Matching Networks That Survive Reality

01 / 10

Why the perfect nominal match failed

The center-frequency S11 dip looks excellent. Why did delivered power and board-to-board pass rate get worse?

The schematic used ideal L and C values at one frequency. The board used finite-Q 0402 parts, pads, vias, a load that moves with frequency, and component values drawn from a process. A deep dip at 2.450 GHz can coexist with dissipation, high circulating current, an edge-of-band miss, and a design that is painful to tune.

The cure is not automatically “more sections.” It is a complete decision contract: what must be optimized, across which states and planes, with which model, and against which evidence threshold. Only then can topology, nominal values, sensitivity, and tuning be compared.

R1 radio plane: Specified 50 + j0 Ω
R2 load plane: 30 − j20 Ω at 2.450 GHz
Declared band: 2.400–2.500 GHz
Local pass rule: RL ≥ 10 dB at all 41 points

Think about itWhich omission can a deeper center-frequency S11 dip repair by itself: component heating, antenna efficiency, or tolerance distribution?

Answer

None. S11 at R1 describes input reflection under its reference conditions. It neither partitions component dissipation nor measures radiated efficiency, and one nominal trace contains no component distribution.

Common misconceptionS11 at chart center alone defines success.

Success belongs to the declared engineering objective. A center point says nothing about the rest of the band, dissipative loss, component ratings, tuning access, antenna radiation, PA stress under large signal, or manufacturing variation.

Common misconceptionA wider sweep with one perfect dip is a broadband match.

Sweep width is only the width of the plot. Broadband performance requires the specified return-loss threshold to hold across the required operating band.

Illustrative engineering case · fixture node-load-table/1.0.0

The load trajectory is synthetic, not measured antenna data.

Five versioned knots from 2.400 to 2.500 GHz are linearly interpolated and include 30 − j20 Ω at center. No extrapolation is permitted. This lesson makes no claim about radiation efficiency, PA safety, temperature, enclosure state, or user loading.

02 / 10

Define the objective and both planes

What would another engineer need to reproduce before “make it 50 Ω” becomes testable?

GoalName the optimized quantityPower transfer, reflection, noise/gain optimum, harmonic load, or device stress are different problems.
Band and stateGive endpoints and operating stateHere: 2.400–2.500 GHz, small signal, 1 mW available-power normalization.
ModelsDefine source and load versus frequencyR1 is a real 50 Ω interface; R2 is synthetic table v1 with stated interpolation.
PlanesBracket what the network includesPads, feed, fixture, and calibration movement cannot be silently assigned to both sides.
EnvironmentName missing conditional dataTemperature, bias, power, enclosure, user proximity, and assembly state may move values.
AcceptanceState criterion and evidence classRL ≥ 10 dB at all 41 samples is this lesson’s local design rule, not a universal standard.

Definition · e^+jωt; inductors are +jωL and capacitors are −j/(ωC). R1 and R2 map to the portfolio’s radio-port and antenna-feed planes.

Source available power: 1.000 mW normalization at R1
Load-delivered power: Re{VL IL*} at R2
Prohibited shortcut: No radiated-efficiency inference

03 / 10

Conjugate match or specified interface?

Is the desired input always the complex conjugate of the load?

No. Start with the source model. For an RMS Thevenin voltage V_S and Z_S = R_S + jX_S, the accepted load power is maximized by presenting Z_S^* to that source, provided the system is linear and R_S > 0. The maximum is the source available power |V_S|²/(4R_S).

$I=\frac{V_{\mathrm S}}{Z_{\mathrm S}+Z_{\mathrm{in}}}$ RMS phasors; current flows into the network at R1.

$P_{\mathrm{del}}=|I|^2\operatorname{Re}\{Z_{\mathrm{in}}\}$ Accepted real power at the selected input, before its internal split is examined.

$Z_{\mathrm{in,opt}}=Z_{\mathrm S}^*\qquad P_{\mathrm{avs}}=\frac{|V_{\mathrm S}|^2}{4R_{\mathrm S}}$ Maximum-available-power result under this Thevenin model.

The recurring radio instead declares a real 50 Ω port contract at R1. That is the synthesis target in the Tradebench. A device data sheet, load-pull result, noise optimum, or stability constraint can demand a different target; those device-specific simultaneous decisions belong to Path 04.

Think about itIf a matching network delivers the source’s maximum available RF power, has it necessarily maximized battery efficiency?

Answer

No. The theorem starts at an RF Thevenin source and says nothing about DC input power, bias, waveform back-off, harmonic terminations, thermal behavior, or load-dependent active-device efficiency.

Common misconceptionMaximum power transfer always maximizes efficiency.

In the textbook resistive case, half the generated power is dissipated in the source resistance at maximum transfer. Real transmitter efficiency is a larger, device- and waveform-dependent problem. Do not replace it with a small-signal conjugate-match slogan.

Common misconceptionConjugate match and a fixed 50 Ω interface are always the same objective.

