Interactive model assumptions and evidence

Compare the round trip against the edge.

Topology

One uniform line.

Reference planes

Both ends.

Equations

• 2*Td > tr/3
• tpd=84.7*sqrt(Dk_eff) ps/in

Units

ps, in

Validity

One driver, one receiver, uniform line, lossless.

Numerics

Exact arithmetic.

Known limitations

One driver, one receiver, uniform line, lossless.

Evidence

check-numbers.py — tpd for stripline and microstrip recomputed.

Show the two loss terms crossing, and where the crossover moves.

Topology

One uniform line; attenuation per unit length only.

Reference planes

Per inch — a rate, not a channel.

Equations

• a = kc*sqrt(f) + kd*f
• delta=sqrt(rho/(pi*f*mu))

Units

Hz, dB/in, um

Validity

A FIT, not a geometry solve. Roughness shown as shape, not a calibrated prediction.

Numerics

Direct evaluation.

Known limitations

A FIT, not a geometry solve. Roughness shown as shape, not a calibrated prediction.

Evidence

check-numbers.py — skin depth at 1 and 10 GHz recomputed.

Say where a coupled digital aggressor lands in a converter's output — as spurs, as offset, or as raised noise — and what that does to SNR, for a SAR and a delta-sigma converter fed the same disturbance.

Topology

Two converters, one aggressor. SAR: ideal bipolar sample-and-hold, then N bit trials spread over the first half of the sample period, each comparing the held input with a DAC scaled by the reference at that trial's own instant. Delta-sigma: discrete-time, second order, one bit, reference multiplying the feedback DAC, followed by sinc-cubed decimation of length 3·OSR−2. The aggressor is a 50%-duty trapezoid with linear edges through a single RC pole, in periodic steady state.

Reference planes

Injection at the converter input or at its reference, chosen explicitly. Observation is the decimated output record for the delta-sigma and the code record for the SAR; the modulator bitstream is separately observable. The aggressor is evaluated in the time domain at the exact instants the converter uses it — the sampling instant for the input path, each bit trial for the SAR reference, each modulator clock for the delta-sigma.

Equations

• aggressor: 50%-duty trapezoid, 10-90% rise/fall, full ramp = that / 0.8, through one pole at the path bandwidth
• flat coupling: v = G·y(t); capacitive: v = G·(f_bw/f_ref)·(x - y), i.e. G/(2*pi*f_ref)·dy/dt with G stated at f_ref = 1 MHz
• fold: f_out = |f - round(f/fs)·fs|
• SAR trial j: c <- c + 2^(N-1-j) if h >= (Vref + d(t_j))·(2c/2^N - 1)
• delta-sigma, for the implemented ordering: X1 = (U-V)/(1-z^-1), X2 = (X1-V)/(1-z^-1), V = z^-1·X2 + E, giving V = z^-1·U + (1-z^-1)^2·E because (1-z^-1)^2 + z^-1(2-z^-1) = 1
• sinc-cubed: three cascaded length-OSR moving averages; |H(f)| = |sin(pi·f·OSR/fmod)/(OSR·sin(pi·f/fmod))|^3
• SNR excludes the fundamental, input harmonics 2-6 and DC, each +/-4 bins; interference counts as noise

Units

Hz, seconds, volts, dBFS, dB, LSB, ppm

Validity

Ideal comparator, DAC, sample-and-hold and clocks. One periodic trapezoidal aggressor. Oversampling is capped at 256 so a run stays interactive. Every parameter is checked against a declared domain before a run — enums, a finite positive reference, whole-number resolution 6-24 and oversampling 8-1024, an input below Nyquist and at or below full scale, finite non-negative noise, swing, phase, ppm, coupling and bandwidth, a positive rise/fall time — and a value outside it is refused with its reason. A reference driven to zero or below is refused, and so is an overloaded loop.

Numerics

SAR 65,536 samples at 16 bits. Delta-sigma output records are chosen from the oversampling ratio to hold the cost near a million modulator samples: 16,384 outputs at OSR 32 and 64, 8,192 at 128, 4,096 at 256, after a warm-up of 4,096 + 8·OSR modulator samples plus a full decimator kernel. Coherent input: an odd number of cycles in a power-of-two record, kept two lobes clear of DC and Nyquist. 4-term Blackman-Harris window, lobes summed over +/-4 bins. Deterministic noise seed. One run is synchronous and cannot be cancelled part way: measured at 14 ms for the SAR and 38-89 ms for the delta-sigma, so a dragged control coalesces to one run.

