Real Components, Models & Operating Regions

01 / 10Opening failure

Failure: one nominal value, three RF behaviors

The node team chose a “1 pF” DC-block capacitor for 2.450 GHz. One review called it a near-perfect open, another called it a low-impedance resonator, and a third saw an inductor at 10 GHz. All three descriptions can be correct—but only after frequency, parasitics, conditions, and plane are attached.

Think about itAt 100 MHz, 2.45 GHz, first series resonance, and 10 GHz, what sign should Im Z have?

Answer

Negative below resonance, zero at the first-order series resonance, and positive above it under the portfolio e^+jωt convention. At 2.45 GHz this fixture is still strongly capacitive, not ideal and not yet resonant.

Derived from p04-m01-cap-series-rlc-v1 · e^+jωt · R1 package terminals · 0 V DC · 25 °C · Illustrative, not measured.

Definition

A usable component model is a conditional mapping. It maps a declared stimulus to voltage, current, waves, heat, or another response for a stated frequency, environment, operating state, topology, and reference plane.

The first-order fixture uses C = 1.000 pF, ESR = 0.350 Ω, and ESL = 0.800 nH at the two R1 package terminals. It deliberately excludes pads, board, fixture, dispersion, nonlinear behavior, and higher modes. Those omissions are part of the model, not fine print. Pozar provides the impedance, resonance, network, and plane foundations used here. [1]

Common misconceptionThe nominal value defines RF behavior.

The nominal value fixes one parameter under stated test conditions. It cannot determine loss, package inductance, resonance, bias response, fixture coupling, power handling, or evidence quality.

The correction starts by separating six statements that data sheets often place near each other.

02 / 10Read the claim

Value, tolerance, rating, and operating condition

Treat each number as a sentence with a subject, population, condition, and consequence. A bare number invites the wrong inference.

Think about itA 25 V, ±0.05 pF capacitor is used at 10 V. Does that establish its 2.45 GHz capacitance or lifetime?

Answer

No. The 25 V rating is a stress boundary, while ±0.05 pF is a bounded parameter tolerance under a named measurement condition. Neither supplies the RF impedance, DC-bias curve, statistical population, derating law, thermal state, or lifetime evidence.

Six claims that must not be collapsed into one “spec”
Statement class	What it can support	What it cannot support alone	Record with
Nominal value	Target parameter and test point	RF behavior, spread, or stress	Unit, frequency, bias, temperature, method
Bounded tolerance	Allowed parameter interval	A probability distribution	Population, lot/temperature scope, test condition
Characterized distribution	Observed population under a protocol	Guaranteed production limits outside that protocol	Sample count, lots, statistic, confidence, conditions
Absolute maximum	A do-not-reach stress boundary	Normal performance just below it	Waveform, duration, duty, thermal plane, simultaneous limits
Recommended condition	Intended operating envelope	Qualification or lifetime by itself	Mode, supply, drive, bias, temperature endpoints
Derating guidance	A stated margin policy	An unstated failure law	Authority, quantity, slope, endpoint, applicable mechanism

Checked example · electrical and thermal planes

Two current numbers, two meanings

A 10 nH synthetic inductor record separates 60 mA DC at the package terminals from 8 mA RF RMS in the same branch. Copper loss is screened from the package to the local board-temperature plane; ambient-to-airflow and junction concepts are not silently substituted. The screen can report margin, not lifetime.

Common misconceptionTolerance and rating describe the same uncertainty.

Tolerance concerns the value delivered by a defined population and test. Rating concerns stress. Operating at half a voltage rating does not imply half the capacitance error, half the temperature rise, or twice the life.

Murata’s first-party MLCC material shows why condition matters: high-permittivity dielectrics can lose effective capacitance with DC bias, whereas the selected GJM C0G family is intended for high-Q RF use and should not be assigned a fictitious large bias coefficient. [4]

03 / 10Model decision

Choose the model, not just the part

Model order is not a quality rank. The right question is the cheapest representation that can resolve the decision at hand.

01
Ideal value
Use when parasitics and condition dependence are demonstrably below the error budget.
Fast · analytic · weakest envelope
02
Lumped equivalent
Add named R, L, C, loss, leakage, or controlled condition coefficients.
Interpretable · useful around first resonance
03
S/Y/Z data
Preserve complex frequency response, Zref, ports, plane, fixture state, bias, temperature, and drive.
Finite-domain · interpolation policy required
04
Behavioral / empirical
Map control state, power, temperature, or measured tables when linear networks are insufficient.
Population and residuals matter
05
Nonlinear / time-domain
Use for harmonics, compression, switching, surge, memory, or transient damage mechanisms.
More conditions · more ways to misuse it

Decision tree

Name the decision.Point impedance? Resonance? Loss? Stress? Harmonics? Transient?
Declare the span and stimulus.Frequency, waveform statistic, bias, power, temperature, duty.
Align the plane.Package, pad, board, or fixture; port order and real Zref.
Set an error criterion.Magnitude, phase, root position, stress class, or discriminating observable.
Try models from simplest upward.Stop at the first representation that passes and has adequate evidence.

Go deeperWhy more parameters can make a worse engineering model

A high-order fit can reduce residuals on one sparse curve while introducing unphysical poles, active behavior, or explosive extrapolation outside the fit band. Decimal precision records arithmetic; it does not create independent evidence. Check limits, passivity/plausibility, parameter sensitivity, and a held-out measurement.

Common misconceptionA higher-order model is automatically more accurate.

Accuracy belongs to a declared observable and domain. A first-order RLC can be the most adequate model at one decision point; a poorly constrained 20-parameter fit can be less credible everywhere else.

04 / 10Locate the model

Package, pad, fixture, and reference plane

A complex value answers “voltage and current where?” A Touchstone record answers “waves where, normalized how, and with which structure removed?”

Informative plane map. IEEE 370 addresses interconnect/test-fixture characterization and validation; detailed calibration and de-embedding procedure belongs to Path 08. [2]

Package-terminal model

Generic component/reference-design RF plane. Includes only the elements named by the package fixture.

P local

Pad / board model

Adds land pattern, solder, via/return, local trace, and possibly coupling. These are board-specific.

F1/F2

Fixture-port record

Adds connector/coupon access unless a documented transformation removes it to the declared DUT plane.

