RF Journey

01 / 10

Failure: CW passes, the burst fails

The CW output looks clean. Does the same requested average guarantee a clean burst?

The fictional battery node targets +10.000 dBm average conducted QPSK-like power at its PA module’s forward-wave output, R1-out. To expose the trap, temporarily ask the linear PA model for +17.000 dBm. Keep gain, saturation, bias, temperature, load and average-power definition fixed; change only the waveform.

Think about itWhich +17 dBm case fails the lesson’s 5% EVM reject boundary: true CW, pulse-shaped QPSK, or 16QAM?

Answer

16QAM. The pinned QPSK record gives 2.669393% and passes the local ≤3% accept screen. 16QAM gives 5.659673% and is rejected. True CW has zero residual after common gain/phase correction. A burst does not fail simply because it is a burst; the sample distribution matters.

Opening trap · same +17 dBm requested linear-output average, canonical Rapp/DC, VSWR 2 ∠0°, 25 °C
Waveform	Actual mean (dBm)	EVM proxy (%)	Peak current (mA)
True CW	16.828344	0.000000	52.442021
QPSK	16.683933	2.669393	77.849113
16QAM	16.436585	5.659673	82.030261

The 16QAM input peak is 4.845859 dB above its own mean. Its compressed output is only 16.436585 dBm average, with 0.563415 dB mean compression and 82.030261 mA peak current. Its worse adjacent proxy, −32.965874 dBc, still passes the local −30 dBc limit: EVM is the binding rejection. For QPSK the corresponding input PAPR is 3.795180 dB and output is 16.683933 dBm. Equal averages do not imply equal peak demands.

VSWR 2 also returns one ninth of the modeled forward power. At reflection phase 0°, the voltage factor is 4/3 while the current factor is 2/3. A phase change exchanges those stresses even though delivered power stays fixed in this proxy. None of these factors is a transistor ruggedness measurement.

Common misconceptionThe same average power means the same fidelity and peak stress.

One average hides the amplitude population. A high-PAPR record visits the nonlinear region more often; a different load changes voltage/current factors. Carry waveform, population and plane with every number.

Bring the Path 02 EVM and waveform contract, Path 03 power-wave planes, and the 04.4 active-device evidence record. By the end, you will choose a node backoff, quantify its energy and mismatch, and request the evidence this memoryless model cannot supply. First make every power definition explicit.

02 / 10

Gain, P1dB, saturation, and operating backoff

Let x and y be complex root-power waves in √W at R1-in and R1-out. Their squared magnitudes are carrier-cycle-averaged envelope power in watts. We use e^+jωt and I+jQ. The normalized 1 Ω envelope algebra makes |x|² a power; it does not imply a physical 1 Ω PA load. The declared RF wave reference is real 50 Ω.

\begin{aligned}\operatorname{mean}|w[n]|^2&=1\\x[n]&=\sqrt{\frac{P_{\mathrm{request}}}{G_{0,\mathrm{power}}}}\,w[n]\\P_{\mathrm{in}}&=\operatorname{mean}|x|^2,\quad P_{\mathrm{out}}=\operatorname{mean}|y|^2\end{aligned}

Definition / derived scaling. Prequest is the average output a linear PA would produce [W]; G0,power=10^(G0,dB/10). w is dimensionless, normalized on exactly 2048 samples.

Power statistics at a named plane; review RF Fundamentals Power for voltage/RMS conversions
Quantity	Definition / condition
Small-signal gain G0	Pout/Pin as drive tends to zero; 20 dB means ×100 in power, ×10 in root-power amplitude.
Compressed gain	10 log₁₀(Pout/Pin); mean gain for this exact waveform.
Compression / achieved error	Prequest,dBm − Pout,dBm. Input is fixed by the linear request; no iterative target solver.
Psat	Asymptotic SAMPLE-envelope power, 100 mW here. Not a promised modulated average.
PEP / sampled envelope peak	Maximum carrier-cycle envelope power; finite sampled maximum may miss continuous-time peaks. Not instantaneous v(t)i(t).
Stationary on-state average	Mean over 2048 valid samples, before the artificial command ramp.
Burst / time average	Ramp-weighted mean over commanded Ton / complete Ton+Toff cycle. Idle periods and gates change the result.

At +10 dBm requested output and 20 dB gain, the input is −10 dBm = 0.100000 mW. The QPSK output is 9.996939 dBm, so its compressed gain is 19.996939 dB. Raising the requested value spends headroom; it never instructs this model to find more drive until the achieved average reaches an impossible target.

\begin{aligned}C(P)&=\frac{10}{p}\log_{10}\left[1+\left(\frac P{P_{\mathrm{sat}}}\right)^p\right]\\P_{\mathrm{linear,1dB}}&=P_{\mathrm{sat}}\left(10^{p/10}-1\right)^{1/p}\end{aligned}

Derived CW compression for a Rapp PA. P is the linearly requested output power [W], Psat in the same unit; p is dimensionless.

The exact default CW 1 dB point is 19.993125 dBm requested output, 18.993125 dBm actual output, and −0.006875 dBm input. It is not “Psat minus 1 dB” at the input. Input backoff here is Psat,dBm−G0,dB−Pin,dBm; output backoff is Psat,dBm−Pout,dBm. The default QPSK values are 10.000000 and 10.003061 dB.

Go deeperWhy +10 dBm CW is too gentle to be the opening failure

P/Psat = 0.1. With p = 3, compression is (10/3)log₁₀(1.001) = 0.001446925 dB. As drive tends to zero the denominator tends to one; as drive grows without bound the output envelope approaches Asat. All three limits are independently tested.

Common misconceptionP1dB is a safe modulated operating point, and requested output equals achieved output.

P1dB names a waveform and gain-deviation condition. It gives no universal EVM allowance. Our request sets the linear reference and input drive; the actual nonlinear output is separately calculated.

Now inspect what happens to each sample, before collapsing the waveform to a single average.

03 / 10

AM–AM and AM–PM reshape the waveform

AM–AM describes output amplitude as a function of input amplitude. AM–PM describes input-amplitude-dependent phase shift. A constant rotation or gain can be removed by the declared reference fit; sample-dependent distortion remains. The Rapp model’s 1991 provenance and MathWorks’ published amplitude equation establish the model family; this lesson adds its own disclosed saturating phase law.

\begin{aligned}v&=\frac{G_{0,\mathrm{amplitude}}|x|}{A_{\mathrm{sat}}}\\A_{\mathrm{out}}&=\frac{G_{0,\mathrm{amplitude}}|x|}{(1+v^{2p})^{1/(2p)}}\\\phi(v)&=\frac{\phi_{\mathrm{max,deg}}\pi}{180}\frac{v^2}{1+v^2}\\y&=A_{\mathrm{out}}\exp[j(\arg x+\phi)]\end{aligned}

Definition: Aout and Asat [√W], G0,amplitude=√G0,power; v dimensionless, φ in radians. φmax is entered in degrees and converted exactly once. At x=0, y=0 and phase is N/A.

At v = 1, amplitude is 2^−1/6Asat for p = 3, and the added phase is 1.5° for φmax = 3°. Larger samples receive both more compression and more phase rotation. The two laws use instantaneous envelope only: they cannot remember a preceding high-power sample.

Think about itDoes 0 dB PAPR guarantee low adjacent-channel power?

Answer

No. The deliberately unfiltered phase-hopping fixture has constant magnitude but abrupt phase steps. Its adjacent proxies are −14.688991 and −14.706449 dBc even before the PA. They stay the same at +10 and +17 dBm because a memoryless PA gives each constant-magnitude sample the same complex multiplier. Reducing drive cannot repair intrinsic waveform occupancy.

Go deeperReproduce the exact waveform and observation population

PRBS-9 x^9+x^5+1; state 0x1FF reset per fixture; emit bit0, feedback bit0 XOR bit4, shift right into bit8; consume continuously; b0 MSB. Each fixture restarts the generator. QPSK uses consecutive bit pairs; 16QAM consumes consecutive I and Q pairs. CW is exactly w[n] = 1+j0. Phase-hopping repeats the same 256 QPSK symbols eight times each without filtering.

