RF Journey

01 / 10

Failure: minimum noise, unstable amplifier

The illustrative 2.450 GHz gateway has 1.500 dB of filter loss before its LNA. A co-located transmitter puts a −25.000 dBm blocker at the LNA input. An engineer selects the source impedance with the lowest noise figure, then connects an output network chosen independently. The two choices close a feedback loop.

We continue the conditional evidence pack from 04.1 and the passive port discipline from 04.2. Review Path 03 two-port networks and noise and cascades if the reference planes or noise factor are unfamiliar. The preceding filter lesson, 04.3, is planned; its assumed 1.5 dB loss is disclosed here.

Definition / illustrative plane map. ΓS and ΓL look outward from the package into the external source and load. Γin and Γout look into the LNA with the opposite termination attached. Supply modes require a larger model.

Think about itWould you choose the 0.750 dB minimum-noise source over a 1.272 dB matched source before checking the load?

Answer

At ΓS = Γopt = 0.950∠90°, the isolated noise equation gives 0.750000 dB. But Γout = −0.329421 − j0.961734, or 1.016588∠−108.907772°. Its magnitude above one permits a passive load to close the output loop. The optimum alone is insufficient.

Choose ΓL = 1/Γout = −0.31875824313240125 + j0.9306047176263108. This passive critical load makes the loop denominator vanish. The solver retains the reciprocal at full precision; the displayed 0.983683∠108.907772° is only a label.

\begin{aligned}\Gamma _{\mathrm{S}} &= \Gamma _{\mathrm{opt}}, \Gamma _{\mathrm{L}} = \frac{1}{\Gamma _{\mathrm{out}}} \\ &1 - \Gamma _{\mathrm{out}}\Gamma _{\mathrm{L}} \approx 0\qquad |D| < 10^{-9}\end{aligned}

Derived from the frozen nominal 2.450 GHz fixture at R1, port 1 input / port 2 output, 25 °C, real 50 Ω. This singularity belongs to the one-frequency linear model.

At that critical point, Γin ≈ 1.052632∠−90°, and both loop magnitudes equal one within numerical precision. Gain, operating NF and cascade NF are unavailable. With the same source and ΓL = 0 instead, the output loop magnitude is zero and |D| = 0.558359. The matched load has not been shown to oscillate: the conservative risk is that another passive load can close the loop.

Fixture, not a part recommendation. Every value and contour in the map is synthetic. Model p04-m04-lna-twoport-v1 describes a settled small-signal state, not a transistor topology or physical matching network. The case remains Illustrative even when its equations are exact.

Illustrative canonical fixture · p04-m04-lna-fixtures/1.0.0 · nominal bias, 2.450 GHz
Parameter	Polar / scalar	Rectangular / definition
S11	0.650000 ∠ -60.000000°	0.325000 − j0.562917
S21	4.500000 ∠ 70.000000°	1.539091 + j4.228617
S12	0.120000 ∠ 25.000000°	0.108757 + j0.050714
S22	0.550000 ∠ -45.000000°	0.388909 − j0.388909
Fmin	0.750000 dB	1.188502227 linear factor
Rn	4.000000 Ω	Equivalent noise resistance
Γopt	0.950000 ∠ 90.000000°	0.000000 + j0.950000
Reference	50.000 Ω; T0 = 290.000 K	Real power-wave normalization; device 25 °C is a separate condition
Bias / current	Nominal settled, 3.0 V / 15.000 mA	Synthetic operating state; no physical bias circuit
Planes / drive	R1 port 1 input, port 2 output	Infinitesimal small-signal linearization; illustrative characterization drive −60 dBm available CW, no finite-power guarantee
Input P1dB / IIP3	−10.000 / 0.000 dBm	Only nominal / 2.450 GHz / 25 °C / ΓS=ΓL=0 / real 50 Ω / R1 input / single unmodulated CW tone supports a scalar input-P1dB screen (−10 dBm). Other tuple values are unqualified context proxies.

Inspect all 15 frozen bias/frequency knots and interpolation rules

p04-m04-lna-twoport-v1 · p04-m04-lna-fixtures/1.0.0. Authored teaching tuples, not vendor data or process/temperature samples. Low/high tuples independently specified, not scaled nominal matrices. Exact knots retained. Between knots: linear Cartesian S and Γopt, linear noise factor Fmin, Rn in Ω, current in mA, compression/intercept powers in mW. No phase interpolation or extrapolation. All tuples use the same R1 planes, 50 Ω, T0 = 290 K, infinitesimal drive and 25 °C. The following compression/intercept columns are context proxies; only the canonical nominal P1dB row supports the live margin.

Illustrative knot ledger · S magnitudes are linear, phases degrees; Fmin dB; Rn Ω; current mA; input P1dB/IIP3 dBm
Bias / GHz	S11	S21	S12	S22	Fmin / Rn	Γopt	mA / P1 / IP3
low / 0.5	0.72 ∠ -25°	5.1 ∠ 140°	0.07 ∠ 12°	0.61 ∠ -20°	1.1 / 7	0.62 ∠ 38°	7 / -17 / -7
low / 1	0.68 ∠ -41°	4.8 ∠ 118°	0.085 ∠ 18°	0.56 ∠ -32°	0.95 / 6	0.7 ∠ 60°	7.5 / -16 / -6
low / 2.45	0.59 ∠ -67°	3.7 ∠ 78°	0.1 ∠ 31°	0.49 ∠ -51°	0.9 / 5.2	0.82 ∠ 84°	8 / -14 / -4
low / 4	0.53 ∠ -102°	2.8 ∠ 24°	0.13 ∠ 49°	0.47 ∠ -89°	1.25 / 7	0.72 ∠ 121°	8.3 / -13 / -3
low / 6	0.64 ∠ -149°	1.9 ∠ -43°	0.16 ∠ 72°	0.6 ∠ -135°	1.9 / 10	0.63 ∠ 164°	8.6 / -12 / -2
nominal / 0.5	0.79 ∠ -22°	6.4 ∠ 138°	0.085 ∠ 8°	0.69 ∠ -17°	0.85 / 5.5	0.72 ∠ 42°	14 / -13 / -3
nominal / 1	0.73 ∠ -37°	5.8 ∠ 114°	0.1 ∠ 15°	0.62 ∠ -28°	0.7 / 4.7	0.84 ∠ 66°	14.5 / -12 / -2
nominal / 2.45	0.65 ∠ -60°	4.5 ∠ 70°	0.12 ∠ 25°	0.55 ∠ -45°	0.75 / 4	0.95 ∠ 90°	15 / -10 / 0
nominal / 4	0.56 ∠ -96°	3.5 ∠ 12°	0.145 ∠ 45°	0.5 ∠ -83°	1.05 / 5.8	0.83 ∠ 130°	15.4 / -9 / 1
nominal / 6	0.61 ∠ -143°	2.4 ∠ -58°	0.18 ∠ 65°	0.58 ∠ -127°	1.65 / 8.8	0.71 ∠ 173°	15.8 / -8 / 2
high / 0.5	0.67 ∠ -19°	6.8 ∠ 131°	0.065 ∠ 6°	0.61 ∠ -14°	0.9 / 4.8	0.65 ∠ 35°	23 / -9 / 1
high / 1	0.61 ∠ -34°	6.2 ∠ 106°	0.075 ∠ 11°	0.54 ∠ -24°	0.72 / 4.1	0.73 ∠ 56°	23.5 / -8 / 2
high / 2.45	0.55 ∠ -55°	5.1 ∠ 61°	0.09 ∠ 21°	0.46 ∠ -39°	0.82 / 3.6	0.8 ∠ 81°	24 / -6 / 4
high / 4	0.48 ∠ -91°	4 ∠ 4°	0.12 ∠ 39°	0.43 ∠ -77°	1.12 / 5.2	0.76 ∠ 119°	24.7 / -5 / 5
high / 6	0.57 ∠ -138°	2.7 ∠ -66°	0.15 ∠ 62°	0.52 ∠ -119°	1.75 / 8	0.67 ∠ 160°	25.5 / -4 / 6

First distinguish the power ratios. Otherwise “more gain” can mean that the denominator changed while delivered power did not.

02 / 10

Which gain do you mean?

