Harmonic Balance: Periodic Steady State and Mode Semantics

The mathematical Fourier-domain solve for nonlinear periodic steady states, its mode semantics, and its evidence limits.

Node role: Method.

Reader task: Formulate and audit a finite Fourier-domain steady-state solve, including pump/signal mode meaning, truncation, branch, stability, source, and port conditions.

Harmonic balance (HB) answers a focused mathematical question: which spectral coefficients satisfy a circuit’s truncated periodic or quasiperiodic steady-state equations under declared drives? It solves directly for those coefficients rather than stepping through time.

The answer is a steady-state candidate. HB does not show that a transient reaches it, that it is the only algebraic branch, or that it is dynamically stable. A small nonlinear residual also does not prove that the selected Fourier basis was large enough.

ImportantUse the method that matches the question

Use HB when a periodic or multi-tone quasiperiodic steady-state candidate, or a small-signal conversion response around one, answers the design question. Use a transient or other appropriate dynamical method when reachability, turn-on, switching time, pulses, chaos, non-periodic noise trajectories, or initial-condition history is the design question.

The decision path

flowchart LR
    Operating["Declare tones and basis<br/>solve nonlinear candidate"] --> Need{"Need weak-signal<br/>response?"}
    Need -->|"no"| Coefficients["Inspect strong-tone<br/>coefficients"]
    Need -->|"yes"| Linearize["Linearize about<br/>the candidate"]
    Coefficients --> Gates{"Evidence gates"}
    Linearize --> Gates
    Gates -->|"pass"| Decision["Human design decision"]
    Gates -->|"fail"| Revise["Revise model, basis,<br/>initialization, or method"]

Before reading an HB result, fix the nonlinear element semantics in Josephson Current, Phase, Energy, and Inductance, the observable in Network Trace Views, and the wave reference in Port Reference Impedance Semantics. All Fourier signs follow Dynamics Conventions and the Flux-to-Mode Symbol Bridge.

1. Replace A Waveform With A Finite Spectrum

The finite Fourier-domain nonlinear formulation is reorganized from (El-Rabaie et al. 1988). This page uses a multi-index lattice to make periodic and multi-tone meanings explicit. The derivative, tuple-frequency, and source- superposition equations below are this page’s algebra under the stated phasor convention, not claims that the source prescribes every implementation tuple.

The independent strong-drive angular frequencies, Fourier-lattice index, and retained lattice are

\[ \boldsymbol{\omega}_p := \begin{bmatrix}\omega_{p,1}&\cdots&\omega_{p,P}\end{bmatrix}^{\mathsf T}, \qquad \mathbf m:=(m_1,\ldots,m_P)\in\mathbb Z^P, \qquad \mathcal K\subset\mathbb Z^P. \]

HB approximates each circuit unknown by a finite Fourier lattice,

\[ \mathbf x(t) \approx \sum_{\mathbf m\in\mathcal K} \mathbf X_{\mathbf m} e^{-i\mathbf m\cdot\boldsymbol{\omega}_p t}, \]

Each \(\mathbf m\) selects one pump-frequency combination; \(\mathcal K\) keeps only the combinations included in the finite model. One commensurate fundamental gives the familiar periodic Fourier series. Several independent tones use the same lattice notation for a multi-tone quasiperiodic response. Thus differentiation multiplies coefficient \(\mathbf X_{\mathbf m}\) by \(-i\mathbf m\cdot\boldsymbol\omega_p\). A solver that stores the conjugate \(e^{+i\omega t}\) coefficients is equivalent only after its mode indices, coefficients, and derivative signs are translated together.

Substituting this expansion into the circuit differential-algebraic equations and matching the retained Fourier components produces a nonlinear algebraic system,

\[ \mathbf F(\mathbf X; \boldsymbol{\omega}_p,\mathbf u,\mathcal K)=\mathbf 0. \]

The Josephson nonlinearity couples coefficients at different tuples: mixing in time becomes convolution across the Fourier lattice. HB therefore solves all retained harmonics together; it is not a collection of independent linear frequency sweeps.