They coincide only when the source reference is real 50 Ω under the relevant state. A fixed interoperable interface, an optimum noise impedance, and a large-signal PA load are distinct contracts even if each is sometimes displayed on a 50 Ω-normalized chart.

04 / 10

Choose an L-section topology

Which side may carry the shunt branch, and which algebraic root becomes a physical L or C?

For a simple transformation between real resistances, place the shunt branch on the high-resistance side. The minimum two-reactance transformation then has loaded Q = √(R_high/R_low − 1). Choosing the positive or negative root gives the familiar low-pass or high-pass orientation. A complex load adds a reactance that must be included in the root algebra before L and C values are assigned.

Four named roots · source-to-load order · default complex load result
Form	Branch condition	Default result	Physical sequence
Input-shunt A	R_L ≤ R₀; +√ root	Valid	Input shunt C 1.06 pF → series L 2.89 nH
Input-shunt B	R_L ≤ R₀; −√ root	Valid	Input shunt L 3.98 nH → series C 14.45 pF
Load-shunt A	1/G_L ≥ R₀; +√ root	Rejected	Default 1/G_L = 43.33 Ω < 50 Ω
Load-shunt B	1/G_L ≥ R₀; −√ root	Rejected	Same negative-radicand condition

Derived · independently checked center-frequency trace

Candidate A reaches 50 + j0 Ω before finite Q is added.

With Z_L = 30 − j20 Ω, choose X_total = +√[30(50−30)] = +24.494897 Ω. Therefore X_series = 24.494897 − (−20) = +44.494897 Ω, or 2.890442 nH at 2.45 GHz. The intermediate impedance is 30 + j24.494897 Ω, whose admittance is 0.020000 − j0.016329932 S. A shunt +j0.016329932 S capacitor, 1.060812 pF, cancels the susceptance and leaves exactly 0.020000 S: Z_in = 50 + j0 Ω.

Go deeperThe two closed-form branch conditions

Input-shunt roots use X_total = ±√[R_L(R₀−R_L)], X_series = X_total−X_L, and B_shunt = X_total/(R_LR₀). Load-shunt roots first write Y_L = G_L + jB_L, then use B_total = ±√(G_L/R₀−G_L²), B_shunt = B_total−B_L, and X_series = B_total/(G_L²+B_total²). Every root is re-evaluated by complex ABCD algebra; negative radicands become reasons, not NaN.

05 / 10

Loaded Q links bandwidth and stress

Why can the same resistance transformation narrow response and raise internal voltage or current?

Reactive branches exchange stored energy each cycle. As the resistance ratio increases, the simple L network requires a higher loaded Q, so circulating reactive voltage/current grows relative to delivered real power. The same stored-energy concentration usually makes response more frequency-selective and more sensitive to value error.

Resistance stepR_high/R_low fixes Q for the simple L transformation.

Stored energyL and C exchange more reactive energy per delivered watt.

StressBranch voltage/current can exceed the matched 1 mW baseline.

SensitivityFrequency and component changes move the cancellation more sharply.

For the real 30-to-50 Ω step, Q = √(50/30−1) = 0.8165. The recurring load is complex, so this number only orients the resistance transformation; the −j20 Ω cancellation changes individual component values and stress. Likewise, bandwidth ≈ f₀/Q is a narrowband orientation for specified resonant forms—not an exact return-loss bandwidth for arbitrary matching networks.

Voltage baseline: √(1 mW · 50 Ω) = 223.6 mV RMS
Current baseline: √(1 mW / 50 Ω) = 4.472 mA RMS
Reported stress: Max branch RMS / stated baseline

Common misconceptionComponent Q affects only bandwidth, not loss or stress.

Component Q defines stored-to-dissipated behavior under a stated model. Finite series resistance spends real power, changes the input impedance, and changes the branch currents and voltages that determine heating and rating margin.

06 / 10

Real components spend power and stop being ideal

What has to be checked before an ideal 1.06 pF or 3.98 nH symbol becomes an 0402 part choice?

Loss modelQ, ESR, and DCR at use frequencyThe Tradebench uses series R_s(f)=|X(f)|/Q for every reactive branch, including the actual series impedance before a shunt conversion.
Operating regionSelf-resonance and parasiticsCapacitor ESL and inductor winding capacitance can reverse the intended behavior around SRF.
MountingPackage, pads, vias, and ground inductanceA nominal part model at its terminals does not automatically include the selected footprint or return path.
Conditional dataBias, temperature, frequency, and powerCapacitance, inductance, Q, and loss are conditional. One catalog headline is not the circuit model.
RatingsVoltage, current, temperature riseNormalized stress identifies where to investigate; it is not an absolute rating or derating claim.
Evidence upgradeVendor equivalent circuit or S-parametersUse first-party data at application frequency, then include the actual mounting structure and uncertainty.

Source available power1 mW from the declared Thevenin source; the maximum power available under conjugate loading.