QPSK 00,01,11,10 -> (1+j),(-1+j),(-1-j),(1-j)/sqrt(2); 16QAM successive I/Q bit pairs 00,01,11,10 -> -3,-1,+1,+3, /sqrt(10). The shaped signals place symbol m at index 8m, convolve 2048 samples with 65 symmetric RRC taps, and retain indices 32 through 2079 inclusive from the 2112-sample linear convolution. Tap energy is one; the retained waveform is then separately normalized to mean |w|² = 1. Group-delay trimming is a pinned finite record, not proof that every edge effect vanished.

\begin{aligned}h(t) &= \frac{[\sin (\pi t(1-\alpha)) + 4\alpha t \cos (\pi t(1+\alpha))]}{[\pi t(1-(4\alpha t)^{2})]} \\ h(0) &= 1 + \alpha (\frac{4}{\pi }-1)\end{aligned}

Local finite RRC definition: α=.35, t=k/8 symbol periods, k=−32…32. Normalize Σh²=1 before convolution.

At t = ±1/(4α), use (α/√2)[(1+2/π)sin(π/(4α))+(1−2/π)cos(π/(4α))]. Fs = 8Rs; no absolute Rs is assigned in this module. This is a stated local variation from Path 02’s physical-rate case. The statistics fixture is never stretched to match the burst-time controls.

\begin{aligned}c&=\frac{\sum_n y[n]\operatorname{conj}(x[n])}{\sum_n|x[n]|^2}\\\mathrm{EVM}_{\mathrm{proxy}}&=100\sqrt{\frac{\sum_n|y[n]-cx[n]|^2}{\sum_n|cx[n]|^2}}\,\%\end{aligned}

Derived sample-domain EVM proxy, all 2048 valid oversampled pairs at R1. c fits one common complex gain. No timing recovery, matched filter, symbol sampling, equalizer or DPD.

This local sample proxy differs from Path 02’s receiver decision-plane EVM. Rohde & Schwarz’s EVM guidance motivates declaring the normalization, reference and processing; it does not supply this lesson’s thresholds. The finite-record spectrum uses a periodic Hann window, 2048 samples plus 6144 zeros, and an 8192-point forward DFT. Main |f/Rs| ≤ 0.675; lower adjacent −2.175…−0.825; upper +0.825…+2.175. Include bin centers on boundaries and keep the two side ratios separate.

\begin{aligned}P_B&=\frac{\sum_{k\in B}|Y_W[k]|^2}{8192\sum_n h_W[n]^2}\\\text{Adjacent-side proxy}&=10\log_{10}\left(\frac{P_{\mathrm{adj,side}}}{P_{\mathrm{main}}}\right)\end{aligned}

Derived absolute band power [W]; hW[n]=.5[1−cos(2πn/2048)]. YW is the unnormalized forward DFT, exp(−j2πkn/8192). Signed q=k below 4096, else k−8192; f=qFs/8192.

Show the identical undistorted-reference calculation without subtracting it. At +10 dBm QPSK the lower/upper reference leakage is −51.508116/−51.514008 dBc; the PA output gives −51.071344/−51.195645 dBc. Ratios below 10⁻¹³ display “<−130 dBc numerical floor.” Zero padding samples the finite spectrum more finely; it does not create a longer observation or establish physical rejection below that floor.

Common misconceptionA memoryless EVM or adjacent-power proxy proves compliance.

A technology standard defines its own waveform, receiver, bands, gates and limits. Our model includes none of those procedures. PAPR and CCDF also require the stated oversampling and observation population; a short finite record cannot certify rare peaks.

04 / 10

Two-tone IM3 is a clue, not the modulated answer

In a weak cubic nonlinearity, two equal input tones produce nearby products at 2f1−f2 and 2f2−f1. Their output power rises about 3 dB per 1 dB increase in each tone, while the fundamental rises 1 dB. Extrapolating those lines defines an intercept that the real amplifier never needs to reach.

\begin{aligned}P_{\mathrm{IM3,out}}&\approx3P_{\mathrm{tone,out}}-2\mathrm{OIP3}\\\mathrm{OIP3}&\approx P_{\mathrm{tone,out}}+\frac{P_{\mathrm{tone,out}}-P_{\mathrm{IM3,out}}}{2}\end{aligned}

Derived small-signal dB extrapolation: output power PER TONE and output-referred OIP3 in dBm, at the same plane and declared tone conditions. Not a total two-tone power or a Rapp fit.

Separate illustrative two-tone fixture · 2.4495 and 2.4505 GHz, 1 MHz spacing, 1 kHz integration around each resolved tone/product, CW, 25 °C, 3.3 V, matched 50 Ω R1 ports
Per-tone output	Assumed OIP3	Predicted each IM3 product	Check
−5 dBm	+30 dBm	−75 dBm	70 dB below each tone; deep small-signal region assumed.
0 dBm	+30 dBm	−60 dBm	5 dB higher tone predicts 15 dB higher product.

This is an arithmetic teaching fixture, not an intercept extracted from our p = 3 Rapp curve. The Rapp expansion does not have a universal cubic coefficient at every p; assigning a convenient OIP3 would invent evidence. For a real PA, record tone spacing, per-tone drive, integration bandwidth, bias, temperature, duty, terminations and the input/output reference planes before using the equation.

Common misconceptionOIP3 predicts every modulated waveform.

The intercept summarizes an extrapolation under one two-tone condition. Compression bends the lines; waveform statistics and electrical or thermal memory alter the error. A modulated result requires its own stimulus and observation contract.

The nearby products belong to a fundamental-band envelope model. Harmonics require a wider physical boundary.

05 / 10

Harmonics depend on the output network too

A nonlinear device can generate content near 2fc, 3fc and higher harmonics as well as intermodulation near fc. The output network determines which impedances the device sees at those frequencies. A network that looks like 50 Ω at the fundamental connector can transform to a very different optimum load at the die.

DieNonlinear terminal V/I and fundamental/harmonic loads
Package + output networkParasitics, matching, resonances and loss
R1-out · moduleForward / reflected root-power waves
R2 · antenna feedRequires an explicit network transformation from R1

In our Rapp model y is only the near-carrier complex envelope. Pout = mean |y|² includes main, adjacent and other represented baseband content before partitioning. It excludes physical RF harmonics. Never infer harmonic emissions, terminal voltage waveforms, output-network loss, or dissipated heat by relabeling that envelope power.

Think about itCould a filter reduce conducted second-harmonic output yet change device stress?

Answer

Yes. Rejection at the external port does not state the impedance presented back to the nonlinear device. A reflected harmonic can reshape terminal voltage/current. The required evidence includes harmonic terminations at the correct internal plane, not just external attenuation.

Cripps’ second-edition chapter sequence explicitly connects amplifier modes, harmonic terminations and load pull. The present lesson uses those distinctions for interpretation; device-terminal synthesis and harmonic balance remain outside the tradebench. Next, state exactly which RF power enters an efficiency ratio.

06 / 10

Efficiency, PAE, burst duty, and battery energy

Drain or collector efficiency compares RF output to DC input. PAE additionally debits the RF drive. Compute both from linear watts at named boundaries. A data-sheet value may use fundamental CW output or a defined channel; our forward-power proxy uses total represented near-carrier envelope power, so it cannot be substituted for either without qualification.

\begin{aligned}P_{\mathrm{DC}}[n]&=V_{\mathrm{DD}}I_q+\frac{|y[n]|^2}{\eta_{\mathrm{add}}}\\\eta&=\frac{\operatorname{mean}|y|^2}{\operatorname{mean}P_{\mathrm{DC}}}\\\mathrm{PAE}&=\frac{\operatorname{mean}|y|^2-\operatorname{mean}|x|^2}{\operatorname{mean}P_{\mathrm{DC}}}\end{aligned}

Illustrative DC law: VDD [V], Iq and Isleep [A], Pout,envelope [W], ηadd dimensionless. On-state average uses a ratio of means, not a mean of instantaneous efficiencies.

