RF Journey

01 / 10

Failure: more PA power makes the product worse

The node raises its PA command. Why might fewer messages arrive?

Continue the fictional 2.450 GHz node from 05.1’s requirements record. The normal mode maps 20 kbit/s uncoded, with no framing overhead, into Gray QPSK at 10 ksymbol/s. An α=0.35 RRC produces 80 kcomplex-sample/s at D3. The 256-symbol interval is 25.6 ms. These values describe the mapped burst; they do not establish service throughput or battery lifetime.

Raising average drive pushes more waveform peaks into compression. Extra current can pull the supply below its minimum. A mismatched antenna accepts less forward power, while heat and interference can grow. The wanted receiver may see a stronger but less useful signal, and a nearby receiver can become the victim described in 05.4.

Think about itA PA supplies +10 dBm and the filter, switch and feed lose 2 dB. Has the +10 dBm product-feed request been met?

Answer

No. In the linear matched lineup, R2 receives +8 dBm. This module’s +10 dBm request is at R2, so the hypothetical linear PA output must be +12 dBm. The actual compressed output is then calculated from a fixed drive; it is not forced back to the request.

Opening failure ledger · mechanisms, not invented field-test results
Change	What can fail	First useful evidence
More drive	Peaks compress; EVM and adjacent power increase	Matched waveform test at R1-PA-in/out, declared processing and bandwidth.
Lower battery voltage	Instantaneous rail drops below the operating allocation	Time-aligned RF, current and loaded-rail capture.
Changed antenna installation	Accepted power and PA load can change	R2 complex reflection and network transformation; OTA pattern/gain.
Hot or repeated operation	Internal temperature and bias may drift	Transient thermal path, initial state and actual PA bias dependence.
Higher coexistence power	A victim loses headroom	Coupling, victim waveform quality and operating overlap.

Common misconceptionExtra PA power always increases reliable range.

Power is one allocation among fidelity, supply, antenna, channel and coexistence conditions. No propagation or receiver-quality model here turns a dBm increase into a guaranteed distance.

The task is to choose an operating mode and identify its binding condition. Begin by naming the planes, before adding any gains.

02 / 10

Conducted, accepted, and radiated power use different planes

Does power at an antenna connector mean power accepted by the antenna, or power radiated in every direction?

Illustrative QPSK stationary mean, real 50 Ω wave references. R2 forward is before mismatch. D3 → A0 digital codes → A1 analog baseband precede R0; their voltage and full-scale allocations remain open.

At real 50 Ω RF wave references, forward and reverse waves carry separate nonnegative powers. Their difference is net accepted power at that plane. Use carrier-cycle-averaged envelope power for |x|² and |y|², with x and y in √W. The normalized envelope algebra does not assign a physical 1 Ω load to the PA.

\begin{aligned}|\Gamma| &= \frac{\mathrm{VSWR}-1}{\mathrm{VSWR}+1} \\ P_{\mathrm{reflected}} &= P_{\mathrm{forward}} |\Gamma|^{2} \\ P_{\mathrm{accepted}} &= P_{\mathrm{forward}} (1-|\Gamma|^{2})\end{aligned}

Derived, fixed-forward model at R2. VSWR is dimensionless; powers in W; no source re-reflection or two-port load transformation is inferred.

Independent fixed-forward anchor · EXACT +10.000 dBm at R2, VSWR 2; separate from compressed system default
Quantity at R2 / S0	Derived value	Condition
Forward at R2	10.000000000 mW = 10.000000000 dBm	Fixed incident wave; not the 9.987889752 dBm actual-system proxy.
Reflected at R2	1.111111111 mW = 0.457574906 dBm	\|Γ\|=1/3, reflected fraction=1/9.
Accepted at R2	8.888888889 mW = 9.488474776 dBm	Accepted fraction=8/9; mismatch loss=0.511525224 dB.
Conditional directional EIRP	11.488474776 dBm	Illustrative +2 dBi accepted-power gain, same +z co-polar direction, 2.450 GHz.
Matched limit, Γ=0	Reflected power exactly 0 W	Logarithm has absent/−∞ limit. No arbitrary floor or NaN.
TRP	Unknown	Pattern, polarization integration and antenna-loss evidence required.

\begin{aligned}&\mathrm{EIRP}(\mathrm{direction}, \mathrm{polarization}) \\ &= P_{\mathrm{accepted,dBm}} + G_{\mathrm{accepted,dBi}} \\ &= P_{\mathrm{forward,dBm}} + G_{\mathrm{realized,dBi}}\end{aligned}

Single-port real-reference convention. Gaccepted includes the antenna’s radiation losses but excludes feed mismatch. A supplied realized gain already includes that mismatch at its specified incident reference.

Think about itCan +2 dBi accepted-power gain and +2 dBi realized gain be interchanged when VSWR is 2?

Answer

No. For this same antenna and reference, +2 dBi accepted gain corresponds to realized gain 2−0.511525224 = 1.488474776 dBi. Then 10+1.488474776 matches 9.488474776+2. Adding +2 to forward power describes a different gain assumption; subtracting mismatch again from realized gain counts it twice.

Ansys HFSS’s realized-gain definition explicitly uses incident power; the port convention matters. Our single real-reference port is intentionally narrower than arbitrary complex/multiport antenna normalization. IEEE 145-2025 is the current antenna terminology reference; full pattern analysis belongs to Path 06.

Common misconceptionConducted connector power equals accepted power, EIRP or TRP.

Each crosses a different boundary. Direction and polarization are necessary for the gain used here. The +2 dBi placeholder is neither realized gain by default nor a radiation-efficiency value.

The 2 dB post-PA loss is a matched scalar forward-wave model. Keeping Γ at R2 is deliberate: load-pull, ruggedness and mismatch-induced PA distortion remain unknown without a two-port transformation and large-signal evidence.

03 / 10

Build the transmitter gain and loss lineup

Which gain/loss rows produce the requested reference, and which rows describe actual hardware evidence?

Start at R0 with an illustrative −7 dBm modulator output. A 1 dB drive-filter loss gives −8 dBm at R1-PA-in. The frozen 20 dB small-signal gain would produce +12 dBm linearly; compression lowers it. The post-PA allocation totals 2 dB and moves the forward wave to R2.

Nominal transmitter lineup · QPSK, 2.450 GHz, real 50 Ω wave references; all entries illustrative
Row / output plane	Gain or loss / nominal level	Bandwidth and operating condition	Tolerance type / evidence
D3 → A0 → A1	Digital samples → DAC codes → analog baseband	80 ksample/s D3; A0 rate/full scale and A1 volts are separate placeholders	Unknown converter, reconstruction response and drive calibration. No dBm arithmetic at digital planes.
Modulator → R0	−7 dBm allocated RF output	2.450 GHz ±6.75 kHz ideal support; linear IQ/LO assumed	Illustrative exact setpoint, not guaranteed calibrated output.
Drive filter → R1-PA-in	−1.000 dB → −8.000 dBm	Flat scalar passband over represented waveform, nominal bias/25 °C	Bounded example ±0.1 dB; not applied to the frozen PA fixture.
PA → R1-PA-out	+20.000 dB small-signal → +12 dBm linear command	Matched Rapp, Psat +20 dBm, p=3, φmax=3°, stationary record	Coefficients fixed, production/temperature evidence unknown. Actual mean 11.987889752 dBm.
Output filter	−1.000 dB → 10.987889752 dBm	Flat scalar forward loss across the represented near-carrier spectrum	Bounded row ±0.2 dB example; actual response and harmonic power handling unknown.
TX switch	−0.500 dB → 10.487889752 dBm	Settled TX state; no switching transient model	Bounded row ±0.1 dB example; temperature/correlation evidence unknown.
Coupler mainline	−0.200 dB → 10.287889752 dBm	Mainline insertion loss; coupled detector branch not a gain row	Bounded row ±0.1 dB example; directivity/calibration unknown.
Feed → R2	−0.300 dB → 9.987889752 dBm forward	Total post-PA loss 2.000 dB; matched scalar transfer	Bounded row ±0.1 dB example; R2 mismatch separately applied.
Antenna → S0	Accepted power + optional directional gain	Same frequency, direction, polarization and gain reference	Illustrative or unknown; TRP and channel excluded.

\begin{aligned}P_{\mathrm{PA,linear,dBm}} &= P_{\mathrm{R2,request,dBm}} + L_{\mathrm{post,dB}} \\ P_{\mathrm{PA,input,dBm}} &= P_{\mathrm{PA,linear,dBm}} - 20 \\ P_{\mathrm{R2,forward,dBm}} &= P_{\mathrm{PA,actual,dBm}} - L_{\mathrm{post,dB}}\end{aligned}

Gains/losses use signed dB at compatible planes. Independent powers add in W or mW. Requested linear output sets input scaling only once.

This is a named system variant of 04.5. Its +10 dBm request was at the PA output. Here +10 dBm is at R2, commanding +12 dBm at the PA. With no automatic drive correction, the checked outputs are 11.987889752 and 9.987889752 dBm; the EVM proxy is 0.243227180%.