Source superposition and generated mixing

Linear forcing adds at its injection coordinates; nonlinear constitutive terms multiply waveforms and therefore convolve their spectra. This page’s expansion below shows why the number of drive tones, Fourier axes, and nonlinear mixing order are three different quantities.

Declared sources and nonlinear mixing enter the lattice in different ways. For sources indexed by \(\ell\), the forcing vector at tuple \(\mathbf m\) is

\[ \mathbf u_{\mathbf m} = \sum_{\ell}\mathbf B_{\ell} I_{\ell,\mathbf m}, \]

where \(\mathbf B_{\ell}\) is the oriented injection vector for source \(\ell\). Source coefficients scalar-add only when they share the same oriented injection coordinate. Different locations or orientations contribute distinct injection vectors and must not be collapsed into one scalar current. This is ordinary source superposition; it does not by itself create coefficients at undeclared combinations. New combination frequencies arise when nonlinear constitutive terms convolve the retained coefficients while solving the coupled HB equations.

The all-zero tuple

\[ \mathbf 0=(0,\ldots,0) \]

is the DC mode for any number of pump axes. A declared DC bias belongs there; it is not a separate DC coordinate for each axis. Multiple pump axes instead represent independent, noncommensurate fundamental frequencies. A drive at an exact commensurate higher tone belongs at the corresponding higher integer mode of one common fundamental. For example, drives at \(\omega_0\) and \(2\omega_0\) use one axis with tuples \((1)\) and \((2)\), not two independent axes.

The lattice is ambiguous if two distinct retained tuples label exactly the same physical frequency:

\[ \mathbf m\ne\mathbf n, \qquad (\mathbf m-\mathbf n)\cdot\boldsymbol\omega_p=0. \]

Such an exact tuple-frequency collision requires the basis to be consolidated or reparameterized before solving. Near-commensurability may be reported as a conditioning or frequency-resolution diagnostic, but this page creates no universal closeness tolerance or acceptance Gate.

To name the generated processes, expand a nonlinear branch current about its declared operating point \(x_0\):

\[ I(x_0+\xi) = I_0+g_1\xi+\frac{g_2}{2}\xi^2+\frac{g_3}{6}\xi^3+\cdots. \]

A quadratic current term \(g_2\xi^2/2\), equivalently a cubic term in the local energy expansion, enables three-wave mixing (3WM). A cubic current term \(g_3\xi^3/6\), equivalently a quartic energy term, enables four-wave mixing (4WM). Bias and symmetry determine which coefficients are present; both may coexist. One Fourier-lattice nonlinear balance and its one linearized small-signal solve include every retained coupling generated by the selected nonlinear model. They are not separate 3WM and 4WM solvers, and the number of declared pump tones is not the mixing order.

Symbol Meaning What must be recorded
\(\mathbf x(t)\) Node fluxes, branch variables, or other circuit unknowns. Variable definition and sign convention.
\(\boldsymbol{\omega}_p\) Independent strong-drive angular frequencies. Values, ordering, and Hz-to-rad/s boundary.
\(\mathbf m\) Integer coordinate in the pump-frequency lattice. Tuple ordering and the physical frequency it labels.
\(\mathcal K\) Finite retained mode set. Harmonic limits and intermodulation truncation.
\(\mathbf X_{\mathbf m}\) Complex steady-state coefficient. Normalization and whether negative-frequency partners are implicit.
\(\mathbf u\) Source amplitudes and biases. Physical unit, calibration, phase, port, and source model.

2. Separate The Nonlinear Operating Point From The Small Signal

First solve the finite nonlinear balance; only then linearize about that particular branch if a weak probe is needed. The method follows the HB construction (El-Rabaie et al. 1988); translated-band notation here is this page’s explicit organization, not a current package-output claim.

Pumped microwave design may use one or both of two related solves.

Strong-tone nonlinear candidate

The first solve finds \(\mathbf X^*\) such that \(\mathbf F(\mathbf X^*)=\mathbf 0\) for the declared pumps and DC biases. Stop here when the strong-tone coefficients themselves are the requested observable.