The nominal supply overhead is 3.3 V × 20 mA = 66 mW even at vanishing RF output. At the +10 dBm QPSK point, average current is 26.729263 mA, peak current is 36.062513 mA, efficiency is 11.329037% and PAE is 11.215667%. With positive Iq, 0 ≤ η < ηadd and PAE = η−Pin/PDC ≤ η. Excessive input drive can make PAE negative; show that signed result and reject it.

\begin{aligned}I_{\mathrm{avg}} &= \frac{\operatorname{mean} P_{\mathrm{DC}}}{V_{\mathrm{DD}}} \\ I_{\mathrm{peak}} &= \frac{[V_{\mathrm{DD}} I_{q} + \frac{\max |y[n]|^{2}}{\eta _{\mathrm{add}}}]}{V_{\mathrm{DD}}}\end{aligned}

Derived peak-current proxy, all 2048 stationary on-state samples. No analog supply response or current filtering.

For energy only, apply a post-Rapp amplitude command r(t): rise linearly for Tr, hold one, fall linearly over the final Tr. The same Iq remains active for all Ton. Integrating r² gives Ton−4Tr/3. Tr = 0 is a valid rectangular command; Toff = 0 is valid continuous cycling. Neither is a settling claim.

\begin{aligned}E_{\mathrm{cycle}}&=V_{\mathrm{DD}}I_q T_{\mathrm{on}}+\frac{P_{\mathrm{out}}}{\eta_{\mathrm{add}}}\left(T_{\mathrm{on}}-\frac{4T_r}{3}\right)\\&\quad+V_{\mathrm{DD}}I_{\mathrm{sleep}}T_{\mathrm{off}}\\\delta_{\mathrm{cmd}}&=\frac{T_{\mathrm{on}}}{T_{\mathrm{on}}+T_{\mathrm{off}}}\\\delta_{\mathrm{RF}}&=\frac{T_{\mathrm{on}}-4T_r/3}{T_{\mathrm{on}}+T_{\mathrm{off}}}\end{aligned}

Derived energy bookkeeping in seconds and joules. Tr≤min(100 µs,0.2Ton); on-state sample statistics are not assigned a physical duration.

For Ton = 1.000 ms, Toff = 9.000 ms and Tr = 10.000 µs, commanded duty is 10%, while RF-energy-equivalent duty is 9.866667%. The default QPSK cycle consumes 88.207479 µJ: 66.000000 µJ quiescent, 21.910479 µJ added-RF contribution, and 0.297000 µJ sleep energy. Burst energy excludes the sleep term; long-term current divides the full cycle energy by supply voltage and cycle duration.

Think about itDoes the higher-PAE +17 dBm point necessarily use less battery energy per identical burst?

Answer

No. True CW at +17 dBm has 27.548585% PAE but consumes 171.928220 µJ per default cycle, compared with 88.207479 µJ for +10 dBm QPSK. Those fixtures transmit different RF energy. A battery-life comparison also needs delivered payload, retries, regulator efficiency, other circuit loads, voltage and temperature.

Common misconceptionHigher PAE always gives longer battery life; short duty removes thermal and supply concerns.

PAE is an on-state ratio. On-time, idle current, supply conversion, peak current and retransmissions change total energy. A short burst can still cause droop, startup error or high peak dissipation. This bookkeeping law has no electrothermal dynamics.

Go deeperCanonical tables at two backoffs, available without JavaScript

Illustrative / simulated · p04-m05-pa-rapp-v1

2.450 GHz · 25 °C reference · QPSK · 2048 on-state samples · Fs = 8Rs, no absolute symbol rate. R1-in to R1-out forward power, with Γ supplied at R1-out. Seed 0x1FF; 20 dB gain, 20 dBm asymptotic Psat, p = 3, AM–PM parameter 3°. DC fixture: 3.3 V / 20 mA / ηadd 0.45 / sleep 10 µA.

VSWR 2.00 ∠0°; Ton 1.000 ms, Toff 9.000 ms, each command ramp 10.000 µs. The sample record has no physical burst duration. No memory, harmonics, thermal or supply dynamics, die stress, load pull, ruggedness, or compliance result.

Frozen +10 and +17 dBm cases · canonical parameters; only waveform and requested power change · R1-out · simulated p04-m05-pa-rapp-v1; common gain, Psat, DC and load conditions
Waveform / request	Actual / compression (dBm / dB)	Input / output PAPR (dB)	EVM (%) / status
True CW / 10.0 dBm	9.998553 / 0.001447	0.000000 / 0.000000	0.000000 / accept: All bounded numeric screens pass
Phase-hopping constant envelope / 10.0 dBm	9.998553 / 0.001447	0.000000 / 0.000000	0.000000 / reject: Worse-side adjacent proxy
QPSK / 10.0 dBm	9.996939 / 0.003061	3.795180 / 3.778460	0.153473 / accept: All bounded numeric screens pass
16QAM / 10.0 dBm	9.993082 / 0.006918	4.845859 / 4.812196	0.267236 / accept: All bounded numeric screens pass
True CW / 17.0 dBm	16.828344 / 0.171656	0.000000 / 0.000000	0.000000 / accept: All bounded numeric screens pass
Phase-hopping constant envelope / 17.0 dBm	16.828344 / 0.171656	0.000000 / 0.000000	0.000000 / reject: Worse-side adjacent proxy
QPSK / 17.0 dBm	16.683933 / 0.316067	3.795180 / 2.656298	2.669393 / accept: All bounded numeric screens pass
16QAM / 17.0 dBm	16.436585 / 0.563415	4.845859 / 3.206715	5.659673 / reject: Sample-domain EVM proxy

Frozen +10 and +17 dBm cases · canonical parameters; only waveform and requested power change · adjacent proxies and DC bookkeeping; no measured emissions or battery prediction
Waveform / request	Adjacent lower / upper (dBc)	Reference leakage lower / upper (dBc)	Average / peak I (mA)	η / PAE (%) / cycle energy (µJ)
True CW / 10.0 dBm	<−130 dBc numerical floor / <−130 dBc numerical floor	<−130 dBc numerical floor / <−130 dBc numerical floor	26.731764 / 26.731764	11.332188 / 11.218828 / 88.215622
Phase-hopping constant envelope / 10.0 dBm	−14.688991 dBc / −14.706449 dBc	−14.688991 dBc / −14.706449 dBc	26.731764 / 26.731764	11.332188 / 11.218828 / 88.215622
QPSK / 10.0 dBm	−51.071344 dBc / −51.195645 dBc	−51.508116 dBc / −51.514008 dBc	26.729263 / 36.062513	11.329037 / 11.215667 / 88.207479
16QAM / 10.0 dBm	−50.892870 dBc / −50.508710 dBc	−50.753515 dBc / −51.553987 dBc	26.723288 / 40.361107	11.321510 / 11.208115 / 88.188027
True CW / 17.0 dBm	<−130 dBc numerical floor / <−130 dBc numerical floor	<−130 dBc numerical floor / <−130 dBc numerical floor	52.442021 / 52.442021	27.838190 / 27.548585 / 171.928220
Phase-hopping constant envelope / 17.0 dBm	−14.688991 dBc / −14.706449 dBc	−14.688991 dBc / −14.706449 dBc	52.442021 / 52.442021	27.838190 / 27.548585 / 171.928220
QPSK / 17.0 dBm	−38.396041 dBc / −38.885381 dBc	−51.508116 dBc / −51.514008 dBc	51.381008 / 77.849113	27.483800 / 27.188215 / 168.473562
16QAM / 17.0 dBm	−33.758433 dBc / −32.965874 dBc	−50.753515 dBc / −51.553987 dBc	49.643677 / 82.030261	26.870803 / 26.564873 / 162.816813

Command amplitude ramp for energy only. Post-Rapp amplitude multiplier r(t). Quiescent current remains active throughout Ton. This diagram is a command, not device settling or a replay of the 2048-sample fixture.