Use RMS voltage V and current I into each port, time dependence e^+jωt, and real Z0 = Zref = 50 Ω. The power waves a and b have units √W. At the device, a travels inward and b outward; net input power is |a|² − |b|². ΓS and ΓL look out from R1 into the source and load.

\begin{aligned}a &= \frac{V + Z_{0}I}{2\sqrt{Z_{0}}} \\ b &= \frac{V - Z_{0}I}{2\sqrt{Z_{0}}} \\ b_{1} &= S_{11}a_{1} + S_{12}a_{2}\qquad b_{2} = S_{21}a_{1} + S_{22}a_{2}\end{aligned}

Definitions for real positive Z0; V in volts RMS, I in amperes RMS. Complex normalization needs different power-wave treatment.

Definitions · small signal, fixed frequency, bias, temperature and R1 planes
Gain	Power ratio	Termination / existence condition
Transducer GT	PL / Pavs	Delivered load power over available source power; use the actual ΓS and ΓL.
Available GA	Pavn / Pavs	Maximum available output power for the chosen source. Passive output conjugate match requires \|Γout\| < 1 and stable operation.
Operating GP	PL / Pin	Delivered load power over positive net power entering port 1; fixed ΓL, \|Γin\| < 1.
Maximum available MAG	Maximum GT over source and load	Stable simultaneous conjugate match, at this frequency. Bilateral K > 1 and \|Δ\| < 1; not the original conditional fixture.
Maximum stable MSG	\|S21\| / \|S12\| boundary limit	K = 1 limiting result for a stabilized network; not an achievable optimum asserted for an unchanged K < 1 device.

\begin{aligned}D &= (1-S_{11}\Gamma _{\mathrm{S}})(1-S_{22}\Gamma _{\mathrm{L}}) - S_{12}S_{21}\Gamma _{\mathrm{S}}\Gamma _{\mathrm{L}} \\ G_{\mathrm{T}} &= \frac{(1-|\Gamma _{\mathrm{S}}|^{2})|S_{21}|^{2}(1-|\Gamma _{\mathrm{L}}|^{2})}{|D|^{2}}\end{aligned}

Derived bilateral gain. Pavs is source available power, PL is net delivered load power, all in watts. S and Γ are dimensionless complex wave ratios; D is dimensionless.

\begin{aligned}G_{\mathrm{A}} &= \frac{|S_{21}|^{2}(1-|\Gamma _{\mathrm{S}}|^{2})}{[|1-S_{11}\Gamma _{\mathrm{S}}|^{2}(1-|\Gamma _{\mathrm{out}}|^{2})]} \\ G_{\mathrm{P}} &= \frac{|S_{21}|^{2}(1-|\Gamma _{\mathrm{L}}|^{2})}{[|1-S_{22}\Gamma _{\mathrm{L}}|^{2}(1-|\Gamma _{\mathrm{in}}|^{2})]}\end{aligned}

Derived for passive conjugate availability and positive net input power, respectively. Each denominator must be positive and the operating solution nonsingular.

For ΓS = ΓL = 0, GT = 4.5² = 20.25, so 10 log₁₀(GT) = 13.064250 dB. GA = 14.628808 dB and GP = 15.448730 dB at their stated power boundaries. S21 describes excitation with the other port matched; its squared magnitude equals GT only in this matched case. The gain definitions and the full feedback derivation agree with Niknejad, slides 54–70.

Go deeperCheck the maximum-gain limits without dividing by S12 = 0

\mathrm{MAG} = \frac{|\frac{S_{21}}{S_{12}}|}{[K + \sqrt{K^{2}-1}]}

Derived bilateral MAG for K > 1, |Δ| < 1; the rationalized form avoids subtracting nearly equal large terms.

In a strictly unilateral, individually stable two-port, MAG = |S21|²/[(1−|S11|²)(1−|S22|²)]. For S11 = 0.2, S22 = 0.3, S21 = 2 and S12 = 0, simultaneous conjugate terminations give 4/(0.96 × 0.91). K is undefined as a finite ratio here; do not print infinity as a result. If both reflections are zero but S12 = 0.1, S21 = 2, the bilateral checks give K = 2.6, μ = μ′ = 5 and MAG = 4.

Common misconceptionThe data-sheet gain is the gain for any source and load.

It belongs to a stated matching circuit, stimulus and operating state. Changing a termination changes both mismatch and bilateral feedback.

03 / 10

Reverse transmission makes terminations matter

A wave leaving port 2 returns as a₂ = ΓLb₂. Solving the port-2 equation gives b₂ = S21a₁/(1−S22ΓL). Substitute that into port 1 to see why its reflection changes when the load moves. Repeat from the other end for Γout.

\begin{aligned}\Gamma _{\mathrm{in}} &= S_{11} + \frac{S_{12}S_{21}\Gamma _{\mathrm{L}}}{1-S_{22}\Gamma _{\mathrm{L}}} \\ \Gamma _{\mathrm{out}} &= S_{22} + \frac{S_{12}S_{21}\Gamma _{\mathrm{S}}}{1-S_{11}\Gamma _{\mathrm{S}}}\end{aligned}

Derived at R1. Γin = b₁/a₁ with ΓL attached; Γout = b₂/a₂ with the source generator suppressed and ΓS attached. All quantities are complex.

The feedback term contains S12S21, whose magnitude is 0.54 here. A reverse transmission magnitude of 0.12 is not enough information to neglect it. ΓL can also make 1−S22ΓL small; phase determines whether the returned wave adds or subtracts.

Derived comparison · nominal 2.450 GHz · ΓS = 0.750∠90°, ΓL = 0.700∠105° · same power boundaries
Model	GT (dB)	What changed
Full bilateral	16.624453	Keep the reverse feedback term in D.
Unilateral approximation	11.776172	Set S12 = 0 only in the transfer solution; this is a comparison, not the fixture.
Critical load with ΓS = Γopt	Unavailable	Actual bilateral loop denominator is singular. A finite unilateral answer would conceal the risk.

Think about itIf you improve output return loss, must input return loss stay unchanged?

Answer

No. Γin depends on ΓL through reverse transmission. Even at one frequency, you must recompute the source and load together. If S12 = 0 exactly, Γin reduces to S11 and Γout to S22; that limiting case is included in the model tests.

The useful next question is whether every passive termination is allowed, or only a restricted region.

04 / 10

Conditional and unconditional stability

An unconditionally stable small-signal two-port admits all passive source and load terminations at the stated frequency under its model assumptions. A conditionally stable one requires restricted terminations. Neither statement covers omitted supply ports, nonlinear drive, or transients.

\begin{aligned}\Delta &= S_{11}S_{22} - S_{12}S_{21} \\ K &= \frac{1-|S_{11}|^{2}-|S_{22}|^{2}+|\Delta|^{2}}{2|S_{12}S_{21}|} \\ \mu &= \frac{1-|S_{11}|^{2}}{[|S_{22}-\Delta \cdot \operatorname{conj}(S_{11})|+|S_{12}S_{21}|]} \\ \mu ' &= \frac{1-|S_{22}|^{2}}{[|S_{11}-\Delta \cdot \operatorname{conj}(S_{22})|+|S_{12}S_{21}|]}\end{aligned}

Derived one-frequency tests. Δ is the complex determinant, K is Rollett's factor; conj denotes complex conjugation. K requires nonzero |S12S21|.

We pin μ as the input/source robustness measure and μ′ as output/load robustness; the formula, rather than a software label, identifies each. For a stable underlying two-port, K > 1 with |Δ| < 1 is the usual unconditional test. The μ test is a geometric equivalent; both directional measures are reported. These are frequency-local quantities, independent of which external Γ point you select.

Derived canonical invariants · independently reproduced with Python complex arithmetic
Quantity	Result	Decision
Δ / \|Δ\|	−0.0454637 − j0.8832636 / 0.8844329	Determinant magnitude below one alone is insufficient.
K	0.9789089	Does not pass K > 1.
μ / μ′	0.8741151 / 0.9403716	Does not pass unconditional stability.
ΓS = ΓL = 0	\|Γin\| = 0.65; \|Γout\| = 0.55	Matched point is finite and passes the local stability screen.

