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Symmetry and the Form of Nonlinear Behavioral Models: A Tutorial for Microwave Engineers
Nicholas B. Tufillaro
TL;DR
Behavioral-model structure is often treated as a modeling choice, although the tutorial identifies a gap in explaining why these models share a form. It derives the form from clock-origin invariance and reports that the general solution is exact within its stated setting. The tutorial also checks an index interpretation against published measurement data.
Problem
Behavioral-model functional forms are usually presented as modeling choices, leaving their shared structure unexplained.
Source: §1 · p. 2Method
The tutorial derives behavioral-model structure from symmetry constraints, using the invariance of measurable quantities to arbitrary choices such as the time origin.
Source: §2 · p. 4, §5.2 · p. 8Results
The general solution is exact for time-invariant two-port responses in single-tone periodic steady state, and the mixing-order identity matches all nine tabulated rows.
Source: §5.2 · p. 8, §10.4 · p. 15Takeaways & Limitations
The mixing-order identity provides an arithmetic check on the interpretation of the model indices using a route independent of the published data.
Source: §10.4 · p. 15The tutorial’s scope is fundamental load-pull under single-tone CW excitation and periodic steady state; Theorem 1 additionally assumes a memoryless load-side nonlinearity.
Source: §13 · p. 21
Abstract
from arXiv · showhide
Behavioral models of nonlinear microwave devices -- the Cardiff model, X-parameters, the higher-order describing functions of the mechanical-systems literature -- all share a functional form that is usually presented as a modeling choice. This tutorial shows that the form follows from a single physical statement, that nothing physical depends on where the clock is started. It develops the consequences of that statement assuming phasors and harmonic balance but no group theory, and uses them to give the Cardiff model's three exponents physical interpretations. The magnitude exponent $m$ turns out to be the order of the device's load-side nonlinearity; the phase exponent $n$ is set by the drive-side harmonic; and the conjugate index $r$ obeys $r_{\max}=\lfloor K/2\rfloor$, where $K$ is the degree of the load-side nonlinearity. The familiar restriction $r\le1$ is therefore a statement about the device, exact whenever $K\le3$, and it can be tested on a bench. The final sections show that a tailored A-pull measurement displays this decomposition directly: each spectral cluster's half-width is the order of the nonlinearity that produced it. The tutorial is pedagogical: the material overlaps largely with a companion paper (arXiv:2609.35828), which states and proves the theorems in full; this tutorial starts at a more elementary level and is meant as a gentler introduction to the results treated in more detail there.
1 What this tutorial is for
Behavioral-model structure is usually presented as a modeling choice, but the tutorial argues that invariance to the clock’s starting point determines the form up to arbitrary smooth coefficient functions. It illustrates the symmetry method by showing why an odd-symmetric device driven by one tone has no even harmonics.
1 What this tutorial is for
- Clock-origin invariance determines the form of every harmonic-grid behavioral model up to arbitrary smooth coefficient functions.The tutorial presents model structure as a theorem rather than a modeling choice.Source: §1 · p. 2
- For a two-port in single-tone periodic steady state, the form is complete; a large CW-fit residual points to conditions outside that setting.Examples include another tone or port, harmonic injection, or a measurement outside steady state.Source: §1 · p. 3
- The Cardiff indices count load-side wave factors, the supplied drive-side harmonic, and conjugate pairs; r ≤1 is exact for load-side nonlinearity up to cubic.The restriction also has a measurable signature.Source: §1 · p. 3
- The tutorial assumes phasors, harmonic balance, and the binomial theorem, but not group theory; it is a gentler introduction to results proved fully in companion paper [1].Its symmetry argument eliminates candidate terms before fitting.Source: §1 · p. 3, §2 · p. 4
2 A symmetry argument you already use
- With an odd-symmetric memoryless device driven by one tone, shifting time by half a period forces every even output harmonic to vanish.The argument uses symmetry rather than an expansion or a particular form for the device function.Source: §2 · p. 3
3 The symmetry behind behavioral models
Re-zeroing the clock rotates each harmonic phasor according to its harmonic index, giving a circle of phase transformations. Reference-plane shifts and independent tone phases extend this symmetry to products of circles, while the tutorial focuses on one harmonic grid.