━ Command multiplier (solid)

Burst and cycle bookkeeping · DC law v1; RF time Ton − 4Tr/3
Quantity	Value	Definition
Commanded / RF-energy duty	10.000000 / 9.866667 %	Ton / cycle; RF equivalent time / cycle
Quiescent energy	66.000000 µJ	3.3 V × 20 mA × full Ton
Added RF-related DC energy	21.910479 µJ	mean Pout / ηadd × (Ton − 4Tr/3)
Sleep energy	0.297000 µJ	VDD Isleep Toff
Burst / cycle DC energy	87.910479 / 88.207479 µJ	On energy / on plus sleep energy
Burst / cycle average RF	9.859716 / 0.985972 mW	R1-out; ramp-weighted on / full-cycle average
Full-cycle average DC current	2.672954 mA	Ecycle / (VDD × cycle duration), not on-state current

You now have the definitions needed for the tradebench. Commit one operating point, inspect its binding constraints, then use the remaining sections to identify evidence that could overturn it.

Class 1 Learn decision model · p04-m05-pa-tradebench-v1

PA Backoff & Load Tradebench

Choose a requested linear-output average and inspect the actual waveform, load, current, and energy. Apply changes to commit one coherent result.

Select the +17 dBm trap, then compare true CW, phase-hopping, QPSK and 16QAM. Identify the cause of each rejection.
Reduce requested power to add backoff. Watch EVM improve and PAE fall. Backoff cannot remove the phase-hopping waveform’s intrinsic adjacent occupancy.
Return to the +10 dBm node, sweep load phase, and name the hardware evidence needed before adopting it.

Numbers use the units below. Times are entered in µs and serialized as integer ns; 1 ms = 1000 µs. Blank, nonfinite, off-step, or out-of-range values are rejected. Presets restore every model, DC, load and timing field.

Case and stimulus

Case preset

WaveformFrozen fixture; QPSK is the node default.

DC bookkeeping presetVDD / Iq / ηadd / sleep. Synthetic fixtures, not bias recommendations.

PA model, supplied load and burst command

Requested linear-output average (dBm)-10 to 30 dBm; step 0.1; default 10. Defines input drive via G0; actual output is never solved to meet this request.

Small-signal gain (dB)5 to 35 dB; step 0.1; default 20.

Asymptotic sample-envelope Psat (dBm)0 to 40 dBm; step 0.1; default 20.

Rapp smoothness p (unitless)1 to 10 unitless; step 0.1; default 3.

Maximum AM–PM parameter (°)0 to 20 °; step 0.1; default 3.

Load VSWR at R1-out (ratio)1 to 6 ratio; step 0.01; default 2.

Load reflection phase at R1-out (°)-180 to 180 °; step 1; default 0.

Burst on-time (µs)10 to 100000 µs; step 1; default 1000.

Burst off-time (µs)0 to 1000000 µs; step 1; default 9000.

Command rise / fall time (each edge) (µs)0 to 100 µs; step 1; default 10. Also ≤20% of on-time. This is a command, not settling.

Applied case: node · QPSK

ACCEPT · All bounded numeric screens pass

Requested linear-output average 10.000000 dBm → actual forward average 9.996939 dBm. This point meets every bounded numeric screen; hardware evidence is still required. These are illustrative lesson limits, never wireless emission limits.

Selected-plane stress is 1.333333×; worst over all phases is 1.333333× at 0° (tie rule applied). A permitted phase does not prove robustness to a changing antenna.

Actual average / sampled peak · R1-out: 9.996939 / 13.775400 dBm
Input average · R1-in: −10.000000 dBm
Gain: small signal / compressed: 20.000 / 19.996939 dB
Compression / achieved-average error: 0.003061 dB
Input backoff / output backoff from Psat: 10.000000 / 10.003061 dB
CW P1dB: input / requested / actual output: −0.006875 / 19.993125 / 18.993125 dBm
Sample EVM proxy · common gain removed: 0.153473%
Input / output envelope PAPR: 3.795180 / 3.778460 dB
Forward efficiency / PAE · ratio of means: 11.329037 / 11.215667%
Average / peak on-state DC current: 26.729263 / 36.062513 mA
DC energy per commanded burst / full cycle: 87.910479 / 88.207479 µJ
Forward / accepted / reflected · R1-out: 9.992955 / 8.882627 / 1.110328 mW

Illustrative / simulated · p04-m05-pa-rapp-v1

Evidence view

Selected sample (0–2047)Observation index only; no change to the model.

Linear reference and compressed envelope. 128 samples around selected sample 1276; selected marker and peak table. Full statistics use all 2048. Power is a near-carrier forward envelope at R1-out.

┄ Linear-output reference (dashed)
━ Compressed output (solid)

On-state current proxy follows compressed peaks. Iq + |y[n]|²/(ηadd VDD); no supply dynamics or device settling. Same sample interval as above.

━ DC current proxy (solid)

Selected, mean and worst sample · exact full-record statistics
Population / sample	Linear / output (mW)	AM–PM (°)	Current (mA)
Selected n=1276	23.961723 / 23.852832	0.579898	36.062513
Peak output n=1276	23.961723 / 23.852832	0.579898	36.062513
2048-sample mean	10.000000 / 9.992955	Not an averaged phase metric	26.729263

Inspect every accept / inspect / reject rule

p04-m05-local-criteria-v1: reject takes precedence; otherwise inspect if any accept limit fails. The stress rule tests max voltage/current at the selected phase. The worst phase sweep is reported separately. Hardware evidence below remains a separate requirement.

Illustrative local requirement card; all axes reported independently
Axis	Applied value	Accept	Reject	Result
Achieved error / compression	0.003061 dB	≤ 1	> 3	accept
Sample-domain EVM proxy	0.153473 %	≤ 3	> 5	accept
Worse-side adjacent proxy	−51.071344 dBc	≤ -30	> -25	accept
Average on-state DC current	26.729263 mA	≤ 100	> 150	accept
Peak on-state DC current	36.062513 mA	≤ 200	> 250	accept
Forward PAE	11.215667 %	≥ 10	< 0	accept
Selected-plane max voltage/current factor	1.333333 ×	≤ 1.5	> 1.65	accept
Requested drive above Psat	−10.000000 dB	≤ 6 (also meet compression screen)	> 6	accept

Compare all four waveforms at this requested average

Applied common-average comparison · R1-out · simulated p04-m05-pa-rapp-v1; common gain, Psat, DC and load conditions
Waveform / request	Actual / compression (dBm / dB)	Input / output PAPR (dB)	EVM (%) / status
True CW / 10.0 dBm	9.998553 / 0.001447	0.000000 / 0.000000	0.000000 / accept: All bounded numeric screens pass
Phase-hopping constant envelope / 10.0 dBm	9.998553 / 0.001447	0.000000 / 0.000000	0.000000 / reject: Worse-side adjacent proxy
QPSK / 10.0 dBm	9.996939 / 0.003061	3.795180 / 3.778460	0.153473 / accept: All bounded numeric screens pass
16QAM / 10.0 dBm	9.993082 / 0.006918	4.845859 / 4.812196	0.267236 / accept: All bounded numeric screens pass