Go deeperCircle equations, a known-point test, and the straight-line limit

\begin{aligned}C_{\mathrm{S}} &= \frac{\operatorname{conj}(S_{11}-\Delta \cdot \operatorname{conj}(S_{22}))}{|S_{11}|^{2}-|\Delta|^{2}} \\ r_{\mathrm{S}} &= \frac{|S_{12}S_{21}|}{\operatorname{abs}(|S_{11}|^{2}-|\Delta|^{2})} \\ C_{\mathrm{L}} &= \frac{\operatorname{conj}(S_{22}-\Delta \cdot \operatorname{conj}(S_{11}))}{|S_{22}|^{2}-|\Delta|^{2}} \\ r_{\mathrm{L}} &= \frac{|S_{12}S_{21}|}{\operatorname{abs}(|S_{22}|^{2}-|\Delta|^{2})}\end{aligned}

Derived circle geometry, dimensionless Γ coordinates; CS/rS in the source plane, CL/rL in the load plane. Magnitudes on the radius denominator keep radii nonnegative.

Here CS ≈ 0.002301 − j0.560785, rS ≈ 1.501161; CL ≈ 0.194945 − j0.158958, rL ≈ 1.125653. Both origins lie inside their circles and on the stable side. If a circle denominator has magnitude ≤ 10⁻¹², the implementation draws its implicit locus instead of dividing. For the source plane that locus is |S22−ΔΓS|² − |1−S11ΓS|² = 0; exchange the port indices for the load plane.

The map refines crossings against this exact equation. Magnitudes ≤ 1−10⁻⁶ are stable, ≥ 1+10⁻⁶ unstable, and the open band between is boundary/inspect. A loop magnitude of one is not by itself phase closure; the complex denominator tests that separately.

Common misconceptionK > 1 in the receive band proves the assembled LNA cannot oscillate.

K does not include a supply resonance absent from the model. Check frequency coverage, bias, temperature, state transitions, layout feedback and drive level. A small-signal file cannot certify all those modes.

05 / 10

Noise parameters and noise circles

Noise matching asks which source impedance minimizes added noise. The standard four real parameters are minimum noise factor Fmin, equivalent noise resistance Rn in Ω, and the two real coordinates of Γopt. Their frequency, bias, temperature and de-embedding planes must match the S-parameters. The reference temperature T0 = 290 K is distinct from device temperature 25 °C.

F=F_{\min}+\frac{4R_n}{Z_0}\frac{|\Gamma_{\mathrm S}-\Gamma_{\mathrm{opt}}|^2}{(1-|\Gamma_{\mathrm S}|^2)|1+\Gamma_{\mathrm{opt}}|^2}

Definition / derived standard four-parameter model, real positive Z0 = 50 Ω and passive |ΓS| < 1. F and Fmin are linear factors; NF = 10 log₁₀ F in dB. Rn/Z0 is dimensionless.

The extra term is nonnegative. At ΓS = Γopt it vanishes exactly, giving NF = 0.750000 dB. With ΓS = 0, convert Fmin to 10^0.75/10 first, add the term using Rn = 4 Ω and Γopt = j0.95, then take 10 log₁₀: NF = 1.272028 dB. Rn measures sensitivity away from the optimum; it is not a physical series resistor to add to a circuit.

Think about itCan the noise equation remain finite when the connected amplifier has no finite steady-state gain?

Answer

Yes. At the critical candidate it still algebraically returns Fmin, but that does not validate a noisy operating state. The map suppresses NF and cascade NF whenever its stability or conditioning screen fails, including the exact critical load.

Go deeperTurn a permitted noise factor into a source-plane circle

\begin{aligned}N &= \frac{(F_{\mathrm{target}}-F_{\mathrm{min}})|1+\Gamma _{\mathrm{opt}}|^{2}Z_{0}}{4R_{n}} \\ C_{\mathrm{N}} &= \frac{\Gamma _{\mathrm{opt}}}{1+N} \\ r_{\mathrm{N}} &= \frac{\sqrt{N(N+1-|\Gamma _{\mathrm{opt}}|^{2})}}{1+N}\end{aligned}

Derived by completing the square. Pick Ftarget ≥ Fmin in linear units; N is dimensionless, CN and rN use source Γ coordinates only.

At Ftarget = Fmin, N = 0 and the circle collapses to Γopt. A target below Fmin has no locus. Larger allowed NF admits a region that can intersect gain and stable-side constraints. The four-parameter interpretation is also described in Maury Microwave’s noise-parameter paper; this lesson draws its own synthetic contours.

Common misconceptionThe best input S11 is the lowest-noise input.

S11 is the matched-output input reflection of the device. Γopt is a source termination defined by its noise correlations. Even a conjugate source match to Γin need not minimize noise, and Γin itself changes with ΓL.

06 / 10

Trade gain, noise, input match, and stability

Start with a feasible region. Then decide which improvement is worth spending match margin, current or evidence on. Simultaneous conjugate matching solves ΓS = conj(Γin) and ΓL = conj(Γout) together, when such a stable passive solution exists. Noise matching fixes ΓS = Γopt instead. A 50 Ω input match and robustness to all passive loads are different objectives again.

Derived four frozen candidates · p04-m04-lna-twoport-v1 · nominal 3 V / 15 mA · 2.450 GHz · 25 °C · real 50 Ω R1 · 1.5 dB matched pre-loss, −25 dBm blocker
Quantity / condition	Matched	Minimum-noise / critical-load rejected	Gain-oriented	Stable compromise
ΓS at R1 input	0.000000 ∠ 0.000000°	0.950000 ∠ 90.000000°	0.750000 ∠ 90.000000°	0.750000 ∠ 90.000000°
ΓL at R1 output	0.000000 ∠ 0.000000°	0.983683 ∠ 108.907772°	0.700000 ∠ 105.000000°	0.000000 ∠ 0.000000°
Γin magnitude / phase	0.650000 ∠ -60.000000°	1.052632 ∠ -90.000000°	0.854892 ∠ -89.620051°	0.650000 ∠ -60.000000°
Γout magnitude / phase	0.550000 ∠ -45.000000°	1.016588 ∠ -108.907772°	0.714394 ∠ -104.757744°	0.714394 ∠ -104.757744°
Source / load loop magnitudes	0.000000 / 0.000000	1.000000 / 1.000000	0.641169 / 0.500075	0.487500 / 0.000000
\|D\| (full precision solve)	1.000000000e+0	6.206335383e-17	3.135190775e-1	6.271215842e-1
GT / GA / GP (dB)	13.064250 / 14.628808 / 15.448730	unavailable / unavailable / unavailable	16.624453 / 16.628210 / 17.013321	13.526996 / 16.628210 / 15.448730
LNA NF (dB)	1.272028	unavailable	0.805834	0.805834
Cascade NF (dB)	2.772028	unavailable	2.305834	2.305834
Cascade condition	matched 290 K attenuator	Suppressed: singular or unstable screen	hypothetical ideal tuner after matched 290 K attenuator	hypothetical ideal tuner after matched 290 K attenuator
Blocker margin (dB), screen only	15.000	Inspect: P1dB condition mismatch	Inspect: P1dB condition mismatch	Inspect: P1dB condition mismatch
Current proxy (mA)	15.000	15.000	15.000	15.000
Local decision	ACCEPT: all local axes pass	REJECT: singular; stable-side and loop boundary	INSPECT: blocker evidence missing; input match > 0.80	INSPECT: blocker evidence missing

The gain-oriented point has 16.624453 dB GT and 0.805834 dB NF, but |Γin| = 0.854892 misses this lesson’s 0.80 input-match limit. Keeping the same source and returning the load to zero yields the stable compromise: 13.526996 dB GT, the same NF, |Γin| = 0.65 and |Γout| = 0.714394. Both changed-source cases still need suitable compression data.

\begin{aligned}F_{\mathrm{total}}&=L+\frac{F_{\mathrm{LNA}}-1}{1/L}=L F_{\mathrm{LNA}}\\\mathrm{NF}_{\mathrm{total}}&=L_{\mathrm{dB}}+\mathrm{NF}_{\mathrm{LNA}}\end{aligned}

Derived cascade subfixture only: matched available loss L = 10^(lossdB/10), physical attenuator temperature Tp = T0 = 290 K; its gain is 1/L and noise factor L.

The matched baseline therefore becomes 2.772028 dB after 1.500 dB loss. For ΓS ≠ 0 we explicitly insert an ideal noiseless, lossless tuner after the matched attenuator. The stable compromise then has a hypothetical 2.305834 dB cascade NF. A separate 2.250000 dB minimum-noise sum would require that ideal tuner to present Γopt and a stable nonsingular load; it is unavailable for the critical-load candidate. The conservative map does not certify that Γopt network.