Load-delivered powerRe{V_LI_L*} at R2; transducer gain is this quantity divided by source available power.

Reflected fraction|Γ|² at the real 50 Ω R1 interface; Γ = (Z_in−50)/(Z_in+50).

Component dissipationReal power entering the network at R1 minus real power delivered at R2.

Component-loss penalty10 log₁₀(P_del,lossless/P_del,lossy) under identical source/load conditions; residual mismatch remains separate.

Common misconceptionAn ideal L and C stay ideal at any frequency, bias, and power.

Murata’s capacitor guidance shows ESR and ESL shaping impedance and a transition from capacitive to inductive behavior above self-resonance. Coilcraft likewise emphasizes application-frequency L, Q, DCR, SRF, fixture, and current conditions. Constant Q = 80 is therefore a transparent teaching proxy, not a vendor model.

07 / 10

Lines, stubs, and transformers are alternatives

When is replacing lumped reactance with geometry a better bargain—and what new constraints appear?

L sectionSmall and tuneableTwo parts, selectable DC behavior, and easy stuffing changes; finite Q, SRF, pad/ground parasitics, and tolerance can dominate.
Quarter-wave transformerOne real-to-real line sectionZ_t=√(R₁R₂) at λ/4; useful at high frequency but long, narrowband, and not a direct solution for arbitrary complex loads.
Series line + open stubTwo geometric degrees of freedomMain line rotates to unit conductance; an open shunt stub cancels susceptance. Layout, dispersion, radiation, discontinuities, and trim access become first-order.
Transformer / balunRatio, balance, isolation, or DC pathCan solve more than impedance ratio, but bandwidth, magnetizing/leakage terms, loss, power handling, and available ratios must be checked.

Derived · declared local line interpretation

Both open-stub solutions are retained in [0, 0.5λ_g).

With Z₀ = 50 Ω and ε_eff = 3.4, the model places a shunt open-circuited stub at the input junction, followed by a lossless main-line section to the 30 − j20 Ω load. At 2.45 GHz the solutions are d = 0.17913λ_g with ℓ_stub = 0.10039λ_g (11.89 mm / 6.66 mm), and d = 0.48486λ_g with ℓ_stub = 0.39961λ_g(32.18 mm / 26.52 mm). Physical lengths remain fixed during a sweep.

Go deeperWhy an open stub supplies shunt susceptance

For a lossless open-circuited stub, Z_in = −jZ₀cot(βℓ), so Y_in = jY₀tan(βℓ). The main line is advanced from the load until its normalized input admittance has conductance one. The stub length is then chosen so its normalized susceptance is equal and opposite. Adding λ/2 repeats each impedance, which is why one half-wavelength interval contains the distinct solutions.

08 / 10

Bandwidth has a physical trade bound

Can enough lossless matching sections make an arbitrary reactive load perfectly matched across any band?

More sections can shape multiple reflection minima and trade ripple against bandwidth, but they add stored energy, component sensitivity, layout area, loss mechanisms, and tuning burden. Passive causal matching of a specified reactive load has integral constraints; a perfect isolated dip spends the available matching “area” without guaranteeing a useful band.

Definition · one specific Bode–Fano load class

Positive R in parallel with C, behind a passive lossless matching network

∫₀^∞ ln(1/|Γ(ω)|) dω ≤ π/(RC)

If |Γ| is held no greater than Γ_m over a band Δω, then Δω ln(1/Γ_m) ≤ π/(RC) is an orientation bound. It trades match depth against width for that exact load and network class. Equality generally demands an idealized high-order realization; a practical finite network may fall short.

This lesson deliberately does not plug the synthetic antenna’s center impedance into that formula. A five-knot impedance table has not been proven equivalent to one parallel RC over the relevant spectrum or shown to satisfy the causal model needed by the bound.

Common misconceptionMore matching sections create bandwidth for free.

They redistribute reflection and add realizability costs. Loss can also make S11 look better while reducing delivered power, so bandwidth must be paired with the power ledger.

Common misconceptionBode–Fano gives one universal maximum bandwidth.

The integral, weighting, and right-hand side depend on the exact RC/RL series/parallel load class and the passive lossless-network assumptions. It is a conditional trade bound, not a universal antenna bandwidth calculator.

09 / 10

Tolerance turns one point into a distribution

What does a repeatable 500-sample sweep reveal—and what can it never establish?

01Nominal + cornersCheck low/high combinations first; corners expose directional failures but do not assign probability.

02Low-discrepancy samplesHalton bases 2 and 3, skip 16, map source-to-load branches independently into bounded-uniform ±5% scales.

03All-frequency passOne sample passes only if all 41 inclusive frequencies meet the local return-loss target.

04Evidence gapCorrelations, lot drift, pad geometry, temperature, bias, power, assembly, and measurement uncertainty remain outside the proxy.

Deterministic low-discrepancy sampling covers the assumed box more evenly than an equally small unstructured random draw and produces byte-stable regression fixtures. It is not random Monte Carlo. The resulting fraction is conditional on the chosen distribution, independence, sample count, sequence, grid, and load model.