Murata’s MLCC measurement note explicitly distinguishes calibration, land patterns, jigs, and extraction of capacitor-alone data. Coilcraft’s 0402DC model note similarly states that its curve-fit model comes from de-embedded measurements on a named fixture and excludes the eventual PCB. Those details are evidence, not administrative metadata. [5] [6]

Common misconceptionA good package model automatically includes pads, vias, enclosure, and assembly variation.

It includes only what its declared planes and extraction state include. Moving the value to a board schematic without adding the board can produce false correlation disputes.

Boundary: this module names the transformation and uncertainty. Calibration, fixture removal, uncertainty evaluation, and de-embedding procedure are deliberately deferred to Path 08.

05 / 10Frequency is an input

Frequency dependence: SRF, Q, loss, dispersion, and modes

A first resonance is a useful landmark. It is not a certificate of broadband validity.

Derived · p04-m01-cap-series-rlc-v1

Z = R + j(ωL − 1/ωC)

At first-order series resonance: Im Z = 0 ⇒ ω₀L = 1/(ω₀C)

f_SRF = 1 / (2π√(LC)) = 5.626976976 GHz

Units: H·F = s², so 1/√(LC) has rad/s dimensions. The 2π converts angular frequency to hertz.

Net port ratioQ_series = |Im Z| / Re Z

Defined only when the displayed series topology has Re Z > 0. It becomes zero at exact series resonance.

Stored-energy resonance valueQ₀ = ω₀L/R = 80.812204

A separately named first-order resonator quantity. It must not be relabelled as port Q at resonance.

Shunt port ratioQ_parallel = |Im Y| / Re Y

Used only for a declared shunt fixture with Re Y > 0; otherwise Q is suppressed.

At low frequency the canonical reactance approaches −j/(ωC); at high frequency it approaches +jωL. The extended fixture adds frequency-dependent loss, a dielectric-loss branch, and a small parallel capacitance, so its numeric root shifts slightly to 5.626968839 GHz. No display-rounded root feeds later arithmetic.

Think about itIf a data sheet lists one SRF, may you treat it as fixture-, bias-, power-, and temperature-independent?

Answer

No. The quoted SRF belongs to a model or measurement under specific conditions. Plane parasitics, material dispersion, bias, heating, drive, mounting, and higher modes can move or obscure the observed crossing.

Server-rendered canonical record · print / no-JavaScript evidence

1 pF model-card baseline

The live inspector initializes from this same fixture and display contract. No client calculation is required to read or print the default conclusion.

Fixture: 1.0 pF canonical (cap-1p0)
Displayed model: lumped RLC · p04-m01-component-validity-v1
Frequency: 2.45 GHz
DC voltage: 0 V
Board temperature: 25 °C
Package-terminal RF voltage (RMS): 1 V RMS
Reference plane: R1 source-side and load-side package terminals

Derived · first-order illustrative fixturep04-m01-cap-series-rlc-v1

Z(f)=R_{\mathrm{ESR}}+j\left(2\pi fL_{\mathrm{ESL}}-\frac1{2\pi fC}\right)

C = 1.000 pF · R = 0.350 Ω · L = 0.800 nH · e^+jωt · R1 package terminals

Simulated · illustrative synthetic referencep04-m01-cap-extended-v1

Z_{\mathrm{extended}}=\frac1{1/Z_{\mathrm{branch}}+j\omega C_{\mathrm{par}}}

Z_{\mathrm{branch}}=R(f,T)+j\omega L(T)+\frac1{\omega C(V,T)(\tan\delta(T)+j)}

\begin{aligned}C(V,T)&=1\,\mathrm{pF}\left[1-0.015\left(\frac V{16\,\mathrm V}\right)^2\right]\\&\quad\times\left[1+0.00002\frac{T-25\,{}^\circ\mathrm C}{{}^\circ\mathrm C}\right]\end{aligned}

\begin{aligned}R(f,T)&=0.35\,\Omega\left[1+0.0039\frac{T-25\,{}^\circ\mathrm C}{{}^\circ\mathrm C}\right]\\&\quad\times\left[1+0.08\left(\sqrt{\frac f{2.45\,\mathrm{GHz}}}-1\right)\right]\end{aligned}

L(T)=0.8\,\mathrm{nH}\left[1+0.00001\frac{T-25\,{}^\circ\mathrm C}{{}^\circ\mathrm C}\right]

\tan\delta(T)=0.002\left[1+0.002\frac{T-25\,{}^\circ\mathrm C}{{}^\circ\mathrm C}\right]

C_{\mathrm{par}}=0.005\,\mathrm{pF}

Four independently checked first-order rows · full precision retained before display rounding
Frequency	Rectangular Z	\|Z\|	Phase	Behavior
100 MHz	0.35 − j1591.046776 Ω	1591.0468146 Ω	-89.987396°	Capacitive
2.45 GHz	0.35 − j52.646158 Ω	52.6473215 Ω	-89.6190942°	Capacitive
5.626976976 GHz	0.35 − j7.10543e-15 Ω	0.35 Ω	-1.163174e-12°	First-order series resonance
10 GHz	0.35 + j34.349988 Ω	34.3517712 Ω	89.4162202°	Inductive

Analytic first-order SRF5.626976976 GHz

$f_{\mathrm{SRF}}=\frac1{2\pi\sqrt{L_{\mathrm{ESL}}C}}$

Specified extended crossing5.626968838751 GHz

Prompt anchor · Re Z = 0.421002231535 Ω · Im Z ≈ 0.

Exact stop-rule midpoint5.626968838591 GHz

4097 log-frequency nodes + bisection; first bracket satisfying the frozen stop rule. Difference from the higher-convergence crossing: 0.159595 Hz.

Stored-energy resonator Q080.8122036

ω0L/R = 1/(ω0CR), distinct from port |Im Z|/Re Z.