Go deeperA bounded corner is not yield

The four post-PA example bounds sum to ±0.5 dB if all signs align. At a fixed +12 dBm PA command, a total-loss sweep 1.5/2/2.5 dB yields R2 stationary means 10.487889752/9.987889752/9.487889752 dBm. The interaction treats an optional total-loss bound as one correlated aggregate, without independently reseeding or repairing drive at each corner.

If four independent zero-mean standard uncertainties were instead 0.2/0.1/0.1/0.1 dB, their RSS would be 0.264575131 dB. Those are different definitions. Correlated terms require covariance: u² = Σuᵢ² + 2Σρᵢⱼuᵢuⱼ. A synthetic corner sweep has no probability, population or manufacturing yield. Tolerance on drive before a nonlinear PA must be propagated through the nonlinear model, not simply added at its output.

The lineup locates power. To decide how much drive is usable, keep the waveform’s peaks attached to that average.

04 / 10

Waveform PAPR sets PA backoff

Why is the same average request more demanding for the 16QAM record?

The inherited 04.5 PA model uses PRBS-9 seed 0x1FF, fixed Gray mapping and 65 unit-energy RRC taps. Retain convolution indices 32…2079 and normalize mean |w|² to one on those 2048 samples. At eight samples/symbol, the QPSK finite-record input PAPR is 3.795180 dB; 16QAM is 4.845859 dB. The 256 symbols remain a small population.

\begin{aligned}v&=\frac{\sqrt{G_0}|x|}{\sqrt{P_{\mathrm{sat}}}}\\A_{\mathrm{out}}&=\frac{\sqrt{G_0}|x|}{(1+v^{2p})^{1/(2p)}}\\\phi(v)&=\phi_{\max}\frac{v^2}{1+v^2}\end{aligned}

R1-PA-in x and R1-PA-out y in √W. G0 is linear power gain 100, Psat=0.1 W, p=3. φmax=3° is converted to radians inside exp; x=0 maps exactly to y=0.

Independent inherited checks · linear request at PA, no system-loss or DC substitutions
Waveform / PA request	Actual PA mean	Fitted oversampled EVM	Input / output PAPR
QPSK / +10 dBm	9.996939311 dBm	0.153473377 %	3.795180 / 3.778460 dB
QPSK / +12 dBm	11.987889752 dBm	0.243227180 %	3.795180 / 3.730096 dB
CW / +17 dBm	16.828343527 dBm	0.000000000 %	0.000000 / 0.000000 dB
QPSK / +17 dBm	16.683933451 dBm	2.669393232 %	3.795180 / 2.656298 dB
16QAM / +17 dBm	16.436585286 dBm	5.659672970 %	4.845859 / 3.206715 dB

The +17 dBm PA stress preset is separate from the normal request: with 2 dB loss it requests +15 dBm at R2. QPSK gives 2.669393% fitted EVM, while 16QAM gives 5.659673% and crosses the local 3% screen. A compressed output can have lower PAPR because its peaks have been flattened; that is not a fidelity improvement.

Common misconceptionQPSK has constant magnitude, so its PAPR is always 0 dB.

Ideal symbol points have constant magnitude. The pulse-shaped trajectory between them does not. True CW has 0 dB PAPR in this record; QPSK’s filtered envelope has a different population.

Go deeperRead the CCDF and backoff at the correct plane

PAPR is 10log10(Ppeak/Pmean). The CCDF counts samples strictly above a threshold relative to each plane’s own mean, over exactly 2048 samples. Its exact peak row therefore has zero exceedances. Input backoff is Psat−G0−Pin in dB; output backoff uses actual mean PA output. The nominal values are 8.000000 and 8.012110 dB. Neither number establishes a safe continuous-time peak or a rare-event probability.

The next question is what errors remain after the observation receiver fits a reference. A low EVM result is meaningful only with that processing declared.

05 / 10

Allocate EVM across the transmitter

Can several “small” transmitter errors fit inside one fidelity allocation?

The PA proxy fits one common complex gain to all 2048 oversampled pairs. It removes the best constant gain/phase factor before reporting a residual. It does not recover timing, match-filter, sample ideal symbols, equalize the channel or apply DPD. 02.7’s processing contract and Rohde & Schwarz’s EVM guide explain why reference definition and normalization must travel with the result.

\begin{aligned}c &= \frac{\sum (y \operatorname{conj}(x))}{\sum |x|^{2} } \\ \mathrm{EVM} &= 100 \sqrt{\frac{\sum |y-c x|^{2}}{\sum |c x|^{2}}} \%\end{aligned}

x = input reference, y = nonlinear output at compatible sample indices. A common fit changes the error population; this is a sample-domain PA proxy, not a standards measurement.

Think about itDo independent 2%, 3% and 4% RMS errors total 9% EVM?

Answer

Only an aligned coherent example sums that way. Independent, identically referenced zero-mean errors give √(2²+3²+4²) = 5.385164807%. Deterministic aligned error vectors can total 9%. A label such as “LO” or “PA” does not prove independence.

Transmitter fidelity allocation · common observation plane R2 with declared processing
Contributor	Required allocation evidence	Current state
D3 baseband / A0 codes	Numerical scaling, quantization, DAC images and full-scale convention	Unknown physical implementation; ideal waveform fixture only.
A1 / R0 I/Q paths	Gain/quadrature imbalance, offsets, image/LO leakage versus frequency and temperature	Unknown; no invented I/Q percentage.
LO phase noise / drift	SSB mask, integrated offset band, tracking loop and common phase/frequency removal	Unknown; one phase-noise marker cannot supply RMS EVM.
Filter flatness / group delay	Complex H(f), occupied spectrum, reference timing and equalization	Unknown; scalar loss fixture does not model ripple or ISI.
PA	Matched waveform amplitude/phase law and reference fit	0.243227180% nominal local proxy; already simulated, never added twice.
Mismatch / antenna interface	Transformed complex load and nonlinear response at the PA	Unknown distortion; scalar accepted-power arithmetic is not an EVM term.
Observation receiver / instrument	Residual noise, gain/phase/timing correction, bandwidth, calibration uncertainty	Unknown; measurement error is not automatically a DUT allocation.

Go deeperFlatness, group delay and small random phase error

Group delay is τg = −dφ/dω. Constant delay can be removed by reference alignment; variation across the occupied spectrum reshapes the pulse. Scalar insertion loss contains no phase or delay information. For small residual zero-mean phase jitter σφ in radians, exp(jφ)≈1+jφ gives an EVM contribution ≈100σφ% under the same normalization. With an SSB phase-noise convention, converting an integrated mask to phase variance requires its sideband factor and integration limits; no mask or tracking bandwidth is supplied here, so there is no numeric LO allocation.

RSS follows from the cross terms in E[|Σeᵢ|²] vanishing. Correlated or deterministic terms leave cross terms. Refit the common reference consistently before combining contributors; subtracting a measured total from an unrelated simulated percentage cannot isolate one physical error.

Common misconceptionEVM allocations add in dB, or a clean fitted EVM proves clean emitted power.

Add compatible error powers only under justified correlation assumptions. A CW with one constant complex multiplier can have zero fitted EVM despite compression. An observation fit does not remove emitted LO leakage or a harmonic.

That makes spectrum a separate allocation, with its own reference and integration bounds.

06 / 10

Track occupied bandwidth, regrowth, harmonics, and spurs

Which unwanted energy is actually represented by this spectrum?

The ideal raised-cosine total null-to-null support is (1+α)Rs = 13.5 kHz, or ±6.75 kHz in signed complex baseband. Finite filtering and a finite record produce leakage outside that ideal support. Necessary, occupied, channel, −x dB and noise-equivalent bandwidth are different quantities; none can be substituted just because one number is available.

\begin{aligned}P_B&=\frac{\sum_{k\in B}|\mathrm{DFT}(yH)[k]|^2}{8192\sum_nH[n]^2}\\\text{Adjacent proxy}_{\mathrm{side}}&=10\log_{10}\left(\frac{P_{\mathrm{side}}}{P_{\mathrm{main}}}\right)\end{aligned}

R1-PA-out near-carrier power, periodic Hann H over N=2048 samples, NFFT=8192 with 6144 zeros. Include bins whose centers lie within each band; no fractional weighting.

Two-sided integration ledger · Fs=80 ksample/s; signed offsets from 2.450 GHz
Band / mechanism	Bounds or observation	What the result can establish
Main	−6.75…+6.75 kHz	Reference denominator from included FFT bins.
Lower adjacent	−21.75…−8.25 kHz	Lower side/main proxy; own ≤−35 dBc local exercise screen.
Upper adjacent	+8.25…+21.75 kHz	Upper side/main proxy; do not sum both sides for this screen.
Spectral regrowth	Difference in nonlinear vs undistorted reference behavior	Near-carrier memoryless redistribution, including finite reference leakage.
RF harmonics	2fc, 3fc and real output-network terminations	Unknown: the complex-envelope model does not compute them.
LO leakage / images / clock spurs	Actual A0/A1/R0 chain and operating clocks	Unknown: require amplitudes and actual filtering at stated planes.
Burst splatter / transient chirp	Amplitude and phase versus time at RF on/off	Unknown: stationary FFT is not recomputed from the command-energy ramp.