Linearized signal and idler response

When a weak-signal response is needed, the second step linearizes the circuit around \(\mathbf X^*\). A probe at angular frequency \(\omega_s\) can then appear in translated bands

\[ \omega_{\mathrm{out}}(\mathbf m) = \omega_s+\mathbf m\cdot\boldsymbol{\omega}_p. \]

The resulting linear system yields mode-to-mode quantities such as scattering or impedance response. An unpumped linear response does not intrinsically require a nonlinear operating-point solve.

WarningThe same tuple notation appears in two different contexts

For a strong source, (0,) denotes the zero-frequency source when DC terms are enabled, and (1,) denotes the first pump fundamental on a single pump axis. For a linearized response, output mode (0,) denotes the untranslated signal band, while another tuple denotes a frequency translated by that tuple’s pump combination. Always label both the tuple and its physical frequency; never infer one meaning from the other context.

For two non-commensurate pumps, (1, 0) and (0, 1) select different strong-tone axes. A linearized output tuple \((m_1,m_2)\) instead labels \(\omega_s+m_1\omega_{p,1}+m_2\omega_{p,2}\).

Preserve the signed physical frequency

A negative tuple component does not by itself mean a negative physical frequency. Compute the full expression first. If the result is negative, do not silently replace it by its absolute value.

The signed frequency and the implementation’s treatment of negative-frequency partners must remain attached to the mode and result metadata.

3. Treat Harmonic Truncation As Part Of The Model

The finite Fourier approximation (El-Rabaie et al. 1988) leaves out coefficients. This is why a small algebraic residual does not establish truncation adequacy. The crop formula below is an explicit engineering basis choice, not a physical mixing-order law or a universal search envelope.

An infinite Fourier lattice is replaced by a finite one. The truncation is therefore a modeling choice, not merely a performance setting.

Choice Role in the finite model Common mistake
Pump-axis count Number of independent strong-drive frequencies. Confusing it with the number of physical sources or ports.
Pump-harmonic limits Harmonics retained in the nonlinear operating point. Increasing Newton iterations while leaving an inadequate basis fixed.
Modulation-harmonic limits Signal/idler bands retained by the linearized problem. Assuming they must equal the pump-harmonic limits.
Intermodulation-order limit Restricts allowed combinations of multi-tone indices. Omitting a conversion path that carries the observable of interest.
DC and mixing-term choices Select which static and nonlinear interactions enter the solve. Changing them without treating the result as a different model.

Mixing order versus intermodulation truncation

Per-axis harmonic limits and an intermodulation crop answer different questions. Pump-harmonic bounds limit the strong-tone operating-point indices on each axis; modulation-harmonic bounds independently limit the translated signal/idler indices in the linearized problem.

For a multi-axis lattice, an optional intermodulation crop may additionally retain only tuples satisfying

\[ \operatorname{order}(\mathbf m) := \lVert\mathbf m\rVert_1 = \sum_{j=1}^{P}|m_j| \le K_{\mathrm{IM}}. \]

This \(L_1\) order is a mode-set truncation rule. It is not the physical mixing order: 3WM and 4WM describe nonlinear interaction terms, whereas \(\operatorname{order}(\mathbf m)\) describes how far one retained tuple lies from the all-zero mode. A model may use only per-axis bounds, or combine them with this crop, but it must record the choice and treat a change as a different finite HB model.

SCQ_Design’s promotion policy requires the selected observables to be stable under a declared basis-refinement study.

4. Make Sources, Units, Ports, And Results Unambiguous

The following identity card should accompany every inspectable HB artifact:

Field Required declaration
Frequency boundary Whether each input is \(f\) in hertz or \(\omega=2\pi f\) in rad/s, and where conversion occurs.
Strong sources Source tuple, port, complex amplitude, physical unit, phase convention, and calibration.
DC bias Whether DC is enabled and how a zero-mode source is represented.
Ports Physical terminal, reference plane, positive direction, and reference impedance.
Mode response Input tuple/port and output tuple/port, plus their physical frequencies.
Output family Whether the value is \(S\), \(Z\), quantum efficiency, commutation evidence, or another declared family.
Circuit identity Netlist/model revision, parameters, nonlinear-element convention, and dependency versions.