Think about itIf 500 of 500 deterministic samples pass, is production yield proven to be 100%?

Answer

No. It only says every sampled point in the stated two-variable tolerance model met the numerical criterion. Production yield needs defensible process distributions, correlations, environmental/power behavior, assembly effects, measurement uncertainty, and real evidence.

Common misconceptionA nominal tolerance sweep proves production yield.

Synthetic component bounds are a design-screening model. Call the result a deterministic pass fraction or yield proxy, preserve the exact assumptions, and plan the measurements that can replace those assumptions.

Class 1 interaction · future Class 3 candidate

Matching Network Tradebench

Select a topology, inspect what survives the band and deterministic tolerance sequence, then record what the model still cannot prove.

1 · Source and load contract

Versioned seeded table Direct complex values

Synthetic node-load table v1.0.0; linear interpolation only, with no extrapolation.

Source R Ω · 1–500Source X Ω · −500–500

Illustrative synthetic R2 load · 5 exact knots
Frequency	Load impedance
2.400 GHz	26.00 − j31.00 Ω
2.425 GHz	28.00 − j25.50 Ω
2.450 GHz	30.00 − j20.00 Ω
2.475 GHz	33.00 − j14.50 Ω
2.500 GHz	36.00 − j8.00 Ω

2 · Band, topology, and reality proxies

Center frequency GHz · 0.1–10Fractional bandwidth % · 0.1–30Topology named source-to-load orderComponent Q 10–500 · center proxyBounded tolerance % · 0–20Low-discrepancy samples integer · 50–5000Return-loss target dB · 6–20

Calculated endpoints: 2.4000–2.5000 GHz. The return-loss threshold is a local design criterion, not a standard.

Default calculation ready.

The complete default result remains visible without JavaScript. Enable JavaScript only to commit different inputs, copy the memo, or change the focused candidate.

Simulated · illustrative small-signal evidence

2.400–2.500 GHz · R1 target 50.0 Ω · R2 synthetic load

Available source power is normalized to 1 mW. Pass means RL ≥ 10.0 dB at every one of 41 band points for a deterministic tolerance sample.

All valid candidates · explicit auto order: pass fraction, worst-band RL, component-loss penalty, stress, then ID
Candidate	Nominal values	Center Zin / \|Γ\|	Worst RL / \|Γ\|	Delivered-power band / minimum G_T	Max dissipation / component-loss penalty	Normalized V / I stress	Tolerance pass
[1] Input shunt C → series L · branch A	E1: shunt C 1.06 pF; E2: series L 2.89 nH	49.67 − j0.88 Ω / 0.0094	13.28 dB / 0.2167 @ 2.4000 GHz	0.926–0.972 mW / 0.9265	0.0293 mW / 0.13 dB	1.19× / 1.34×	500/500 · 100.0%
[2] Input shunt L → series C · branch BAuto selection	E1: shunt L 3.98 nH; E2: series C 14.45 pF	49.51 + j0.08 Ω / 0.0050	14.11 dB / 0.1970 @ 2.5000 GHz	0.945–0.988 mW / 0.9447	0.0168 mW / 0.06 dB	1.19× / 1.35×	500/500 · 100.0%
[3] Input shunt open stub → series line A	main line 11.89 mm (0.1791 λg); open stub 6.66 mm (0.1004 λg)	50.00 − j0.00 Ω / 0.0000	12.91 dB / 0.2263 @ 2.4000 GHz	0.949–1.000 mW / 0.9488	0.0000 mW / 0.00 dB	1.09× / 1.49×	500/500 · 100.0%
[4] Input shunt open stub → series line B	main line 32.18 mm (0.4849 λg); open stub 26.52 mm (0.3996 λg)	50.00 − j0.00 Ω / 0.0000	12.17 dB / 0.2464 @ 2.4000 GHz	0.939–1.000 mW / 0.9393	0.0000 mW / 0.00 dB	1.24× / 1.34×	384/500 · 76.8%

2 rejected branches · show the algebraic reasons

Load-shunt branch A: Load-shunt forms require the load parallel resistance 1/Gload ≥ Rtarget. Here it is 43.33333333333333 Ω, below 50 Ω, so the square root is negative.
Load-shunt branch B: Load-shunt forms require the load parallel resistance 1/Gload ≥ Rtarget. Here it is 43.33333333333333 Ω, below 50 Ω, so the square root is negative.

Simulated · nominal branch values · 41-point inclusive grid. Higher is better. Values above 40 dB are clipped visually, never in the table.