Extended Z at 2.450 GHz0.476056009414 − j52.433450050857 Ω

|Z| 52.435611120305 Ω · -89.47981201534°

Extended reference as a reciprocal equal-50 Ω two-port · Port 1 source side · Port 2 load side · R1 package terminals · 2.450000 GHz
Quantity	Rectangular value	Magnitude / screen
S11 = S22	0.217762807322 − j0.408210636436	0.462662473032
S21 = S12	0.782237192678 + j0.408210636436	0.882344008485
Passivity	largest singular value	σmax = 1 within float64 roundoff

Point comparison against p04-m01-cap-extended-v1
Candidate	Magnitude error	Phase error	Nearby resonance agreement	Criterion
Ideal C	23.88756396%	0.520187985°	No root represented	Fails magnitude
Constant series RLC	0.403753%	0.139282176°	1.44611266659e-6 relative to the specified crossing (prompt-rounded anchor 1.446116×10⁻⁶); 1.44614102906e-6 to the exact stop-rule midpoint	First passing model
Synthetic S-data	0%	0°	Reference root	Reference representation

Local decisionvalid—lumped RLC is the simplest adequate model

Binding evidence: RLC passes the 2.0% magnitude and 2.0° phase criteria, represents the nearby resonance when required, stays inside the recommended state, and uses complete fixture metadata. Unperformed hardware evidence remains separate.

Operating envelope · lesson fixture, not a manufacturer rating
Quantity	Supported	Recommended	Absolute screen
Frequency	1 MHz–20 GHz	10 MHz–12 GHz	Domain, not stress rating
DC voltage	0–16 V	0–10 V	16 V
Board temperature	−40–125 °C	−40–85 °C	125 °C
RF voltage	0–10 V RMS	0–5 V RMS	10 V RMS

Plane map · the package model does not silently include the board
Plane	Included	Excluded / transformation
R1 package terminals	Electrode/package R, L, C, dielectric branch, Cpar	Pads, vias, coupon, connector, board and enclosure
F1/F2 fixture ports	R1 model plus each disclosed 0.035 Ω + jω0.10 nH access section	Any structure not named by fixture v1.0.0

Component-card takeaway: model, function, frequency, condition, drive statistic, plane, Zref, port order, fixture state, evidence class, revision, interpolation, limits, checks, unresolved evidence, prohibited inference, and one falsification measurement travel together.

Common misconceptionSRF is a universal number independent of fixture and operating condition.

SRF is a root of a particular conditional response. Always preserve the root definition, topology, plane, domain, and conditions.

06 / 10Condition the model

Bias and temperature change the model

The nominal part can keep its label while effective C, L, R, leakage, Q, and resonance change. Each curve therefore carries its condition tags.

Named synthetic effects · bounded by each versioned fixture · not universal material laws
Fixture family	Operating input	Declared response	What remains absent
Capacitor	DC voltage + board temperature	C changes quadratically with V; R, L, tanδ use named T/f coefficients	Large-signal RF nonlinearity and production distribution
Inductor	DC current + board temperature	L decreases monotonically with current; R rises with T/f; Cpar carries a small T term	Transient saturation waveform, core loss map, board coupling
Ferrite	DC current + board temperature	Frozen complex-Z knots scale with separate bounded R and X factors	An inferred ideal L or RLC—the scalar \|Z\| nominal cannot supply one
ESD device	Reverse voltage + board temperature	Small-signal branch capacitance, R, and L use bounded coefficients	Snapback, surge clamp, harmonics, damage and lifetime

Think about itA lumped model passes at 0 V and 25 °C. What should happen when bias or temperature changes?

Answer

Re-evaluate the same observable at the new state and carry the updated metadata. If the simplified model omits the condition dependence, its comparison error or envelope rule can move from valid to inspect; the nominal label does not overrule that result.

Vendor model libraries reinforce the point. Murata’s current SimSurfing tools tag model data by DC bias and temperature, while Coilcraft warns that small-signal RF models do not automatically establish current-saturation behavior. [4] [6]

Class 1 embedded decision model · Illustrative synthetic

Model Validity Inspector

Commit one condition set. The inspector compares model families at the same component, state, topology, Zref, and plane, then exposes the binding rule.

p04-m01-component-validity-v1p04-m01-fixtures/1.0.0e^+jωt · 50 Ω real

1 · Component and closed fixture

Component presetfour component classesNominal / library presettwo complete fixtures

2.450 GHz series DC block. This is a fixed instructional fixture, not a purchasable part or statistical population.

2 · Model and inclusion plane

Fixture-port view is defined only for synthetic S-data through: Each fixture side adds 0.035 Ω + jω·0.10 nH between its fixture port and package terminal. Select synthetic S-data to enable the fixture-port transformation.

3 · Frequency and operating state

FrequencyGHz · 0.001–20 · 1 MHz typed step

DC voltageV · 0–16

Board temperature°C · −40–125

Package-terminal RF voltage (RMS)V RMS · 0–10

Log-frequency cursor draft only; commit to announce and calculate

The SRF/root preset preserves the full float64 root even though typed frequency uses a 1 MHz step. RMS, available, incident, accepted, delivered, and dissipated quantities remain distinct.

Update 1. Canonical calculation ready. Valid: lumped RLC is the simplest adequate model.

The complete canonical result, equations, tables, evidence pack, and decision remain below without JavaScript. JavaScript is needed only to commit different conditions, copy the card, or print from this control.

valid

valid—lumped RLC is the simplest adequate model

This is a local model-validity teaching result, not a purchasing, safety, qualification, or lifetime decision.

Fixture: 1.0 pF canonical (cap-1p0)
Displayed model: lumped RLC · p04-m01-component-validity-v1
Frequency: 2.45 GHz
DC voltage: 0 V
Board temperature: 25 °C
Package-terminal RF voltage (RMS): 1 V RMS
Reference plane: R1 source-side and load-side package terminals
Complex impedance: 0.35 − j52.646158 Ω
|Z|: 52.647321 Ω
Phase: -89.619094°
Port Qseries: 150.417594
Analytic lumped-model resonance: 5.626976976 GHz
First reference resonance: 5.626968839 GHz
Distance from first resonance: 56.459684% · below the root
Selected-model agreement: 0.403753% magnitude · 0.139282° phase
Simplest adequate model: lumped RLC
Evidence: Illustrative synthetic · p04-m01-fixtures/1.0.0

Same state · same plane · same 50 Ω normalization

Which is the simplest adequate model?