Amplifying harder after the loss cannot recover the SNR lost before the LNA. If the real filter is mismatched, lossy at another temperature, or connected to a lossy tuner, use a compatible noise-wave network model before replacing this subfixture. General receiver allocation belongs to Path 05.

Illustrative local requirement card · p04-m04-local-criteria-v1. Accept requires stable loops and stable sides, |D| > 10⁻⁶, GT ≥ 10 dB, NF ≤ 1.50 dB, both port reflections ≤ 0.80, current ≤ 20 mA and a supported blocker margin ≥ 10 dB. Reject takes precedence for invalid/domain input, singular |D| ≤ 10⁻⁹, unstable states, undefined/nonpositive GT, NF > 3 dB, current > 30 mA or supported margin < 0 dB. A near-singular denominator or other unmet accept limit is inspect; gain is suppressed near singularity. Hardware evidence gaps remain a separate list.

Now test the sequence below. Predict a winner before changing the controls; after each result, name the binding criterion.

Class 1 · embedded decision model · p04-m04-lna-map-v1

LNA Gain–Noise–Stability Map

Choose source, load and bias; explain which constraint binds. Start from the matched baseline, then compare the apparent noise winner with a stable compromise.

Predict the minimum-NF winner, then select the critical-load preset.
Set only ΓL magnitude to 0 and apply. Distinguish a conservative load risk from this selected loop.
Use numeric or keyboard controls to set ΓS = 0.750∠90°, ΓL = 0. Compare the stable compromise.
Change pre-loss between 0 and 1.5 dB, then compare all three bias states. Record the missing blocker evidence.

Named candidate presetEach named preset atomically restores nominal bias, 2.450 GHz, 1.5 dB loss and −25 dBm blocker, including its exact source and load.

Contour plane / view

Operating state

Bias stateThree independent sets of knots; no interpolation between bias states.

Frequency (GHz)0.500–6.000 GHz; 0.001 GHz keyboard step. Exact knots: 0.500, 1.000, 2.450, 4.000, 6.000. Between knots: declared interpolation.

ΓS · R1 input source · real 50 Ω

ΓS entry modeBoth representations stay synchronized on Apply or mode change. Presets retain full precision; steps govern manual increments.

ΓS magnitude0.000–0.990; dimensionless; step 0.010.

ΓS phase (°)−180° to +180°; step 1°. Zero magnitude has no physical phase.

ΓL · R1 output load · real 50 Ω

ΓL entry modeBoth representations stay synchronized on Apply or mode change. Presets retain full precision; steps govern manual increments.

ΓL magnitude0.000–0.990; dimensionless; step 0.010.

ΓL phase (°)−180° to +180°; step 1°. Zero magnitude has no physical phase.

Gateway subfixtures

Pre-LNA loss (dB)0.0–5.0 dB; step 0.1 dB. Matched attenuator at 290 K, then an ideal tuner if ΓS ≠ 0.

Blocker at LNA input R1 (dBm)−60.0…+10.0 dBm; step 0.5 dB. Single CW screening only; does not change the small-signal matrix.

Results reflect the applied point.

JavaScript is unavailable. The matched baseline, fixture, four candidate rows, contour summaries and written operating-point decision remain available below and in the lesson.

Illustrative local screen · p04-m04-local-criteria-v1ACCEPT — nominal bias at 2.450 GHz

The matched point passes every local synthetic criterion. Hardware evidence remains a separate inspection request.

Transducer gain GT · Derived: 13.064250 dB
LNA noise figure · Derived: 1.272028 dB
Cascade noise figure · Derived: 2.772028 dB
Blocker margin · screen only: 15.000 dB

matched 290 K attenuator: 1.500 dB irreversible pre-loss at 290 K. For ΓS ≠ 0, no realizable tuner bandwidth, loss, or stability has been designed. Device: 25 °C, real 50 Ω, settled nominal bias. Exact 2.450 GHz knot.

Inspect exact reflections, gain definitions and every decision axis

Derived exact selected point · R1 package planes · full bilateral power-wave solution
Quantity	Value	Condition / meaning
ΓS / ΓL	0.000000 + j0.000000 / 0.000000 + j0.000000	0.000000 ∠ 0.000000° / 0.000000 ∠ 0.000000°
ZS / ZL (Ω)	50.000000 + j0.000000 / 50.000000 + j0.000000	Z = 50(1+Γ)/(1−Γ); exact Γ=1 is open, not a finite impedance
Normalized zS / zL	1.000000 + j0.000000 / 1.000000 + j0.000000	Dimensionless Z / 50 Ω
Passive-bound distances	1.000000 / 1.000000	1−\|ΓS\| / 1−\|ΓL\|; UI also enforces \|Γ\| ≤ 0.990
Γin	0.325000 − j0.562917	0.650000 ∠ -60.000000°
Γout	0.388909 − j0.388909	0.550000 ∠ -45.000000°
1−Γin ΓS / 1−Γout ΓL	1.000000 + j0.000000 / 1.000000 + j0.000000	Complex loop denominators; unity-loop magnitude alone is not a phase-closure proof
D	1.000000 + j0.000000; \|D\| = 1.000000000e+0	(1−S11ΓS)(1−S22ΓL)−S12S21ΓSΓL
GA / GP	14.628808 / 15.448730 dB	Available gain needs passive output conjugate; operating gain needs positive net input power
MAG / MSG	unavailable / unavailable dB	MAG: one-frequency unconditional stability. MSG: K=1 boundary limit; not a promised gain for the original K<1 network.
Δ / \|Δ\|	-0.045464 − j0.883264 / 0.8844329	Independent of external ΓS/ΓL for this frozen S matrix
K / μ / μ′	0.9789089 / 0.8741151 / 0.9403716	One-frequency unconditional-stability test not satisfied
Distance from Γopt	0.950000	Γopt = 0.950000 ∠ 90.000000°; Fmin 0.750000 dB; Rn 4.000000 Ω
Source circle CS / rS	0.002301 − j0.560785 / 1.501161	Inside contains Γ=0 (stable); test exact \|Γout\| at the chosen point
Load circle CL / rL	0.194945 − j0.158958 / 1.125653	Inside contains Γ=0 (stable); test exact \|Γin\| at the chosen point
IIP3 context	0.000 dBm, no IM3 prediction	Context only: nominal 0 dBm input intercept, synthetic equal CW tones centered at 2.450 GHz, 1 MHz spacing, −40 dBm per tone, matched 50 Ω, 25 °C. One blocker gives no IM3 estimate. Other bias/frequency values are context proxies only.

Every local decision axis · reject takes precedence; no hidden aggregate score
Axis	Result	Value and rule
Denominator conditioning	ACCEPT	\|D\| = 1.000000e+0. reject ≤ 1e−9; inspect (gain suppressed) ≤ 1e−6; accept > 1e−6
Source loop \|Γin ΓS\|	ACCEPT	0.000000. stable ≤ 1−1e−6; unstable ≥ 1+1e−6; intervening band: inspect
Load loop \|Γout ΓL\|	ACCEPT	0.000000. same unity band; loop magnitude is a sufficient screen, not a full dynamical proof
Source-plane stable side \|Γout\|	ACCEPT	0.550000. conservative passive-load robustness; failing this does not prove the selected matched load oscillates
Load-plane stable side \|Γin\|	ACCEPT	0.650000. conservative passive-source robustness; test exact defining function, not a shading convention
Transducer gain	ACCEPT	13.064250 dB. defined positive GT and ≥ 10.0 dB to accept
Noise figure	ACCEPT	1.272028 dB. accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair
Input match	ACCEPT	0.650000 ∠ -60.000000°. accept \|Γin\| ≤ 0.80 at the selected ΓL
Output match	ACCEPT	0.550000 ∠ -45.000000°. accept \|Γout\| ≤ 0.80 at the selected ΓS
Bias current proxy	ACCEPT	15.000 mA. accept ≤ 20.0 mA; reject > 30.0 mA; no battery-life or thermal inference
Blocker compression screen	ACCEPT	15.000 dB (screen only). ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic

GT: 10 / 13 / 16 dB
NF: 1 / 1.5 / 3 dB
Stability boundary
× Conservative unstable side
◆ Selected · M matched · C compromise · G gain · X critical

Simulated / Illustrative · p04-m04-lna-twoport-v1. nominal, 2.450 GHz, R1, 50 Ω, 25 °C, T0 = 290 K. ΓL held at 0.000000 ∠ 0.000000°. Exact selected point 0.000000 ∠ 0.000000° is evaluated independently of the 0.025 Cartesian grid. Noise and gain gaps include rejected or ill-conditioned cells; dense crosses indicate a conservative termination region, not observed oscillation. Only candidates with the same held termination/bias/frequency are marked.

