Clarke-Park Transformation Simulator: abc to dq0

The amplitude-invariant Clarke transform is

The power-invariant form uses $k=\sqrt{2/3}$ for the αβ rows and $k_0=1/\sqrt3$ for the sum row. These two complete matrices follow the MathWorks Clarke Transform equations (https://www.mathworks.com/help/simscape-electrical/ref/clarketransform.html). TI SPRA524, pp. 4–5, uses power-invariant stationary coordinates called d and q; these correspond to α and β here.

Park is a rotation, with α along phase a, d at θ counterclockwise from α, and q 90° ahead of d:

This sign convention and its inverse agree with Microchip AN1078 Rev. B, pp. 8–10. Inverse Park rotates by +θ; inverse Clarke restores the common component $i_0/(3k_0)$ and divides the αβ projections by $3k/2$.

For amplitude-invariant quantities,

For power-invariant quantities the weights are both 1. With voltage and current transformed using the same convention, the same weights apply to instantaneous power: $p=\frac32(v_di_d+v_qi_q)+3v_0i_0$ for amplitude normalization, or $p=v_di_d+v_qi_q+v_0i_0$ for power normalization. The displayed norm is in A², not watts.

The waveform generator adds positive sequence $I\cos(u+\varphi)$, reversed b–c negative sequence of peak $rI$, and equal DC current $I_c$. It calculates their phase sum analytically as $3I_c$ to avoid floating-point cancellation. θ = frame ratio × u + initial d-axis angle. The full-cycle curves are computed independently of the playing input.

Reference cases reproduce the JET published numerical vector, the UPC Figure 4 current levels, and TI basic space-vector coordinates. The UPC Eq. (7) prints sine where Eq. (1) and Figure 4 use cosine; this tool follows Eq. (1) and Figure 4. The JET Eq. (1.13) omits a minus sign on phase c in one line; Eqs. (1.14)–(1.15) use the correct negative value. No transform coefficients have been adjusted to these typographical errors.

Ideal, simultaneous, real current samples; no filtering, sampling delay or quantization. The waveform positive sequence is a → b → c with cosine references separated by 120°. The negative sequence has the same phase-a starting phase and reversed b–c ordering; its amplitude ratio is 0–1. The zero component added by the generator is DC, not a triplen harmonic.

Angles are electrical degrees. The frame is user-defined and need not coincide with a motor rotor. The 2D drawing shows only the αβ or dq part; 0 is a separate scalar. Plot lengths share a fixed full-cycle scale during animation; the light circle is a guide, and the grey locus is the actual path.

Inverse currents and transform results retain full numerical precision. Only error-reporting steps are rounded to 10⁻¹⁰ A or A² to prevent insignificant cancellation noise appearing as a unit-scale discrepancy. Source graph readings have explicitly looser tolerances than printed numerical reference vectors.

The complete three-component transform is valid for balanced and unbalanced real abc currents, including a nonzero sum. A three-wire isolated circuit requires a zero sum; the input condition allows 10⁻⁹ A numerical tolerance. For nonzero sum use a neutral return or treat the inputs as independent signals.

The selectable integer frame frequency ratios −2…2 keep the one-cycle animation loop continuous. Generated signal frequency is 0.01–100000 Hz; the transform is algebraic and this range does not imply controller or sensor bandwidth. Phase entries and common current are limited to ±100000 A; positive-sequence peak is 0–100000 A. These are interface bounds, not equipment ratings. Snapshot curves sweep θ while holding abc fixed. Period and physical-time steps apply only to waveform mode.

DC dq values require a balanced sinusoid and a frame rotating synchronously with that sequence. Negative sequence produces twice-frequency ripple in a positive synchronous frame. No dynamic motor, PWM, FOC controller or torque calculation is included.