At +17 dBm PA drive, 16QAM’s lower/upper adjacent proxies are −33.758433/−32.965874 dBc. Both exceed this module’s local −35 dBc criterion. Its EVM also fails. By contrast, QPSK at the same command stays below −35 dBc on each side. This is a model comparison, not a named waveform conformance claim.

Common misconceptionOne CW marker or a synthetic regrowth plot proves emissions compliance.

A CW test has no information about this modulated population or its bursts. The proxy omits harmonics, actual spurs and settling. Named waveform limits belong to Path 07, measurement to Path 08, and market requirements to Path 09.

Reference leakage is shown beside output leakage and is never subtracted. Side/main ratios below 10⁻¹³ display “<−130 dBc numerical floor”; deeper floating-point values are not physical rejection. A post-PA flat loss reduces absolute main and adjacent powers equally, leaving their ratio unchanged.

Now move from stationary statistics into a time ledger. That boundary changes what the model can claim.

07 / 10

Burst ramp, settling, and transient chirp matter

Does a 25.6 ms RF-on command necessarily deliver all 256 symbols?

The default cycle allocates 2.000 ms startup, 25.600 ms commanded RF on, and 972.400 ms sleep. Each 0.200 ms linear-amplitude ramp is inside the on interval. This is an energy-screening schedule: the first and last mapped symbols may be attenuated. It does not prove intact payload, PA settling, PLL lock or filter-tail capture.

Default command-time ledger · right-open intervals, separate from waveform samples
Cycle interval	Command / current law	Evidence limit
0…2.000 ms	Startup, 5 mA; RF command absent	No PLL/PA time-response data.
2.000…2.200 ms	Linear amplitude rise 0→1	Power scales as r², not r; symbols not protected by this schedule.
2.200…27.400 ms	Amplitude one; waveform-moment demand	Stationary statistics used for energy only.
27.400…27.600 ms	Linear amplitude fall 1→0	No RF chirp, phase settling or burst FFT inferred.
27.600…1000.000 ms	Sleep, 5 µA	Other circuitry, retries and service traffic omitted.

\begin{aligned}\int r^{2}d t &= T_{\mathrm{on}} - \frac{4T_{\mathrm{r}}}{3} \\ \int r^{4}d t &= T_{\mathrm{on}} - \frac{8T_{\mathrm{r}}}{5}\end{aligned}

r is an ideal post-Rapp amplitude factor. Tr is each ramp duration, in seconds, with 0≤2Tr≤Ton. Rectangular Tr=0 and triangular 2Tr=Ton are explicit limits.

At nominal timing these integrals are 25.333333333 and 25.280000000 ms. A ramp is linear in amplitude, so its RF energy follows a squared ramp. Quiescent current still runs through all 25.6 ms. RF-energy-equivalent duty is 2.533333333%, while commanded duty is 2.560000000%.

Go deeperAn explicitly payload-preserving schedule costs extra time

Keep the 25.6 ms data interval at full command. Add 0.2 ms rise before it and 0.2 ms fall after it, plus 2 ms startup and a separately allocated 0.8 ms filter-tail hold. The total occupied schedule is 28.8 ms, leaving 971.2 ms in a fixed 1 s cycle. The 0.8 ms tail here is a separate illustrative guard, not a measured filter requirement; its current and RF state must be specified before calculating its energy. More generally Tsleep=Tcycle−Tstartup−2Tr−Tdata−Ttail. Do not stretch the symbol clock or silently drop overhead.

Common misconceptionA smoother command ramp proves that the PA and PLL have settled.

The command is applied after the Rapp output for energy bookkeeping. Actual settling depends on amplitude and phase response, bias, supply and loop dynamics. Transient chirp is frequency changing with time: Δf(t)=(1/2π)dφ/dt. This model has no measured φ(t) startup record.

A real power controller must observe these boundaries with a known sensing path. A scalar detector output cannot fill in missing RF evidence.

08 / 10

Power control needs sensing and compensation

What power does the detector actually sense, and how quickly can it report a change?

A coupled sample is meaningful only after calibration to the mainline plane. Forward and reverse directionality, frequency response, attenuation, waveform dependence, temperature and response time all enter that transfer. A loop calibrated on CW at one temperature cannot be assumed accurate on short amplitude-varying bursts across voltage and load changes.

Annotated detector/coupler calibration request · no calibration supplied for this node
Element / plane	Required transfer or condition	Unknown / next evidence
Forward sample at R2	Coupling and additional attenuation, direction arrow, mainline insertion loss	Calibrate detector volts→forward dBm at R2 over level/frequency.
Reverse sample / directivity	Isolation from forward wave; magnitude and phase of unwanted leakage	Finite directivity mixes waves. Measure residual/error versus Γ phase.
Detector input	Log/RMS/envelope response, impedance, safe input range, waveform crest factor	Unknown detector calibration; no accepted-power inference from one scalar forward sample.
Temperature and supply	Calibration surface versus temperature and rail voltage	Cold/hot/low-voltage characterization; retain bounded vs statistical uncertainties.
Timing / control range	Rise/fall, averaging/filter delay, settling, loop stability and gain-control monotonicity	Synchronize detector, command and RF; no unverified loop implemented.
Uncertainty / reference plane	Power-reference calibration, cable/coupler loss and correlated error terms	Store uncertainty, direction and calibration date with every reading.

The ADL5513 Rev. B separates measurement and controller modes and shows a directional-coupler path with loop filtering. Its main specification conditions include 5 V, 25 °C, real 50 Ω and CW drive; its example loop is not this node. Mini-Circuits’ compression-measurement note explains why finite coupler directivity lets reflected energy contaminate an incident-power sample.

Think about itCan a perfectly calibrated forward detector tell you accepted antenna power when the load changes?

Answer

Not by itself. Accepted power is forward minus reverse at the same plane. You need reverse-wave or complex-reflection evidence with adequate directivity, plus a consistent reference transformation. Calibration of one scalar path does not make it directional in both senses.

Power-control modes can request nominal, reduced coexistence or constrained low-battery operation. That is a policy decision tied to link and thermal allocations. Here the request is open-loop: gain is fixed, drive is set once, and actual nonlinear output is reported separately. Failed supply allocation suppresses delivered-RF claims.

With the sensing gaps recorded, compare instantaneous current, accumulated energy and temperature. They answer three different questions.

09 / 10

Duty, heat, current peaks, supply droop, and coexistence

Can two transmitters have the same average current and different battery heating or reset risk?

\begin{aligned}I(t) &= I_{q} + \frac{P_{\mathrm{RF,out}} r^{2}(t)}{\eta _{\mathrm{add}} V_{\mathrm{oc}}} \\ V_{\mathrm{load}}(t) &= V_{\mathrm{oc}} - I(t) R_{\mathrm{batt}} \\ Q &= \int I d t\qquad E_{\mathrm{source}} = V_{\mathrm{oc}} Q \\ E_R &= R_{\mathrm{batt}} \int I^{2}d t\qquad E_{\mathrm{load}} = E_{\mathrm{source}}-E_R\end{aligned}

Prescribed demand at PA load boundary; Iq=.020 A, ηadd=.45, startup=.005 A, sleep=.000005 A. Voc is fixed during each cycle. The rail check does not feed back into the RF waveform.

Let μ₁=E[Pout] and μ₂=E[Pout²] over the exact waveform. With a=1/(ηadd Voc), the on-time current-square integral is Iq²Ton + 2Iq a μ₁∫r²dt + a²μ₂∫r⁴dt. Replacing μ₂ by μ₁² erases waveform variation and underestimates battery heating. Peak current instead uses the actual maximum output sample at r=1.

Independent nominal waveform-moment energy check · 3.0 V, 2 Ω, 1 s schedule
Boundary / statistic	Checked result	Interpretation
Charge / cycle-average current	0.823445881 mC / 0.823445881 mA	Same number only because this cycle is exactly 1 s.
Battery source	2.470337644 mJ	Energy drawn at the open-circuit source boundary.
Internal resistance loss	0.052503356 mJ	Battery heat, using E[I²], not E[I]².
Load terminals	2.417834287 mJ	Source minus resistor loss; this is not all RF output.
Instantaneous peak current	47.635340581 mA	Waveform peak plus quiescent demand.
Minimum loaded voltage	2.904729319 V	Clears the local 2.2 V allocation at nominal inputs.

Think about itTrace A draws 20 mA continuously. Trace B draws 80 mA for 25% of the time and zero otherwise. Same average: same heating and droop?

Answer

Both average 20 mA, but at 3 V/2 Ω over 1 s, source energy is 60 mJ in both. Trace A loses 0.8 mJ in the resistor and has a 2.96 V rail. Trace B loses 3.2 mJ and reaches 2.84 V during its pulses. Mean current fixes charge; its square and peak are separate constraints.