Known limitations

No clock jitter or aperture uncertainty, no supply or ground bounce as a separate path, no DAC mismatch, no continuous-time or multi-bit modulator, no anti-alias filter beyond the single path pole. DNL and INL are named as mechanisms and are NOT measured: there is no static ramp or histogram test here, and THD is not a substitute for one. Interference that folds inside an excluded lobe is excluded with it and does not reduce the reported SNR, however large it is; diagnostics.masked names any such component and measurements.maskedCount counts them. The excluded bins carry no measurable noise of their own, so the measured average density stands in for them: exact for white noise, and otherwise bounded by the replaced fraction, which is under 0.2% of bins and so under 0.01 dB. The reference-ripple mixing shown is a property of this loop and does not transfer unchanged to a continuous-time or multi-bit modulator. Nothing here is a compliance measurement.

Evidence

check-models.js, eight named suites, 47 assertions, none of which re-runs the model's own formulas. The loop's transfer function is checked by inverting its claim: the double integral of (V - z^-1·U) must BE the quantization error, bounded by the reference plus the overload threshold and consistent with the decision that produced it, which holds at 7.73 V against a 42.5 V bound with no sign violations and fails at 130 V for any other delay. Spur levels come from the trapezoid's Fourier series through a least-squares tone fit to the output record, agreeing within 0.2 dB. Ideal SAR SNR is checked against 6.02N + 1.76 at 10, 12 and 16 bits; the delta-sigma against 15.05 dB per doubling and the closed-form 1-bit second-order limit. Folds are checked against a remainder-and-mirror construction. The decimator is checked against the properties of three cascaded boxcars. Scenario prose is guarded separately by tests/check-adc-integration.js (56 values) and the page's closed forms by check-numbers.py (21). Twelve planted faults in mutate.js, each required to fail the assertion that claims it.

Show a peak appearing between two capacitor banks, and what damps it.

Topology

Two RLC branches in parallel, plus the VRM branch.

Reference planes

One port at the load.

Equations

• Zpeak ~ L/(R*C)
• fSRF=1/(2*pi*sqrt(LC))

Units

Hz, ohm

Validity

Lumped; no plane modes.

Numerics

Direct complex evaluation; explicit no-peak state when none exists.

Known limitations

Lumped; no plane modes.

Evidence

check-models.js — PDN (anti-resonance exists and is damped by R; no phantom peak when flat).

Show the measured region and the extrapolated region as different things.

Topology

Dual-Dirac: two impulses separated by DJ, convolved with a Gaussian.

Reference planes

Receiver decision point.

Equations

• BER(t)=0.5[Q((t-DJ/2)/s)+Q((UI-DJ/2-t)/s)]
• TJ=DJpp+2*Q(BER)*RJrms

Units

UI, s

Validity

Dual-Dirac is a FIT, not a measurement of the deterministic width.

Numerics

erfc below x=3, asymptotic series above; explicit empty result when no interval meets the target.

Known limitations

Dual-Dirac is a FIT, not a measurement of the deterministic width.

Evidence

check-models.js — Bathtub (closed eye returns no interval; crossings match the analytic Q inverse).

Separate the three functions a clock-recovery loop defines: input jitter onto the recovered clock, the residual phase error at the decision, and the input jitter the receiver can withstand.

Topology

Second-order type-2 loop. No phase-detector model, no VCO model, no data pattern.

Reference planes

Phase at the receiver's decision instant, relative to the incoming data.

Equations

• H(s) = (2*zeta*wn*s + wn^2) / (s^2 + 2*zeta*wn*s + wn^2)
• residual = COMPLEX 1 - H(f), never 1 - |H(f)|
• tolerance = margin / |1 - H(f)|, capped by a declared tracking ceiling
• f3dB: u^2 - (2 + 4*zeta^2)*u - 1 = 0 with u = (w/wn)^2, so f3dB = fn*sqrt(root)
• steady-state sinusoid: recovered phase = H(f) times input phase; residual = input - recovered

Units

Hz, seconds, UI peak-to-peak, dimensionless transfer

Validity

A teaching loop with declared parameters: natural frequency, damping and sampling margin. The sinusoidal experiment is the steady-state small-signal response after settling. The low-frequency tracking ceiling is DECLARED, not derived — a real one comes from the phase detector's range and the VCO's tuning. fn is the NATURAL frequency and is reported separately from the closed-loop -3 dB bandwidth, which is 1.58x to 3.37x higher depending on damping.