Power in dBm is not a source current. Converting between available power, voltage, and current requires a declared source and impedance convention. The same numerical current can represent a different drive after the port model or reference plane changes.

Likewise, an array named S or Z is not self-describing. Its mode ordering, port ordering, reference impedance, and frequency axis are part of the value. Use Port-Termination Compensation only when the desired observable explicitly removes a known physical shunt; it does not replace the HB solve or the scattering-wave reference.

WarningKeep HB mode and port normalization attached

JosephsonCircuits byte amplitudes use a solver-native frequency scaling; they are not directly physical photon-flux amplitudes. A comparison with classical power waves must apply the frequency-dependent conversion owned by Port Reference Impedance Semantics while preserving mode tuples, signed physical frequencies, and the zero-mode exclusion. A physical port-coordinate transform separately follows the power-conjugate coordinate contract.

Neither operation turns a native matched-wave \(S\) submatrix into a zero-current Schur reduction.

5. Read A Mode-To-Mode Result

For a single pump, suppose a weak signal enters input port \(a\) at \(f_s\). An entry labeled

\[ S_{(m),b\leftarrow(0),a}(f_s) \]

describes the linearized response at output port \(b\) in the band \(f_s+m f_p\), from the untranslated input band at port \(a\). In particular:

Output tuple Frequency label Reading question
(0,) \(f_s\) What remains in the original signal band?
(1,) \(f_s+f_p\) How much reaches the upper translated band?
(-1,) \(f_s-f_p\) What is the signed value, and which package convention governs it? Do not use \(|f_s-f_p|\).

Do not call every nonzero translated response an “idler” without declaring the mixing process. Do not compare two tuple-indexed traces until their physical frequencies, input/output roles, and normalizations agree.

6. The HB Evidence Ladder

The algebraic model, selected branch, adequacy of its truncation, and dynamical stability are separate conclusions. The retained review ladder below is an engineering workflow, not a set of physical guarantees of HB and not a newly activated universal acceptance policy. No numerical refinement threshold is introduced by this page.

The solver’s algebraic termination criterion is necessary. For example, the JosephsonCircuits.jl reference documents convergence through a relative or absolute residual test involving ftol. That test says the retained nonlinear system was solved accurately enough; it does not test whether the retained system is an adequate physical approximation.

The official reference directly defines the algebraic residual criterion, and the supporting tutorial directly establishes the finite Fourier truncation and nonlinear algebraic system. The remaining acceptance rows below are explicit SCQ_Design review policy. They are conservative engineering inferences from the finite model and the intended design use; they are not outputs or guarantees of JosephsonCircuits.jl or El-Rabaie et al.

Gate Minimal evidence Authority What a failure means
Algebraic solve Termination status, residual norm, tolerance, iterations, and finite requested outputs. Official solver criterion plus SCQ_Design finite-output gate. The chosen discrete HB problem was not solved or did not return the requested artifact.
Pump-basis refinement Repeat with larger pump-harmonic limits and compare promoted observables. SCQ_Design policy inferred from finite truncation. The nonlinear candidate is truncation-sensitive.
Signal/idler refinement Repeat with a larger modulation basis or intermodulation order. SCQ_Design policy inferred from finite truncation. A conversion path or loading effect may be missing.
Frequency resolution Refine the probe grid around poles, notches, gain peaks, or bifurcations. SCQ_Design review policy. The reported feature may be a sampling artifact.
Branch sensitivity Record continuation order and repeat from justified initial states or sweep directions where multistability is plausible. SCQ_Design review policy when branch selection matters. The result may be one of several algebraic branches.
Physical baseline Compare pump-off/linear limits with an analytic circuit, independent solver, or trusted measurement. SCQ_Design review policy. Units, ports, modes, or model lowering may be wrong.
Stability evidence Apply a stability analysis or time-domain cross-check when operating-point stability affects the decision. SCQ_Design review policy; no stability is inferred from HB residual alone. An algebraic candidate may not be physically observable.
Provenance Preserve the complete identity card and refinement table. SCQ_Design review policy. The result cannot be reproduced or audited.