Separate rectangular prescribed-current anchor · not the PA-derived demand
Input / boundary	Independent result
3.0 V, 2 Ω; startup 5 mA ×2 ms; TX 80 mA ×25.6 ms; sleep 5 µA ×972.4 ms	Tr=0; exactly 1 s.
Charge / average	2.062862 mC / 2.062862 mA
Source / resistor / load energy	6.188586000 / 0.327780049 / 5.860805951 mJ
Minimum loaded voltage	2.840000 V

The low-battery preset has Voc=2.2 V and Rbatt=6 Ω. Any positive current already lowers the rail below the local 2.2 V minimum. Its calculated PA waveform remains a conditional requested-waveform diagnostic; it cannot be labeled delivered RF. If the prescribed demand would make terminal voltage or heat balance impossible, the ledger suppresses load/thermal results rather than displaying negative delivered energy or refrigeration.

Heat has memory even when the PA proxy does not

\begin{aligned}P_{\mathrm{heat}} &= P_{\mathrm{load}} + P_{\mathrm{RF,in}} r^{2} - P_{\mathrm{RF,out}} r^{2} \\ \frac{C_{\mathrm{th}} d\Delta T}{d t} &= P_{\mathrm{heat}} - \frac{\Delta T}{R_{\mathrm{th}}} \\ \Delta T_{\mathrm{next}} &= P_{\mathrm{heat}} R_{\mathrm{th}} + (\Delta T_{\mathrm{start}}-P_{\mathrm{heat}} R_{\mathrm{th}})e^{-\frac{\Delta t}{\tau }}\end{aligned}

One-pole illustrative PA thermal boundary, Rth=100 K/W, Cth=.020 J/K, τ=2 s. Default includes input RF power using the same synthetic r² command; post-PA loss is never PA junction heat.

A separate constant 150 mW heat pulse for 25.6 ms from ambient gives 0.190776426 K rise. The default waveform-moment model gives a first-cycle peak of 25.099694061 °C and periodic peak of 25.254819615 °C at 25 °C ambient. The hot preset shifts ambient to 70 °C and still clears the synthetic 85 °C threshold; that clearance does not supply real bias or thermal evidence.

Go deeperNumerical thermal contract and physically meaningful boundaries

Each constant interval uses the exact exponential step. Each command ramp is split into 64 equal intervals with midpoint waveform-moment heat; every startup/ramp/plateau/sleep boundary is included. Repeating with 128 ramp intervals checks a ≤0.01 K difference. For the affine cycle map ΔTnext=aΔTstart+b, solve ΔTperiodic=b/(1−a) and verify a residual ≤10⁻⁶ K. This avoids an arbitrary iteration count when τ is long.

The TI thermal-network discussion relates transient impedance to device construction and packaging. This lesson’s single R/C pair is synthetic; it does not establish actual junction temperature. The Energizer handbook likewise conditions pulse voltage on resistance, discharge state and temperature. Its cell data are not the 3 V/2 Ω model coefficients. Regulators, decoupling, chemistry and other circuitry remain outside this model.

Common misconceptionLow duty cycle removes current peaks, heat accumulation and startup risk.

Low duty reduces some integrated terms, but the largest instantaneous current still causes droop. Startup repeats, sleep still consumes charge, and a slow thermal path remembers prior cycles. A short-burst average is not a battery-life result.

Now try the five modes. In coexistence mode, a 0 dBm R2 request reduces RF energy and potential interference, while spending 10 dB of transmit link allocation. No unmodeled range guarantee survives that trade.

Class 1 · one operating-mode decision

TX Allocation & Energy Ledger

Choose a mode, predict its binding constraint, then inspect planes, waveform and time. Commit edits explicitly. Every named mode restores its full defining inputs; the PA command never follows an output-target solver.

Operating mode and linear power request

Operating modeEach named mode restores waveform, loss, supply, gain evidence, timing, thermal state and tolerance.

Stationary waveformQPSK default. Closed 2048-sample records, Rs=10 ksymbol/s for both modulated choices; CW has no mapped bits.

Requested linear R2 forward power (dBm)-10…18 dBm; step 0.5. Reset: 10 dBm.

Post-PA matched forward loss (dB)0…6 dB; step 0.1. Reset: 2 dB.

Frozen PA command presetRestores all nominal inputs; stress sets R2 request +15 dBm with 2 dB loss. G0=20 dB, Psat=20 dBm, p=3, AM–PM max=3° in both.

Current-demand modelRestores nominal timing/supply. Anchor additionally sets Tr=0 and separates RF from current evidence. Iq=20 mA, ηadd=.45, startup=5 mA, sleep=5 µA are fixed.

R2 mismatch, antenna evidence and bounded loss

R2 load and conditional S0 gain

R2 load VSWR (:1)1…5 :1; step 0.1. Reset: 1 :1.

Directional gain evidenceUnknown by default; a supplied number is illustrative, not antenna measurement.

Illustrative directional antenna gain (dBi)-10…10 dBi; step 0.1. Reset: 2 dBi.

Gain power basisSame direction/polarization; default accepted. Do not subtract mismatch again from realized gain.

Post-PA loss uncertaintyFixed PA command; total-loss offsets −bound/0/+bound. No statistical independence or yield claim.

Bound on total post-PA loss (±dB)0…1 ±dB; step 0.1. Reset: 0 ±dB.

Burst schedule and supply

Command timing · separate from the symbol population

Commanded RF on-time Ton (ms)0.1…100 ms; step 0.1. Reset: 25.6 ms.

Each amplitude ramp Tr (inside Ton) (ms)0…5 ms; step 0.1. Reset: 0.2 ms.

Startup before RF on (ms)0…20 ms; step 0.1. Reset: 2 ms.

Complete cycle (ms)30…10000 ms; step 0.1. Reset: 1000 ms.

2Tr ≤ Ton and startup + Ton ≤ cycle. Changing Ton never changes the 10 ksymbol/s clock or proves that 256 data symbols survive. Filters, tails and retries require separate allocations.

Prescribed demand and instantaneous supply screen

Open-circuit source Voc (V)1.8…4.2 V; step 0.01. Reset: 3 V.

Battery internal resistance (Ω)0…20 Ω; step 0.1. Reset: 2 Ω.

Vload = Voc − I R. This supply screen never changes PA coefficients or increases drive.

Thermal start and one-pole assumptions

PA-plane heat · thermal estimate only

Ambient temperature (°C)-20…85 °C; step 1. Reset: 25 °C.

One-pole thermal resistance (K/W)10…500 K/W; step 1. Reset: 100 K/W.

One-pole thermal capacitance (J/K)0.001…1 J/K; step 0.001. Reset: 0.02 J/K.

Initial thermal stateDefault ambient; both peaks always shown. Periodic peak controls the repeated-operation thermal allocation.

Inspect committed command time (ms)0…1000 ms, step 0.1. Key-time buttons select exact event boundaries; model inputs and committed energy do not change.

Inspect hardware evidence · Nominal

Hardware evidence remains open; supplied numeric screens clear.

Matched-model RF estimate. Numeric clearance does not establish measured RF delivery.

Linear command · R2 → PA → PA input: 10.000000 → 12.000000 → −8.000000 dBm
Actual nonlinear stationary proxy · PA → R2: 11.987890 → 9.987890 dBm
R2 burst average · ramps inside Ton: 9.942413 dBm
PA fitted oversampled EVM: 0.243227 %
Battery-source cycle energy: 2.470338 mJ
Instantaneous peak current / minimum rail: 47.635341 mA / 2.904729 V

Every local exercise constraint · p05-m05-tx-screens-v1 · no normative limits
Axis / source	Committed value	Local allocation	Decision / remaining evidence
R2 burst-average forward	9.942413 dBm	9…11 dBm (normal telemetry)	CLEAR · Includes the command ramps inside Ton; stationary power is reported separately.
Fitted oversampled EVM proxy	0.243227 %	≤3 %	CLEAR · Matched memoryless PA only; total transmitter EVM remains unknown.
Lower adjacent proxy	−50.494406 dBc	≤−35 dBc each	CLEAR · Stationary near-carrier side/main integration; not burst splatter or a named emission mask.
Upper adjacent proxy	−50.761604 dBc	≤−35 dBc each	CLEAR · Stationary near-carrier side/main integration; not burst splatter or a named emission mask.
Battery-source cycle energy	2.470338 mJ	≤10 mJ / declared cycle	CLEAR · 1000.0 ms cycle; default 1 s exercise. Payload, retries and lifetime unknown.
Instantaneous loaded supply	2.904729 V	≥2.2 V	CLEAR · Actual waveform peak in prescribed demand; supply does not feed back into the Rapp coefficients.
Periodic temperature estimate	25.254820 °C	≤85 °C	CLEAR · First burst and selected start shown separately. One-pole PA heat estimate; not measured junction temperature.
Remaining hardware evidence	Unknown	Detector / PA / antenna / supply tests required	INSPECT · Real I/Q, LO, filter amplitude/group delay, mismatch distortion, harmonics, spurs, memory, settling, detector calibration, antenna efficiency/pattern, actual supply/thermal feedback, range, legal emissions, delivered payload/retries and battery lifetime remain unknown.