Candidate order is fixed: ideal → lumped RLC → synthetic S-data
Candidate	Complex Z	Magnitude error	Phase error	Resonance represented?	Local result
ideal value	0 − j64.961201 Ω	23.887564%	0.520188°	Yes / not required here	Does not passFails at least one local point-agreement criterion.
lumped RLCfirst adequate	0.35 − j52.646158 Ω	0.403753%	0.139282°	Yes / not required here	PassPasses the local 2% magnitude, 2° phase, and nearby-resonance criteria at this declared state.
synthetic S-data	0.476056 − j52.43345 Ω	0%	0°	Yes / not required here	PassPasses the local 2% magnitude, 2° phase, and nearby-resonance criteria at this declared state.

Lesson criterion: e_Z ≤ 2.0%, wrapped phase error ≤ 2.0°, and any reference resonance within 20% of the selected frequency must be represented. Agreement is necessary, never sufficient.

Magnitude + phase · selected marker + boundaries

See where a point agreement stops being a model agreement.

Synthetic reference lumped RLC recommended boundaries. Hatched regions remain supported but are outside the recommended frequency envelope. Committed fixture: 1.0 pF canonical · ESR 0.35 Ω · ESL 0.80 nH (p04-m01-fixtures/1.0.0); 0 V; 25 °C board; 1 V RMS; R1 package-terminal plane; Illustrative synthetic evidence. Curves are calculated from the versioned illustrative synthetic fixture, not measured hardware.

Exact text equivalent for selected, resonance, boundary, and worst-disagreement conclusions
Key point	Frequency	Reference Z / \|Z\| / phase	Selected Z / \|Z\| / phase	Magnitude / phase error	Why exposed
Selected point	2.45 GHz	0.476056 − j52.43345 Ω\|Z\| 52.435611 Ω · -89.479812°	0.35 − j52.646158 Ω\|Z\| 52.647321 Ω · -89.619094°	0.403753% / 0.139282°	Committed cursor
Recommended lower boundary	10 MHz	31.835502 − j15836.200269 Ω\|Z\| 15836.232268 Ω · -89.884818°	0.35 − j15915.444044 Ω\|Z\| 15915.444048 Ω · -89.99874°	0.500193% / 0.113922°	Envelope boundary
Recommended upper boundary	12 GHz	0.425454 + j47.905478 Ω\|Z\| 47.907367 Ω · 89.491163°	0.35 + j47.055667 Ω\|Z\| 47.056969 Ω · 89.573842°	1.775089% / 0.082679°	Envelope boundary
First reference resonance	5.626968839 GHz	0.421002 − j1.59998e-9 Ω\|Z\| 0.421002 Ω · -2.17747e-7°	0.35 − j8.18060e-5 Ω\|Z\| 0.35 Ω · -0.013392°	16.865046% / 0.013392°	4097-node scan plus bisection
Worst plotted magnitude disagreement	20 GHz	0.471128 + j98.290225 Ω\|Z\| 98.291354 Ω · 89.72537°	0.35 + j92.573218 Ω\|Z\| 92.573879 Ω · 89.783378°	5.816864% / 0.058008°	Magnitude maximum within the displayed 65-interval log sweep
Worst plotted phase disagreement	5.7996428 GHz	0.420226 + j1.710597 Ω\|Z\| 1.761458 Ω · 76.198011°	0.35 + j1.709987 Ω\|Z\| 1.745438 Ω · 78.432474°	0.909442% / 2.234462°	Wrapped-phase maximum within the displayed 65-interval log sweep

R1 package plane ↔ F1/F2 fixture ports

Moving the plane changes the network.

Each fixture side adds 0.035 Ω + jω·0.10 nH between its fixture port and package terminal. Selected R1 package-terminal plane for 1.0 pF canonical · ESR 0.35 Ω · ESL 0.80 nH (p04-m01-fixtures/1.0.0) at 2.45 GHz, 0 V, 25 °C board, and 1 V RMS. Illustrative synthetic evidence; this is an illustrative inclusion map, not a measured geometry. Board routing, vias, enclosure, and assembly variation remain excluded.

Included in the package model

electrode/package ESR and ESL
condition-dependent dielectric branch
0.005 pF parallel term

Still excluded

PCB pads and vias at R1
connector launch at R1
nonlinear and higher modes
assembly variation

Electrical + simple package-to-board thermal screen

Stress ledger—not a lifetime model.

Every margin retains its quantity, unit, limit class, and plane
Quantity	Value	Limit	Upper utilization / margin	Plane or endpoint
DC voltage	0 V	16 Villustrative absolute screen	0% / 16 V	R1 package terminals
Package-terminal RF voltage (RMS)	1 V RMS	10 V RMSillustrative absolute screen	10% / 9 V RMS	R1 package terminals
Board temperature	25 °C	125 °Csupported board-temperature ceiling	N/A / 100 °C	Entered temperature is the board temperature immediately beneath the package (illustrative thermal plane).
Estimated package temperature	25.0164486 °C	125 °Cillustrative thermal screen; not lifetime	N/A / 99.9835514 °C	Package-to-board θ = 95 K/W

Package RF loss: 0.17314318 mW · DC loss: 0 mW · total modeled package loss: 0.17314318 mW. Estimated package temperature: 25.016449 °C using the named package-to-board endpoint. Fixture-access loss, when selected, is not charged to the package thermal endpoint. No DC-loss term is present in this synthetic fixture; that is not evidence of zero leakage in a real part.

Reject > inspect > valid

The decision is inspectable, rule by rule.

pass
model-plane-supportSelected model and reference plane are explicitly supported.
pass
operating-absoluteDC voltage remains below the 16 V illustrative absolute screen.
pass
drive-absolutePackage-terminal RF voltage (RMS) remains below the 10 V RMS illustrative absolute screen.
pass
board-temperature-absoluteBoard temperature remains below the disclosed absolute screen.
pass
thermal-screenThe simple package-to-board thermal screen remains below its ceiling; it does not predict lifetime.
pass
sdata-passivitySynthetic S-data passes the local passivity screen (σmax 1).
pass
recommended-frequencyFrequency is inside the recommended lesson envelope.
pass
recommended-operatingDC voltage is inside the recommended lesson envelope.
pass
recommended-temperatureBoard temperature is inside the recommended lesson envelope.
pass
recommended-drivePackage-terminal RF voltage (RMS) is inside the recommended lesson envelope.
pass
frequency-boundaryFrequency two-sided data-boundary margin m=0.1225 is greater than 0.10.
pass
temperature-boundaryTemperature two-sided data-boundary margin m=0.3939 is greater than 0.10.
pass
operating-proximityDC voltage upper utilization u=0% is below 90%; the safe zero endpoint is not a lower-edge warning.
pass
drive-proximityPackage-terminal RF voltage (RMS) upper utilization u=10% is below 90%.
pass
model-agreementPasses the local 2% magnitude, 2° phase, and nearby-resonance criteria at this declared state.

