ps_per_mm=PS_PER_MM_STRIPLINE):
    return skew_budget_ps / ps_per_mm

for budget in (2, 5, 10, 20):
    print(f"{budget:2d} ps budget -> {allowed_mismatch_mm(budget):.2f} mm on stripline")

The number that matters is not the mismatch in millimetres but the mismatch in time at the receiver, which includes the package and connector. Two pairs that each meet a 0.8 mm rule can still fail if one has its length compensation placed at the far end of the route and the other at the near end, because the compensation’s own shape adds a local impedance discontinuity whose effect depends on where it sits.

Where length matching is added and why the shape matters

The standard technique is to add length to the shorter conductor with a serpentine or a trombone. Both are meanders, and both are discontinuities: the parallel sections of the meander couple to each other, changing the local impedance, and the tight turns add capacitance.

A meander whose segments are closer together than a few times the trace width couples to itself. The coupled sections behave as a short coupled line with its own odd and even mode impedances, and the net effect is a section whose delay is not the delay of the same length of straight trace. That is why meanders are specified with a minimum spacing between segments, usually at least three trace widths, and why the accumulated length in a meander is not a reliable predictor of accumulated delay.

The placement matters as much as the shape. Compensating at the point where the mismatch occurs is better than accumulating it at one end, because the common-mode pulse generated by the skew is then present only over the short section between the mismatch and its compensation rather than over the whole route. For a pair that must escape a connector with a fixed pinout, the mismatch at the breakout is unavoidable; compensating immediately after the breakout limits the length over which the pair runs with an imbalance.

A related error is compensating the wrong thing. Length matching to within 0.1 mm on the board while the package’s own routing introduces 2 mm of mismatch inside the device is a common outcome when the package is treated as ideal. The interface’s total skew budget must include the package’s contribution, and the board’s tolerance should be set after that is known.

Asymmetry beyond length

Length is the obvious asymmetry and not the only one. A pair whose two conductors have different widths, different dielectric heights, or different distances to a plane also has an imbalance, and the imbalance shows up as mode conversion just as a length difference does.

The most common geometric asymmetry is a pair routed around a corner: the inner conductor’s path is shorter than the outer one’s, and the compensation is usually placed after the corner. The second most common is a pair whose return vias are placed asymmetrically at a layer change, so one conductor’s return path is longer than the other’s. Both produce a local common-mode component without any length mismatch in the routed trace.

A third is the presence of a nearby conductor. A pair routed close to a wide aggressor has one conductor closer to it than the other, so the coupled noise arrives unequally and becomes differential. The fix is orientation rather than spacing: route the pair so its centreline faces the aggressor, or move it away, as discussed in Crosstalk: Mutual Capacitance, Mutual Inductance, and Coupling Length.

Terminating a pair

A differential pair needs a termination that matches its odd-mode impedance, which is why the resistor across the pair at the receiver is $2 Z_{odd}$ — the differential impedance — rather than twice the single-ended value. For a pair with 50 Ω single-ended lines and moderate coupling, the differential impedance is usually in the 85 to 100 Ω range and the resistor is 100 Ω.

That resistor terminates the odd mode. The even mode is left open at the far end unless a separate path to ground is provided, and the common-mode energy generated by any asymmetry reflects from the open and travels back. Some designs add a split termination — two resistors to a centre node, with a capacitor from that node to ground — which terminates the odd mode with the series combination and the even mode through the capacitor. The capacitor’s value sets the frequency above which the common-mode termination is effective, and its own inductance sets the frequency above which it is not.

The centre node must be bypassed to the pair’s reference plane, not to a distant ground, or the common-mode return path is the same long detour that Return Current Paths and Why Loop Area Sets Trace Inductance describes for single-ended nets.

Why pairs are routed tightly coupled

The coupling between the two conductors of a pair is what makes the odd-mode impedance lower than the single-ended impedance, and it is also what makes the pair tolerant of its environment. A tightly coupled pair has most of its field between its own conductors, so an external aggressor couples less into it and the pair radiates less.

Loosely coupled pairs — conductors spaced several widths apart — have field that extends further and therefore couple more to the environment, but they are more tolerant of impedance variation and easier to route through dense regions. The trade is between immunity and routability, and the choice is usually fixed by the stackup: the gap that gives the target differential impedance at the available width is the gap the design uses.

What must not vary is the gap along the route. A pair that spreads to pass a via or a keep-out changes its differential impedance over that section, and the resulting reflection is the same discontinuity discussed in Via Transitions, Stubs, and Backdrilling. When a spread is unavoidable, keeping it short relative to the edge length limits the damage.