Applied common-average comparison · adjacent proxies and DC bookkeeping; no measured emissions or battery prediction
Waveform / request	Adjacent lower / upper (dBc)	Reference leakage lower / upper (dBc)	Average / peak I (mA)	η / PAE (%) / cycle energy (µJ)
True CW / 10.0 dBm	<−130 dBc numerical floor / <−130 dBc numerical floor	<−130 dBc numerical floor / <−130 dBc numerical floor	26.731764 / 26.731764	11.332188 / 11.218828 / 88.215622
Phase-hopping constant envelope / 10.0 dBm	−14.688991 dBc / −14.706449 dBc	−14.688991 dBc / −14.706449 dBc	26.731764 / 26.731764	11.332188 / 11.218828 / 88.215622
QPSK / 10.0 dBm	−51.071344 dBc / −51.195645 dBc	−51.508116 dBc / −51.514008 dBc	26.729263 / 36.062513	11.329037 / 11.215667 / 88.207479
16QAM / 10.0 dBm	−50.892870 dBc / −50.508710 dBc	−50.753515 dBc / −51.553987 dBc	26.723288 / 40.361107	11.321510 / 11.208115 / 88.188027

Evidence still needed

waveform and emissions at matched observation planes
ruggedness versus load magnitude AND phase
harmonic terminations and emissions
temperature and thermal transients
supply impedance/droop and peak current
electrical/thermal memory and ramp settling
calibrated device load-pull data