Show evidence, domain, and prohibited-use metadata

Evidence class / source: Illustrative synthetic. Exact lesson coefficients and knots frozen in this fixture module after first-party model-document review; no manufacturer samples are copied.
Supported frequency: 1 MHz–20 GHz
Condition domain: 0–16 V; -40–125 °C; 0–10 V RMS
Interpolation / extrapolation: The disclosed continuous extended formula is evaluated directly in float64; the frozen frequency grid is an audit grid, not an interpolated substitute. Extrapolation: Rejected.
Port contract: 50 Ω real; Port 1 source-side, Port 2 load-side; both physical port currents point into the network.
Hardware evidence outstanding: Package-terminal complex-Z sweep at 0 V, 25 °C, and 1.0 V RMS with exact points at 2.450000 GHz and around 5.62697 GHz. No real-part qualification, lot distribution, lifetime, or assembly evidence has been performed by this lesson.

Do not identify this synthetic fixture as a purchasable part or measured population.
Do not infer surge, harmonic, aging, safety, qualification, or lifetime behavior.
Do not reuse the result outside its stated frequency, condition, plane, or drive domain.

Copyable / printable cumulative artifact

Component model card

RF JOURNEY · COMPONENT MODEL CARD
Model family: p04-m01-component-validity-v1
Fixture version: p04-m01-fixtures/1.0.0
Display contract: p04-m01-display/1.0.0
Decision: valid—lumped RLC is the simplest adequate model
Fixture: 1.0 pF canonical (cap-1p0)
Displayed model: lumped RLC · p04-m01-component-validity-v1
Frequency: 2.45 GHz
DC voltage: 0 V
Board temperature: 25 °C
Package-terminal RF voltage (RMS): 1 V RMS
Reference plane: R1 source-side and load-side package terminals
Complex impedance: 0.35 − j52.646158 Ω
|Z|: 52.647321 Ω
Phase: -89.619094°
Port Qseries: 150.417594
Analytic lumped-model resonance: 5.626976976 GHz
First reference resonance: 5.626968839 GHz
Distance from first resonance: 56.459684% · below the root
Selected-model agreement: 0.403753% magnitude · 0.139282° phase
Simplest adequate model: lumped RLC
Evidence: Illustrative synthetic · p04-m01-fixtures/1.0.0
Canonical fixture models: p04-m01-cap-series-rlc-v1; p04-m01-cap-extended-v1
Function: 2.450 GHz series DC block
Zref / ports: 50 Ω real; Port 1 source-side, Port 2 load-side; both physical port currents point into the network.
Interpolation: The disclosed continuous extended formula is evaluated directly in float64; the frozen frequency grid is an audit grid, not an interpolated substitute.
Falsification measurement: Package-terminal complex-Z sweep at 0 V, 25 °C, and 1.0 V RMS with exact points at 2.450000 GHz and around 5.62697 GHz.
Hardware evidence outstanding: Package-terminal complex-Z sweep at 0 V, 25 °C, and 1.0 V RMS with exact points at 2.450000 GHz and around 5.62697 GHz. No real-part qualification, lot distribution, lifetime, or assembly evidence has been performed by this lesson.
Prohibited inference: Do not identify this synthetic fixture as a purchasable part or measured population. Do not infer surge, harmonic, aging, safety, qualification, or lifetime behavior. Do not reuse the result outside its stated frequency, condition, plane, or drive domain.

RF JOURNEY · COMPONENT MODEL CARD
Model family: p04-m01-component-validity-v1
Fixture version: p04-m01-fixtures/1.0.0
Display contract: p04-m01-display/1.0.0
Decision: valid—lumped RLC is the simplest adequate model
Fixture: 1.0 pF canonical (cap-1p0)
Displayed model: lumped RLC · p04-m01-component-validity-v1
Frequency: 2.45 GHz
DC voltage: 0 V
Board temperature: 25 °C
Package-terminal RF voltage (RMS): 1 V RMS
Reference plane: R1 source-side and load-side package terminals
Complex impedance: 0.35 − j52.646158 Ω
|Z|: 52.647321 Ω
Phase: -89.619094°
Port Qseries: 150.417594
Analytic lumped-model resonance: 5.626976976 GHz
First reference resonance: 5.626968839 GHz
Distance from first resonance: 56.459684% · below the root
Selected-model agreement: 0.403753% magnitude · 0.139282° phase
Simplest adequate model: lumped RLC
Evidence: Illustrative synthetic · p04-m01-fixtures/1.0.0
Canonical fixture models: p04-m01-cap-series-rlc-v1; p04-m01-cap-extended-v1
Function: 2.450 GHz series DC block
Zref / ports: 50 Ω real; Port 1 source-side, Port 2 load-side; both physical port currents point into the network.
Interpolation: The disclosed continuous extended formula is evaluated directly in float64; the frozen frequency grid is an audit grid, not an interpolated substitute.
Falsification measurement: Package-terminal complex-Z sweep at 0 V, 25 °C, and 1.0 V RMS with exact points at 2.450000 GHz and around 5.62697 GHz.
Hardware evidence outstanding: Package-terminal complex-Z sweep at 0 V, 25 °C, and 1.0 V RMS with exact points at 2.450000 GHz and around 5.62697 GHz. No real-part qualification, lot distribution, lifetime, or assembly evidence has been performed by this lesson.
Prohibited inference: Do not identify this synthetic fixture as a purchasable part or measured population. Do not infer surge, harmonic, aging, safety, qualification, or lifetime behavior. Do not reuse the result outside its stated frequency, condition, plane, or drive domain.

07 / 10Do not merge power nouns

Power, heating, stress, and reliability margin

RF power and component stress become reviewable only after waveform statistics, terminations, duty, electrical plane, and thermal endpoints are explicit.