Numerics

Direct complex evaluation on 340 log-spaced points from 1 kHz to 1 GHz; 181 analytic sinusoidal samples over three periods at the selected frequency.

Known limitations

Not a lock or acquisition transient, cycle-slip model, BER simulation, or compliance jitter-tolerance mask. A mask carries a BER target, a defined pattern, the phase detector's own characteristic and usually a measured low-frequency corner. This model has none of those, and the dashed part of the tolerance curve is an assumption drawn as one. No noise model: the panel makes no claim about how much of the loop's own reference noise reaches the recovered clock, because that needs a spectrum and an injection point and neither is here. The tracking ceiling is a DECLARED assumption, not a derived result, and where it binds the tolerance no longer scales with the sampling margin.

Evidence

check-models.js — CDR loop: the analytic asymptotes (|1-H| -> 1 far above the loop so tolerance equals the bare margin; |1-H| -> (f/fn)^2 far below), H(0) = 1 exactly, the complex complement distinguished from 1 - |H|, and type-2 peaking present at every damping. N3-1: the -3 dB bandwidth is checked against a closed form solved by hand from |H|^2 = 1/2 and, independently, against where the swept |H| crosses 1/sqrt(2) with log-interpolation between grid samples. The residual peak is asserted to EXCEED unity whenever the loop is underdamped, and the worst tolerance to fall below the sampling margin with it. Three assertions record that a fixed axis limit of 3, and a tolerance floor of half the margin, would clip the extrema the damping control exists to produce.

Show why near-end saturates and far-end accumulates.

Topology

Single coupled pair, uniform section, weak coupling.

Reference planes

Both ends of the victim.

Equations

• Kb=(Lm/L0+Cm/C0)/4
• Kfe proportional to (Lm/L0-Cm/C0)

Units

ps, mV

Validity

Weak coupling, ratios <= 0.30. One aggressor, not a bus.

Numerics

Closed form superposition over the coupled length.

Known limitations

Weak coupling, ratios <= 0.30. One aggressor, not a bus.

Evidence

check-models.js — Crosstalk (FEXT cancels when the ratios are equal; NEXT saturates past tr/2tpd).

Separate what each equaliser block can and cannot reach.

Topology

Shared channel impulse, then CTLE and/or DFE.

Reference planes

Receiver input and slicer input.

Equations

• H(s)=(1+s/wz)/((1+s/wp1)(1+s/wp2))
• DFE subtracts h[k]*a[n-k] for k>0

Units

UI, V

Validity

Ideal or decision-directed DFE as selected; no adaptation dynamics.

Numerics

Half-open tap window; CTLE normalised to unity peak gain.

Known limitations

Ideal or decision-directed DFE as selected; no adaptation dynamics.

Evidence

check-models.js — CTLE (causality, normalisation) and DFE (exact cancellation, window width).

Build an eye from bits and state what the run supports.

Topology

Shared channel impulse plus Gaussian jitter.

Reference planes

Receiver input.

Equations

• superposition of 858 symbols; rule of three for the evidence floor

Units

ps, mV

Validity

Fixed sampling phase; no adaptation. 858 symbols evidences 3.5e-3, not a compliance BER.

Numerics

Seeded LFSR and seeded Gaussians — reproducible.

Known limitations

Fixed sampling phase; no adaptation. 858 symbols evidences 3.5e-3, not a compliance BER.

Evidence

check-models.js — Evidence suite (13 assertions) guards the BER floor claim.

Compare four impairments on one consistent channel, as counterfactuals rather than a decomposition.

Topology

Two-term loss law, one through-section discontinuity, source and load mismatch, up to four FEXT aggressors.

Reference planes

Transmitter output to receiver input. No package.

Equations

• S21_disc=2/(2cos+j(r+1/r)sin)
• S11_in=S11_disc*e^-j2wTd1 + S21_disc^2*GL*H^2*e^-j2wTd
• echo = Gs.GL * H^2 * direct, delayed by 2*Td (first term of a geometric series in Gs.GL)

Units

s, V, ohm, dB

Validity

FIRST ORDER in the reflection coefficients: one source-load round trip is kept and the (Gs.GL)^2 terms and beyond are dropped. The relative size of what is dropped is |Gs.GL|, measured at 0.06% to 0.71% of the retained echo across the released presets and 6.3% at the controls' extremes — so it is accurate here, but it is an expansion and not a cascade. Single-ended; fixed CTLE+DFE, not adapting. For a network with every internal re-reflection included exactly, see labChannel, which cascades ABCD matrices instead.