3.1 Where you start the clock
- Re-zeroing the clock leaves the physical waveform unchanged while rotating the k-th harmonic phasor by kθ, where θ = ω0τ.A full-period shift leaves every phasor unchanged, so θ lies on a circle.Source: §3.1 · p. 4, Figure 1 · p. 5
3.2 This set of transformations is a circle group
- These phase shifts form a circle group, and harmonic k has weight k because it transforms by the factor e^jkθ.The weight describes how fast the harmonic rotates relative to the fundamental.Source: §3.2 · p. 5
3.3 More circles: reference planes and extra tones
- Reference-plane shifts and independent phases of multiple tones add one circle per choice; the tutorial restricts its main argument to one harmonic grid.The two-tone M2S parameters of Gibiino et al. are an example of the corresponding two-torus setting.Source: §3.3 · pp. 5–6
4 What survives the rotation: invariants
The section identifies quantities unchanged by a common phase rotation and establishes that they exhaust the smooth invariants. It also relates the Cardiff index constraint to smoothness at zero load wave and briefly describes invariant counting.
4.1 The question
- The invariant question is which functions of the two weight-one incident waves remain unchanged when both undergo a common phase rotation.The section calls functions unchanged by every group element invariants.Source: §4.1 · p. 6
4.2 The answer for a circle
- The magnitudes of the incident waves and their relative phase survive common rotation, because the phases cancel in weight-balanced products.The invariant condition for a monomial is that its total weight is zero.Source: §4.2 · p. 6
4.3 Why that is the complete list
- For smooth device responses, the same finite set of generators exhausts the invariants, so any added physical term is already a function of them.The smooth-function extension is attributed to Schwarz.Source: §4.3 · pp. 6–7
- Smoothness at zero load wave requires m = |n| + 2r, explaining the Cardiff index relation as a regularity condition rather than a convention.The relative phase is undefined when the load wave vanishes, so |A2,1|^m must cancel that singularity.Source: §4.3 · p. 7
4.4 An aside: counting the invariants
- For two weight-one variables and their conjugates, the Molien series counts invariants, with three degree-two generators and one additional invariant appearing at most linearly.The additional invariant is Im(A2,1 conjugate(A1,1)); no later result depends on this counting aside.Source: §4.4 · pp. 7–8
5 What the output must do: equivariance
A device response transforms with the harmonic weight of its output, rather than remaining invariant. Factoring out a carrier yields the general equivariant form, and gauge fixing leaves phase-reference-independent device data.
5.1 The output is not invariant
- The output Bp,h is a harmonic-h phasor, so it rotates with weight h; such a map is equivariant, with invariance as the h = 0 case.Source: §5.1 · p. 8
5.2 Factoring an equivariant
- An equivariant of weight h factors into a carrier of weight h times an arbitrary invariant function.The tutorial gives this as the general solution, not an approximation, for time-invariant two-port responses in single-tone periodic steady state.Source: §5.2 · p. 8
5.3 Gauge fixing, and why three communities wrote the same factor
- Choosing the carrier fixes a gauge; after absolute phase is normalized out, the coefficient functions are device data independent of the phase reference.The factor corresponds to the phase normalization of X-parameters, the virtual harmonics generator, and the Cardiff phase-vector construction.Source: §5.3 · p. 9
6 Writing the exponents as monomials
The Cardiff indices can be rewritten as exponents of a monomial in the load wave and its conjugate. In this lattice, r ≤1 keeps an L-shaped region, excluding interior monomials that first appear beyond total degree 3.
6.1 Two ways to write the same term
- The exact identity |A|^m(∠A)^n = A^aĀ^b gives a=(m+n)/2 and b=(m−n)/2, with r indexing conjugate pairs.A term with r=0 is a pure power of A or Ā; each additional r contributes another factor |A|^2.Source: §6.1 · p. 9
6.2 What r ≤1 keeps and what it discards
- On the (a,b) lattice, r=min(a,b), so r≤1 keeps the first two rows and columns and excludes the interior where a≥2 and b≥2.Figure 3 identifies A^2Ā^2=|A|^4 as the first excluded monomial.Source: §6.2 · p. 10, Figure 3 · p. 10
- Through total degree 3 the restriction excludes no monomials; beyond that, it retains 4D−2 terms against (D+1)(D+2)/2 total terms through degree D.The restriction therefore removes many coefficients at high order and almost none at low order.Source: §6.2 · p. 10
7 Where r ≤1 comes from
For a memoryless device with polynomial load-side dependence and open-loop load voltage, the maximum conjugate-pair index is set by the load-side degree. The resulting bound explains when r≤1 is exact and remains applicable across several model variations.