[1]Input shunt C → series L · branch A
[2]Input shunt L → series C · branch B
[3]Input shunt open stub → series line A
[4]Input shunt open stub → series line B

Chart-equivalent nominal frequency data · return loss / delivered power
Frequency	Synthetic load	[1] RL / mW	[2] RL / mW	[3] RL / mW	[4] RL / mW
2.4000 GHz	26.00 − j31.00 Ω	13.28 dB / 0.927	14.14 dB / 0.945	12.91 dB / 0.949	12.17 dB / 0.939
2.4025 GHz	26.20 − j30.45 Ω	13.73 dB / 0.932	14.62 dB / 0.949	13.36 dB / 0.954	12.65 dB / 0.946
2.4050 GHz	26.40 − j29.90 Ω	14.21 dB / 0.936	15.12 dB / 0.953	13.84 dB / 0.959	13.16 dB / 0.952
2.4075 GHz	26.60 − j29.35 Ω	14.71 dB / 0.940	15.65 dB / 0.957	14.35 dB / 0.963	13.70 dB / 0.957
2.4100 GHz	26.80 − j28.80 Ω	15.24 dB / 0.944	16.21 dB / 0.960	14.89 dB / 0.968	14.28 dB / 0.963
2.4125 GHz	27.00 − j28.25 Ω	15.80 dB / 0.947	16.81 dB / 0.964	15.47 dB / 0.972	14.88 dB / 0.968
2.4150 GHz	27.20 − j27.70 Ω	16.41 dB / 0.951	17.46 dB / 0.967	16.08 dB / 0.975	15.53 dB / 0.972
2.4175 GHz	27.40 − j27.15 Ω	17.05 dB / 0.954	18.15 dB / 0.970	16.75 dB / 0.979	16.23 dB / 0.976
2.4200 GHz	27.60 − j26.60 Ω	17.75 dB / 0.957	18.89 dB / 0.972	17.46 dB / 0.982	16.97 dB / 0.980
2.4225 GHz	27.80 − j26.05 Ω	18.51 dB / 0.959	19.71 dB / 0.975	18.23 dB / 0.985	17.78 dB / 0.983
2.4250 GHz	28.00 − j25.50 Ω	19.33 dB / 0.961	20.60 dB / 0.977	19.08 dB / 0.988	18.66 dB / 0.986
2.4275 GHz	28.20 − j24.95 Ω	20.24 dB / 0.963	21.59 dB / 0.979	20.02 dB / 0.990	19.63 dB / 0.989
2.4300 GHz	28.40 − j24.40 Ω	21.26 dB / 0.965	22.70 dB / 0.981	21.06 dB / 0.992	20.72 dB / 0.992
2.4325 GHz	28.60 − j23.85 Ω	22.39 dB / 0.967	23.97 dB / 0.983	22.25 dB / 0.994	21.93 dB / 0.994
2.4350 GHz	28.80 − j23.30 Ω	23.70 dB / 0.968	25.45 dB / 0.984	23.61 dB / 0.996	23.33 dB / 0.995
2.4375 GHz	29.00 − j22.75 Ω	25.22 dB / 0.969	27.21 dB / 0.985	25.22 dB / 0.997	24.98 dB / 0.997
2.4400 GHz	29.20 − j22.20 Ω	27.04 dB / 0.970	29.42 dB / 0.986	27.18 dB / 0.998	26.98 dB / 0.998
2.4425 GHz	29.40 − j21.65 Ω	29.31 dB / 0.971	32.36 dB / 0.987	29.71 dB / 0.999	29.54 dB / 0.999
2.4450 GHz	29.60 − j21.10 Ω	32.27 dB / 0.971	36.77 dB / 0.987	33.26 dB / 1.000	33.12 dB / 1.000
2.4475 GHz	29.80 − j20.55 Ω	36.34 dB / 0.972	45.86 dB / 0.988	39.31 dB / 1.000	39.21 dB / 1.000
2.4500 GHz	30.00 − j20.00 Ω	40.51 dB / 0.972	46.08 dB / 0.988	308.76 dB / 1.000	307.08 dB / 1.000
2.4525 GHz	30.30 − j19.45 Ω	37.02 dB / 0.972	36.71 dB / 0.988	38.91 dB / 1.000	39.24 dB / 1.000
2.4550 GHz	30.60 − j18.90 Ω	32.64 dB / 0.971	32.23 dB / 0.988	32.93 dB / 0.999	33.30 dB / 1.000
2.4575 GHz	30.90 − j18.35 Ω	29.51 dB / 0.971	29.31 dB / 0.987	29.46 dB / 0.999	29.86 dB / 0.999
2.4600 GHz	31.20 − j17.80 Ω	27.18 dB / 0.970	27.14 dB / 0.987	27.00 dB / 0.998	27.45 dB / 0.998
2.4625 GHz	31.50 − j17.25 Ω	25.34 dB / 0.969	25.43 dB / 0.986	25.11 dB / 0.997	25.59 dB / 0.997
2.4650 GHz	31.80 − j16.70 Ω	23.82 dB / 0.967	24.01 dB / 0.985	23.58 dB / 0.996	24.09 dB / 0.996
2.4675 GHz	32.10 − j16.15 Ω	22.54 dB / 0.966	22.81 dB / 0.984	22.29 dB / 0.994	22.84 dB / 0.995
2.4700 GHz	32.40 − j15.60 Ω	21.44 dB / 0.964	21.76 dB / 0.983	21.18 dB / 0.992	21.77 dB / 0.993
2.4725 GHz	32.70 − j15.05 Ω	20.47 dB / 0.962	20.84 dB / 0.981	20.20 dB / 0.990	20.83 dB / 0.992
2.4750 GHz	33.00 − j14.50 Ω	19.60 dB / 0.960	20.01 dB / 0.980	19.34 dB / 0.988	20.00 dB / 0.990
2.4775 GHz	33.30 − j13.85 Ω	18.72 dB / 0.958	19.16 dB / 0.978	18.46 dB / 0.986	19.14 dB / 0.988
2.4800 GHz	33.60 − j13.20 Ω	17.93 dB / 0.955	18.39 dB / 0.975	17.67 dB / 0.983	18.37 dB / 0.985
2.4825 GHz	33.90 − j12.55 Ω	17.22 dB / 0.952	17.70 dB / 0.973	16.95 dB / 0.980	17.68 dB / 0.983
2.4850 GHz	34.20 − j11.90 Ω	16.57 dB / 0.949	17.06 dB / 0.971	16.30 dB / 0.977	17.06 dB / 0.980
2.4875 GHz	34.50 − j11.25 Ω	15.97 dB / 0.946	16.47 dB / 0.968	15.71 dB / 0.973	16.49 dB / 0.978
2.4900 GHz	34.80 − j10.60 Ω	15.41 dB / 0.942	15.93 dB / 0.965	15.15 dB / 0.969	15.96 dB / 0.975
2.4925 GHz	35.10 − j9.95 Ω	14.89 dB / 0.938	15.43 dB / 0.962	14.64 dB / 0.966	15.48 dB / 0.972
2.4950 GHz	35.40 − j9.30 Ω	14.41 dB / 0.935	14.96 dB / 0.959	14.17 dB / 0.962	15.03 dB / 0.969
2.4975 GHz	35.70 − j8.65 Ω	13.96 dB / 0.931	14.52 dB / 0.955	13.72 dB / 0.958	14.62 dB / 0.965
2.5000 GHz	36.00 − j8.00 Ω	13.54 dB / 0.926	14.11 dB / 0.952	13.30 dB / 0.953	14.23 dB / 0.962