Numerics

4096-point FFT, 32 samples/UI, 420 symbols.

Known limitations

An expansion truncated at first order, not an exact network. It also uses the two-term magnitude loss law with a minimum phase, not the causal material model that labChannel now uses, so its phase is reconstructed rather than derived.

Evidence

check-models.js — Networks and Impairment chain: the through-section loses at most 0.0481 dB and never notches, the echo is Gs.GL rather than GL^2, and the truncated series is bounded by |Gs.GL| of the retained term over the whole released range.

Show a channel's memory as specific earlier bits.

Topology

Shared channel impulse; cursor and neighbours.

Reference planes

Receiver sampling instant.

Equations

• y[n]=h0*a[n] + sum h[k]*a[n-k]

Units

UI, mV

Validity

Linear time-invariant channel; no equaliser unless stated.

Numerics

Shared kit channel; cursor at the single-bit peak.

Known limitations

Linear time-invariant channel; no equaliser unless stated.

Evidence

check-models.js — CTLE and DFE suites share this channel model.

Show how TJ depends on the BER you name.

Topology

As bathtub.

Reference planes

Receiver decision point.

Equations

• Q(BER) by bisection on the Gaussian tail integral

Units

ps, UI

Validity

Gaussian tails assumed; a periodic component is fitted badly.

Numerics

Same erfc treatment as the bathtub.

Known limitations

Gaussian tails assumed; a periodic component is fitted badly.

Evidence

check-models.js — Bathtub suite, shared model.

Show one network in six domains at once, and link a post-cursor back to the bit that caused it.

Topology

ABCD cascade: causal lossy line, optional mismatched section, optional open stub, causal lossy line. 50 ohm source and load.

Reference planes

Both ports referenced to a real 50 ohm. No package, no connector, no die. Transport delay de-embedded as an exact integer sample rotation and reported separately.

Equations

• z'(f)=Rs(f)(1+j)+jwL, y'(f)=jw*Cvac*eps_r(f)
• gamma=sqrt(z'y'), Zc=sqrt(z'/y')
• eps_r(f) = Djordjevic-Sarkar wideband Debye, decades 1e4 to 1e13
• A=D=cosh(gamma*l), B=Zc*sinh(gamma*l), C=sinh(gamma*l)/Zc
• S21=2/(A+B/Z0+C*Z0+D)

Units

Hz, s, ohm, V. Loss referenced to a FIXED 8 GHz belonging to the board, not to the symbol rate.

Validity

Single-ended, no crosstalk, CTLE only, ideal terminations. Loss is a budget at one frequency split between two mechanisms, not an arbitrary measured curve. Validity is per measurement, not per page: an impulse response that has not decayed inside its record marks the IMPULSE and PULSE panels and names them; it does not invalidate the S-parameters, and it is not used to infer the eye's accuracy. The eye's accuracy is measured directly instead - every released preset is within 0.35 mV of its value at four times the record.

Numerics

4096-point FFT at 32 samples/UI. Pre-wavefront energy below 1e-5 of the total and falling as the record is refined at fixed bandwidth. Propagation loss meets its budget to 1e-6 dB; |S21| differs from it by up to 5e-3 dB, which is the port mismatch of a lossy line whose Zc is not its nominal z. The tail diagnostic measures |h| at a quarter-record or more from the response PEAK. Measuring a fixed last eighth of the circular record instead reported 3.8462% on a matched lossless line with no reflector anywhere — the sidelobes of a band-limited fractional delay, falling as 1/n, wrapping in behind a peak that sits at sample 0. Impulse exports are dimensionless discrete convolution weights; normalized SBR exports are dimensionless too. The panel converts SBR weights to volts using its transmitted amplitude. Raw and active impulse records share dt and removed transport delay tdBulk; absolute display labels add tdBulk without changing the samples. The early view is a display window, and the full FFT record remains available. Frequency inspection selects the nearest native sample in linear Hz and links S21, S11 and group delay; exported group delay is in seconds. It does not interpolate wrapped phase or rebuild the model.

Known limitations

Not a field solver, not coupled-line, not correlated against measurement. The eye height is the extremal gap between the two transmitted populations over one PRBS7 period at the decision instant — not a percentile, not a BER contour, and not comparable with a compliance eye height.