7.1 Setting up
- A degree-p load-side term has a+b=p, while the drive-side harmonic sets a−b=h−ℓ.The setup assumes a memoryless device with polynomial load-side dependence and load voltage set by the incident wave.Source: §7.1 · p. 11
7.2 The theorem
- For load-side polynomial degree K and incident-wave-set load voltage, every response term satisfies r=min(a,b)≤⌊K/2⌋.The proof follows because r conjugate pairs require at least 2r load-side factors, while the degree is at most K.Source: §7.2 · p. 11
7.3 What Theorem 1 says
- The r≤1 restriction is exact if and only if K≤3; at fourth order, |A|^4 is the first newly available term.The bound is attained at the fundamental when the drive side has a linear term, and it also holds for non-separable responses and after recentering.Source: §7.3 · pp. 11–12
7.4 Numerical check
- For K=2,…,7, brute-force expansions of separable and non-separable responses observed rmax=⌊K/2⌋ at every tested order.The cubic example has only a constant and |A|^2 term at the fundamental; raising K to 4 admits |A|^4.Source: §7.4 · p. 12, §7.3 · p. 12
8 What each exponent means
The Cardiff indices correspond to physical features of the response: load-side order, drive-side harmonic, and conjugate-pair count. For loaded devices, self-consistency replaces the finite-degree bound with a measurable coefficient-decay stopping rule.
8 What each exponent means
- The magnitude exponent m equals load-side order p, the phase exponent n equals h−ℓ, and m+|n−h|=p+|ℓ| is the mixing order.The mixing-order identity matches the quantity tabulated in the A-pull identification literature.Source: §8 · p. 12
- The r≤1 cutoff is a device property when load-side nonlinearity is at most cubic, making its physical basis measurable rather than merely a fitting choice.The indices also count load-side factors, drive-side harmonic imbalance, and conjugate pairs.Source: §8 · pp. 12–13
9 The loaded device: the decay in r
- Under load, self-consistency gives the response infinitely many orders, so truncate r where the fitted coefficient decay reaches the accuracy floor.For a quadratic intrinsic device, the open-loop expansion has no terms beyond r=1, while the closed-loop coefficients are present at every r and decay; the decay depends on load-wave amplitude and loop gain.Source: §9 · p. 13, Figure 4 · p. 14
10 Seeing the decomposition in a measurement
Ordinary load-pull sweeps combine Cardiff terms, whereas tailored A-pull modulation and spectral analysis separate them by exponent. Cluster half-widths reveal load-side nonlinearity order, and the simulations and published measurements illustrate how this can inform model-order decisions and detection limits.
10.1 Why an ordinary load-pull sweep does not show this
- In an ordinary load-pull sweep, every term contributes at every point, so least-squares recovery depends on the assumed basis and can assign energy to the wrong exponent.Individual (m, n) terms cannot be seen separately in the sweep; only their sum is observed.Source: §10.1 · p. 14
10.2 The method: modulate, then look at the spectrum
- Tailored A-pull modulates the load wave in amplitude and phase, then FFTs the response: cluster centers encode n, half-widths encode m, and rate spacing prevents overlap.An ordinary load-pull sweep combines terms at each measured point, so its least-squares fit cannot show individual (m, n) terms.Source: §10.2 · p. 14, §10.1 · p. 14
10.3 The cluster width has a physical interpretation
- The widest cluster has half-width K, the load-side nonlinearity order; measuring it gives rmax = floor(K/2) without an intervening fit.The half-width is the lowest load-side order able to produce that cluster.Source: §10.3 · p. 15
10.4 What the published data shows
- The nine published GaN exponent pairs all have r ≤ 1 and maximum m = 3, while the computed mixing-order identity matches the reported value in all nine rows.The table lists the reported and computed columns for comparison.Source: §10.4 · p. 15, Table 2 · p. 16
10.5 What the same table says about the terms that are missing
- A one-step extrapolation places missing (4, 0) at 1.3 × 10^-3 relative to (0, 0), between the spurious and noise floors, though neither estimate merits much weight alone.The estimate comes from an absolute coefficient of about 1.8 × 10^-3 referred to the (0, 0) coefficient of 1.379; the closed-loop calculation gives 1.1 × 10^-3.Source: §10.5 · p. 16
10.6 The experiment that would settle it
- If the spurious floor reaches −60 dBc, (4, 0) is predicted to appear first at roughly −58 dBc; no new terms by −60 dBc would confirm r ≤ 1 and effective K ≤ 3 for that device.The predicted level is an extrapolation; the ordering is structural.Source: §10.6 · pp. 16–17
10.7 A simulated run-through
- At a −60 dBc floor, the synthetic device yields the measured pattern; its first fourth-order line, (4, 0), is at −59.6 dBc, followed by (4, 2) at −61.1 dBc and (5, 1) at −66.1 dBc.With the loop opened, (4, 0) is at −300 dBc for K = 3 and −109 dBc for K = 4.Source: Figure 6 · p. 17, §10.7 · pp. 17–18
10.8 Stage two: a compact transistor model in a circuit simulator
- In ASM-HEMT, the spectrum follows Cardiff clusters; at ’s floors it gives K_eff = 2 and rmax = 1, while (4, 0) is the strongest r = 2 term at both drives.The r decay is not geometric; one-step extrapolation underestimates (4, 0) by 16 dB and 6 dB at the softer and harder drives, respectively.Source: §10.8 · p. 19
11 What to take to the bench
The bench checklist maps Cardiff coefficients to device behavior and gives practical tests for model order, truncation, and scope. It also distinguishes persistent CW residuals from missing terms in the complete representation.