01Available
P_avs from the source under its conjugate-match definition.
02Incident
|a₁|² at the 50 Ω plane. It equals P_avs only because the power fixtures pin Γ_S = 0.
03Accepted
P_inc(1 − |S₁₁|²), after source-side reflection.
04Delivered
P_inc|S₂₁|² with Γ_L = 0.
05Dissipated
Accepted minus delivered for the disclosed passive two-port.

Electrical screen

Voltage/current uses linear upper utilization. A dBm-controlled fixture converts input and both endpoints to watts before utilization. Stress at or beyond a declared absolute maximum rejects.

Thermal screen

The synthetic package-to-board θ maps calculated dissipation to an estimated package temperature. It does not solve ambient airflow, transient junction temperature, or coupled electrothermal behavior.

Reliability boundary

Derating can express local margin. Without an applicable failure mechanism, acceleration model, qualification protocol, and population evidence, it cannot predict life.

Common misconceptionElectrical derating automatically guarantees thermal or lifetime margin.

A voltage ratio cannot prove temperature rise, core loss, surge survival, solder fatigue, dielectric wear, or lifetime. Keep each stress and thermal endpoint separate.

Nexperia’s low-capacitance protection data separate normal small-signal capacitance and insertion-/return-loss evidence from working voltage, TLP, surge, and IEC ESD evidence. A small-signal shunt model cannot be promoted into a clamp or survival model. [7]

08 / 10Audit the evidence

Typical, guaranteed, characterized, and simulated evidence

Smoothness, file size, and decimal places do not determine claim strength. Population, method, conditions, revision, and limits do.

Guaranteed / bounded

A stated production or qualification limit under named conditions. Strong for that claim; silent elsewhere.

Characterized

Observed samples or plots under a protocol. Preserve lots, sample count, statistic, and residuals.

Typical

A representative value or curve. Useful for design orientation; not a safe production boundary.

Simulated / fitted

A consequence of assumptions, parameters, and solver/model version. Validate against independent evidence.

Illustrative synthetic

A disclosed teaching fixture. Reproducible here, but never manufacturer or measured data.

What must travel with a model or data file
Field	Audited record	Failure if absent
Identity + revision	Source, part/family, model/fixture/file version, issue date, lifecycle check	Silent supersession or wrong population
Complex convention	RI/MA/DB format, frequency units, e^+jωt, Zref per port	Sign, scale, and phase errors
Ports + planes	Number/order/orientation, terminal names, included fixture, extraction state	Model attached to the wrong physical boundary
Stimulus + environment	Bias, current, RF drive/waveform, terminations, temperature meaning, duty	Conditional data treated as unconditional truth
Population + label	Typical, guaranteed, characterized, simulated, measured, or illustrative	False confidence and false precision
Domain + interpolation	Frequency/condition range, exact knots, interpolation, prohibited extrapolation	Invented behavior between or beyond evidence

Common misconceptionAn S-parameter file is unconditional part truth.

It is a conditional complex network record. Without ports, Zref, planes, fixture state, bias, temperature, drive, range, interpolation, and revision, it is incomplete.

Common misconceptionTypical means safe production limit.

Typical describes a representative observation or estimate. Only an applicable guaranteed limit can bound the stated production claim—and only under its conditions.

09 / 10Try to break it

Validate with limits and one discriminating measurement

Validation is not admiring a smooth curve. It is looking for the cheapest observation that makes competing models disagree.

01
Dimensions and conventions
Check H, F, Ω, S, Hz/rad·s⁻¹, RMS/peak, dB/linear, e^+jωt, port direction, and thermal endpoints.
02
Limiting cases
Evaluate low/high frequency, zero bias/current, 25 °C baseline, open/short/match, and exact supported endpoints.
03
Cross-model residuals
Compare |Z|, wrapped phase, complex error near zero, resonance location, and envelope class—never one unexplained score.
04
Plausibility and passivity
For passive S-data, reject σ_max > 1 + 10⁻¹⁰. Check positive loss terms and topology recovery.
05
Sensitivity
Move one declared coefficient or condition at a time. Rank the observables that change the decision, not merely those with many digits.
06
Discriminating measurement
Specify plane, complex observable, conditions, exact frequencies/span, and a numerical rejection threshold.

Pinned falsification request

Measure package-terminal complex Z, not only nominal C.

At R1, sweep the real part and imaginary part at 0 V DC, 25 °C, and 1.0 V RMS. Include exact points at 2.450000 GHz and around 5.62697 GHz. Reject the lumped model if the de-embedded measurement exceeds 2.0% magnitude or 2.0° phase error, or moves the first resonance by more than 20%.

This test is requested, not fabricated as already performed. Fixture characterization and uncertainty must accompany the eventual measurement.

Think about itWhich measurement best distinguishes ideal C from the RLC model in this decision?

Answer

A low-frequency capacitance measurement is cheap but weak because both models converge there. Complex Z near 2.45 GHz plus a sweep through the predicted first resonance exposes ESR, ESL, sign/phase, and root location in one targeted record.

Common misconceptionA curve match at one point validates the model everywhere.

Two models may share an omission or fit one point for the wrong reason. A passing point comparison remains bounded by the fixture, conditions, plane, evidence, and required falsification measurement.

10 / 10Cumulative artifact

Build the component evidence pack

The node review now carries four audited records forward. Each retains what was decided, what supports it, what could falsify it, and what no one may infer.

Record 01

RF inductor

Function / mode: Gate-bias feed isolation near 2.450 GHz; continuous receive duty.
Supply, bias, drive, span, temperature: 3.3 V rail; 60 mA DC; 8 mA RF RMS; 2.30–2.60 GHz; board temperature −40 to +85 °C.
Chosen model: Frequency-dependent lumped package model at R1, cross-checked against a synthetic complex-Z/S record.
Fixture / model version: p04-m01-fixtures/1.0.0 · ind-10n fixture record
Interpolation / extrapolation: Closed-form continuous extended model inside 1 MHz–20 GHz and the stated current/temperature domain; generated complex samples use no table interpolation; reject extrapolation.
Ports / planes / inclusions: Port 1 source-side package terminal; Port 2 load-side package terminal; 50 Ω S-data. Pads, vias, and board return excluded.
Source / revision / evidence label: Illustrative synthetic fixture informed by Coilcraft 0402DC model documentation, Document 158 (2021-07-28), not copied device data.
Nominal, tolerance, rating, recommendation: Nominal 10 nH; tolerances and production distribution unresolved; lesson recommendation 300 mA DC / 180 mA RF RMS; absolute screens are synthetic.
Checks passed: Units/sign, low-frequency limit, first-resonance scan, passivity, current/temperature sensitivity, and cross-model residual passed in-fixture.
Measurement that could falsify it: De-embedded package-terminal Z sweep at 0 and 300 mA, 25 °C, spanning 2.45 GHz and the first resonance.
Unresolved / prohibited inference: No lot distribution, nonlinear saturation waveform, board coupling, qualification, or lifetime evidence. Do not infer rated-part suitability.