Evidence

docs/channel-model.md is the contract; every clause has assertions. check-models.js — Causal line (44 assertions): lossless Zc and delay exact, loss budget exact, agreement with an independently derived S21 to 1e-12, pre-wavefront causality with measured convergence, and eps' monotone with eps'' flat. Lab B assembled (60 assertions): the arrival is retained for every preset and both extremes, one sampling phase across four views, populations labelled by transmitted symbol, an ideal channel delivering the full swing with zero errors, and S-parameters unchanged by a rate-only change. mutate.js: nine planted defects each caught by the assertion that claims it.

Say in amps which capacitor bank supplies the load at each frequency, and which two make each peak.

Topology

Five-stage ladder: VRM, bulk, board, package, die. Series R-L between stages, shunt R-L-C at each.

Reference planes

Board and die load nodes on one five-stage ladder; observe either node. Frequency branch-share panels retain unit die excitation.

Equations

• Zout_vrm=R(1+jf/fbw)
• ladder reduction from the VRM end
• M di/dt + R i + vC = -Br I - Bl dI/dt; dvC/dt = diag(1/C)i
• Y(f) v(f) = i(f); Zij = vi / ij
• rail deviation dVi = -sum_j Zij Ij for positive withdrawal

Units

Hz, ohm, A, V, s

Validity

Lumped five-stage RL/series-RLC network with zero initial incremental state. No plane cavity modes or spatial distribution. The browser transient is causal; the finite observation interval can omit later behavior and timestep resolution still matters.

Numerics

420 log-spaced frequency points with nodal transfer impedances. Two-load browser transients use an eight-state causal ladder model and a small matrix exponential with first-order-held sampled currents; analytic raised-cosine derivatives recover inductive voltage. Nominal record 32768 × 200 ps; edge refinement halves dt and doubles sample count at fixed duration. Physical event times are rounded to samples. Signed contributions are solved separately and summed; no repeating-record tail is introduced. The legacy single-load API and explicit test-only periodic option retain FFT convolution for regression comparisons. All board/die complex transfer pairs and full-precision time traces remain exported. Frequency inspection retains nearest-native-Hz semantics.

Known limitations

The causal transient starts from zero incremental state and approximates current between samples linearly. A quiet pre-event baseline follows from causality and is not a timestep convergence certificate. Six selected two-load corners are compared with an independent causal trapezoidal reference; this is not proof for every parameter combination. Reported extrema cover the declared observation interval, not all future time. The VRM remains a first-order R+jwL teaching model, without compensation, phase margin or nonlinear behavior. The topology has one die and one board load; scalar target impedance remains a single-load heuristic. Frequency branch-current magnitudes can exceed the injected current because phasors circulate. Legacy FFT-only helper results retain periodic-record limitations.

Evidence

check-models.js — PDN ladder (21 assertions): DC limit is the resistive path, 5 GHz limit is the die ESL reactance, the biggest peak matches the closed-form VRM-against-bulk resonance, and |Z| agrees with K.cascadeS — a separate code path — to 1e-6 ohm. Lab C transient (N2-4): the reference solver is itself validated against an analytical DC limit — capacitors open, inductors shorted, 7.8 mOhm at the die — which it reaches to under a microvolt from rest; 200 ps and 400 ps are asserted to differ; every requested edge is asserted to be realised exactly; droop is asserted monotone in edge rate and to fall 3.36% across the control. Causal two-load path: tests/check-pdn-multi.js compares six corners at both observation nodes, checks timestep and record extension separately, and exercises closed-form causal/periodic references and targeted numerical faults.

Make the current half of a travelling wave visible, and account for where the energy goes.

Topology

As reflection, with variable length and 24 launched waves.

Reference planes

Driver node, load node, and a movable probe along the line.

Equations

• V/I=+Z0 forward, -Z0 backward
• E=Td*int(0.5 v^2/Z0 + 0.5 i^2 Z0)

Units

ps, V, A, ohm, pJ

Validity

Lossless, uniform, no mid-line discontinuity.

Numerics

Trapezoid integration with 900 time intervals and max(220, ceil(64 Td/tr)) spatial intervals, capped at 65536. The energy history uses 121 uniformly spaced times plus finite-edge boundary transition landmarks; straight segments join computed samples. History is cached until physical settings change. Cursor energies are recomputed at the actual cursor time; the residual is shown on a separate aligned scale. Energy integrals begin before the source edge. Energy-history UI updates debounce rapid physical edits, then yield between samples with a generation guard. Pending histories are hidden and omitted from exports; instantaneous calculations remain current. Completed histories use the same sample grid and quadrature as the synchronous model.

