A trace is a waveguide long before anyone intends it to be. Where that threshold lies, why the industry settled on fifty ohms when the physics points at two other numbers, and what an instrument is really showing you when it draws an impedance profile.
A wire, in the sense the schematic means, is a component with the same voltage everywhere along it at every instant. Nothing on a circuit board is that. A copper trace carries a disturbance at a finite speed, so while the disturbance is in transit the two ends genuinely hold different voltages, and the question is never whether that is true but whether it matters over the timescale you care about.
The timescale that matters is the rise time of the edge, not the clock frequency. A hundred-megahertz clock with a two-hundred-picosecond edge is a fast signal; it contains energy well past ten gigahertz whatever its repetition rate is. The useful comparison is between the rise time and the round trip — the time for the edge to reach the far end and for the reflection to come back — because if the reflection returns while the driver is still moving, the driver and the line settle together and the net behaves like a lumped circuit. If it returns afterwards, the far end has already made a decision based on a voltage that is about to change.
The rule most houses use is that a net may be treated as lumped while the round trip is below about a sixth of the rise time. The sixth is a choice, not a theorem: it corresponds to holding the reflection-induced error to a few per cent. What is not a choice is the scaling. The critical length is proportional to rise time and inversely proportional to the square root of the dielectric constant, so halving the edge rate halves the length at which the problem begins, regardless of which threshold anyone prefers.
Model a short length of the line as a series resistance and inductance with a shunt conductance and capacitance, all per unit length, and let that length go to zero. What comes out is a pair of coupled first-order equations relating the gradient of the voltage to the current and the gradient of the current to the voltage. These are the telegrapher's equations, and their sinusoidal steady-state solution is a wave travelling in each direction:
$$\frac{\partial V}{\partial z} = -(R + j\omega L)\,I,$$
$$\frac{\partial I}{\partial z} = -(G + j\omega C)\,V.$$
Eliminating one variable gives a wave equation whose solutions propagate with a complex propagation constant $\gamma = \alpha + j\beta = \sqrt{(R+j\omega L)(G+j\omega C)}$, where $\alpha$ is attenuation in nepers per metre and $\beta$ is phase shift in radians per metre. The ratio of voltage to current in a single travelling wave is the characteristic impedance
$$Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}.$$
Those four parameters are where the geometry enters. Nothing else about the cross-section matters to the signal: two lines with the same $R$, $L$, $G$ and $C$ per unit length behave identically however differently they are drawn. That is why the next three decks are, in effect, four separate investigations into how a cross-section and a material set those four numbers.
Set $R$ and $G$ to zero and the algebra collapses: $Z_0 = \sqrt{L/C}$, real and frequency-independent, and $\beta = \omega\sqrt{LC}$, so every frequency travels at the same speed and a pulse keeps its shape. This is the picture most people carry, and it is a good approximation at the frequencies where digital design used to live.
$R$ grows as $\sqrt{f}$ once current crowds into a skin thinner than the conductor, and $G$ grows roughly as $f$. Both are negligible at a hundred megahertz and neither is at ten gigahertz. Once they matter, $Z_0$ acquires an imaginary part, the velocity becomes frequency-dependent, and the pulse spreads. Deck 03 is about exactly that.
A fifty-ohm line made of copper measures a fraction of an ohm end to end with a meter, and yet it presents fifty ohms to a driver. The two statements are not in conflict because they are about different things. The meter measures the series resistance of the metal. The driver sees the ratio of voltage to current in the wave it has just launched, and that ratio is set by how much charge the line demands per unit length and how quickly the wavefront advances — that is, by $L$ and $C$, not by $R$.
The consequence worth internalising is that a line consumes energy from the driver before anything has reached the far end. The driver has no way of knowing what is at the other end; it simply charges the capacitance in front of the wavefront, and the current it must supply to do so is exactly $V/Z_0$. Only when the reflection returns does the far end's existence become known. A transmission line is, in this sense, a device that makes a distant load look like a fixed resistor for as long as the round trip lasts.
The characteristic impedance is a property of the line. The input impedance is a property of the line together with whatever terminates it, and equals the characteristic impedance only when the termination does. Conflating the two is how a half-wavelength of badly terminated cable ends up described as "fifty ohms" in a report, and it is the reason the impedance profile in slide 08 has to be read with care.
Fifty ohms is so universal that it is easy to assume it was derived. It was negotiated. Take a coaxial line whose outer radius is fixed — the connector shell and the hole in the bulkhead decide that — and vary the inner conductor. Two quantities have optima, and they are not in the same place.
Conductor loss per unit length goes as the surface resistance divided by the perimeter of each conductor, summed over both and divided by the impedance, which makes it proportional to $(x+1)/\ln x$ with $x = b/a$. Power handling is limited by breakdown at the inner conductor, where the field is strongest; holding the peak field fixed makes the transmissible power proportional to $\ln x / x^2$. Each has a single, well-defined optimum, and the two disagree by a factor of two and a half.