Record 02

DC-block capacitor

Function / mode: Series DC block between the radio and feed chain at 2.450 GHz; continuous small signal.
Supply, bias, drive, span, temperature: 0 V DC across the selected element; 1.0 V RF RMS; 2.450 GHz decision point; 25 °C board plane.
Chosen model: p04-m01-cap-series-rlc-v1; validated locally against p04-m01-cap-extended-v1.
Fixture / model version: p04-m01-fixtures/1.0.0 · p04-m01-cap-series-rlc-v1 · p04-m01-cap-extended-v1
Interpolation / extrapolation: Closed-form continuous first-order and extended models inside 1 MHz–20 GHz and the stated voltage/temperature domain; generated complex samples use no table interpolation; reject extrapolation.
Ports / planes / inclusions: R1 package terminals, Port 1 source-side and Port 2 load-side; 50 Ω when represented as S-data. Board pads/fixture excluded.
Source / revision / evidence label: Derived from an Illustrative fixture; Murata GJM/SimSurfing documentation informs evidence labels only.
Nominal, tolerance, rating, recommendation: 1.000 pF nominal; tolerance/distribution not supplied by the fixture; recommended ≤10 V DC, ≤5 V RF RMS, −40 to +85 °C; synthetic absolute screens 16 V, 10 V, 125 °C.
Checks passed: Analytic SRF and Q0, asymptotes, numeric-root cross-check, S/ABCD round trip, passivity, and 2%/2° point agreement passed.
Measurement that could falsify it: Package-terminal complex-Z sweep at 0 V, 25 °C, 1.0 V RMS at 2.450000 GHz and around 5.62697 GHz.
Unresolved / prohibited inference: No measured part, fixture uncertainty, production distribution, nonlinear, qualification, or lifetime evidence. Do not call the local valid result a purchase approval.

Record 03

Low-capacitance ESD device

Function / mode: Shunt protection at the antenna-side node; normal operation only, not an ESD strike.
Supply, bias, drive, span, temperature: 0–3.3 V reverse bias; −10 dBm source available power; matched 50 Ω source/load; 2.30–2.60 GHz; board-ground temperature −40 to +85 °C.
Chosen model: Bias-tagged small-signal shunt Y/S-data fixture; transient clamp behavior deliberately absent.
Fixture / model version: p04-m01-fixtures/1.0.0 · esd-0p25 fixture record
Interpolation / extrapolation: Closed-form continuous small-signal shunt model inside 1 MHz–20 GHz and the stated bias/temperature domain; generated complex samples use no table interpolation; reject extrapolation.
Ports / planes / inclusions: R1 protected signal node to package ground; fixture-port view includes only disclosed shunt pad admittance. Board ground inductance excluded.
Source / revision / evidence label: Illustrative synthetic fixture informed by Nexperia PESD5V0R1BDSF-Q data sheet v.1 (2025-07-02); no values copied into the fixture.
Nominal, tolerance, rating, recommendation: 0.25 pF nominal fixture; synthetic recommended 3.3 V / +10 dBm; pulse and working-voltage manufacturer ratings are a different evidence population.
Checks passed: Bias monotonicity, shunt topology recovery, S passivity, accepted/delivered/dissipated power ledger, and boundary rules passed.
Measurement that could falsify it: Bias-tagged package-terminal shunt-Y sweep at 0 and 3.3 V, 25 °C; a separate pulse test is required for protection performance.
Unresolved / prohibited inference: No clamp waveform, surge survival, nonlinear distortion, assembly return path, distribution, or lifetime evidence. Do not infer ESD qualification from S-data.

Record 04

SAW-filter model package

Function / mode: Post-PA ISM 2.4 GHz filtering in the gateway transmit path.
Supply, bias, drive, span, temperature: Synthetic audit state: 2.450 GHz, 25 °C, +20 dBm input at terminal 1; supported 2.2–2.7 GHz, −20 to +85 °C, and −30 to +25 dBm input. The Revision K conditional endurance statement is +25.0 dBm input for 5000 h at +50 °C using IEEE 802.11 b/g/n/ax (RU242), applied to terminal 1 in the Tx passband; duty factor is not stated and remains unresolved.
Chosen model: p04-m01-saw-2g45-audit-v1; nonselectable frozen matched-circuit two-port table with its matching circuit and terminal map.
Fixture / model version: p04-m01-saw-2g45-audit-v1 · fixture pack p04-m01-fixtures/1.0.0
Interpolation / extrapolation: Real and imaginary coordinates of S11/S22 and S21/S12 are interpolated linearly in frequency; exact knots are returned unchanged; extrapolation outside 2.2–2.7 GHz is rejected.
Ports / planes / inclusions: Terminal 1 PA side, terminal 4 antenna side, terminals 2/3/5 grounded; 50 Ω real; Reciprocal two-port S-data in rectangular real/imaginary coordinates under e^(+jωt). Both ideal 9.1 nH shunt matches are included in the matched-circuit planes; bare-package recovery is prohibited.
Source / revision / evidence label: Murata SAFQA2G45MA0G0A data sheet Revision K informs the source-field audit. The frozen S table is separately labeled Illustrative synthetic; no official downloadable S2P was located and no vendor plot is reproduced.
Nominal, tolerance, rating, recommendation: +25.0 dBm input / 5000 h / +50 °C is a conditional data-sheet statement tied to the stated IEEE 802.11 waveform and terminal-1 Tx-passband direction, not an unconditional synthetic-fixture rating. Revision K also states 3 V DC between terminals at 25 ± 2 °C and 10 MΩ minimum resistance. Procurement lifecycle state remains unresolved.
Checks passed: Port/termination/temperature/power metadata audit passed; no synthetic curve is promoted to measured evidence.
Measurement that could falsify it: Matched-circuit two-port sweep at cold/room/hot plus the declared waveform power test in terminal-1 direction.
Unresolved / prohibited inference: No downloadable official S2P, stated duty factor for the endurance line, lot distribution, arbitrary-board response, reverse-power capability, or lifetime extrapolation. Do not remove the matching circuit from the claim.