Simulated, stationary 2048 samples, periodic Hann, 8192-point DFT. Every eighth FFT bin is drawn; all bins enter the integrals. Plot clipped at −100 dB/bin for legibility, separate side/main reporting floor −130 dBc. This is neither a PSD nor a burst-settling measurement.

Power and waveform record

Matched-model RF estimate · stationary near-carrier mean except labelled burst quantities
Quantity / plane	Value	Definition / limitation
Forward at R2	9.972154 mW / 9.987890 dBm	Actual PA proxy minus matched post-PA loss; no servo.
Reflected at R2	0.000000 mW / absent (−∞ dBm limit)	\|Γ\| = 0.000000 at R2. PA-plane load unknown.
Accepted at R2	9.972154 mW / 9.987890 dBm	Forward minus reflected; not total radiated power.
Conditional directional EIRP	Unknown	S0, +z co-polar placeholder, 2.450 GHz. Gain/direction evidence not supplied. Stationary; conditional on forward power and the declared gain.
TRP / antenna efficiency	Unknown	Angular/polarization integration and antenna-loss evidence missing.
Input / output backoff	8.000000 / 8.012110 dB	Input uses Psat−G0−Pin; output uses Psat−actual mean, not a safe modulated limit.
Input / output PAPR	3.795180 / 3.730096 dB	R1-PA-in / R1-PA-out, each relative to its own mean; exactly 2048 samples.
R1-PA-out finite peak	37.307710 mW / 15.717986 dBm	Peak index 1276; sampled envelope maximum, not guaranteed continuous-time PEP.
PA power moments	15.804799 mW; 295.456854 mW²	First moment E[P] and second E[P²], before spectral partition. Second moment is used for I²R.
Main / lower / upper band integrals	15.814291 / 0.000141127 / 0.000132706 mW	R1-PA-out, window power normalization; lower/upper ratios referenced to main.
Output lower / upper proxy	−50.494406 dBc / −50.761604 dBc	Bands −21.75…−8.25 / +8.25…+21.75 kHz; main ±6.75 kHz.
Undistorted reference leakage	−51.508116 dBc / −51.514008 dBc	Same window and bands. Retained, never subtracted.
Total transmitter EVM	Unknown	I/Q, LO, filter and mismatch terms not supplied. The simulated PA impairment is not added twice.

Finite-population CCDF key rows

Strict sample exceedance · 2048 samples at each plane; complete .1 dB rows in copied record
Threshold above own mean	PA input count / percent	PA output count / percent
0 dB	1107 / 54.052734 %	1111 / 54.248047 %
1 dB	554 / 27.050781 %	554 / 27.050781 %
2 dB	133 / 6.494141 %	132 / 6.445313 %
3 dB	15 / 0.732422 %	14 / 0.683594 %
4 dB	0 / 0.000000 %	0 / 0.000000 %
5 dB	0 / 0.000000 %	0 / 0.000000 %
12 dB	0 / 0.000000 %	0 / 0.000000 %
Exact peak rows (3.795180 / 3.730096 dB)	0 / 0 %	0 / 0 %

Time, source, load and heat record

Waveform-moment demand: Iq = 20 mA; ηadd = 0.45; startup = 5 mA; sleep = 5 µA. Post-Rapp command r scales PA output and input power by r² for energy bookkeeping. Battery resistor loss and post-PA loss are excluded from PA heat.

Committed charge/energy balance · full cycle, source and load boundaries kept separate
Quantity	Value	Boundary / condition
Timing	2.0 / 25.6 / 972.4 ms	Startup / commanded on / sleep. Ramps inside Ton; mapped payload survival not established.
Amplitude-command integrals	25.333333 / 25.280000 ms	∫r²dt / ∫r⁴dt; time factors, not a waveform spectrum.
Commanded / RF-equivalent duty	2.560000 / 2.533333 %	RF-equivalent duty concerns this synthetic command.
Charge / cycle-average current	0.823446 mC / 0.823446 mA	Q = ∫I dt; Icycle = Q/Tcycle.
On-time / plateau mean current	31.585308 / 31.707258 mA	Includes ramps / stationary waveform mean. Neither is the instantaneous peak.
Battery source	2.470338 mJ	Esource = Voc Q; prescribed demand even if supply fails.
Internal battery resistance	0.052503 mJ	ER = R∫I²dt, using waveform second power moment.
Load-terminal energy	2.417834 mJ	Eload = Esource−ER. Unavailable for negative terminal voltage or impossible heat balance.
PA RF command energy	0.400388 mJ	Near-carrier forward PA output; no harmonic or antenna-energy inference.
Thermal first / periodic peak	25.099694 / 25.254820 °C	One-pole estimate. First cycle starts at ambient; periodic cycle includes prior heat.
Selected thermal start	ambient → peak 25.099694 °C	Starts at declared ambient.
Thermal numerical evidence	0.000000014 K grid difference; 0.000000000 K periodic residual	64 vs 128 ramp panels, ≤.01 K convergence; affine fixed-point residual ≤10⁻⁶ K. Invalid heat: unavailable.

Selected command time: 0.0 ms · Startup; amplitude 0.000, mean demand 5.000000 mA and mean loaded rail 2.990000 V. This averaged timeline cannot replace the 2.904729 V instantaneous minimum screen.

Key command boundaries · right-hand states; cycle endpoint is the left limit
Time / state	Command amplitude	Mean current / mean rail
0.0 ms · Startup	0.000	5.000000 mA / 2.990000 V
2.0 ms · RF command	0.000	20.000000 mA / 2.960000 V
2.2 ms · RF command	1.000	31.707258 mA / 2.936585 V
27.4 ms · RF command	1.000	31.707258 mA / 2.936585 V
27.6 ms · Sleep	0.000	0.005000 mA / 2.999990 V
1000.0 ms · Cycle end (left limit · Sleep)	0.000	0.005000 mA / 2.999990 V

Applied waveform, processing and evidence contract

Committed operating conditions · retained when the form is hidden in print
Input / boundary	Committed value
Operating mode / PA preset / waveform	nominal / nominal / QPSK
R2 linear request / matched post-PA loss	10.000000 dBm / 2.000000 dB
R2 VSWR / directional gain / gain basis	1.000000 / Unknown (numeric field inactive) / accepted
Loss tolerance	Disabled
Startup / on / each ramp / cycle	2.0 / 25.6 / 0.2 / 1000.0 ms
Battery Voc / internal R	3.000000 V / 2.000000 Ω
Ambient / Rth / Cth / initial state	25.000000 °C / 100.000000 K/W / 0.020000 J/K / ambient
Current-demand law	Waveform moments: Iq=20 mA, ηadd=.45; startup=5 mA, sleep=5 µA

evidence: Illustrative engineering case / simulated matched PA and synthetic command energy; not measured.
planes: D3 complex samples → A0 digital DAC codes → A1 analog baseband → R0 RF modulator → R1-PA-in → R1-PA-out → R2 antenna feed → conditional S0 direction/polarization. RF wave references real 50 Ω.
waveform: PRBS-9 0x1FF; emit bit0, feedback bit0 XOR bit4, right shift into bit8; Gray QPSK/16QAM; 256 symbols; 65 unit-energy RRC taps α=.35 at 8 samples/symbol; convolution indices 32…2079; mean |w|²=1 over 2048 samples.
timing: QPSK: 20 kbit/s uncoded/no-overhead mapping, Rs=10 ksymbol/s, D3 Fs=80 ksample/s, 25.6 ms symbol interval. 16QAM at unchanged Rs is a 40 kbit/s mapped variant; CW carries no mapped bits. Command timing is a separate population, not a regenerated RF burst.
evm: c=Σ(y conj(x))/Σ|x|²; EVM=100√(Σ|y−cx|²/Σ|cx|²). 2048 oversampled pairs; common complex gain only, no timing recovery, matched filtering, equalization or DPD.
spectrum: 8192-point forward DFT; 2048-point periodic Hann plus 6144 zeros; band power Σ|YH|²/(8192ΣH²). Main |f|≤6.75 kHz; lower −21.75…−8.25 kHz; upper +8.25…+21.75 kHz. Inclusive bin centers. Each side/main ratio; reference leakage retained. Below 1e−13 → <−130 dBc numerical floor.
ccdf: Strict > exceedance over 2048 samples; 0…12 dB in .1 dB steps plus exact peak. Input and output use their own means. Finite population, no stable tail probability.
energy: Post-Rapp amplitude r(t); I=20 mA+Pout r²/(.45 Voc), startup 5 mA, sleep 5 µA. Exact first/second waveform moments with ∫r²=Ton−4Tr/3 and ∫r⁴=Ton−8Tr/5. Vload=Voc−IR; no RF feedback or hidden drive solver.
thermal: Pheat=Pload+Pin r²−Pout r² at PA planes; battery I²R and post-PA filter loss excluded from PA heat. Waveform moments, not mean(I)². Each ramp split into 64 equal intervals (128 convergence check), exact exponential steps at midpoint heat; all event boundaries included. Constant plateaus/start/sleep analytic. Affine periodic map solved exactly, residual target 1e−6 K; grid check ≤.01 K.
radiation: Gain optional, same illustrative +z co-polar direction at 2.450 GHz; accepted or realized basis explicit. No R2-to-PA Γ transformation, no TRP without angular/polarization integration.
unknown: Real I/Q, LO, filter amplitude/group delay, mismatch distortion, harmonics, spurs, memory, settling, detector calibration, antenna efficiency/pattern, actual supply/thermal feedback, range, legal emissions, delivered payload/retries and battery lifetime remain unknown.
precision: Full-precision inputs/results; half-away-from-zero display, six decimal places (nine for independent anchors), SVG coordinates three decimals. JSON retains full numeric precision and null for absent/unknown quantities.

p05-m05-tx-ledger-v1 · p05-m05-tx-ledger-v1 · p04-m05-prbs9-rrc-v1 · p04-m05-pa-rapp-v1 · p05-m05-tx-screens-v1 · p05-m05-tx-record-json-v1 · p05-m05-display-v1

Applied QPSK; 20 dB gain, +20 dBm asymptotic envelope Psat, p=3, AM–PM maximum 3°. Local coefficients are frozen across voltage/temperature. Actual hardware behavior is unknown.