The other standard impedance is not a second compromise; it is one of the two optima taken neat. A cable with a mostly-air dielectric has its minimum attenuation within a whisker of seventy-five ohms, which is why the impedance that carries television signals down long runs of coaxial cable is the lowest-loss one and the impedance that carries test signals around a laboratory is the one that also handles power.
None of the reasoning on the previous slide applies to a trace on a circuit board. There is no breakdown limit anywhere near the operating voltage, the geometry is not free to vary because the layer thicknesses are set by the stackup, and minimum loss would call for a wide trace that the escape from a ball-grid array cannot accommodate. Boards ended up near fifty ohms single-ended largely because the instruments and connectors were already there.
What high-speed digital design actually standardised on is a differential impedance, and the usual values — eighty-five, ninety, one hundred ohms — come from the standards bodies rather than from physics. They represent a workable compromise between three pressures: a lower impedance needs wider traces and more layers, a higher impedance needs the traces further from the plane which makes them couple to each other, and the driver has to push the current the chosen impedance demands, which costs power on every lane.
| Interface | Differential impedance | What set it |
|---|---|---|
| PCI Express | 85 Ω | A deliberate reduction from 100 Ω to ease escape routing under dense packages |
| USB, Ethernet backplanes, most SerDes | 100 Ω | Inherited from twisted-pair practice and kept for continuity of test equipment |
| DDR and LPDDR command/address | 40–50 Ω single-ended | A parallel bus with a shared reference; see deck 15 |
| Laboratory instrumentation | 50 Ω single-ended | The coaxial compromise on the previous slide |
The practical point is that the number is a target with a tolerance, usually ten per cent, and that tolerance is consumed by things the designer does not control. Deck 05 computes one of them: giving the copper its real thickness rather than the zero thickness every closed-form formula assumes moves a hundred-ohm pair by nearly ten per cent on its own.
A reflection is not an exotic event. When a wave meets an impedance change, some of it continues and some returns, in the proportion set by the reflection coefficient $\Gamma = (Z_L - Z_0)/(Z_L + Z_0)$. There is nothing more to it than that. What makes reflections confusing is that the returned wave meets the source, reflects again, and comes back — so a single mismatch produces an unbounded series of arrivals, each smaller than the last by the product of the two reflection coefficients.
The line below is driven from a source of adjustable impedance into a high-impedance receiver, which is the usual arrangement inside a chip. Watch what the source impedance does to the shape of the settling rather than to its eventual value: every case converges on the same final voltage, and they differ entirely in how they get there.
A source impedance equal to the line impedance is the special case worth knowing: the wave reflects off the open receiver, returns to the source and is absorbed there, so the settling completes in exactly one round trip with no overshoot. That is series termination, and it is the cheapest correct answer for a point-to-point net.
Terminating a line means arranging that a wave arriving somewhere is absorbed rather than returned. There are only a few ways to do it and they differ mainly in where the power goes.
A resistor in series with the driver brings the source impedance up to the line impedance. The launched wave is half amplitude, doubles at the open receiver, and the return is absorbed at the source. It dissipates nothing in the steady state, which is why it dominated CMOS board design, but it works only for a single receiver at the far end: anything tapped off part-way sees the half-amplitude wave first.
A resistor to a reference at the far end absorbs the incident wave outright. It works with multiple loads and gives a clean full-amplitude edge, and it draws direct current whenever the line is driven — which for a hundred-ohm differential pair at four hundred millivolts is a few milliwatts per lane, multiplied by every lane in the system.
A pair of resistors to the two supplies sets both the termination impedance and the idle level; an added series capacitor removes the static current at the cost of a low-frequency droop that the data coding then has to respect. The run-length limits in line codes exist partly to keep that droop bounded.
What every modern SerDes actually uses. The termination is inside the receiver, trimmed against an external precision resistor, so it tracks process and temperature and adds no package parasitics of its own. It is the reason the interesting discontinuities in a modern channel are the vias and the connectors rather than the terminations.
Time-domain reflectometry launches a step into the channel and records what comes back. Because a reflection arrives at a time set by how far away the discontinuity is, and with an amplitude set by how severe it is, the returned waveform can be redrawn as impedance against distance. It is the most directly interpretable measurement in signal integrity and the easiest to over-read.
The conversion uses $Z = Z_0(1+\rho)/(1-\rho)$ applied sample by sample. That relation is exact for the first discontinuity and approximate thereafter, because everything past the first is being probed by a wave that has already been partly reflected and re-reflected. The traces below are computed from a cascade of transmission-line sections, so the model knows the true impedance of each and the plot shows what an instrument would report.
A discontinuity shorter than the rise time of the step does not get reported at its true impedance. The reflection from its start and the reflection from its end overlap and partly cancel, so the instrument shows something shallower and wider than the reality. The table below is the same thirty-eight-ohm section, of varying length, inserted into a fifty-ohm line and probed with a twenty-picosecond edge.