Pack versionp04-m01-fixtures/1.0.0

Four records · one exact condition per claim · R1/local plane map · model/evidence revisions · independent checks · one falsification request each.

The SAW record is the deliberate contrast. Murata’s SAFQA2G45MA0G0A Rev. K response is specified with named terminals, grounded pins, 50 Ω source/load, and external 9.1 nH matching inductors; its temperature and long-duration power statement are tied to specified conditions. No official downloadable S2P was located. The audit therefore keeps a separately named, versioned Illustrative synthetic matched-circuit table; it is never described as Murata measurement data or exposed as an interaction preset. [8]

Ungraded review

Check your understanding

Answer each question in your own words, then reveal the model answer.

01Why is ‘1 pF’ not a complete RF component model?
Model answer
Because the value omits loss, package inductance, frequency, bias, temperature, drive, topology, evidence population, and reference plane. In the pinned RLC example, the same nominal capacitor is capacitive below its first resonance, approximately resistive at resonance, and inductive above it.
02What is the difference between a tolerance and an absolute maximum rating?
Model answer
Tolerance bounds a specified parameter around a nominal value under stated test conditions. An absolute maximum is a stress boundary that must not be reached or exceeded; it does not promise normal performance below it. Neither is automatically a statistical distribution, derating rule, thermal limit, or lifetime model.
03Which plane does the canonical capacitor model describe, and what is excluded there?
Model answer
It describes the two R1 package-terminal planes. The frozen first-order fixture includes constant ESR, ESL, and C, while pads, vias, fixture access, board return path, dielectric dispersion, radiation, nonlinear behavior, assembly spread, and higher modes are excluded.
04Why is Qseries zero at series resonance while the fixture also reports Q0 = 80.8122?
Model answer
They are different quantities. Port Qseries is |Im Z|/Re Z and therefore becomes zero when net reactance cancels. Q0 is the stored-energy series-resonator value ω0L/R = 1/(ω0CR), so it remains finite at the same resonance.
05What single measurement is pinned to falsify the default lumped model?
Model answer
A package-terminal complex-Z sweep at 0 V DC, 25 °C, and 1.0 V RMS containing exact points at 2.450000 GHz and around 5.62697 GHz. The lesson model fails if magnitude or phase exceeds the 2%/2° local criterion, or if the first resonance moves by more than 20%.
06Can a valid result in the inspector qualify a real component for production?
Model answer
No. Valid means that one selected synthetic model passes the disclosed local rules inside its fixture envelope. Purchasing, safety, qualification, surge, nonlinear, distribution, assembly, and lifetime evidence remain separate and unresolved.

Sources and further study

Sources actually used

Accessed 6 September 2026. Manufacturer values below orient the evidence audit; the interaction fixtures remain disclosed Illustrative synthetic data.

[1] D. M. Pozar. Microwave Engineering, 4th ed., Wiley, 2012. Used for impedance, resonance, networks, S-parameters, and reference-plane foundations.
[2] IEEE Std 370-2020. Official active-standard record, approved 2020-09-24 and published 2021-01-08; plus the 2022-01-21 errata. Used for fixture/interconnect characterization and validation orientation. Normative text was not reproduced.
[3] Anton Patyuchenko. “RF Signal Chain Discourse—Part 2: Essential Building Blocks”, Analog Dialogue, Vol. 55 No. 3, July 2021. Used for first-party signal-chain orientation.
[4] Murata. GJM0335C1E1R2WB01 SimSurfing record, updated 2026-08-20, and “The voltage characteristics of electrostatic capacitance”, 2012-11-28. Used for exact-family metadata, evidence labels, and MLCC bias orientation.
[5] Murata. SimSurfing MLCC Measurement Conditions, 2022-09-13. Used for network-analyzer, SOLT/TRL, land/jig, extraction, and frequency-range context.
[6] Coilcraft. 0402DC data sheet, Document 1153, revised 2023-11-09; 0402DC SPICE model note, Document 158, revised 2021-07-28; and inductor-model selection note, Document 1710, revised 2022-01-19. Used for fixture-conditioned Q/SRF, model boundaries, and the small-signal/current-saturation distinction.
[7] Nexperia. PESD5V0R1BDSF-Q data sheet, v.1, released 2025-07-02. Used to distinguish small-signal capacitance and scalar insertion-/input-return-loss evidence from voltage, TLP, surge, and IEC ESD evidence; the sheet does not provide a complex S-parameter table or file.
[8] Murata. SAFQA2G45MA0G0A product record and linked data sheet, data-sheet Revision K (current document checked 2026-09-06). Used for terminal orientation, grounded pins, 50 Ω + 9.1 nH matching conditions, temperature, and conditional power/lifetime statement. Current lifecycle label and official downloadable S2P remain unresolved.

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Next · 04.2Passive Routing, Sampling & Control

Carry valid component models into terminated passive multiport networks.

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Real Components, Models & Operating Regions

Failure: one nominal value, three RF behaviors

Value, tolerance, rating, and operating condition

Two current numbers, two meanings

Choose the model, not just the part

Ideal value

Lumped equivalent

S/Y/Z data

Behavioral / empirical

Nonlinear / time-domain

Decision tree

Package, pad, fixture, and reference plane

Package-terminal model

Pad / board model

Fixture-port record

Frequency dependence: SRF, Q, loss, dispersion, and modes

1 pF model-card baseline

Bias and temperature change the model

Which is the simplest adequate model?

See where a point agreement stops being a model agreement.

Moving the plane changes the network.

Included in the package model

Still excluded

Stress ledger—not a lifetime model.

The decision is inspectable, rule by rule.

Component model card

Power, heating, stress, and reliability margin

Electrical screen

Thermal screen

Reliability boundary

Typical, guaranteed, characterized, and simulated evidence

Validate with limits and one discriminating measurement

Measure package-terminal complex Z, not only nominal C.

Build the component evidence pack

Check your understanding

Sources actually used