For basis-refinement evidence, SCQ_Design may record a dimensionless change such as

\[ \epsilon_M = \frac{|M_{\mathrm{refined}}-M_{\mathrm{base}}|} {\max(|M_{\mathrm{refined}}|,M_{\mathrm{floor}})}, \]

with a human-selected \(M_{\mathrm{floor}}\) and acceptance threshold. The floor prevents a harmless near-zero quantity from producing an unbounded relative error. This metric is SCQ_Design policy, not a formula from the cited tutorial; its floor and threshold must not be chosen after seeing the result merely to force a pass.

Plausible output Hidden failure Required response
Smooth gain or \(S\)-parameter curve Basis too small to contain important mixing products. Run and publish a basis-refinement comparison.
Tiny nonlinear residual Convergence to the wrong or unstable branch. Check continuation, initial-state sensitivity, and stability as needed.
Correct-looking center frequency Hz supplied where rad/s was expected, with another scale error compensating. Audit the explicit \(2\pi\) boundary and an independent baseline.
Reasonable pump response dBm-to-current conversion assumed an undeclared impedance. Preserve the source/calibration model.
Correct baseband trace Input/output tuple labels were silently flattened or reordered. Store tuple and port maps with the artifact.
Missing output treated as zero The requested result family was disabled or absent. Fail fast; absence is not physical zero.
Optimizer rejects a point Solver infrastructure failure was classified as poor physics. Separate execution failure from physical rejection.

The last distinction follows Auditable Scientific Optimization: a singular solve may describe a candidate-specific numerical problem, but an unknown exception, malformed intent, or missing artifact must not be hidden inside a finite optimization penalty.

Ready For A Design Decision

Use these retained engineering questions to interpret the requested result, not as a promise that any current package performs all checks. The method’s limits remain: no transient reachability, uniqueness, or dynamical stability follows from a solved Fourier balance alone.

An HB result is ready for Human review only when a reviewer can answer all of these questions from the artifact and linked documentation:

  1. What steady-state question was asked, and why is HB the right method?
  2. Which pumps, DC biases, source units, ports, and reference planes define it?
  3. What does every input and output mode tuple mean in physical frequency?
  4. Which nonlinear, DC, and mixing terms were enabled?
  5. Did the nonlinear residual and every requested output pass?
  6. Are the promoted observables stable under harmonic and frequency refinement?
  7. Could another algebraic branch or an unstable state change the decision?
  8. Does a pump-off, analytic, independent-solver, or measurement baseline agree?

Connections

References

  • El-Rabaie, Fusco, and Stewart’s tutorial (El-Rabaie et al. 1988) derives the Fourier-domain nonlinear algebraic formulation and iterative solution context. See the supporting paper page.
  • JosephsonCircuits.jl documentation documents the package’s harmonic-balance workflow and examples.
  • JosephsonCircuits.jl reference documents hbnlsolve, hblinsolve, hbsolve, harmonic controls, outputs, and nonlinear convergence controls used by the Workbench implementation.
  • The official package hbsolve.jl source shows the signed mode-frequency construction and negative-frequency conjugation used by the current implementation.
Field Value
Status Source-backed
Semantic role Define the finite HB model, the distinction between operating-point and linearized solves, and the evidence needed before a steady-state result supports a design decision.
Used by Nonlinear periodic-response workflows.
Implementation boundary Solver-specific APIs, pump-off profiles, artifact schemas, and executable evidence belong to the implementing child repository.
Open review question Can a reader reconstruct every mode’s signed physical frequency and distinguish algebraic convergence, truncation convergence, branch selection, and dynamical stability?

Bibliography

El-Rabaie, S., V. F. Fusco, and C. Stewart. 1988. “Harmonic Balance Evaluation of Nonlinear Microwave Circuits—a Tutorial Approach.” IEEE Transactions on Education 31 (3): 181–92. https://doi.org/10.1109/13.2310.