The track that forks

How long a track may be and the heavier input that allows a longer track measured the longest single track a fast edge may cross before it overshoots, with its load at the far end; the load spread along the track spread that load along it. All three assumed a track with two ends. Real nets branch: a clock goes to two chips, a reset to four, an enable to a row of buffers, and the schematic draws each as a single node. The layout cannot. It must choose a topology — branch at the driver, branch partway along, or run one track past every input in turn — and the earlier essays named the question without measuring it: whether the longest branch is set by the trunk, by each branch’s own ring, or by the two branches’ rings interacting.

The setting is the earlier one: fifty-ohm track on FR-4, a ten-ohm driver, one-nanosecond edges, inputs of two picofarads, and one per cent of overshoot at any input as the limit. The network — trunk, fork, branches — is marched by its characteristics, each segment carrying its two waves and each junction settling its own voltage.

Where the fork goes #

Run a trunk from the driver to a fork, and two equal branches from the fork to an input each. Bisect the longest branch that keeps both inputs inside one per cent, for trunks from nothing to a millimetre.

Forked at the driver, each branch may be 2.33 millimetres. One track carrying both inputs at its end may be only 1.05 — so forking at the source more than doubles the distance to each input. Move the fork away from the driver and the branches shrink fast: 1.89 millimetres with 0.2 of trunk, 1.43 with 0.4, 1.01 with 0.6, 0.57 with 0.8, and 0.11 with a full millimetre. Every millimetre of trunk costs more than a millimetre of branch; with a millimetre of trunk the net reaches its inputs no further than a single track would have.

The reason the star wins is the driver’s mismatch. A ten-ohm driver into a fifty-ohm track is a source five times too strong, and every wave returning to it is reflected with a large negative coefficient; that reflection, arriving back at the far end, is what makes the overshoot. Two fifty-ohm branches leaving the driver together are, to the driver, a single twenty-five-ohm line, and a ten-ohm source is only two and a half times too strong for it. The returning waves are reflected less, the staircase at each input is gentler, and each branch can be longer before its first peak exceeds one per cent. A trunk undoes that: the driver sees a fifty-ohm trunk, as badly matched as a single track, and the trunk’s own inductance rings with both inputs’ capacitance at once.

The reach, and the copper laid #

Two numbers matter for a layout: how far from the driver the inputs may be, and how much copper the net uses.

The star reaches 2.33 millimetres to each input and lays 4.65 millimetres of copper. With a millimetre of trunk the reach falls to 1.11 and the copper to 1.22. A single track with both inputs at its end reaches 1.05, and a daisy chain — one input halfway along, one at the end — reaches 1.40 to its far input. The order is star, daisy chain, tee with a long trunk, single end, and the star’s lead over the next best is two-thirds again.

The copper is worth a sentence of its own, because layout tools and design reviews both treat less copper as better. The star lays nearly four times the copper of the long-trunk tee and reaches more than twice as far; measured by what the net is for — getting a clean edge to every input — the copper is the cheapest thing in it. Copper costs area and a little capacitance to neighbours, and both are second-order beside the factor of two in reach.

The daisy chain’s advantage over the single end is the previous essay’s spread load at its smallest: two taps rather than one, the load no longer all at the end. Its disadvantage against the star is that the driver still sees one fifty-ohm track. The star changes what the driver sees, and that is a larger effect than anything the spacing of the inputs along a single track can do.

Equal branches, or not at all #

A star of equal branches is a choice a layout can make deliberately, and it matters, because unequal branches are much worse than either branch alone would suggest.

Across 225 pairs of branch lengths, 73 keep both inputs inside one per cent, and the allowed region is a spur along the diagonal rather than a square. A 2.2-millimetre branch overshoots 0.91 per cent beside an equal partner and 1.7 per cent beside a short one — adding copper to the short branch, up to the long one’s length, improves the long branch. That is the reverse of the single-track intuition that less copper is always better, and it has a clean reason.