Local operating-mode record · same content as Copy

{
  "versions": {
    "model": "p05-m05-tx-ledger-v1",
    "fixture": "p05-m05-tx-ledger-v1",
    "waveform": "p04-m05-prbs9-rrc-v1",
    "pa": "p04-m05-pa-rapp-v1",
    "rules": "p05-m05-tx-screens-v1",
    "serialization": "p05-m05-tx-record-json-v1",
    "display": "p05-m05-display-v1"
  },
  "metadata": {
    "evidence": "Illustrative engineering case / simulated matched PA and synthetic command energy; not measured.",
    "planes": "D3 complex samples → A0 digital DAC codes → A1 analog baseband → R0 RF modulator → R1-PA-in → R1-PA-out → R2 antenna feed → conditional S0 direction/polarization. RF wave references real 50 Ω.",
    "waveform": "PRBS-9 0x1FF; emit bit0, feedback bit0 XOR bit4, right shift into bit8; Gray QPSK/16QAM; 256 symbols; 65 unit-energy RRC taps α=.35 at 8 samples/symbol; convolution indices 32…2079; mean |w|²=1 over 2048 samples.",
    "timing": "QPSK: 20 kbit/s uncoded/no-overhead mapping, Rs=10 ksymbol/s, D3 Fs=80 ksample/s, 25.6 ms symbol interval. 16QAM at unchanged Rs is a 40 kbit/s mapped variant; CW carries no mapped bits. Command timing is a separate population, not a regenerated RF burst.",
    "evm": "c=Σ(y conj(x))/Σ|x|²; EVM=100√(Σ|y−cx|²/Σ|cx|²). 2048 oversampled pairs; common complex gain only, no timing recovery, matched filtering, equalization or DPD.",
    "spectrum": "8192-point forward DFT; 2048-point periodic Hann plus 6144 zeros; band power Σ|YH|²/(8192ΣH²). Main |f|≤6.75 kHz; lower −21.75…−8.25 kHz; upper +8.25…+21.75 kHz. Inclusive bin centers. Each side/main ratio; reference leakage retained. Below 1e−13 → <−130 dBc numerical floor.",
    "ccdf": "Strict > exceedance over 2048 samples; 0…12 dB in .1 dB steps plus exact peak. Input and output use their own means. Finite population, no stable tail probability.",
    "energy": "Post-Rapp amplitude r(t); I=20 mA+Pout r²/(.45 Voc), startup 5 mA, sleep 5 µA. Exact first/second waveform moments with ∫r²=Ton−4Tr/3 and ∫r⁴=Ton−8Tr/5. Vload=Voc−IR; no RF feedback or hidden drive solver.",
    "thermal": "Pheat=Pload+Pin r²−Pout r² at PA planes; battery I²R and post-PA filter loss excluded from PA heat. Waveform moments, not mean(I)². Each ramp split into 64 equal intervals (128 convergence check), exact exponential steps at midpoint heat; all event boundaries included. Constant plateaus/start/sleep analytic. Affine periodic map solved exactly, residual target 1e−6 K; grid check ≤.01 K.",
    "radiation": "Gain optional, same illustrative +z co-polar direction at 2.450 GHz; accepted or realized basis explicit. No R2-to-PA Γ transformation, no TRP without angular/polarization integration.",
    "unknown": "Real I/Q, LO, filter amplitude/group delay, mismatch distortion, harmonics, spurs, memory, settling, detector calibration, antenna efficiency/pattern, actual supply/thermal feedback, range, legal emissions, delivered payload/retries and battery lifetime remain unknown.",
    "precision": "Full-precision inputs/results; half-away-from-zero display, six decimal places (nine for independent anchors), SVG coordinates three decimals. JSON retains full numeric precision and null for absent/unknown quantities."
  },
  "input": {
    "mode": "nominal",
    "waveform": "qpsk",
    "paPreset": "nominal",
    "request": 10,
    "loss": 2,
    "vswr": 1,
    "gainKnown": false,
    "gain": 2,
    "gainBasis": "accepted",
    "on": 25.6,
    "ramp": 0.2,
    "startup": 2,
    "cycle": 1000,
    "voc": 3,
    "resistance": 2,
    "ambient": 25,
    "rth": 100,
    "cth": 0.02,
    "demand": "waveform",
    "thermalStart": "ambient",
    "toleranceOn": false,
    "tolerance": 0
  },
  "pa": {
    "gainDb": 20,
    "psatDbm": 20,
    "p": 3,
    "phaseMaxDeg": 3
  },
  "demand": {
    "iqA": 0.02,
    "etaAdd": 0.45,
    "startupA": 0.005,
    "sleepA": 0.000005,
    "prescribedA": 0.08
  },
  "result": {
    "status": "inspect",
    "binding": "Hardware evidence remains open; supplied numeric screens clear",
    "deliveredRfSupportedByScreen": true,
    "conditionalPaRequestDbm": 12,
    "driveDbm": -8,
    "conditionalPaActualDbm": 11.987889751698063,
    "conditionalR2": {
      "rho": 0,
      "forwardW": 0.0099721539649943,
      "reflectedW": 0,
      "acceptedW": 0.0099721539649943,
      "forwardDbm": 9.987889751698063,
      "reflectedDbm": null,
      "acceptedDbm": 9.987889751698063,
      "eirpDbm": null,
      "trp": null
    },
    "burstForwardDbm": 9.942413474190857,
    "inputPaprDb": 3.7951804319837597,
    "outputPaprDb": 3.7300961457077007,
    "evmPercent": 0.24322717999503046,
    "lowerDbc": -50.49440609129768,
    "upperDbc": -50.76160393407164,
    "referenceLowerDbc": -51.50811601072544,
    "referenceUpperDbc": -51.51400798037514,
    "chargeC": 0.0008234458812173609,
    "sourceJ": 0.0024703376436520826,
    "resistorJ": 0.0000525033563865159,
    "loadJ": 0.0024178342872655666,
    "peakA": 0.047635340581093505,
    "averageA": 0.0008234458812173609,
    "minV": 2.904729318837813,
    "sleepMs": 972.4000000000001,
    "firstTempC": 25.099694054257025,
    "periodicTempC": 25.254819596542518,
    "selectedTempC": 25.099694054257025,
    "convergenceK": 1.3787627428651916e-8,
    "periodicResidualK": 0
  },
  "axes": [
    {
      "axis": "R2 burst-average forward",
      "value": "9.942413 dBm",
      "limit": "9…11 dBm (normal telemetry)",
      "status": "clear",
      "reason": "Includes the command ramps inside Ton; stationary power is reported separately."
    },
    {
      "axis": "Fitted oversampled EVM proxy",
      "value": "0.243227 %",
      "limit": "≤3 %",
      "status": "clear",
      "reason": "Matched memoryless PA only; total transmitter EVM remains unknown."
    },
    {
      "axis": "Lower adjacent proxy",
      "value": "−50.494406 dBc",
      "limit": "≤−35 dBc each",
      "status": "clear",
      "reason": "Stationary near-carrier side/main integration; not burst splatter or a named emission mask."
    },
    {
      "axis": "Upper adjacent proxy",
      "value": "−50.761604 dBc",
      "limit": "≤−35 dBc each",
      "status": "clear",
      "reason": "Stationary near-carrier side/main integration; not burst splatter or a named emission mask."
    },
    {
      "axis": "Battery-source cycle energy",
      "value": "2.470338 mJ",
      "limit": "≤10 mJ / declared cycle",
      "status": "clear",
      "reason": "1000.0 ms cycle; default 1 s exercise. Payload, retries and lifetime unknown."
    },
    {
      "axis": "Instantaneous loaded supply",
      "value": "2.904729 V",
      "limit": "≥2.2 V",
      "status": "clear",
      "reason": "Actual waveform peak in prescribed demand; supply does not feed back into the Rapp coefficients."
    },
    {
      "axis": "Periodic temperature estimate",
      "value": "25.254820 °C",
      "limit": "≤85 °C",
      "status": "clear",
      "reason": "First burst and selected start shown separately. One-pole PA heat estimate; not measured junction temperature."
    },
    {
      "axis": "Remaining hardware evidence",
      "value": "Unknown",
      "limit": "Detector / PA / antenna / supply tests required",
      "status": "inspect",
      "reason": "Real I/Q, LO, filter amplitude/group delay, mismatch distortion, harmonics, spurs, memory, settling, detector calibration, antenna efficiency/pattern, actual supply/thermal feedback, range, legal emissions, delivered payload/retries and battery lifetime remain unknown."
    }
  ],
  "corners": [
    {
      "loss": 2,
      "paCommand": 12,
      "r2StationaryDbm": 9.987889751698063,
      "r2BurstDbm": 9.942413474190857,
      "powerClear": true
    }
  ],
  "keyTimes": [
    {
      "timeMs": 0,
      "state": "Startup",
      "amplitude": 0,
      "currentA": 0.005,
      "meanV": 2.99,
      "rfForwardW": 0
    },
    {
      "timeMs": 2,
      "state": "RF command",
      "amplitude": 0,
      "currentA": 0.02,
      "meanV": 2.96,
      "rfForwardW": 0
    },
    {
      "timeMs": 2.2,
      "state": "RF command",
      "amplitude": 1,
      "currentA": 0.031707258469106346,
      "meanV": 2.9365854830617875,
      "rfForwardW": 0.0099721539649943
    },
    {
      "timeMs": 27.400000000000002,
      "state": "RF command",
      "amplitude": 0.9999999999999964,
      "currentA": 0.03170725846910626,
      "meanV": 2.9365854830617875,
      "rfForwardW": 0.00997215396499423
    },
    {
      "timeMs": 27.6,
      "state": "Sleep",
      "amplitude": 0,
      "currentA": 0.000005,
      "meanV": 2.99999,
      "rfForwardW": 0
    },
    {
      "timeMs": 1000,
      "state": "Cycle end (left limit · Sleep)",
      "amplitude": 0,
      "currentA": 0.000005,
      "meanV": 2.99999,
      "rfForwardW": 0
    }
  ],
  "ccdf": {
    "input": [
      {
        "thresholdDb": 0,
        "count": 1107,
        "percent": 54.052734375,
        "peak": false
      },
      {
        "thresholdDb": 0.1,
        "count": 1068,
        "percent": 52.1484375,
        "peak": false
      },
      {
        "thresholdDb": 0.2,
        "count": 1011,
        "percent": 49.365234375,
        "peak": false
      },
      {
        "thresholdDb": 0.3,
        "count": 962,
        "percent": 46.97265625,
        "peak": false
      },
      {
        "thresholdDb": 0.4,
        "count": 900,
        "percent": 43.9453125,
        "peak": false
      },
      {
        "thresholdDb": 0.5,
        "count": 847,
        "percent": 41.357421875,
        "peak": false
      },
      {
        "thresholdDb": 0.6,
        "count": 786,
        "percent": 38.37890625,
        "peak": false
      },
      {
        "thresholdDb": 0.7,
        "count": 731,
        "percent": 35.693359375,
        "peak": false
      },
      {
        "thresholdDb": 0.8,
        "count": 667,
        "percent": 32.568359375,
        "peak": false
      },
      {
        "thresholdDb": 0.9,
        "count": 600,
        "percent": 29.296875,
        "peak": false
      },
      {
        "thresholdDb": 1,
        "count": 554,
        "percent": 27.05078125,
        "peak": false
      },
      {
        "thresholdDb": 1.1,
        "count": 494,
        "percent": 24.12109375,
        "peak": false
      },
      {
        "thresholdDb": 1.2,
        "count": 458,
        "percent": 22.36328125,
        "peak": false
      },
      {
        "thresholdDb": 1.3,
        "count": 408,
        "percent": 19.921875,
        "peak": false
      },
      {
        "thresholdDb": 1.4,
        "count": 379,
        "percent": 18.505859375,
        "peak": false
      },
      {
        "thresholdDb": 1.5,
        "count": 326,
        "percent": 15.91796875,
        "peak": false
      },
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  },
  "nextTests": [
    "Calibrate forward/reverse coupler and detector versus frequency, waveform, temperature, mismatch phase and response time.",
    "Measure PA waveform power/EVM/emissions and settling across supply, temperature and transformed load.",
    "Measure R2 antenna impedance and direction/polarization gain; integrate pattern for TRP.",
    "Capture simultaneous RF envelope, current and loaded rail under actual battery state and burst schedule."
  ]
}