Learner decision

Focus the evidence for one candidate

Candidate for sensitivity, corners, and memo [1] Input shunt C → series L · branch A [2] Input shunt L → series C · branch B [3] Input shunt open stub → series line A [4] Input shunt open stub → series line B

Dominant modeled tradeoff: Bandwidth margin is the main discriminator: every deterministic sample passes, while nominal worst-band return loss is 14.11 dB.

Sensitivity ranking · one branch at low/nominal/high bounded value
Rank / branch	Low worst RL	Nominal	High worst RL	Low-to-high span
1 · E1	13.47 dB	14.11 dB	13.83 dB	0.36 dB
2 · E2	13.98 dB	14.11 dB	13.99 dB	0.01 dB

Best, median, and worst sampled corners
Corner	Sequence index	Branch scales	Worst RL	Pass
best	36	1.01719 / 1.04753	14.19 dB	Yes
median	318	1.04473 / 1.02092	13.93 dB	Yes
worst	495	0.95010 / 1.04657	13.37 dB	Yes

Copyable / printable artifact

Match decision memo

MATCH DECISION MEMO · ILLUSTRATIVE SMALL-SIGNAL EVIDENCE
Model matching-tradebench/1.0.0; fixture node-load-table/1.0.0; grid linear-inclusive-41/1.0.0; sequence halton-2-3-skip-16/1.0.0.
Objective: transform the synthetic R2 load to the radio's specified 50.0 Ω R1 interface from 2.400 to 2.500 GHz; local criterion RL ≥ 10.0 dB at all 41 samples; 1 mW available source power; small signal.
Selected candidate: Input shunt L → series C · branch B. E1: shunt L 3.98 nH; E2: series C 14.45 pF.
Nominal evidence: center Zin 49.51 + j0.08 Ω; worst-band RL 14.11 dB at 2.5000 GHz; minimum delivered power 0.945 mW.
Dissipation: maximum component-loss penalty 0.06 dB relative to the otherwise identical lossless network. This is not an unexplained insertion-loss claim.
Stress proxy: max voltage 1.19× and current 1.35× relative to √(1 mW·R0) and √(1 mW/R0). Verify actual ratings with vendor data.
Tolerance proxy: 500/500 (100.0%) deterministic Halton samples pass. Independent bounded-uniform ±5.0% branch scaling is assumed; this is not measured factory yield.
Tuning access: Keep the populated series and shunt 0402 pads, add a DNP shunt pad at the opposite side to preserve a π-style tuning option, use a short ground-via pair, and retain probeable R1/R2 structures.
First lab evidence: establish/calibrate a one-port VNA reference plane at the accessible fixture, characterize the fixture/coupon, then verify or de-embed to R1 and R2 before comparing the impedance trajectory. Detailed calibration execution belongs to Path 08.
Prohibited inferences: this result does not prove antenna efficiency, radiated performance, PA ruggedness, harmonic compliance, noise optimum, large-signal behavior, or production yield.