Read or manually copy the complete PA record

{
  "metadata": {
    "model": "p04-m05-pa-rapp-v1",
    "tradebench": "p04-m05-pa-tradebench-v1",
    "waveform": "p04-m05-prbs9-rrc-v1",
    "dcModel": "p04-m05-dc-energy-v1",
    "mismatchModel": "p04-m05-fixed-forward-v1",
    "criteria": "p04-m05-local-criteria-v1",
    "evidence": "Illustrative engineering case / Simulated; no measured PA data",
    "carrier": "2.450 GHz; no band authorization implied",
    "temperature": "25 °C reference; no temperature model",
    "inputPlane": "R1-in: matched PA module RF input, root-power wave x [sqrt(W)]",
    "outputPlane": "R1-out: PA module forward-wave output y [sqrt(W)], real 50 ohm wave reference",
    "loadPlane": "Gamma supplied at R1-out, already transformed from R2 antenna feed; no R2-to-R1 inference",
    "phasor": "exp(+j omega t); I+jQ; RF reconstruction I cos - Q sin",
    "normalization": "mean |w|^2=1 across 2048 valid samples; x=sqrt(Prequest/G0power)w; |x|^2, |y|^2 in W. Normalized 1 ohm envelope algebra is not device impedance.",
    "saturation": "Asymptotic sample-envelope Psat=Asat^2; P1dB is a CW model point, not a burst limit",
    "rapp": "v=sqrt(G0power)|x|/sqrt(Psat); Aout=sqrt(G0power)|x|/(1+v^(2p))^(1/(2p))",
    "ampm": "phase=(phiMaxDeg*pi/180)*v^2/(1+v^2); zero input returns y=0 and phase N/A",
    "prbs": "PRBS-9 x^9+x^5+1; state 0x1FF reset per fixture; emit bit0, feedback bit0 XOR bit4, shift right into bit8; consume continuously; b0 MSB",
    "mapping": "QPSK 00,01,11,10 -> (1+j),(-1+j),(-1-j),(1-j)/sqrt(2); 16QAM successive I/Q bit pairs 00,01,11,10 -> -3,-1,+1,+3, /sqrt(10)",
    "record": "256 symbols; Fs=8Rs, no absolute Rs or physical duration; CW w=1; hopping repeats each QPSK symbol 8 samples; RRC QPSK/16QAM 65 symmetric unit-energy taps, alpha=.35, k=-32..32, t=k/8; linear convolution 2112; retain indices 32..2079; normalize once on valid population",
    "ccdf": "Strict > threshold; 0..12 dB step .1 plus exact peak row; input and output each relative to their OWN mean; population 2048",
    "evm": "c=sum(y conj(x))/sum|x|^2; 100 sqrt(sum|y-cx|^2/sum|cx|^2), all 2048 oversampled pairs. Common complex gain only; no timing recovery, matched filter, symbol sampling, equalizer or DPD",
    "fft": "Periodic Hann .5(1-cos(2pi*n/2048)), 6144 zero padding, 8192 forward DFT exp(-j2pi*kn/8192); q=k if k<4096 else k-8192; f=q*8Rs/8192; PB=sum|YW|^2/(8192*sum hW^2)",
    "bands": "Main |f/Rs|<=.675; lower [-2.175,-.825]; upper [.825,2.175], inclusive bin centers, no fractional weights. Each adjacent/main separately, worse side is decision; reference leakage shown, never subtracted",
    "floor": "Side/main <1e-13 reports <-130 dBc numerical floor, not physical rejection",
    "power": "Pout is total near-carrier sampled envelope power BEFORE spectral partition, not total physical RF including harmonics",
    "dc": "PDC=VDD*Iq+Pout/etaAdd; eta=mean(Pout)/mean(PDC); PAE=(mean(Pout)-mean(Pin))/mean(PDC); Ipeak=(VDD*Iq+max|y|^2/etaAdd)/VDD; stationary on-state ratios",
    "energy": "Post-Rapp command amplitude r: linear 0..1 for Tr, plateau, linear 1..0 for Tr; RF time=Ton-4Tr/3. Ecycle=VDD*Iq*Ton+mean(Pout)/etaAdd*(Ton-4Tr/3)+VDD*Isleep*Toff; deltaCmd=Ton/(Ton+Toff); deltaRF=(Ton-4Tr/3)/(Ton+Toff). No rerun of PA along ramp; not settling",
    "mismatch": "rho=(VSWR-1)/(VSWR+1); Gamma=rho exp(j*phaseDeg*pi/180); Pref=Pforward*rho^2; Paccepted=Pforward*(1-rho^2); voltage/current factors |1+Gamma|,|1-Gamma|; selected stress=max(factors); no feedback into Rapp",
    "sweep": "-180..180 degrees inclusive, step 1; ties smallest absolute phase then negative phase, tolerance 1e-12",
    "omitted": [
      "harmonics",
      "die voltage/current stress",
      "physical load pull",
      "source re-reflection",
      "memory",
      "electrothermal feedback",
      "supply droop",
      "device settling",
      "DPD",
      "ruggedness",
      "battery life",
      "compliance"
    ],
    "neededEvidence": [
      "waveform and emissions at matched observation planes",
      "ruggedness versus load magnitude AND phase",
      "harmonic terminations and emissions",
      "temperature and thermal transients",
      "supply impedance/droop and peak current",
      "electrical/thermal memory and ramp settling",
      "calibrated device load-pull data"
    ],
    "rounding": "6 decimals, half away from zero; SVG coordinates 3 decimals; full internal precision"
  },
  "applied": {
    "case": "node",
    "waveform": "qpsk",
    "dc": "nominal",
    "requestDbm": 10,
    "gainDb": 20,
    "psatDbm": 20,
    "p": 3,
    "phaseMaxDeg": 3,
    "vswr": 2,
    "phaseDeg": 0,
    "onNs": 1000000,
    "offNs": 9000000,
    "rampNs": 10000
  },
  "dc": {
    "vdd": 3.3,
    "iq": 0.02,
    "etaAdd": 0.45,
    "sleep": 0.00001
  },
  "result": {
    "status": "accept",
    "binding": [
      "All bounded numeric screens pass"
    ],
    "requestedDbm": "10.000000",
    "actualDbm": "9.996939",
    "peakDbm": "13.775400",
    "inputDbm": "−10.000000",
    "compressionDb": "0.003061",
    "p1": {
      "requestedW": 0.099841827,
      "requestedDbm": 19.9931252,
      "actualDbm": 18.9931252,
      "inputDbm": -0.0068748
    },
    "inputPaprDb": "3.795180",
    "outputPaprDb": "3.778460",
    "evmPercent": "0.153473",
    "adjacentLower": "−51.071344 dBc",
    "adjacentUpper": "−51.195645 dBc",
    "referenceLower": "−51.508116 dBc",
    "referenceUpper": "−51.514008 dBc",
    "mainMw": "9.999513",
    "lowerMw": "0.000078",
    "upperMw": "0.000076",
    "avgCurrentMa": "26.729263",
    "peakCurrentMa": "36.062513",
    "efficiencyPercent": "11.329037",
    "paePercent": "11.215667",
    "burstEnergyUj": "87.910479",
    "cycleEnergyUj": "88.207479",
    "commandedDuty": "0.100000",
    "rfDuty": "0.098667",
    "forwardMw": "9.992955",
    "acceptedMw": "8.882627",
    "reflectedMw": "1.110328",
    "voltageFactor": "1.333333",
    "currentFactor": "0.666667",
    "allPhaseWorst": "1.333333",
    "worstPhaseDeg": 0
  },
  "screens": [
    {
      "axis": "Achieved error / compression",
      "value": "0.003061",
      "unit": "dB",
      "accept": "≤ 1",
      "reject": "> 3",
      "status": "accept"
    },
    {
      "axis": "Sample-domain EVM proxy",
      "value": "0.153473",
      "unit": "%",
      "accept": "≤ 3",
      "reject": "> 5",
      "status": "accept"
    },
    {
      "axis": "Worse-side adjacent proxy",
      "value": "−51.071344",
      "unit": "dBc",
      "accept": "≤ -30",
      "reject": "> -25",
      "status": "accept"
    },
    {
      "axis": "Average on-state DC current",
      "value": "26.729263",
      "unit": "mA",
      "accept": "≤ 100",
      "reject": "> 150",
      "status": "accept"
    },
    {
      "axis": "Peak on-state DC current",
      "value": "36.062513",
      "unit": "mA",
      "accept": "≤ 200",
      "reject": "> 250",
      "status": "accept"
    },
    {
      "axis": "Forward PAE",
      "value": "11.215667",
      "unit": "%",
      "accept": "≥ 10",
      "reject": "< 0",
      "status": "accept"
    },
    {
      "axis": "Selected-plane max voltage/current factor",
      "value": "1.333333",
      "unit": "×",
      "accept": "≤ 1.5",
      "reject": "> 1.65",
      "status": "accept"
    },
    {
      "axis": "Requested drive above Psat",
      "value": "−10.000000",
      "unit": "dB",
      "accept": "≤ 6 (also meet compression screen)",
      "reject": "> 6",
      "status": "accept"
    }
  ],
  "loadKeyRows": [
    {
      "phaseDeg": -180,
      "voltage": "0.666667",
      "current": "1.333333"
    },
    {
      "phaseDeg": -90,
      "voltage": "1.054093",
      "current": "1.054093"
    },
    {
      "phaseDeg": 0,
      "voltage": "1.333333",
      "current": "0.666667"
    },
    {
      "phaseDeg": 90,
      "voltage": "1.054093",
      "current": "1.054093"
    },
    {
      "phaseDeg": 180,
      "voltage": "0.666667",
      "current": "1.333333"
    }
  ],
  "inputCcdf": [
    {
      "thresholdDb": "0.000000",
      "count": 1107,
      "percent": "54.052734",
      "peak": false
    },
    {
      "thresholdDb": "0.100000",
      "count": 1068,
      "percent": "52.148438",
      "peak": false
    },
    {
      "thresholdDb": "0.200000",
      "count": 1011,
      "percent": "49.365234",
      "peak": false
    },
    {
      "thresholdDb": "0.300000",
      "count": 962,
      "percent": "46.972656",
      "peak": false
    },
    {
      "thresholdDb": "0.400000",
      "count": 900,
      "percent": "43.945313",
      "peak": false
    },
    {
      "thresholdDb": "0.500000",
      "count": 847,
      "percent": "41.357422",
      "peak": false
    },
    {
      "thresholdDb": "0.600000",
      "count": 786,
      "percent": "38.378906",
      "peak": false
    },
    {
      "thresholdDb": "0.700000",
      "count": 731,
      "percent": "35.693359",
      "peak": false
    },
    {
      "thresholdDb": "0.800000",
      "count": 667,
      "percent": "32.568359",
      "peak": false
    },
    {
      "thresholdDb": "0.900000",
      "count": 600,
      "percent": "29.296875",
      "peak": false
    },
    {
      "thresholdDb": "1.000000",
      "count": 554,
      "percent": "27.050781",
      "peak": false
    },
    {
      "thresholdDb": "1.100000",
      "count": 494,
      "percent": "24.121094",
      "peak": false
    },
    {
      "thresholdDb": "1.200000",
      "count": 458,
      "percent": "22.363281",
      "peak": false
    },
    {
      "thresholdDb": "1.300000",
      "count": 408,
      "percent": "19.921875",
      "peak": false
    },
    {
      "thresholdDb": "1.400000",
      "count": 379,
      "percent": "18.505859",
      "peak": false
    },
    {
      "thresholdDb": "1.500000",
      "count": 326,
      "percent": "15.917969",
      "peak": false
    },
    {
      "thresholdDb": "1.600000",
      "count": 288,
      "percent": "14.062500",
      "peak": false
    },
    {
      "thresholdDb": "1.700000",
      "count": 247,
      "percent": "12.060547",
      "peak": false
    },
    {
      "thresholdDb": "1.800000",
      "count": 206,
      "percent": "10.058594",
      "peak": false
    },
    {
      "thresholdDb": "1.900000",
      "count": 165,
      "percent": "8.056641",
      "peak": false
    },
    {
      "thresholdDb": "2.000000",
      "count": 133,
      "percent": "6.494141",
      "peak": false
    },
    {
      "thresholdDb": "2.100000",
      "count": 111,
      "percent": "5.419922",
      "peak": false
    },
    {
      "thresholdDb": "2.200000",
      "count": 99,
      "percent": "4.833984",
      "peak": false
    },
    {
      "thresholdDb": "2.300000",
      "count": 79,
      "percent": "3.857422",
      "peak": false
    },
    {
      "thresholdDb": "2.400000",
      "count": 69,
      "percent": "3.369141",
      "peak": false
    },
    {
      "thresholdDb": "2.500000",
      "count": 59,
      "percent": "2.880859",
      "peak": false
    },
    {
      "thresholdDb": "2.600000",
      "count": 47,
      "percent": "2.294922",
      "peak": false
    },
    {
      "thresholdDb": "2.700000",
      "count": 34,
      "percent": "1.660156",
      "peak": false
    },
    {
      "thresholdDb": "2.800000",
      "count": 27,
      "percent": "1.318359",
      "peak": false
    },
    {
      "thresholdDb": "2.900000",
      "count": 18,
      "percent": "0.878906",
      "peak": false
    },
    {
      "thresholdDb": "3.000000",
      "count": 15,
      "percent": "0.732422",
      "peak": false
    },
    {
      "thresholdDb": "3.100000",
      "count": 12,
      "percent": "0.585938",
      "peak": false
    },
    {
      "thresholdDb": "3.200000",
      "count": 4,
      "percent": "0.195313",
      "peak": false
    },
    {
      "thresholdDb": "3.300000",
      "count": 3,
      "percent": "0.146484",
      "peak": false
    },
    {
      "thresholdDb": "3.400000",
      "count": 3,
      "percent": "0.146484",
      "peak": false
    },
    {
      "thresholdDb": "3.500000",
      "count": 3,
      "percent": "0.146484",
      "peak": false
    },
    {
      "thresholdDb": "3.600000",
      "count": 2,
      "percent": "0.097656",
      "peak": false
    },
    {
      "thresholdDb": "3.700000",
      "count": 1,
      "percent": "0.048828",
      "peak": false
    },
    {
      "thresholdDb": "3.795180",
      "count": 0,
      "percent": "0.000000",
      "peak": true
    },
    {
      "thresholdDb": "3.800000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "3.900000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.000000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.100000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.200000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.300000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.400000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.500000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.600000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.700000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.800000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "4.900000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.000000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.100000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.200000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.300000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.400000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.500000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.600000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.700000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.800000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "5.900000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.000000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.100000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.200000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.300000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.400000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.500000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.600000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.700000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.800000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "6.900000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.000000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.100000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.200000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.300000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.400000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.500000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.600000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.700000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.800000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "7.900000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.000000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.100000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.200000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.300000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.400000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.500000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.600000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
    },
    {
      "thresholdDb": "8.700000",
      "count": 0,
      "percent": "0.000000",
      "peak": false
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}

07 / 10

PA classes and waveform compatibility

A class describes a mode of operation, not an automatic efficiency or waveform-quality guarantee. For conventional tuned amplifiers, conduction angle states how long device current flows in each carrier cycle. Bias, drive, load and the output network determine the useful result. Steer’s class discussion supplies the orientation below.