Inspect contour levels, stable regions and boundary coordinates

Simulated source-plane contour summary · 4925 passive grid samples · step 0.025 · held termination printed above
Locus	Segments	Meaning
GT = 10 dB	335	Exact equation refines each grid-edge crossing; straight segments between crossings; gaps never bridge unavailable cells
GT = 13 dB	252	Exact equation refines each grid-edge crossing; straight segments between crossings; gaps never bridge unavailable cells
GT = 16 dB	0	No supported contour segments in this view; not an optimum or absence-of-risk claim
NF = 1 dB	135	Exact equation refines each grid-edge crossing; straight segments between crossings; gaps never bridge unavailable cells
NF = 1.5 dB	255	Exact equation refines each grid-edge crossing; straight segments between crossings; gaps never bridge unavailable cells
NF = 3 dB	359	Exact equation refines each grid-edge crossing; straight segments between crossings; gaps never bridge unavailable cells
\|Γout\| = 1	71	Exact equation refines each grid-edge crossing; straight segments between crossings; gaps never bridge unavailable cells

Simulated stable-side counts · exact reflection-function test, not a hidden score
Region	Grid points
stable	4869
boundary	0
unstable	56
pole	0

Derived stability-boundary key points · refined Cartesian edge crossings; inspect within 1e−6 of unity
Key / Γ re + j im	\|Γout\| or \|Γin\|	Region
B1: -0.435885 + j0.875000	1.000000	Boundary / inspect
B2: 0.369718 + j0.894718	1.000000	Boundary / inspect
B3: -0.227284 + j0.922716	1.000000	Boundary / inspect
B4: 0.308762 + j0.908762	1.000000	Boundary / inspect
B5: -0.114147 + j0.935853	1.000000	Boundary / inspect
B6: 0.050000 + j0.939618	1.000000	Boundary / inspect

Inspect every Cartesian grid sample

Rows 1–20 of 4925; ordered by increasing imaginary coordinate, then real coordinate. The exact selected point is 0.000000 ∠ 0.000000° and is calculated separately. Download the complete labelled grid to inspect all points.

Simulated contour grid · current view and held termination · no hardware inference
Γ re / im	GT / NF dB	Reflection / region	Local status and binding reasons
-0.150 / -0.975	-6.802563 / 13.895440	0.846085 / stable	reject: Transducer gain: inspect; -6.802563 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 13.895440 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.846085 ∠ -32.744957°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.125 / -0.975	-5.781275 / 12.952169	0.844265 / stable	reject: Transducer gain: inspect; -5.781275 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 12.952169 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.844265 ∠ -32.414809°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.100 / -0.975	-5.080426 / 12.319964	0.842554 / stable	reject: Transducer gain: inspect; -5.080426 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 12.319964 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.842554 ∠ -32.078577°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.075 / -0.975	-4.592142 / 11.891153	0.840955 / stable	reject: Transducer gain: inspect; -4.592142 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.891153 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.840955 ∠ -31.736451°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.050 / -0.975	-4.262487 / 11.611949	0.839473 / stable	reject: Transducer gain: inspect; -4.262487 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.611949 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.839473 ∠ -31.388643°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.025 / -0.975	-4.062834 / 11.453827	0.838111 / stable	reject: Transducer gain: inspect; -4.062834 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.453827 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.838111 ∠ -31.035384°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
0.000 / -0.975	-3.978908 / 11.402565	0.836873 / stable	reject: Transducer gain: inspect; -3.978908 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.402565 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.836873 ∠ -30.676927°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
0.025 / -0.975	-4.006366 / 11.453827	0.835764 / stable	reject: Transducer gain: inspect; -4.006366 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.453827 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.835764 ∠ -30.313545°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
0.050 / -0.975	-4.149582 / 11.611949	0.834787 / stable	reject: Transducer gain: inspect; -4.149582 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.611949 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.834787 ∠ -29.945531°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
0.075 / -0.975	-4.422861 / 11.891153	0.833946 / stable	reject: Transducer gain: inspect; -4.422861 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.891153 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.833946 ∠ -29.573199°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
0.100 / -0.975	-4.854863 / 12.319964	0.833244 / stable	reject: Transducer gain: inspect; -4.854863 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 12.319964 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.833244 ∠ -29.196884°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
0.125 / -0.975	-5.499552 / 12.952169	0.832685 / stable	reject: Transducer gain: inspect; -5.499552 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 12.952169 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.832685 ∠ -28.816937°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
0.150 / -0.975	-6.464836 / 13.895440	0.832272 / stable	reject: Transducer gain: inspect; -6.464836 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 13.895440 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.832272 ∠ -28.433729°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.275 / -0.950	-7.787614 / 14.702304	0.852231 / stable	reject: Transducer gain: inspect; -7.787614 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 14.702304 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.852231 ∠ -34.459067°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.250 / -0.950	-5.710704 / 12.750224	0.849843 / stable	reject: Transducer gain: inspect; -5.710704 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 12.750224 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.849843 ∠ -34.156189°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.225 / -0.950	-4.406841 / 11.560577	0.847544 / stable	reject: Transducer gain: inspect; -4.406841 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 11.560577 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.847544 ∠ -33.846236°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.200 / -0.950	-3.485041 / 10.742871	0.845337 / stable	reject: Transducer gain: inspect; -3.485041 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 10.742871 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.845337 ∠ -33.529298°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.175 / -0.950	-2.795160 / 10.147376	0.843227 / stable	reject: Transducer gain: inspect; -2.795160 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 10.147376 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.843227 ∠ -33.205483°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.150 / -0.950	-2.263894 / 9.701132	0.841219 / stable	reject: Transducer gain: inspect; -2.263894 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 9.701132 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.841219 ∠ -32.874922°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic
-0.125 / -0.950	-1.850349 / 9.363527	0.839317 / stable	reject: Transducer gain: inspect; -1.850349 dB; defined positive GT and ≥ 10.0 dB to accept \| Noise figure: reject; 9.363527 dB; accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair \| Output match: inspect; 0.839317 ∠ -32.537764°; accept \|Γout\| ≤ 0.80 at the selected ΓS \| Blocker compression screen: inspect; inspect—P1dB condition mismatch; ≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic

Hardware evidence — always inspect separately

Out-of-band and between-knot modes: characterize below 0.5 GHz, above 6 GHz and between sparse samples.
All-mode and bias-network stability: include decoupling, supply impedance, startup, shutdown and control transitions.
Large-signal evidence: actual modulated blocker, recovery, two-tone spacing, and worst source/load impedance.
Temperature, process and thermal evidence: characterized populations, guaranteed limits and dissipation; current is only a proxy.
Layout and measured stability: package/board planes, feedback coupling, source/load sweeps and measurement uncertainty.

Local accept remains reachable. It approves only the disclosed lesson criteria, not a physical component or matching network.