Model assumptions

Illustrative small-signal model with e^(+jωt), R1 radio plane, R2 antenna-feed plane, and 1 mW source available power.
The input match target is the declared real source resistance; Γ = (Zin − R0)/(Zin + R0).
L and C use exact ideal frequency dependence; each lumped branch adds series Rs(f) = |X(f)|/Q before series/shunt placement.
The line option is a lossless main line followed at its input by one shunt open-circuited stub; Z0 equals the real interface resistance and εeff = 3.4.
Tolerance uses independent bounded-uniform element/length scale factors from the pinned Halton sequence; it is not measured production yield.
Halton bases attach to source-to-load branch position, so permuting physical branches intentionally permutes which deterministic coordinate drives each branch.
A sample passes only when return loss meets the local target at all 41 inclusive, uniformly spaced band frequencies.

Warnings and prohibited inferences

The constant-Q model omits SRF, package/pad parasitics, bias, temperature, and power dependence. Verify each chosen 0402 part at application frequency.
Do not infer PA ruggedness, harmonic emissions, noise optimum, antenna efficiency, radiated performance, compliance, or production yield.

Versions: matching-tradebench/1.0.0 · node-load-table/1.0.0 · linear-inclusive-41/1.0.0 · halton-2-3-skip-16/1.0.0 · matching-display/1.0.0

10 / 10

Write the node match decision memo

Which candidate would you build first, and what evidence keeps that choice honest?

Under the pinned default fixture, both lumped branches pass all 500 deterministic samples. Candidate B—the input shunt 3.98 nH followed by series 14.45 pF—has the stronger nominal worst-band return loss (14.11 dB) and lower modeled component-loss penalty (0.063 dB) than candidate A (13.28 dB and 0.126 dB). That makes B the explicit auto-ranked first prototype under this model—not a universal winner.

Path 03 capstone artifact · final matching decision

Record the decision so a reviewer can reproduce and challenge it.

Objective, band, small-/large-signal state, source/load definitions, and R1/R2 planes.
Candidate topology and nominal values, including element order and shunt location.
Nominal Γ/return-loss trajectory, delivered power, component dissipation, and component-loss penalty.
Normalized voltage/current stress plus the vendor evidence still needed for absolute ratings.
Deterministic tolerance pass fraction, exact distribution/independence/grid/sequence contract, sensitivity ranking, and corners.
Preserved tuning footprints or stub trim access, ground-return intent, and probeable structures.
First calibration/reference-plane verification and the fixture/de-embedding evidence needed in the lab.
Prohibited inferences: no PA ruggedness, harmonic, NF, antenna-efficiency, compliance, or factory-yield claim.

Common misconceptionA good conducted match proves antenna efficiency or radiated performance.

It proves neither. The input can accept power and still dissipate it in conductors, dielectric, enclosure currents, or loss resistance. Learning Path 06 owns radiation and efficiency; this lesson stops at R2 accepted/delivered power.

Path 04.4 / 04.5 · planned

Simultaneous noise/gain/stability matching, nonlinear load pull, harmonic terminations, PA efficiency, power handling, and ruggedness.

Path 04.2 / future specialty · planned

Automatic or closed-loop tuners and device-specific control.

Path 06 · planned

Antenna radiation, efficiency, pattern, environment, and OTA evidence.

Path 08 · planned

VNA calibration, de-embedding, measurement uncertainty, and execution.

Path 10.2–10.4 · planned

PCB layout/build execution, tuning workflow, and full-wave correlation.

Tools candidate · not implemented

A generalized production-grade matching/tolerance synthesizer remains a future Class 3 opportunity.