Known limitations

Lossless, uniform, no mid-line discontinuity. The energy account is refused rather than reported when it closes to worse than 1%. Across every length and edge the controls allow it now closes to 0.086% or better.

Evidence

check-models.js — Line1D (21 assertions), shared with the reflections panel.

Compare NRZ and PAM4 on the same channel at the same data rate: PAM4 halves the Nyquist frequency and so meets less loss, and pays for it with a third the eye height.

Topology

Two full chains, identical except for the alphabet. Each runs K.channelImpulse at its own Nyquist loss, the shared CTLE at a fixed boost, and the shared DFE with a fixed tap count. NOT an ideal-level comparison.

Reference planes

Transmitter output.

Equations

• loss_NRZ = lossPerGHz * rate/2, loss_PAM4 = lossPerGHz * rate/4
• level spacing penalty = 20*log10(3) = 9.54 dB at equal total swing
• h = ctle(channelImpulse(loss, sps, nfft)); y = applyDFE(conv(levels, h), ...)

Units

dB, V

Validity

A controlled comparison of two alphabets on one loss law, not a system margin. Fixed CTLE boost and fixed DFE taps for both, deliberately, so the comparison is about the alphabet. No adaptation, no jitter, no crosstalk, ideal terminations.

Numerics

FFT channel at 32 samples/UI, minimum phase by real cepstrum; seeded symbol sequence. The 9.54 dB level-spacing figure is exact arithmetic and is checked separately from the chain.

Known limitations

The 20*log10(3) level-spacing penalty is a fact about amplitudes and is exact. Everything else on the panel is the output of a numerical channel, and the two should not be read with the same confidence.

Evidence

check-models.js — DFE and Convergence suites cover the chain's parts: exact cancellation with known symbols, the half-open tap window, gain-aware slicing for PAM4 (M2-7), and grid independence of the channel. The 9.54 dB figure is in check-numbers.py.

Show a capacitor bank's impedance and where mounting inductance takes over.

Topology

N parallel RLC branches against a plane.

Reference planes

One port.

Equations

• Z=R/N+j(w*L/N-1/(w*C*N))

Units

Hz, ohm

Validity

Lumped; no plane modes; no package.

Numerics

Direct complex evaluation.

Known limitations

Lumped; no plane modes; no package.

Evidence

check-models.js — PDN (RLC limits and damping).

Show that where supply noise enters decides whether the loop removes it.

Topology

First-order PLL, three injection points: VCO, reference, output buffer.

Reference planes

Clock output.

Equations

• VCO noise high-passed at the loop bandwidth; reference noise low-passed; buffer noise uncorrected

Units

Hz, V, s

Validity

ILLUSTRATIVE topology — a first-order loop, not a solved PLL, and a linear small-signal sensitivity. What IS independently checked is the behaviour that matters: the three injection points are treated oppositely, and the units are dimensionally correct.

Numerics

Closed-form transfer functions.

Known limitations

ILLUSTRATIVE topology — a first-order loop, not a solved PLL, and a linear small-signal sensitivity. What IS independently checked is the behaviour that matters: the three injection points are treated oppositely, and the units are dimensionally correct.

Evidence

check-models.js — PSIJ suite. The assertions cover units and the three transfer paths; they do not validate the loop order.

Show what a marked capacitance is worth at an operating point, and how much of the answer the derating uncertainty owns.

Topology

One capacitor: ESR(f) in series with ESL_part + L_mount in series with Ceff. No neighbours, no plane.

Reference planes

One port, at the capacitor terminals as mounted. Nothing beyond the mounting loop.

Equations

• Z=ESR(f)+j(w*Ltot-1/(w*Ceff))
• Ceff=cNom*retained
• SRF=1/(2*pi*sqrt(Ltot*Ceff))
• L_mount=length*(mu0/pi)*acosh(s/2r)
• ESR=esrMin*cosh(ln(f/fMin)/2)

Units

Hz, ohm, F, H

Validity

Lumped R+L+C. Not valid above the part body's first structural resonance. The retained fraction is an INPUT - this panel does not know your derating and will not guess it.

Numerics

Closed form at every frequency. The band is the exact envelope over the capacitance range, not a two-endpoint min/max.