The nominal results and all five canonical modes are available in this page without JavaScript. Editing and copying require JavaScript; browser print retains the lesson, equations, results and evidence record.

10 / 10

Baseline the node transmit modes

Which mode can you defend today, and which first test could overturn it?

Load Nominal. Reconcile −8 dBm PA input, +12 dBm linear command and the two actual stationary outputs. Distinguish the ramp-weighted R2 burst average.
Load Stress, compare QPSK and 16QAM, and apply the edit. Identify both EVM and adjacent failures before considering more drive.
Load Low battery. Explain why the waveform calculation survives only as a conditional diagnostic.
Load Hot, then select a periodic thermal start. Compare first and repeated peaks without inventing temperature-dependent PA coefficients.
Load Mismatch. Reconcile forward, reflected, accepted and conditional directional EIRP; switch gain basis and explain the different assumption.
Load Coexistence. Record the energy change and lost link allocation, then copy or print the committed record with its next tests.

Five canonical transmitter modes · QPSK, 2 dB loss, 2 / 25.6 / 972.4 ms, 0.2 ms ramps inside Ton; illustrative only
Mode / power allocation	Stationary PA → R2 forward; R2 accepted	Fidelity / energy / rail	Temperature / decision / first test
Nominal · request 10 dBm at R2	11.987890 → 9.987890 dBm; 9.987890 dBm accepted. Directional EIRP unknown. TRP unknown.	0.243227% fitted EVM; adjacent L/R −50.494406 dBc / −50.761604 dBc; 2.470338 mJ source; 2.904729 V minimum.	25.099694 / 25.254820 °C first/periodic. Hardware evidence remains open; supplied numeric screens clear. Calibrate detector and verify modulated PA power/settling across corners.
Low battery · request 10 dBm at R2	11.987890 → 9.987890 dBm; 9.987890 dBm accepted. Directional EIRP unknown. TRP unknown.	0.243227% fitted EVM; adjacent L/R −50.494406 dBc / −50.761604 dBc; 2.048848 mJ source; 1.853893 V minimum.	25.071397 / 25.182517 °C first/periodic. Instantaneous loaded supply. Supply fails: RF output remains conditional. Capture current/loaded rail and RF together.
Hot · request 10 dBm at R2	11.987890 → 9.987890 dBm; 9.987890 dBm accepted. Directional EIRP unknown. TRP unknown.	0.243227% fitted EVM; adjacent L/R −50.494406 dBc / −50.761604 dBc; 2.470338 mJ source; 2.904729 V minimum.	70.099694 / 70.254820 °C first/periodic. Hardware evidence remains open; supplied numeric screens clear. Synthetic thermal screen clears; obtain real PA thermal/bias data at 70 °C.
Mismatch · request 10 dBm at R2	11.987890 → 9.987890 dBm; 9.476365 dBm accepted. Conditional EIRP 11.476365 dBm (+2 dBi accepted basis). TRP unknown.	0.243227% fitted EVM; adjacent L/R −50.494406 dBc / −50.761604 dBc; 2.470338 mJ source; 2.904729 V minimum.	25.099694 / 25.254820 °C first/periodic. Hardware evidence remains open; supplied numeric screens clear. R2 Γ does not qualify PA load; measure two-port transformation and antenna gain.
Coexistence · request 0 dBm at R2	1.999988 → −0.000012 dBm; −0.000012 dBm accepted. Directional EIRP unknown. TRP unknown.	0.028646% fitted EVM; adjacent L/R −51.484284 dBc / −51.503143 dBc; 1.669809 mJ source; 2.954374 V minimum.	25.079104 / 25.202490 °C first/periodic. Hardware evidence remains open; supplied numeric screens clear. 10 dB lower request spends link allocation. Test the victim and wanted link; range unresolved.

Printable synthesis · TX operating-mode record

Keep the nominal request provisional; retire the supply and antenna gaps

Choose +10 dBm requested linear R2 forward as the normal telemetry allocation. With QPSK and 2 dB matched post-PA loss, actual stationary PA/R2 means are 11.987889752/9.987889752 dBm. The ramp-weighted R2 burst mean is 9.942413474 dBm. Nominal Γ=0 makes accepted equal forward and reflected exactly zero. Directional EIRP and TRP remain unknown because gain/pattern evidence is absent.

The 0.243227180% fitted sample EVM and both adjacent proxies clear the supplied local screens. They establish neither total transmitter EVM nor emissions. The stationary record remains PRBS-9 0x1FF, 256 QPSK symbols, 65-tap α=.35 RRC, 2048 valid samples, 80 ksample/s, common complex-gain fit and the declared Hann/DFT bands.