Informative class orientation · idealized conduction angle; no device design or efficiency promise
Mode	Conduction / mechanism	Waveform consequence
A	360° conduction	Can preserve amplitude with adequate headroom; quiescent power remains at small signal.
AB	More than 180°, less than 360°	Trades quiescent current and distortion; practical amplitude-modulated use requires qualification.
B	180° ideal conduction	Tuned or combined output restores the fundamental; crossover and drive dependence need attention.
C	Less than 180°	Strongly nonlinear current pulses; constant-envelope use with a suitable network, not direct amplitude fidelity.
D / E; harmonic shaping F	Switching or terminal-waveform shaping	D switches states; E shapes switching transitions; F engineers harmonic impedances. These are not one conduction-angle ladder.

Switching and harmonic-shaping modes seek low simultaneous device voltage and current, with strongly constrained terminations. Preserving an amplitude-varying signal may require a broader architecture such as envelope control; that architecture and its bandwidth are additional evidence. The tradebench’s three DC presets are synthetic laws, not class A/AB/B/C selections.

A class label therefore cannot decide whether this node’s QPSK envelope survives. Keep the measured amplitude/phase response, gain, load, thermal state and burst behavior attached. Now examine the load itself.

08 / 10

Load line, load pull, mismatch, and ruggedness

A load line relates terminal current and voltage under an assumed circuit and bias. At RF, reactive and harmonic terminations can turn the dynamic trajectory into a loop. An optimum fundamental load is conditional on the chosen objective—power, efficiency or distortion—and need not equal a small-signal conjugate match.

Load pull varies a load while observing a device at fixed declared drive and operating conditions. A contour joins equal measured or simulated values of one metric. Read its plane, frequency, bias, temperature, harmonic terminations and stimulus before comparing a power contour with an efficiency contour. Their optima may differ. No vendor contour is copied or reconstructed here.

\begin{aligned}\rho &= \frac{\mathrm{VSWR}-1}{\mathrm{VSWR}+1}\qquad \Gamma = \rho \exp (j\phi) \\ P_{\mathrm{reflected}} &= P_{\mathrm{forward}} \rho ^{2} \\ P_{\mathrm{accepted}} &= P_{\mathrm{forward}}(1-\rho ^{2}) \\ \text{Voltage factor} &= |1+\Gamma|\qquad \text{current factor} = |1-\Gamma|\end{aligned}

Derived fixed-forward proxy at R1-out only. All P in W; φ in radians internally. Γ is supplied at this plane; a measured R2 antenna reflection is not automatically the same Γ.

Independent anchor: EXACT +10.000 dBm fixed forward power, separate from slightly compressed default PA output
Quantity / condition	Derived result
VSWR 2	\|Γ\|=1/3; reflected fraction=1/9
Forward / reflected / accepted	10.000000 / 1.111111 / 8.888889 mW
Reflected / accepted levels	+0.457575 / +9.488475 dBm
Phase 0°: voltage / current factor	1.333333 / 0.666667
Phase ±180°: voltage / current factor	0.666667 / 1.333333
Phase ±90°: both factors	√(10/9) = 1.054093
Matched VSWR 1	No reflected power; both factors exactly one

Standing-wave factors at the supplied load plane. Fixed forward wave, VSWR held constant. Voltage and current factors exchange extrema with phase; accepted/reflected powers do not change. Endpoints are equivalent.

━ Voltage |1+Γ| (solid)
┄ Current |1−Γ| (dashed)

Selected and extreme phase rows; no transistor stress or load-pull result
Phase / condition	Voltage ×	Current ×	Accepted / reflected (mW)
0° · selected	1.333333	0.666667	8.882627 / 1.110328
-180°	0.666667	1.333333	8.882627 / 1.110328
-90°	1.054093	1.054093	8.882627 / 1.110328
0°	1.333333	0.666667	8.882627 / 1.110328
90°	1.054093	1.054093	8.882627 / 1.110328
180°	0.666667	1.333333	8.882627 / 1.110328

All-phase worst 1.333333×: choose 0° by smallest |phase| then negative phase. Current maximum is at -180°. R2 antenna Γ is not reused here without a transformation.

The stress screen uses the larger voltage/current factor at the selected phase. The entire −180°…+180° sweep is also shown: the worst factor is 1+ρ. Equal maxima choose the smallest absolute phase, then negative phase. At VSWR 6, the worst factor is 12/7 = 1.714286, exceeding the local 1.65 reject boundary. At the default VSWR 2 it is only 1.333333; that is numeric screening, not survival evidence.

Common misconceptionMismatch only reduces delivered power, and this phase map proves ruggedness.

The fixed-forward model explains external accepted/reflected power and standing-wave factors. It neither changes the Rapp output with Γ nor computes die stress, source re-reflections or load pull. Real feedback can change output, current, stability and terminal trajectories. Ruggedness needs device evidence across phase, duration, power, bias and temperature.

Go deeperRead one real PA’s conditions before transferring a headline

The HMC453ST89 / ST89E data sheet v02.0710, currently linked by Analog Devices, gives a useful contrast. Its 2010–2170 MHz application band uses +5 V, TA = 25 °C and the designated matching circuit. Typical output P1dB is +32.5 dBm, Psat +32.75 dBm, and OIP3 +49 dBm; the intercept footnote specifies 0 dBm input per tone and 1 MHz spacing. Quiescent collector current is 725 mA.

Those are application-circuit RF results, not bare-die data or the 2.450 GHz teaching fixture. Its board guidance calls for a heat sink; the junction-to-ground-paddle thermal resistance is 17.1 °C/W. The table does not pin every compression stimulus gate, saturation criterion, uncertainty or mismatch phase. Record duty/ramp, precise power de-embedding and all-phase ruggedness as missing. Do not transfer this out-of-band example into the node model. Accessed 6 September 2026.

09 / 10

Bias, temperature, memory, supply, and settling

A real PA can give different outputs for identical current input samples because its recent history differs. Bias networks and supply impedance introduce electrical memory; heating changes parameters more slowly. A changing envelope can modulate the supply, which changes amplitude and phase again. A static 25 °C curve contains none of that history.

Hardware evidence request · the memoryless model returns no settling or temperature result
Mechanism	Possible consequence	Evidence needed next
Supply impedance / droop	Envelope current changes rail voltage and RF response	Synchronized current, supply and RF envelope for actual load and burst timing.
Bias / control transitions	Startup or mode change differs from settled operation	Amplitude, phase and spectral behavior versus time after each commanded state change.
Thermal memory	Gain/phase drift within or between bursts	Case/junction boundary, thermal transient evidence, duty history and temperature corners.
Electrical memory	Dependence on recent envelope and tone spacing	Modulated and spacing-dependent response; compare rising/falling envelopes at equal magnitude.
Mismatch dynamics	Load phase interacts with internal output trajectory	Qualified ruggedness and stability over specified VSWR, phase, power and dwell duration.
Production / lifetime	Nominal fit fails to bound a population	Guaranteed conditions or characterized spread; reliability evidence tied to actual stresses.

Think about itCan changing the tradebench’s 10 µs command ramp tell you when the PA settles?

Answer

No. It changes an analytically integrated RF-energy multiplier after the stationary PA. Iq stays active throughout Ton, and the model never runs a transient differential equation. Device settling remains unsupported; request time-resolved RF amplitude/phase and supply/current evidence.