Copyable / printable LNA operating-point model card

{
  "metadata": {
    "model": "p04-m04-lna-twoport-v1",
    "interaction": "p04-m04-lna-map-v1",
    "fixture": "p04-m04-lna-fixtures/1.0.0",
    "criteria": "p04-m04-local-criteria-v1",
    "evidence": "Illustrative synthetic parameters; Simulated contours; Derived equations/results",
    "phasor": "exp(+jωt); RMS V/I; normalized power waves a,b in √W, currents into DUT",
    "zrefOhms": 50,
    "referenceTemperatureK": 290,
    "deviceTemperatureC": 25,
    "planes": "R1 de-embedded package input port 1 and output port 2; a1/a2 into DUT, b1/b2 out",
    "drive": "Infinitesimal small-signal linearization; illustrative characterization drive −60 dBm available CW, no finite-power guarantee",
    "bias": "Three separate settled synthetic bias states at 3.0 V; no bias interpolation or physical bias circuit",
    "frequencyKnotsGHz": [
      0.5,
      1,
      2.45,
      4,
      6
    ],
    "interpolation": "Exact knots retained. Between knots: linear Cartesian S and Γopt, linear noise factor Fmin, Rn in Ω, current in mA, compression/intercept powers in mW. No phase interpolation or extrapolation.",
    "provenance": "Authored teaching tuples, not vendor data or process/temperature samples. Low/high tuples independently specified, not scaled nominal matrices.",
    "compression": "Only nominal / 2.450 GHz / 25 °C / ΓS=ΓL=0 / real 50 Ω / R1 input / single unmodulated CW tone supports a scalar input-P1dB screen (−10 dBm). Other tuple values are unqualified context proxies.",
    "iip3": "Context only: nominal 0 dBm input intercept, synthetic equal CW tones centered at 2.450 GHz, 1 MHz spacing, −40 dBm per tone, matched 50 Ω, 25 °C. One blocker gives no IM3 estimate.",
    "cascade": "Matched attenuator at physical Tp=T0=290 K, then ideal noiseless lossless transformation presenting ΓS; ΓS≠0 is hypothetical ideal tuner. Requires stable nonsingular pair.",
    "criteriaEvidence": "Illustrative local requirements, not a device-selection standard",
    "contourPlane": "source",
    "hardwareInspect": [
      "Out-of-band and between-knot modes: characterize below 0.5 GHz, above 6 GHz and between sparse samples.",
      "All-mode and bias-network stability: include decoupling, supply impedance, startup, shutdown and control transitions.",
      "Large-signal evidence: actual modulated blocker, recovery, two-tone spacing, and worst source/load impedance.",
      "Temperature, process and thermal evidence: characterized populations, guaranteed limits and dissipation; current is only a proxy.",
      "Layout and measured stability: package/board planes, feedback coupling, source/load sweeps and measurement uncertainty."
    ]
  },
  "input": {
    "bias": "nominal",
    "ghz": 2.45,
    "source": {
      "re": 0,
      "im": 0
    },
    "load": {
      "re": 0,
      "im": 0
    },
    "lossDb": 1.5,
    "blockerDbm": -25
  },
  "s": {
    "s11": {
      "re": 0.32500000000000007,
      "im": -0.562916512459885
    },
    "s21": {
      "re": 1.5390906449655097,
      "im": 4.228616793536587
    },
    "s12": {
      "re": 0.10875693444439799,
      "im": 0.05071419140888393
    },
    "s22": {
      "re": 0.3889087296526012,
      "im": -0.38890872965260115
    }
  },
  "noiseParameters": {
    "fminLinear": 1.1885022274370185,
    "rnOhms": 4,
    "gammaOpt": {
      "re": 5.817072295949927e-17,
      "im": 0.95
    }
  },
  "interpolation": "Exact 2.450 GHz knot",
  "display": {
    "gammaSource": "0.000000 ∠ 0.000000°",
    "gammaLoad": "0.000000 ∠ 0.000000°",
    "gammaIn": "0.650000 ∠ -60.000000°",
    "gammaOut": "0.550000 ∠ -45.000000°",
    "delta": "-0.045464 − j0.883264",
    "k": "0.9789089",
    "mu": "0.8741151",
    "muPrime": "0.9403716",
    "denominator": "1.000000000e+0",
    "transducerDb": "13.064250",
    "availableDb": "14.628808",
    "operatingDb": "15.448730",
    "nfDb": "1.272028",
    "cascadeNfDb": "2.772028",
    "cascadeCondition": "matched 290 K attenuator",
    "blockerScreenMarginDb": "15.000",
    "currentMa": "15.000"
  },
  "status": "accept",
  "cause": "The matched point passes every local synthetic criterion. Hardware evidence remains a separate inspection request.",
  "axes": [
    {
      "name": "Denominator conditioning",
      "status": "accept",
      "value": "|D| = 1.000000e+0",
      "rule": "reject ≤ 1e−9; inspect (gain suppressed) ≤ 1e−6; accept > 1e−6"
    },
    {
      "name": "Source loop |Γin ΓS|",
      "status": "accept",
      "value": "0.000000",
      "rule": "stable ≤ 1−1e−6; unstable ≥ 1+1e−6; intervening band: inspect"
    },
    {
      "name": "Load loop |Γout ΓL|",
      "status": "accept",
      "value": "0.000000",
      "rule": "same unity band; loop magnitude is a sufficient screen, not a full dynamical proof"
    },
    {
      "name": "Source-plane stable side |Γout|",
      "status": "accept",
      "value": "0.550000",
      "rule": "conservative passive-load robustness; failing this does not prove the selected matched load oscillates"
    },
    {
      "name": "Load-plane stable side |Γin|",
      "status": "accept",
      "value": "0.650000",
      "rule": "conservative passive-source robustness; test exact defining function, not a shading convention"
    },
    {
      "name": "Transducer gain",
      "status": "accept",
      "value": "13.064250 dB",
      "rule": "defined positive GT and ≥ 10.0 dB to accept"
    },
    {
      "name": "Noise figure",
      "status": "accept",
      "value": "1.272028 dB",
      "rule": "accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair"
    },
    {
      "name": "Input match",
      "status": "accept",
      "value": "0.650000 ∠ -60.000000°",
      "rule": "accept |Γin| ≤ 0.80 at the selected ΓL"
    },
    {
      "name": "Output match",
      "status": "accept",
      "value": "0.550000 ∠ -45.000000°",
      "rule": "accept |Γout| ≤ 0.80 at the selected ΓS"
    },
    {
      "name": "Bias current proxy",
      "status": "accept",
      "value": "15.000 mA",
      "rule": "accept ≤ 20.0 mA; reject > 30.0 mA; no battery-life or thermal inference"
    },
    {
      "name": "Blocker compression screen",
      "status": "accept",
      "value": "15.000 dB (screen only)",
      "rule": "≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic"
    }
  ],
  "criteria": "p04-m04-local-criteria-v1"
}

Illustrative synthetic parameters; Simulated contours; Derived equations/results. Model p04-m04-lna-twoport-v1; fixture p04-m04-lna-fixtures/1.0.0. Limits remain attached to every export.

07 / 10

Compression, IIP3, blockers, and bias current

The small-signal matrix does not change when the blocker control moves. The control only compares a stated CW input power with a separately qualified compression screen.

\text{Margin}=P_{\mathrm{in,1dB}}-P_{\mathrm{blocker}}=-10-(-25)=15\,\mathrm{dB}

Illustrative screen only: input P1dB = −10 dBm, nominal bias, 2.450 GHz single unmodulated CW tone, no modulation bandwidth, ΓS = ΓL = 0, 50 Ω, 25 °C, both powers at R1 input.

That passes the local 10 dB margin requirement. At −15 dBm the margin is 5 dB and needs inspection; above −10 dBm it is negative and rejects. Change source, load, frequency or bias and the number becomes unavailable because its measurement condition no longer matches. It is not legitimate to move an input compression point using only the new small-signal gain.

The nominal IIP3 context is 0 dBm for equal CW tones centered at 2.450 GHz, spaced 1 MHz, at −40 dBm per tone, matched 50 Ω and 25 °C. The gateway has supplied only one blocker. There is no second tone, spacing or in-band product to calculate, so the map produces no IM3 estimate. A two-tone intercept also cannot predict reciprocal mixing, modulated blocking, memory, recovery or damage.

Derived matched-source/load comparison · 2.450 GHz, 25 °C, 3 V · three separately authored synthetic bias states
Bias	GT / NF (dB)	Current proxy	Local result / reason
low	11.364034 / 1.404999	8.000 mA	inspect; compression condition missing
nominal	13.064250 / 1.272028	15.000 mA	accept; matched compression screen qualified
high	14.151404 / 1.157178	24.000 mA	inspect; current > 20 mA; compression condition missing

Higher current need not minimize noise; the high-bias knot has a different S matrix and Γopt as well as current. These settled states are not a battery-life or thermal model. The ADI RF signal-chain discussion provides practical context for evaluating gain, noise and nonlinear behavior together.

