Harmonic Balance

# Harmonic Balance Simulation: Solve Nonlinear RF Steady State

Harmonic balance (HB) finds a nonlinear circuit’s steady-state response by solving for the amplitudes and phases of its spectral components. It is useful for RF power amplifiers, mixers and frequency multipliers, where sinusoidal excitation can produce harmonics, compression and mixing products that linear small-signal analysis cannot predict.

HB solves the settled operating condition directly rather than stepping through the entire startup transient. This makes it especially useful when frequency-dependent passive networks surround nonlinear devices. It does not directly show how long a circuit takes to start or settle.

## 1. A distorted waveform is a collection of tones

Nonlinear waveform and its harmonic spectrumTwo periods of a pure input sine and distorted output. The illustrative output is sin(theta) plus 0.28 sin(2 theta) plus 0.16 sin(3 theta), giving harmonic amplitude ratios 1, 0.28 and 0.16.One tone in. A spectrum out.Nonlinear devices reshape the waveform and create harmonics.INPUT · PURE SINEOUTPUT · DISTORTEDTHE SAME OUTPUT, RESOLVED BY FREQUENCYf₀2f₀3f₀Relative amplitudeFrequencyIllustration: chosen Fourier amplitudes, not a transistor simulation.

*The output in this illustration is a deliberately chosen sum of three sine waves, with amplitude ratios 1 : 0.28 : 0.16. It is not measured data or a simulated transistor characteristic. Spectral stems use linear amplitude, not dBm.*

For a driven single-tone periodic circuit, approximate each node voltage with DC plus a finite Fourier series: v(t) = VDC + Σ[Aₖ cos(kω₀t) + Bₖ sin(kω₀t)], for k = 1 through H. The solver must determine the coefficients consistently with the device models, sources and circuit connections. Harmonic phases matter: a magnitude-only spectrum cannot reconstruct the waveform.

A two-tone mixer or amplifier also creates sum and difference frequencies. Multitone HB retains a selected set of combinations such as m f₁ + n f₂. With incommensurate drives the steady state is quasiperiodic, so a single short common period is not assumed. The number of retained combinations controls both coverage and computational cost.

## 2. Balance currents in two domains

Harmonic balance solver iterationGuess harmonic voltages, transform to time to evaluate nonlinear devices, transform their currents back to frequency and test Kirchhoff current residuals. Update voltage guess until tolerance is met; then verify spectral truncation.Solve the waveform by balancing currentsA common alternating time/frequency implementation of HB.1 · GUESS THE SPECTRUMDC + complex harmonicsUnknown node voltages2 · ENTER TIME DOMAINInverse Fourier transformEvaluate device i(t), q(t)3 · RETURN TO SPECTRUMFourier-transform currentsAdd the linear-network part4 · CHECK CURRENT ERRORKCL at every retained toneResidual within tolerance?No: update guessYES → A steady-state solution of the retained model.Next: increase spectral order and check the answer stays stable.

Linear components are convenient to evaluate at each frequency using their admittance or network models. Nonlinear transistor and diode currents are convenient to evaluate from sampled time-domain voltages. A common HB implementation transforms between those representations, forms the current residual at each node and retained frequency, then updates the voltage coefficients with a Newton-type method.

For charge-based nonlinear devices, charge derivatives contribute to current; the model must preserve that behavior consistently through the time/frequency evaluation. The goal is to satisfy Kirchhoff’s current law (KCL) across the retained spectrum. A small residual means the numerical equations are satisfied to the chosen tolerance. It does not establish model accuracy or physical stability.

## 3. Set up a result you can trust

Choose the spectral coverage deliberately. Include enough harmonics and intermodulation order for the waveform and metric of interest. A compressed waveform needs more spectral detail than a nearly sinusoidal one. Time-domain sampling must also be adequate to limit aliasing in nonlinear evaluations.

Increase the order and compare. Repeat the solution with more retained harmonics and appropriate sampling. Check that output power, efficiency, distortion and relevant voltage/current peaks stop changing materially. Convergence at one truncation order is not a spectral-accuracy check.

Use a credible initial condition. DC operating-point initialization, input-power continuation and frequency sweeps can help the nonlinear solve. If a branch jumps or a solution disappears, investigate dependence on the initial guess and sweep direction; circuits can have multiple steady-state solutions.

Keep models and reference planes consistent. Verify device-model validity over the intended bias, drive, temperature and frequency range. Include the fundamental and harmonic impedances of packages, matching networks and loads. HB can predict how those impedances affect an amplifier; load-pull measurement supplies experimental data against which to compare that prediction.

## 4. Match the analysis to the question

QuestionSuitable starting pointWhat to verify separately
Settled harmonic power or gain compressionDriven HBSpectral order and model validity
Two-tone intermodulation or mixer conversionMultitone HBIncluded frequency combinations and drive levels
Oscillator steady-state frequency and amplitudeAutonomous oscillator HBFrequency unknown, phase condition, startup and stability
Startup, switching events or settling timeTransient analysisTime-step and simulation-duration accuracy
Slowly varying modulated RF envelopeCircuit-envelope analysisEnvelope bandwidth and underlying carrier representation

An autonomous oscillator needs extra treatment because its oscillation frequency is unknown and its absolute phase is arbitrary. A dedicated formulation adds the required frequency and phase constraints. Ordinary driven HB is not sufficient simply because the schematic contains an oscillator.

A converged HB result can support compression, harmonic distortion and efficiency calculations. Phase noise or noise conversion requires an additional analysis around the large-signal operating state. Stability likewise requires separate examination: solving the steady-state equations alone does not prove that the circuit will reach or remain in that state.

The practical finish line is a numerically converged, spectrally adequate solution whose models and operating conditions match the intended chip.

## References

  • [Keysight: Harmonic Balance Simulation](https://edadownload.software.keysight.com/eedl/ads/2011_01/pdf/cktsimhb.pdf) — technical documentation for frequency-domain steady-state analysis and its setup.
  • [Keysight: Genesys Harbec Harmonic Balance Simulation](https://www.keysight.com.cn/cn/zh/content/lib/resources/training-materials/genesys-harbec-harmonic-balance-simulation-1823632.html) — applications and the current-balancing interpretation.
  • [Qucs: Basic Concept](https://qucs.sourceforge.net/tech/node31.html) and [solution algorithm](https://qucs.sourceforge.net/tech/node32.html) — time/frequency partitioning, current residuals and Newton iteration.
  • [Qucs: Autonomous Harmonic Balance](https://qucs.sourceforge.net/tech/node35.html) — dedicated oscillator analysis.

Take harmonic balance further

Ask the copilot about this term, or have our engineers assess it against your process.

Ask CFSGPT about this → Request a technical assessment → Open the CFS app