The model also makes no DPD or crest-factor-reduction claim. Measurement setup, uncertainty and calibrated load pull belong to Path 08; practical bias, layout and thermal realization to Path 10. ADI’s signal-chain overview places the PA within a broader gain/efficiency/linearity trade, but the final device choice needs its complete conditional evidence.

10 / 10

Choose node PA power and backoff

Keep +10.000 dBm requested linear-output average, canonical QPSK, and the supplied VSWR 2 load as the node’s provisional model baseline. It produces 9.996939 dBm actual forward average, 8.882627 mW accepted at the supplied R1-out load, 0.153473% sample EVM, and 88.207479 µJ per default cycle. The target is forward conducted power; do not silently reinterpret it as accepted antenna power.

Decision synthesis · canonical model/DC and timing; all rows are Illustrative
Candidate	Decision	Why / next action
+10 dBm QPSK, VSWR 2 ∠0°	Accept numeric baseline	Meets output/fidelity/current/PAE/factor screens. Request actual modulated and ruggedness evidence.
+17 dBm QPSK, VSWR 2 ∠0°	Accept numeric screen; not chosen for node	2.669393% EVM; 168.473562 µJ/cycle. Extra output is not required by the +10 dBm target.
+17 dBm 16QAM, VSWR 2 ∠0°	Reject	5.659673% EVM >5%. Increase backoff, reassess waveform or obtain a better-supported PA.
+10 dBm QPSK, VSWR 6 ∠0°	reject	1.714286× factor >1.65. Power delivery alone misses this rejection.
Unfiltered phase-hopping, either power	Reject waveform occupancy	Adjacent proxy >−25 dBc already in the reference. Backoff does not fix it.

ACCEPT · All bounded numeric screens pass

Selected-plane stress is 1.333333×; worst over all phases is 1.333333× at 0° (tie rule applied). A permitted phase does not prove robustness to a changing antenna.

Two requirements travel with this choice. First, qualify power, EVM, emissions and supply current on the actual burst across voltage, temperature and timing corners at compatible planes; include ramps and inter-burst history. Second, obtain all-phase mismatch/ruggedness evidence for the assembled output network and define a protection or power-reduction policy for loads outside that qualified region. The model does not select protection thresholds for a real transistor.

Illustrative / simulated · p04-m05-pa-rapp-v1

PA addition to the component evidence pack · provisional node record
Record field	Carry forward
Function / requested operation	Create the 2.450 GHz QPSK-like forward conducted signal. Request +10 dBm from the linear reference; actual 9.996939 dBm.
Model / evidence	p04-m05-pa-rapp-v1; p04-m05-prbs9-rrc-v1; p04-m05-dc-energy-v1; p04-m05-fixed-forward-v1. Simulated / Illustrative, not measured.
Planes and load	R1-in to R1-out; real 50 Ω wave reference, Γ supplied at R1-out, VSWR 2 ∠0°. R2 transformation belongs in the real network record.
Screens / result	p04-m05-local-criteria-v1: accept for bounded numeric values. Worst all-phase factor 1.333333×. Hardware evidence stays separate.
Falsifying evidence	Actual modulated response, peak supply current or phase-dependent mismatch performance outside the promised region.
Unproven	Memory, physical harmonic power, load pull, die V/I, thermal behavior, settling, ruggedness, battery life, lifetime, DPD and compliance.

The tradebench’s copy action exports your applied point, all screen outcomes, exact input units, source/model conventions, key load rows and full CCDF counts. Keep that record beside the earlier passive/filter/LNA decisions. Complete transmitter allocation belongs to Path 05.5; the next Path 04 module introduces mixer frequency conversion.

Ungraded review

Check your understanding

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

01What does +10 dBm requested output mean here, and why is the actual QPSK result different?
Model answer
It is the average output the linear 20 dB PA would produce. Scale x=sqrt(Prequest/100)w, giving −10 dBm input. The Rapp PA produces 9.996939 dBm average; no solver increases drive to force +10 dBm.
02Why can true CW have zero corrected EVM while compressed, and why does phase-hopping still fail?
Model answer
Constant magnitude receives one constant complex multiplier, removed by c. That does not remove average compression. Phase-hopping’s unfiltered phase discontinuities have about −14.7 dBc adjacent occupancy even in the undistorted reference; backoff cannot cure it.
03Where is the default model’s CW P1dB point?
Model answer
Requested linear output 19.993125 dBm; actual output 18.993125 dBm; input −0.006875 dBm with 20 dB gain. The exact power relation is Psat(10^(p/10)−1)^(1/p). It is not a universal modulated limit.
04Why is RF duty 9.866667% when commanded duty is 10%, and what stays on?
Model answer
The 10 µs linear amplitude ramps at each edge integrate in power to Ton−4Tr/3. Iq stays active over the full 1 ms Ton. Include 9 ms sleep energy separately. The model is energy bookkeeping, not settling or battery life.
05At VSWR 2, is +10 dBm forward power also +10 dBm accepted? Can R2 Γ be used directly?
Model answer
No. With fixed 10 mW forward, 1.111111 mW is reflected and 8.888889 mW accepted, or 9.488475 dBm. Γ must already be at R1-out or be transformed through the actual network. The fixed-forward map does not compute die stress or load pull.
06What overturns a numeric accept and what must the node evidence pack request?
Model answer
A measured waveform, supply transient, temperature corner or load-phase result outside the assumed envelope can overturn it. Request actual modulated/emissions and current evidence plus all-phase ruggedness/protection qualification. Numeric accept proves none of memory, harmonics, thermal dynamics, settling, lifetime or compliance.

Sources and further study

Access checked 6 September 2026. Curves and tables on this page are original local derivations or simulations. No vendor data are fitted into the tradebench. Publisher records establish editions and reading scope; openly accessible references cross-check the technical equations.

C. Rapp, Effects of HPA-Nonlinearity on a 4-DPSK/OFDM-Signal, ESA SP-332, 1991, pp. 179–184; DLR bibliographic record (full text unavailable there). Model provenance. MathWorks MemorylessNonlinearity documentation, current web documentation, modified Rapp equation and model limitations; the page records changes through R2024b. Our v-dependent phase law is explicitly local, not a vendor fit.
S. C. Cripps, RF Power Amplifiers for Wireless Communications, 2nd ed., Artech House, 2006, ISBN 9781596930186. Publisher-confirmed reading scope: modes, envelope distortion, bias, harmonic terminations and load pull; full textbook not reproduced.
D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012 (MW-1), Chapters 2, 4, 10 and 12. Network and power-wave reading path. Fixed-forward relations on this page are also derived explicitly and independently tested.
M. Steer, Microwave and RF Design V: Amplifiers and Oscillators, 3rd ed., 2019; accessible chapter versions: §2.5 Classes A/AB/B/C, §4.3 Switching modes, §4.5 Loadpull. Informative class, terminal and load interpretation.
A. Patyuchenko, RF Signal Chain Discourse, Part 2: Essential Building Blocks, Analog Dialogue, July 2021 (CIR-2). PA role and gain/linearity/efficiency tradeoffs.
Rohde & Schwarz, Understanding EVM, 3683.8038.52, Version 01.00, October 2022 (SYS-3). EVM interpretation and normalization only; no local criteria or standards result borrowed.
Analog Devices, HMC453ST89 / HMC453ST89E, data sheet v02.0710, official product-linked document (web listing 16 April 2015). Separate conditional example: RF application circuit, compression/intercept definitions, bias and thermal boundary; missing conditions remain explicit.

Power Amplifiers

Failure: CW passes, the burst fails

Gain, P1dB, saturation, and operating backoff

AM–AM and AM–PM reshape the waveform

Two-tone IM3 is a clue, not the modulated answer

Harmonics depend on the output network too

Efficiency, PAE, burst duty, and battery energy

PA classes and waveform compatibility

Load line, load pull, mismatch, and ruggedness

Bias, temperature, memory, supply, and settling

Choose node PA power and backoff

Check your understanding

Sources and further study