The practical reading is that TDR is trustworthy for features longer than roughly half the spatial extent of the edge and progressively optimistic below that. A via, which is a few tens of thousandths of an inch of discontinuity, is well inside the optimistic region for any realistic instrument, which is why vias are characterised in the frequency domain instead — the subject of deck 04.
A large discontinuity masks what lies behind it. The wave reaching the second feature has already been attenuated by the first, and the reflection from the second is attenuated again on the way back, so its apparent severity is reduced by the square of the first feature's transmission. The third channel in the previous slide's selector shows this: the second discontinuity is genuinely larger than it appears.
Johnson and Graham's most useful organising idea is that a transmission line does not have one behaviour but six, and which one applies depends on frequency, length and the conductor's own parameters. Each region has a boundary that can be computed from the line's per-unit-length parameters.
| Region | What dominates | What the designer sees |
|---|---|---|
| Lumped element | The line is shorter than the signal cares about | A single RC or pi network models it; reflections are invisible |
| RC | Series resistance against shunt capacitance | Diffusive spreading, delay growing as the square of length. On-chip interconnect lives here; boards mostly do not |
| LC | Inductance against capacitance, losses negligible | The textbook transmission line: constant impedance, constant velocity, undistorted pulses |
| Skin effect | Series resistance rising as $\sqrt{f}$ | Loss in decibels growing as $\sqrt{f}$; the first real dispersion |
| Dielectric loss | Shunt conductance rising as $f$ | Loss growing in proportion to frequency; the dominant term for modern channels |
| Waveguide | Geometry comparable with a wavelength | Non-TEM modes appear; the whole two-conductor picture stops applying |
A model that agrees with nothing but itself is not worth much, so the region boundaries computed here are checked against Johnson and Graham's own worked example — a hundred-ohm differential stripline, six-mil traces, half-ounce copper, twenty-mil plate separation, six hundred millimetres long on FR‑4. It is close enough to the backplane channel this series keeps returning to that agreement is worth having for two reasons rather than one.
On this line the skin-effect region begins at twenty-seven megahertz and the dielectric-loss region at four hundred and ninety-eight, a spread of less than a factor of twenty. The two mechanisms therefore overlap across most of the usable band, and the loss never settles into a clean square-root slope before the linear one takes over. Anyone fitting a measured channel to a single power law is fitting the blend rather than either mechanism, which is the subject of deck 03.
Most of the decks that follow are worked against one channel: the twenty-eight-point-eight inches of differential stripline across two line cards and an eighteen-inch backplane that SerDes_Equalisation characterises and equalises. Using the same channel throughout means the numbers in one deck can be set against the numbers in another without an argument about whether the comparison is fair.
In the vocabulary of this deck, that channel is firmly in the overlap of the skin-effect and dielectric-loss regions across its whole band, it is many wavelengths long at Nyquist so the lumped and RC pictures are both useless, and it is comfortably below the onset of waveguide behaviour — a hundred and forty-two gigahertz on Johnson's example, and similar here — so two-conductor transmission-line theory remains valid throughout. Everything this series computes about it is within the region where the equations on slide 02 hold.
Deck 02 asks where the return current in these equations physically flows, because $L$ and $C$ per unit length are properties of a loop rather than of a trace.
Deck 03 takes $R$ and $G$ seriously, which turns the clean travelling wave of slide 02 into the smeared pulse that makes equalisation necessary.
Matrix Methods in Network Parameters is the companion for the S-, Z- and Y-parameter machinery this series uses to cascade these sections together.
| Quantity | Expression | Worth remembering |
|---|---|---|
| Propagation constant | $\gamma=\sqrt{(R+j\omega L)(G+j\omega C)}$ | Real part attenuates, imaginary part delays |
| Characteristic impedance | $Z_0=\sqrt{(R+j\omega L)/(G+j\omega C)}$ | Tends to $\sqrt{L/C}$ once the line is inductive rather than resistive |
| Propagation delay | $t_{pd}=\sqrt{\varepsilon_{\text{eff}}}/c$ | About 170 ps/inch in stripline on a Dk of 4; about 140–150 ps/inch in microstrip, because half the field is in air |
| Reflection coefficient | $\Gamma=(Z_L-Z_0)/(Z_L+Z_0)$ | $+1$ open, $-1$ short, $0$ matched |
| Critical length | $\ell_{\text{crit}}\approx v\,t_r/12$ | Round trip below a sixth of the rise time; a convention, but the scaling is not |
| Minimum-loss coax | $b/a=3.591$ | 76.7 Ω in air — hence 75 Ω cable |
| Maximum-power coax | $b/a=\sqrt{e}=1.649$ | 30 Ω in air; the geometric mean with the above is 48 Ω |
| TDR conversion | $Z=Z_0(1+\rho)/(1-\rho)$ | Exact for the first discontinuity only |
Deck 1 of eleven in Signal Integrity & High-Speed Digital Design. Every figure on this page is computed by si_models/deck01 and embedded as data; nothing is typed in by hand.
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