Known limitations

No temperature coefficient on ESR, although one is measured to exist at high frequency. No mutual coupling to neighbouring parts. The ESR shape between its anchors is illustrative.

Evidence

check-models.js — Real capacitors (11 assertions): SRF scales exactly as one over root C independent of L and ESR, the two-term ESR equals the closed form esrMin*cosh(ln(f/fMin)/2), reactance is zero at SRF so |Z| there is ESR(SRF), the marked and derated curves converge above resonance and differ by exactly the retained fraction below it, the band provably contains the operating point at every frequency, and a zero retained fraction is refused rather than drawn.

Show how a step reflects between two mismatched ends, and why the ringing decays at the rate it does.

Topology

Ideal step source with series Rs, one uniform lossless line of delay Td, one resistive or open load.

Reference planes

Driver output node and load node. No package, no connector.

Equations

• Gs=(Rs-Z0)/(Rs+Z0)
• GL=(RL-Z0)/(RL+Z0)
• a0=Vs*Z0/(Rs+Z0)
• V(x,t) by superposition of 7 launched waves

Units

ps, V, ohm

Validity

Passive terminations. Lossless: late-time amplitudes are optimistic.

Numerics

Closed-form lattice sum. The edge is a CUBIC SMOOTHSTEP, s^2*(3-2s), centred on t = 0 and spanning tr from 0 to 1 — not a raised cosine, which the contract and the module header both used to claim. The two differ by at most 1% pointwise, but the rise-time convention differs: a cubic smoothstep's 10-90% time is 0.608*tr where a raised cosine's is 0.590*tr.

Known limitations

tr is the full 0-100% support of the smoothstep, not a 10-90% time. Multiply by 0.608 to compare with a datasheet 10-90% figure. Lossless, so late-time amplitudes are optimistic.

Evidence

check-models.js — Line1D (21 assertions): Ohm's law at the load and Kirchhoff at the source hold to 1e-12 at every instant; energy balances across five terminations.

Show what passivity, reciprocity and causality each assert, separately.

Topology

Through-section and stub networks.

Reference planes

Two 50 ohm ports.

Equations

• exact ABCD of a mismatched through-section and an open stub

Units

Hz, dB

Validity

Lossless unless stated; two-port only.

Numerics

Closed form.

Known limitations

Lossless unless stated; two-port only.

Evidence

check-models.js — Networks (passivity, and reciprocity checked as a separate property).

Tie the edge in time to the envelope in frequency.

Topology

Trapezoidal clock, one edge rate, one duty cycle.

Reference planes

Driver output.

Equations

• |An| proportional to |sinc(n*pi*tau/T)|*|sinc(n*pi*tr/T)|

Units

Hz, s, dB

Validity

Ideal trapezoid; no channel.

Numerics

Closed form harmonics.

Known limitations

Ideal trapezoid; no channel.

Evidence

check-models.js — Spectrum (a 50% duty square wave has no even harmonics).

Show ground bounce as v=L di/dt across a shared return.

Topology

N drivers sharing one return inductance.

Reference planes

Die-side ground against board ground.

Equations

• v=L*di/dt
• peak slope = (pi/2)*dI/tr for a raised-cosine edge

Units

A/s, H, V

Validity

Single lumped shared inductance; no mutual coupling between returns.

Numerics

Closed form; peak rather than mean slope reported.

Known limitations

Single lumped shared inductance; no mutual coupling between returns.

Evidence

check-numbers.py — the di/dt and mean-slope figures are recomputed from first principles.

Report what a Touchstone 1.0 two-port file contains, and whether this reader can trust it.

Topology

No model. It parses a file and computes properties of what it read.

Reference planes

Whatever the file's own reference planes are — the reader cannot know and does not guess.

Equations

• sigma_max(S) = sqrt( largest eigenvalue of S^H S ), over the FULL complex 2x2
• T = sum |Sij|^2, D = |det S|^2, sigma_max = sqrt( (T + sqrt(max(0, T^2 - 4D))) / 2 )
• IL = -20*log10(|S21|)
• reflection in dB = 20*log10(|S11|); return loss = -20*log10(|S11|)

Units

Hz after conversion from the file's own unit; dB; linear S

Validity

Touchstone 1.0, two-port, S-parameters, MA/DB/RI format, Hz through GHz. Strictly ascending non-negative frequencies. Values validated AFTER unit and format conversion, because a finite 1e308 becomes infinite once multiplied by 1e9.