08 / 10

Bias, decoupling, supply noise, and low-frequency stability

Imagine a package whose RF two-port passes at 2.450 GHz. Its supply choke, capacitor ESL, regulator output impedance and transistor bias response can form an omitted loop at a much lower frequency. The RF file may assume an ideal AC-grounded supply; the real board does not supply one at every frequency.

Informative hardware inspection · no inferred transfer function from the synthetic two-port
Network / state	Possible mechanism	Evidence needed
RF input/output and return path	Unintended coupling alters the loop phase	Board/package reference planes and wider-band network including return paths.
Supply, choke and decoupling	Resonance, capacitor self-resonance and finite supply impedance	Frequency-dependent bias network and small-signal loop analysis beyond the RF passband.
Regulator / baseband bias feedback	Supply ripple or a slow control loop changes amplitude/phase	Supply-to-RF sensitivity, relevant noise spectra and baseband loop evidence.
Startup, shutdown and control transitions	An intermediate bias state differs from the settled model	Time-dependent state sequence, transient current and observed recovery.
Strong blocker and temperature	Nonlinear or heated state changes feedback	Large-signal, thermal and termination evidence for the actual envelope.

ADI’s MAX2648 stability note gives a concrete out-of-band caution: an amplifier intended for 5–6 GHz can retain gain far above its band, while decoupling parasitics change the supply impedance. Its circuit values belong to that device and layout; they are not universal LNA prescriptions.

Think about itWould a clean in-band K sweep remove the need to examine the supply port?

Answer

No. It tests the supplied two-port at its assumed bias termination. The omitted supply network may create another mode. Extend the model boundary and the frequency range before treating the result as a board decision.

Keysight’s amplifier-design example includes component and PCB parasitics and states its stability-analysis bandwidth. The scope of the model is part of the result. Network realization and bring-up belong to Path 10; measurement and calibration belong to Path 08.

09 / 10

Temperature, process spread, and loss before the LNA

The filter, switch and ESD network change both preceding loss and the source impedance seen by the LNA. Temperature and process spread change the LNA itself. A nominal parameter sweep is not a characterized population, and a smooth interpolated curve does not fill a missing mode or temperature corner.

Informative operating-corner evidence pack · each row can falsify the nominal decision
Corner	Carry into the model card	Discriminating evidence
Filter / switch / ESD loss	Maximum loss, temperature and state at antenna-to-R1 planes	Loss and impedance of the assembled front end; no double-counted de-embedding.
Source and load variation	Complex Γ envelopes including phase at R1	Worst permitted cable/antenna/filter and following-stage terminations.
Bias and control state	Voltage, current, transition sequence, settling	Data for each required state, including intermediate states.
Device temperature / process	Population, sample count, limits and uncertainty	Characterized corners or guaranteed limits; typical curves alone do not bound production.
Interpolation / bandwidth	Exact knots and rule; no extrapolation	Data between sparse knots and beyond both ends, especially resonances.

F_{\mathrm{attenuator}} = 1 + \frac{(L-1)T_{\mathrm{p}}}{T_{0}}

Derived matched attenuator extension, not enabled by this fixed-290 K interaction. Tp is physical loss temperature in kelvin; T0 is the noise reference temperature.

Only when Tp = T0 does the attenuator’s noise factor equal L. Only with the declared cascade conditions does its loss add directly to the LNA NF in dB.

Go deeperRead a real model package: ADL5523, separate from the map

The official ADL5523 datasheet, Rev. C (September 2017 revision history), describes a 400 MHz–4 GHz, 3 V/5 V LNA. At 2600 MHz and 5 V its typical application data give 13.2 dB gain, 0.9 dB NF and +21.2 dBm output P1dB at 25 °C with external matching. Typical supply current is 60 mA at 5 V, 30 mA at 3 V. Those are not this lesson’s input-P1dB fixture.

Table 3 is a three-port S-parameter table: port 1 RFIN, port 2 VPOS, port 3 RFOUT, sampled at 0.125 GHz increments through 4 GHz. Dropping the supply port and relabeling S21 would change the network. The 3 × 3 mm LFCSP package and its external matching/decoupling also belong in the boundary record.

The NF note de-embeds to the first input matching component; that is not the bare-package R1 plane used here. The table header does not establish an exact small-signal drive or a complete four-noise-parameter file. Record those as missing before numerical reuse. Figure 61’s source-pull discussion is qualitative evidence, not data copied into our map. Access checked 6 September 2026.

10 / 10

Choose the gateway LNA operating point

Choose the nominal matched point for the current gateway evidence pack. At 2.450 GHz and 25 °C, ΓS = ΓL = 0 gives 13.064250 dB GT, 1.272028 dB LNA NF and 2.772028 dB cascade NF with the 1.500 dB matched pre-loss. Its 15 mA current proxy and qualified 15 dB CW blocker margin pass the illustrative local card.

Reject the minimum-noise/critical-load alternative because its bilateral denominator is singular. Do not select the gain-oriented point: its input reflection exceeds the 0.80 limit, and its blocker evidence is missing. Keep the stable compromise as an inspection candidate: it offers a hypothetical 0.466194 dB cascade improvement, but needs a realized low-loss tuner and compression evidence at the changed source impedance before promotion.

This choice accepts a calculation under named conditions. It still requests wider-band and all-mode stability evidence, bias-network and state-transition analysis, temperature/process bounds, layout feedback checks and measurements at compatible planes. A useful next discriminating check is the assembled network’s termination-dependent stability over the required bias and frequency envelope, followed by blocker characterization at the chosen source/load state. The measurement procedure is deferred.