Equal branches return their reflections to the fork at the same instant and with the same shape, so the fork sees them as one line of half the impedance and nothing distinguishes the two branches from each other. Unequal branches return their reflections at different times: the short branch’s reflection arrives early and is passed into the long branch as a new wave, the long branch’s arrives late and is passed into the short one, and each input sees its own staircase with a second one superimposed out of step. The fork stops being a match for either branch. The layout rule that follows is the one high-speed clock distribution has used for decades — length-match the branches of a star — and the figure gives it a number: with two-millimetre branches, a mismatch of a millimetre costs the net its margin.

The two layouts, in time #

The waveforms make the difference between a star and a tee concrete.

Both inputs are two millimetres of copper from the driver in both layouts. Forked at the driver, each input rises with 0.77 per cent of overshoot. Forked after a millimetre of shared trunk, with a millimetre of branch beyond, each overshoots 1.6 per cent: the trunk carries both inputs’ charging current, its inductance rings with their combined four picofarads, and the ring is twice as large for the same total distance. The extra copper in the star — two millimetres of branch instead of one of trunk and two of branch — is what buys the difference.

More inputs, further reach #

The star’s advantage grows with the number of branches, which is the result least expected from a lumped point of view.

A star of two branches reaches 2.33 millimetres, of three 5.12, of four 11.85. A daisy chain of the same inputs reaches 1.40, 1.96 and 2.54; all inputs at one end, 1.05, 1.26 and 1.47. Every topology reaches further with more inputs, because more capacitance damps the ring the earlier essays measured, but the star’s reach grows fivefold from two inputs to four while the daisy chain’s grows by four-fifths and the single end’s by two-fifths.

The star’s branches are fifty-ohm lines in parallel at the driver: twenty-five ohms for two, 16.7 for three, 12.5 for four. The ten-ohm driver is matched more closely by each branch added, and at four branches the mismatch at the source is nearly gone. That is source termination by topology — the same effect the previous essay found when spread inputs pulled a single line’s impedance down to the driver’s. Five branches, at ten ohms in parallel, would match it exactly.

A junction of three lines #

The fork is a junction of three lines, and its behaviour at the instant a wave arrives is a small calculation worth doing once. A wave travelling along a fifty-ohm line reaches a point where two fifty-ohm lines leave. It sees them in parallel, twenty-five ohms, and is partly reflected with a coefficient of $(25 - 50)/(25 + 50) = -\tfrac13$ and partly transmitted, two-thirds of its voltage continuing into each branch. The receiver that is a branch measured exactly that junction on a bus, where an interior receiver’s stub is the third line, and found it presenting each line with half the impedance it expected.

In a star at the driver there is no incoming fifty-ohm line: the driver itself is the junction, and the twenty-five ohms are what it sees. With a trunk, the trunk’s wave meets the junction, a third of it is reflected back towards the driver, and the driver — a ten-ohm source — reflects that negatively in turn, so the trunk carries a second staircase between driver and fork superimposed on everything the branches send back. That is the mechanism behind the steep cost of trunk in the first figure: a trunk adds a junction mismatch to the source mismatch the star already had, and the two interact.

The same arithmetic says why equal branches are special. A wave returning from one branch meets the fork from the branch side: it sees the trunk and the other branch in parallel, is reflected by a third and transmitted two-thirds into each. If the other branch returns its own wave at the same instant, the two reflections superpose symmetrically and the fork behaves as if a single twenty-five-ohm line had returned a single wave. If not, each branch receives the other’s wave as an unscheduled arrival, and the unequal-branch map is the count of how much that costs.

The schematic node that is a network #

A schematic draws a net as a node: one name, one voltage, connected to everything on it at once. Kirchhoff’s own frequency put a number on when that is true — below the frequency at which a signal crosses the circuit in a negligible fraction of its period — and the staircase in time showed what replaces it above: waves, delays and reflections. A fork is where the schematic’s simplification hides the most, because the schematic has no way to say whether the net is a star, a tee or a chain, and the three differ by a factor of two in how far apart their pins may be.