Keep 2 ms startup, 25.6 ms on with 0.2 ms ramps inside it, and 972.4 ms sleep as the energy-screening schedule. Source/load/resistor energies are 2.470337644/2.417834287/0.052503356 mJ, charge 0.823445881 mC, peak demand 47.635340581 mA and minimum rail 2.904729319 V. First and periodic thermal states stay separate; the numeric thermal screen clears at nominal and hot ambient. Complete payload survival is still unresolved.

Binding condition: hardware evidence remains open for nominal. Low battery fails the instantaneous supply allocation. The +17 dBm-at-PA 16QAM stress fails fidelity and adjacent screens, and its R2 power also exceeds the normal allocation. Mismatch needs transformed PA-load and antenna evidence. Coexistence uses its own 0 dBm allocation; range remains unknown.

First evidence requests attached to the provisional mode record
Owner / test	Condition to pin	What it could overturn
Detector / calibration	Forward and reverse at R2; frequency, waveform, directivity, temperature, response and uncertainty	The inferred power-control accuracy and directionality.
PA / modulated burst	R1 input/output, actual supply, bias, temperature, load transformed through the two-port, settling and harmonics	Matched memoryless EVM, actual power, safe operating region and transient assumptions.
Antenna / installed product	R2 complex Γ; same direction/polarization gain; enclosure, configuration and angular integration	Accepted/radiated relationship, conditional EIRP or any eventual TRP claim.
Supply / battery pulse	Simultaneous current, loaded rail and RF; discharge state, temperature, regulation and decoupling	The 2.2 V support screen, current-demand law and thermal input.
System / service	Framing, ramps, tails, retries, receive/idle energy and payload delivery	Per-message energy, throughput, reliability and battery lifetime.

Use RF Unit Converter for a standalone power conversion and Return Loss & VSWR for matched-reference reflection arithmetic. The lesson ledger stays a bounded Learn model. Antenna reality, named technologies and measurements retain their owning paths.

Carry this record into 05.6: the modulator’s clean reference still depends on converter headroom, reconstruction, clocks and analog interfaces. D3 samples, A0 codes and A1 voltages cannot inherit one another’s power units.

Ungraded review

Check your understanding

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

01Where does the +10 dBm request live, and how is the PA drive set?
Model answer
At R2 as requested linear forward mean. Add 2 dB matched post-PA loss to obtain +12 dBm linear PA command, then subtract 20 dB small-signal gain for −8 dBm PA input. Fixed scaling yields 11.987889752 dBm actual PA and 9.987889752 dBm stationary R2 forward. No output-target solver repairs compression.
02At fixed +10 dBm forward and VSWR 2, how do accepted-power gain and realized gain agree?
Model answer
Reflected is 1.111111111 mW; accepted is 8.888888889 mW or 9.488474776 dBm. Accepted gain +2 dBi yields conditional EIRP 11.488474776 dBm. Equivalent realized gain is 1.488474776 dBi, applied to forward +10 dBm. Keep the same direction/polarization, apply mismatch once, and leave TRP unknown.
03When do 2%, 3% and 4% errors give 5.385164807%, and when can they give 9%?
Model answer
RSS requires independent, zero-mean RMS errors with the same observation/reference normalization and corrections. Aligned deterministic error vectors can sum coherently to 9%. PA EVM already simulated is not added again, and I/Q/LO/filter/mismatch labels do not establish independence.
04Why are the waveform-moment and 80 mA cycle energies different?
Model answer
The former derives sample demand from the actual Rapp output and integrates r² and r⁴, giving 2.470337644 mJ source. The separate rectangular 80 mA anchor prescribes a different current and gives 6.188586000 mJ. Its source equals 5.860805951 mJ load plus 0.327780049 mJ resistor loss. It does not validate the PA waveform.
05What does low battery invalidate, and why does average current not rescue it?
Model answer
Voc=2.2 V with nonzero current and 6 Ω puts the loaded rail below the local 2.2 V minimum. The prescribed demand is unsupported; RF delivery is not established. The conditional waveform estimate may remain. Peak current governs minimum voltage; E[I²] governs resistor heating, while E[I] sets charge.
06What must the final record still leave open?
Model answer
R2-to-PA load transformation/ruggedness, antenna gain/pattern/TRP, total transmitter EVM, actual spurs/harmonics/emissions, PA/PLL settling, supply and thermal feedback, intact payload and retries, range, legal limits and battery lifetime. Attach the first detector, PA, antenna and supply tests at exact planes and operating conditions.

Sources and further study

Access checked 7 September 2026. All numeric limits, PA coefficients, current demand, thermal constants and antenna gain placeholders are local illustrative constraints. The independent NumPy/analytic fixture and equations supply the numerical evidence; no measured or regulatory result is claimed.

W. F. Egan, Practical RF System Design, Wiley–IEEE, 2003, cascade/nonlinearity/phase-noise chapter sequence. Publisher record and contents consulted; full chapters unavailable. D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012, Chapters 2, 4, 7, 10 and 12: publisher-confirmed reading path for waves, networks, couplers and amplifiers; full text not consulted. The displayed power equations are independently derived.
C. Rapp, Effects of HPA-Nonlinearity on a 4-DPSK/OFDM-Signal, ESA SP-332, 1991, pp.179–184: DLR bibliographic provenance; full paper unavailable there. MathWorks MemorylessNonlinearity documentation, current web model definitions, provides the Rapp-family cross-check. The v-dependent AM–PM law is local. Exact waveform, fit, bands and numerical floor are inherited from 04.5; the system energy law is new.
Rohde & Schwarz, Understanding EVM, 3683.8038.52, version 01.00, October 2022, §§3.1–3.7 and §4.3: EVM reference, normalization, interpretation and measurement processing. Current first-party PDF checked. No technology limit or measured-reference result adopted.
Analog Devices ADL5513, Rev. B, July 2023, specification Table 1, Measurement Mode / Controller Mode / Constant Power Operation, pp.3 and 19–21. Specification conditions: 5 V, 25 °C, real 50 Ω, single-ended CW; application loop uses its own frequency and coupler. These categories inform the calibration request, with no transferred detector coefficients or loop dynamics.
Mini-Circuits, Automated Compression Measurements Using Network Analyzers, first-party technical note, 2015; “Effect of Bad Return Loss of DUT” and calibration sections. Used for finite-directivity contamination and mainline calibration. Web note has no pinned revision; no coupler performance value imported.
Analog Devices HMC453ST89/ST89E, data sheet v02.0710, 2010–2170 MHz application conditions and thermal/PCB guidance, the same separate example as 04.5. +5 V, TA=25 °C, application matching circuit; intercept stimulus and thermal boundary are conditional. It is outside this 2.450 GHz fixture. No device power, current, thermal constant or mismatch capability is substituted into the node.
Energizer Lithium Coin Handbook and Application Manual, March 2018, pp.3–5, Pulse Effects / Temperature / Internal Resistance: pulse and loaded-voltage evidence categories. The examples use specified coin cells, currents and temperatures; this lesson’s Voc/Rbatt are independent illustrative coefficients, not a cell model or lifetime claim.
Texas Instruments, How to Drive Resistive, Inductive, Capacitive, and Lighting Loads, SLVAE30E, February 2021, revised March 2021, p.18, Figure 3-13 and transient-thermal-impedance discussion. Foster RC modeling supplies boundary/packaging context; the PA’s single-pole equation and 100 K/W, .020 J/K values are local, not that device’s thermal model.
IEEE 145-2025, Definitions of Terms for Antennas: official record checked active, published 31 March 2026; full standard text unavailable. Ansys HFSS 2026 R1 Help, Peak Realized Gain, incident-power denominator and port-reference caveat. Used alongside explicit single-port power derivation; no pattern, efficiency or normative gain value supplied.

Model/fixture p05-m05-tx-ledger-v1; PA p04-m05-pa-rapp-v1; waveform p04-m05-prbs9-rrc-v1; rules p05-m05-tx-screens-v1; record p05-m05-tx-record-json-v1; display p05-m05-display-v1. Full-precision calculations, six-decimal live display, nine-decimal independent anchors; numeric anchor tolerance ±10⁻⁶ in displayed dB/%/mJ/mA/V, thermal convergence ≤.01 K and periodic residual ≤10⁻⁶ K. Source/load/resistor boundaries and all unknowns are retained in the copied/printed record.

Transmit Power,Fidelity & Efficiency

Failure: more PA power makes the product worse

Conducted, accepted, and radiated power use different planes

Build the transmitter gain and loss lineup

Waveform PAPR sets PA backoff

Allocate EVM across the transmitter

Track occupied bandwidth, regrowth, harmonics, and spurs

Burst ramp, settling, and transient chirp matter

Power control needs sensing and compensation

Duty, heat, current peaks, supply droop, and coexistence

Heat has memory even when the PA proxy does not

Power and waveform record

Time, source, load and heat record

Baseline the node transmit modes

Keep the nominal request provisional; retire the supply and antenna gaps

Check your understanding

Sources and further study

Transmit Power,
Fidelity & Efficiency