11 What to take to the bench
- The largest resolvable m is the order of the device’s load-side nonlinearity at that drive and mismatch.Convert Cardiff exponents to a = (m + n)/2 and b = (m − n)/2, then r = min(a, b).Source: §11 · p. 20
- Treat r ≤1 as a device hypothesis: it is exact for K ≤3, while needing r = 2 terms indicates fourth-order load-side content.Source: §11 · p. 20
- For a loaded device, plot coefficients against r on a log axis; the floor intersection marks truncation and the slope is device data worth reporting.Source: §11 · p. 20
- With A-pull apparatus, cluster widths reveal K by inspection of the spectrum rather than by fitting.The checklist recommends simulating the floor sweep with xp_apull_demo first.Source: §11 · p. 20
- Representation (8) is complete for time-invariant, single-tone steady-state operation, so a persistent CW residual points to another tone, port, harmonic injection, or non-steady-state measurement.Memory appears under modulation, not as a missing CW term.Source: §11 · p. 20
12 Glossary
The glossary names the symmetry concepts used in the tutorial and gives their engineering interpretations, from transformations and weights to invariants and orbits.
12 Glossary
- A group consists of transformations that compose, undo, and include doing nothing; a Lie group has continuous parameters that compose smoothly.Source: §12 · p. 21
- U(1), S1, and SO(2) name the unit circle of phase shifts, while a torus T is several independent circles.An action is the rule describing how each quantity transforms, and a representation collects those actions on all variables.Source: §12 · p. 21
- A harmonic’s weight is its rotation rate under phase shifts, and an orbit is the set swept out by a quantity under the group.For a quantity with fixed magnitude, the orbit is a circle of constant |A|.Source: §12 · p. 21
- An invariant is unchanged by every group element; an equivariant transforms with a prescribed weight.Source: §12 · p. 21
- Generators are the finite list from which everything else is built; gauge fixing chooses a convention that cancels out.Source: §12 · p. 21
13 Scope
The tutorial translates established symmetry mathematics into microwave-engineering interpretations, including a bound and an A-pull spectral interpretation. Its scope is fundamental single-tone CW load-pull in periodic steady state, with a memoryless load-side nonlinearity required for Theorem 1.
13 Scope
- The underlying invariant and equivariant structure is classical, Schwarz’s theorem [9] extends it to smooth functions, and the completeness statement translates the form in the 2010 Cardiff paper.Source: §13 · p. 21
- The tutorial contributes the translation, r = min(a, b), its implied bound, an interpretation of the A-pull spectrum, and two simulations on polynomial and compact transistor models.Source: §13 · p. 21
- The scope is fundamental load-pull under single-tone CW excitation and periodic steady state; Theorem 1 additionally assumes a memoryless load-side nonlinearity.Multi-harmonic and multi-port cases require a torus and more indices, while memory is treated in companion paper [6].Source: §13 · p. 21
A The same result with no group theory
The no-group-theory argument derives the conjugate-index bound by counting load-wave factors. Companion routines implement checks, model demonstrations, and A-pull analysis.
A The same result with no group theory
- A degree-K polynomial load-side nonlinearity yields terms Aa Āb with a + b ≤K; matching Cardiff notation gives m = a + b, n = a −b, and r = min(a, b), hence 2r ≤K.For K = 3, r ≤1 exactly; for K = 4, r ≤2 and the newly available term is A2 Ā2 = |A|4.Source: A The same result with no group theory · p. 22
B Companion MATLAB routines
- The versioned toolkit includes MATLAB routines and two ngspice ASM-HEMT datasets, with xp_selftest checking routines against independently computed values.The archive requires the matlab/ and spice/ folders to remain side by side because xp_apull_asmhemt locates datasets relative to its location.Source: B Companion MATLAB routines · p. 22
- The routines count monomials and Cardiff terms, test completeness by numerical rank, demonstrate loaded-device decay, and analyze tailored A-pull spectra and ASM-HEMT data.The A-pull demo supports a floor sweep, while xp_apull_read analyzes measured or simulated response sequences.Source: B Companion MATLAB routines · p. 23