Numerics

Exact arithmetic on the file's own numbers. No interpolation, no renormalisation, no extrapolation.

Known limitations

An INSPECTOR, not an importer. It cannot feed the channel lab, because that needs renormalisation to the model's reference impedance, interpolation onto its frequency grid and a DC extrapolation — three pieces of numerical work with three error budgets, none written. Does not support Touchstone 2.0 keywords, noise data, ports other than two, Y/Z/H/G parameters, per-port reference impedances, or any repair of a malformed file. Passivity is evaluated only at the frequencies the file contains: it establishes nothing about the gaps between samples, about causality, about measurement quality, or about fitness for a transient transform.

Evidence

check-models.js — Touchstone (30 assertions): refuses a 2.0 file, a non-2-port file, a missing option line, a wrong reference impedance, descending or negative frequencies, and values that overflow after conversion. Accepts MA, DB and RI and agrees between them to 1e-9. Passivity (N1-3): sigma_max(S) is computed from the full complex matrix and cross-checked against power iteration -- a different algorithm -- over 300 random complex matrices to 1e-9, plus four checked-in .s2p fixtures whose true singular values were computed independently: a unitary asymmetric network the retired |S11 +/- S21| shortcut called active at 1.414, an active one it called passive at 0.5, a non-reciprocal isolator it reported as 0.000, and a port-symmetric lossy case where the shortcut was correct and still is. The degenerate case that makes the discriminant negative in floating point is asserted by name.

Show how a UI is consumed by contributions that are not all the same kind.

Topology

A budget, not a circuit.

Reference planes

Controller pin to DRAM pin.

Equations

• UI_ps = 1e6 / rate_MTps (the control is a TRANSFER rate in MT/s)
• budget terms summed as an RSS of the independent contributors

Units

ps

Validity

ILLUSTRATIVE — the contributions are representative magnitudes, not JEDEC values. Training gain is a labelled model, not a specification.

Numerics

Exact arithmetic on the budget terms. At 8533 MT/s, UI = 1e6/8533 = 117.2 ps, which is what the page states.

Known limitations

The contract previously wrote UI = 1/(2*rate), which is the formula for a clock frequency rather than a transfer rate and would have given 58.6 ps. The code was always right; the contract was not. The budget split itself is illustrative — the terms are plausible magnitudes, not a sourced allocation.

Evidence

check-numbers.py — the UI at 8533 MT/s is recomputed. The allocation is not sourced; see docs/claims.md.

Show where a rectangular cavity's first modes land and that position matters.

Topology

Rectangular plane pair, open edges, ideal conductors.

Reference planes

Between the two planes.

Equations

• fmn=(c/(2*sqrt(Dk)))*sqrt((m/a)^2+(n/b)^2)

Units

Hz, m

Validity

ILLUSTRATIVE — solid rectangular planes only. Antipads, splits and cutouts perturb every mode.

Numerics

Closed-form mode frequencies.

Known limitations

ILLUSTRATIVE — solid rectangular planes only. Antipads, splits and cutouts perturb every mode.

Evidence

check-numbers.py — the mode frequencies are recomputed from the formula.

Turn a ripple budget and a current step into an impedance line, and show it against a load spectrum.

Topology

None — this is one division.

Reference planes

The point the ripple is specified at.

Equations

• Ztarget=(Vdd*ripple)/Istep

Units

V, A, ohm

Validity

ILLUSTRATIVE — the impedance division is exact arithmetic, but the current spectrum beside it is synthetic and representative, not measured die demand. Assumes a load independent of the rail.

Numerics

Exact arithmetic.

Known limitations

ILLUSTRATIVE — the impedance division is exact arithmetic, but the current spectrum beside it is synthetic and representative, not measured die demand. Assumes a load independent of the rail.

Evidence

check-numbers.py recomputes the worked budget. The current spectrum is not sourced.

Show the shape of a regulator's recovery and that release is its own event.

Topology

First-order loop response to a current step and release.

Reference planes

Regulator output.

Equations

• tau=1/(2*pi*fc)

Units

s, V

Validity

ILLUSTRATIVE — a real loop is not single-pole; order, damping and peaking change the shape.

Numerics

Analytic first-order response.

Known limitations

ILLUSTRATIVE — a real loop is not single-pole; order, damping and peaking change the shape.

Evidence

Not independently verified. The time-constant arithmetic is checked by check-numbers.py.