Ungraded review

Check your understanding

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

01Why is ‘make it 50 Ω’ not a complete matching objective?
Model answer
It omits the goal, band, source and load models, power state, environment, input/output planes, acceptance criterion, and evidence class. A 50 Ω target can mean a fixed interface, not necessarily a source conjugate or a device-specific optimum.
02When does the conjugate-match condition apply?
Model answer
For a linear Thevenin source ZS = RS + jXS with RS > 0 and a load/network that may be adjusted, maximum available power occurs when the impedance presented to the source is ZS*. This statement does not by itself maximize DC-to-RF efficiency, antenna efficiency, noise performance, or ruggedness.
03What does Q = √(Rhigh/Rlow − 1) tell you here?
Model answer
It is the loaded Q of the stated simple real-resistance L transformation. It orients circulating energy, selectivity, stress, and sensitivity; it is not an exact arbitrary-network bandwidth formula and does not include the extra cancellation needed for a complex load.
04Why is component-loss penalty different from mismatch loss?
Model answer
The lesson compares delivered power through the lossy network with its otherwise identical lossless counterpart under the same source and load. That isolates the teaching model's dissipation. Residual mismatch is separately visible in Γ and return loss.
05What precisely is the distributed candidate?
Model answer
A lossless 50 Ω main-line section from the shunt junction to the load, plus one 50 Ω shunt open-circuited stub at the input junction, with εeff = 3.4 and physical lengths fixed after center-frequency synthesis. Both solutions are reported in [0, 0.5λg).
06Why is Bode–Fano not a universal maximum-bandwidth equation for the synthetic antenna?
Model answer
The displayed bound applies to a specified positive parallel-RC load driven through a passive lossless matching network. The synthetic five-knot antenna table has not been shown to be that causal equivalent circuit over the required range, so inserting one center-frequency R and C would manufacture a false limit.
07What does the 500-sample pass fraction establish?
Model answer
Only a deterministic sensitivity/yield proxy under independent bounded-uniform branch scaling, a pinned Halton sequence, a 41-point grid, the synthetic load, and the local return-loss threshold. It is neither random Monte Carlo nor measured factory yield.
08What is the first lab evidence requested by the memo?
Model answer
Establish a calibrated one-port VNA plane at the accessible fixture, characterize the fixture or coupon, and verify or de-embed to the declared R1 and R2 planes before comparing trajectories. Path 08 owns the calibration procedure and uncertainty budget.

Sources and further study

Accessed 5 September 2026. Definition/Informative sources are paraphrased. Numerical curves and pass fractions are Simulated by matching-tradebench/1.0.0; the node load is Illustrative (node-load-table/1.0.0); the 41-point sweep is linear-inclusive-41/1.0.0; tolerance sampling is halton-2-3-skip-16/1.0.0. No source chart, proprietary part data, or measured antenna trace is reproduced.

Network theory and synthesis

David M. Pozar, Microwave Engineering, 4th ed., matching, transmission-line/stub, network, and bandwidth chapters. Stable theory; not a component/fixture model.
R. M. Fano, MIT RLE Technical Report No. 41, Theoretical Limitations on the Broadband Matching of Arbitrary Impedances (1948), and the 1950 Journal of the Franklin Institute papers. Original realizability/integral-bound treatment.
Georgia Tech ECE notes, Chapter 6: Impedance Matching and Tuning: L matching, single-stub orientation, and explicit RC/RL Bode–Fano load classes. University treatment; ideal network assumptions apply.
Virginia Tech open textbook, Electromagnetics, §3.23: single shunt open/short-stub formulation and worked center-frequency check.

Implementation and future evidence

Analog Devices, Simon Bramble, “Radio Frequency Impedance Matching: Calculations and Simulations” (2021). Informative L/T and complex-load implementation cross-check; the lesson’s equations were independently derived and tested.
Murata, “What are impedance/ESR frequency characteristics in capacitors?”. First-party ESR, ESL, SRF, and application-frequency orientation; no proprietary curve is reused.
Coilcraft, “Testing Inductors at Application Frequencies”. First-party L/Q/DCR/SRF, fixture, frequency, and current context; no specific part is selected.
IEEE, IEEE 370-2020, active at access, published 2021-01-08 with linked errata. Reference-plane/interconnect characterization orientation only; Path 08 owns procedure.
Keysight, “Applying Error Correction to Vector Network Analyzer Measurements”. One-/two-port calibration, systematic/random/drift error, adapters, TRL/SOLT, and uncertainty orientation; no measurement procedure is implemented here.

Path handoff

The matching decision closes the approved Path 03 capstone artifact.

Module 03.5 is implemented and hands its Smith-chart/algebra record into this decision. Return to the path overview for the current availability of every other dependency.

Previous implemented module03.5 · Smith Chart as an Engineering Map

← Return to Transmission Lines & Matching

Matching Networks That Survive Reality

Why the perfect nominal match failed

The load trajectory is synthetic, not measured antenna data.

Define the objective and both planes

Conjugate match or specified interface?

Choose an L-section topology

Candidate A reaches 50 + j0 Ω before finite Q is added.

Loaded Q links bandwidth and stress

Real components spend power and stop being ideal

Lines, stubs, and transformers are alternatives

Both open-stub solutions are retained in [0, 0.5λg).

Bandwidth has a physical trade bound

Positive R in parallel with C, behind a passive lossless matching network

Tolerance turns one point into a distribution

Matching Network Tradebench

Focus the evidence for one candidate

Match decision memo

Model assumptions

Warnings and prohibited inferences

Write the node match decision memo

Record the decision so a reviewer can reproduce and challenge it.

Check your understanding

Sources and further study

Network theory and synthesis

Implementation and future evidence

The matching decision closes the approved Path 03 capstone artifact.

Both open-stub solutions are retained in [0, 0.5λ_g).