Copyable / printable LNA operating-point model card

{
  "metadata": {
    "model": "p04-m04-lna-twoport-v1",
    "interaction": "p04-m04-lna-map-v1",
    "fixture": "p04-m04-lna-fixtures/1.0.0",
    "criteria": "p04-m04-local-criteria-v1",
    "evidence": "Illustrative synthetic parameters; Simulated contours; Derived equations/results",
    "phasor": "exp(+jωt); RMS V/I; normalized power waves a,b in √W, currents into DUT",
    "zrefOhms": 50,
    "referenceTemperatureK": 290,
    "deviceTemperatureC": 25,
    "planes": "R1 de-embedded package input port 1 and output port 2; a1/a2 into DUT, b1/b2 out",
    "drive": "Infinitesimal small-signal linearization; illustrative characterization drive −60 dBm available CW, no finite-power guarantee",
    "bias": "Three separate settled synthetic bias states at 3.0 V; no bias interpolation or physical bias circuit",
    "frequencyKnotsGHz": [
      0.5,
      1,
      2.45,
      4,
      6
    ],
    "interpolation": "Exact knots retained. Between knots: linear Cartesian S and Γopt, linear noise factor Fmin, Rn in Ω, current in mA, compression/intercept powers in mW. No phase interpolation or extrapolation.",
    "provenance": "Authored teaching tuples, not vendor data or process/temperature samples. Low/high tuples independently specified, not scaled nominal matrices.",
    "compression": "Only nominal / 2.450 GHz / 25 °C / ΓS=ΓL=0 / real 50 Ω / R1 input / single unmodulated CW tone supports a scalar input-P1dB screen (−10 dBm). Other tuple values are unqualified context proxies.",
    "iip3": "Context only: nominal 0 dBm input intercept, synthetic equal CW tones centered at 2.450 GHz, 1 MHz spacing, −40 dBm per tone, matched 50 Ω, 25 °C. One blocker gives no IM3 estimate.",
    "cascade": "Matched attenuator at physical Tp=T0=290 K, then ideal noiseless lossless transformation presenting ΓS; ΓS≠0 is hypothetical ideal tuner. Requires stable nonsingular pair.",
    "criteriaEvidence": "Illustrative local requirements, not a device-selection standard",
    "contourPlane": "source",
    "hardwareInspect": [
      "Out-of-band and between-knot modes: characterize below 0.5 GHz, above 6 GHz and between sparse samples.",
      "All-mode and bias-network stability: include decoupling, supply impedance, startup, shutdown and control transitions.",
      "Large-signal evidence: actual modulated blocker, recovery, two-tone spacing, and worst source/load impedance.",
      "Temperature, process and thermal evidence: characterized populations, guaranteed limits and dissipation; current is only a proxy.",
      "Layout and measured stability: package/board planes, feedback coupling, source/load sweeps and measurement uncertainty."
    ]
  },
  "input": {
    "bias": "nominal",
    "ghz": 2.45,
    "source": {
      "re": 0,
      "im": 0
    },
    "load": {
      "re": 0,
      "im": 0
    },
    "lossDb": 1.5,
    "blockerDbm": -25
  },
  "s": {
    "s11": {
      "re": 0.32500000000000007,
      "im": -0.562916512459885
    },
    "s21": {
      "re": 1.5390906449655097,
      "im": 4.228616793536587
    },
    "s12": {
      "re": 0.10875693444439799,
      "im": 0.05071419140888393
    },
    "s22": {
      "re": 0.3889087296526012,
      "im": -0.38890872965260115
    }
  },
  "noiseParameters": {
    "fminLinear": 1.1885022274370185,
    "rnOhms": 4,
    "gammaOpt": {
      "re": 5.817072295949927e-17,
      "im": 0.95
    }
  },
  "interpolation": "Exact 2.450 GHz knot",
  "display": {
    "gammaSource": "0.000000 ∠ 0.000000°",
    "gammaLoad": "0.000000 ∠ 0.000000°",
    "gammaIn": "0.650000 ∠ -60.000000°",
    "gammaOut": "0.550000 ∠ -45.000000°",
    "delta": "-0.045464 − j0.883264",
    "k": "0.9789089",
    "mu": "0.8741151",
    "muPrime": "0.9403716",
    "denominator": "1.000000000e+0",
    "transducerDb": "13.064250",
    "availableDb": "14.628808",
    "operatingDb": "15.448730",
    "nfDb": "1.272028",
    "cascadeNfDb": "2.772028",
    "cascadeCondition": "matched 290 K attenuator",
    "blockerScreenMarginDb": "15.000",
    "currentMa": "15.000"
  },
  "status": "accept",
  "cause": "The matched point passes every local synthetic criterion. Hardware evidence remains a separate inspection request.",
  "axes": [
    {
      "name": "Denominator conditioning",
      "status": "accept",
      "value": "|D| = 1.000000e+0",
      "rule": "reject ≤ 1e−9; inspect (gain suppressed) ≤ 1e−6; accept > 1e−6"
    },
    {
      "name": "Source loop |Γin ΓS|",
      "status": "accept",
      "value": "0.000000",
      "rule": "stable ≤ 1−1e−6; unstable ≥ 1+1e−6; intervening band: inspect"
    },
    {
      "name": "Load loop |Γout ΓL|",
      "status": "accept",
      "value": "0.000000",
      "rule": "same unity band; loop magnitude is a sufficient screen, not a full dynamical proof"
    },
    {
      "name": "Source-plane stable side |Γout|",
      "status": "accept",
      "value": "0.550000",
      "rule": "conservative passive-load robustness; failing this does not prove the selected matched load oscillates"
    },
    {
      "name": "Load-plane stable side |Γin|",
      "status": "accept",
      "value": "0.650000",
      "rule": "conservative passive-source robustness; test exact defining function, not a shading convention"
    },
    {
      "name": "Transducer gain",
      "status": "accept",
      "value": "13.064250 dB",
      "rule": "defined positive GT and ≥ 10.0 dB to accept"
    },
    {
      "name": "Noise figure",
      "status": "accept",
      "value": "1.272028 dB",
      "rule": "accept ≤ 1.50 dB; reject > 3.00 dB; suppressed for unstable/singular pair"
    },
    {
      "name": "Input match",
      "status": "accept",
      "value": "0.650000 ∠ -60.000000°",
      "rule": "accept |Γin| ≤ 0.80 at the selected ΓL"
    },
    {
      "name": "Output match",
      "status": "accept",
      "value": "0.550000 ∠ -45.000000°",
      "rule": "accept |Γout| ≤ 0.80 at the selected ΓS"
    },
    {
      "name": "Bias current proxy",
      "status": "accept",
      "value": "15.000 mA",
      "rule": "accept ≤ 20.0 mA; reject > 30.0 mA; no battery-life or thermal inference"
    },
    {
      "name": "Blocker compression screen",
      "status": "accept",
      "value": "15.000 dB (screen only)",
      "rule": "≥ 10 dB to accept; < 0 dB rejects; only pinned matched nominal CW condition supports arithmetic"
    }
  ],
  "criteria": "p04-m04-local-criteria-v1"
}

The map’s card records your applied point; this fixed card records the lesson’s chosen baseline. Both carry exact terminations, fixture version, criteria, source/noise provenance and the unresolved hardware evidence. Add that conditional record to the Path 04 evidence pack.

Ungraded review

Check your understanding

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

01Why do matched GT, GA and GP differ even for the same S matrix?
Model answer
GT divides delivered load power by available source power. GA allows an output conjugate match for the selected source. GP divides delivered power by net accepted input power. At the canonical matched point they are 13.064250, 14.628808 and 15.448730 dB; their power boundaries differ.
02A source-plane circle crosses the passive disk. How do you identify its stable side?
Model answer
Its defining boundary is |Γout(ΓS)| = 1. Evaluate a known point: at ΓS = 0, Γout = S22. Here |S22| = 0.55, so the side containing the origin is stable. A load-plane circle instead tests Γin and uses |S11| at zero.
03Can you insert Fmin = 0.75 directly into the four-parameter noise equation?
Model answer
No. Convert 0.75 dB to 10^(0.75/10), use Rn/Z0 with both resistances in ohms, and convert the final linear F to dB. At Γopt the extra term vanishes; at the matched source the answer is 1.272028 dB.
04Does |Γout| > 1 mean the matched load oscillates?
Model answer
No. It identifies risk from some passive output termination. With ΓL = 0 the output loop is zero; the exact critical ΓL = 1/Γout closes the complex loop in this one-frequency model. Neither result is an all-frequency hardware proof.
05Why does the stable compromise have no displayed blocker margin or IM3 estimate?
Model answer
Its changed source impedance does not match the qualified nominal P1dB screen. The single blocker also lacks a second tone and spacing needed for an IM3 prediction. Small-signal gain cannot supply the missing nonlinear evidence.
06When may 1.5 dB pre-loss be added to the LNA NF, and what evidence remains?
Model answer
For the matched attenuator at physical 290 K and the declared compatible stable cascade. A nonzero source Γ additionally assumes the explicitly hypothetical ideal lossless tuner. Real mismatch, tuner loss, other temperatures, supply modes, layout, transitions and large-signal operation require further evidence.

Sources and further study

Primary technical sources below were checked on 6 September 2026. Equations and canonical candidates also have independent numerical checks. No vendor contours or S tables are reproduced.

A. M. Niknejad, Scattering Parameters, UC Berkeley, 6 February 2025. Power-wave conventions, bilateral feedback, μ stability and gain definitions; slides 11, 53–70. Our RMS convention is stated explicitly.
Simpson, Ballo, Dunsmore and Ganwani, A New Noise Parameter Measurement Method, Maury 5A-042, March 2013. Four-noise-parameter interpretation; measurement workflow is outside this lesson.
Analog Devices, RF signal-chain discourse: essential building blocks, Part 2 (CIR-2). Receiver amplifier tradeoffs.
Analog Devices, ADL5523 Rev. C datasheet. Conditional vendor example only; specifications, Table 3 port definitions and tuning/source-pull discussion.
Analog Devices, Designing with the MAX2648 5GHz LNA for High-Frequency Stability, 27 September 2002. Out-of-band and decoupling evidence.
Keysight, Practical RF Amplifier Design Using the Available Gain Procedure and ADS EM, publication 5990-3356. Model bandwidth, bias network and PCB parasitics.

For the curriculum textbook reading path, use CIR-1: Gonzalez, Microwave Transistor Amplifiers, 2nd ed., amplifier design; and MW-1: Pozar, Microwave Engineering, 4th ed., Chapters 4, 10 and 12. The linked primary sources above provide directly accessible technical cross-checks.

Low-NoiseAmplifiers

Failure: minimum noise, unstable amplifier

Which gain do you mean?

Reverse transmission makes terminations matter

Conditional and unconditional stability

Noise parameters and noise circles

Trade gain, noise, input match, and stability

Compression, IIP3, blockers, and bias current

Bias, decoupling, supply noise, and low-frequency stability

Temperature, process spread, and loss before the LNA

Choose the gateway LNA operating point

Check your understanding

Sources and further study

Low-Noise
Amplifiers