That makes topology a specification a schematic cannot carry and a layout must. A designer handing a net with four loads to a layout tool with a note saying “keep it short” has specified the wrong thing: “route as a star of equal branches from the driver” is the instruction that buys the factor of two, and “keep it short” applies to the trunk, not the branches. Every model has an edge placed the lumped-element assumption among the four edges every lumped circuit runs on; the fork is the place on a board where that edge is crossed first, at the shortest length, and with the least warning from the schematic.

Series termination, which changes the answer #

Everything above is for a driver whose ten ohms are its own. A series resistor at the driver, chosen to match the line, removes the source mismatch the overshoot comes from, and the resistor at the wrong end measured what that costs on a single net: half the swing travels the line and the far end doubles it back to full, which is exactly right at an open end and wrong everywhere in between. A star of equal branches with a series resistor matched to its parallel impedance behaves the same way at each branch end — but only while the branches are equal, because the match is to the branches in parallel. A series-terminated star is therefore the most demanding of length matching of all the topologies here, and its tolerance to branch mismatch is the question this essay leaves with the most practical weight.

A clock net, worked #

Put the figures to a design. A clock buffer with a ten-ohm output and one-nanosecond edges must reach four flip-flops, each with two picofarads of input, scattered within a centimetre of it. A single route past all four as a daisy chain allows the furthest to be 2.54 millimetres away: not enough. All four on one track’s end, 1.47: worse. A star of four equal branches allows 11.85 millimetres to each, which covers the centimetre with margin, provided the four branches are routed to equal length — the unequal-branch map says a millimetre’s mismatch on two-millimetre branches costs the margin, and on twelve-millimetre branches the tolerance is proportionally larger but still tight.

If the four flip-flops cannot be reached by equal branches — one is near the buffer and three far away — the map says what to do, and it is counter-intuitive: lengthen the short branch to match. The extra copper costs nothing in overshoot and buys the symmetry the fork needs. Clock trees on chips are built this way, with deliberate meanders in the short branches, and the reason is this figure’s diagonal.

How the networks were solved #

Each net is a set of lossless fifty-ohm segments joined at nodes — the driver, the fork, each input — marched by its characteristics: every segment carries a forward and a backward wave delayed by its transit in whole steps, and at each node the arriving waves, the node’s capacitance and, at the driver, the source through its resistance settle the node voltage by the trapezoidal rule. The step is an eighth of the shortest segment’s transit, and the march runs long enough for six ring periods of the whole net’s inductance and capacitance. The single-track limits are checked against the earlier essays’ march, and the branch limits are bisected on the logarithm of the length. The unequal-branch map is the same march at every pair of lengths on a 0.2-millimetre grid; it is symmetric to rounding, as two interchangeable branches must be.

The driver is a voltage ramp behind a fixed resistance; the inputs are pure capacitances with no lead inductance and no clamping diodes; the copper has no loss and the fork is a point. A real fork is a pad and a via, with a few tenths of a picofarad of its own, which the pad the stub comes out of measured as a resonance at its own frequency; on two-millimetre branches it adds a small lumped load at the junction, which the map would show as a slight shift of its diagonal.

Still open: the fork’s own pad, a driver with series termination, and the star of unequal loads #

The pad at the fork. A via and pad at the junction add capacitance exactly where the branches’ reflections meet, and a small capacitance there is a low-pass on the waves passing from one branch into the other. Whether it helps the unequal case — by smearing the out-of-step reflection — or hurts the equal case — by adding a lumped load the driver must ring with — is a march with one more capacitance in it.

Series termination at the driver. A series resistor at the driver chosen to match the star’s parallel impedance — fifteen ohms for two branches — makes the driver a matched source for the star, and the earlier essays found a matched source removing the limit altogether for a single track. For a star the match holds only while the branches are equal, so the question becomes how much branch mismatch a series-terminated star tolerates, which is the unequal-branch map redrawn with a terminated source.

Branches with unequal loads. A clock net often drives inputs of different capacitance, and equal lengths then give unequal branches. Whether the right symmetry is equal length, equal delay to the input’s threshold, or equal ring frequency — and how much each buys — is the question that decides how a clock tree’s branches should be balanced when its leaves are not alike.