CT Saturation Calculator: DC Offset, Remanence & Time

The secondary-referred equivalent circuit follows the PSRC approach described by Xu et al. (2015), attached PDF pp.4–5. SEL’s Sizing Current Transformers for Line Protection Applications explains numerical flux integration and a full-cycle Fourier filter (PDF p.16), and distinguishes the knee from saturation voltage (PDF p.4).

The fault is $i_0(t)=\sqrt{2}I_{sf}[\sin(\omega t+\beta)-\sin\beta\,e^{-t/T_p}]$, where $T_p=(X/R)/\omega$. Secondary resistance is $R_t=R_{ct}+R_{loop}+S_b\,pf/I_{sn}^2$ and $L_b=S_b\sqrt{1-pf^2}/(I_{sn}^2\omega)$.

Fit $I_e(V)=I_{ek}(V/V_k)^S$ through the two entered RMS excitation points. Let $V_{10}$ be its voltage at 10 A RMS, $\lambda_{10}=\sqrt{2}V_{10}/\omega$, $x=\lambda/\lambda_{10}$, and $R_P=\sqrt{\Gamma(S+1/2)/(\sqrt{\pi}\Gamma(S+1))}$. The instantaneous law is $F(x)=(10\,\mathrm{A}/R_P)\,\operatorname{sgn}(x)|x|^S$. This RMS-to-peak correction matters because the exciting current is not sinusoidal.

Integrate

Initial flux is $\lambda(0)=r\sqrt{2}E_{lim}/\omega$. All cases use implicit second-order backward differentiation of $q=x+(L_b/\lambda_{10})(F-i_0)$, with one backward-Euler startup. Each step solves a monotone scalar power-law equation; no explicit RK45 selection or overflow-prone explicit fallback is used. The time grid has at most 1,920 intervals, with an integer number per cycle: $N_{per}=\lfloor1920/n_{cycles}\rfloor$. This gives 320 intervals per cycle at the defaults and 160 at the longest window. Threshold crossings are linearly interpolated between samples.

Knee crossing uses $|\lambda|\ge\sqrt{2}V_k/\omega$. Significant distortion uses $|i_0-i_s|\ge0.1 I_{sf}$, following the explicit threshold described in Xu et al., PDF p.23. The two events need not coincide.

The legacy hypothetical full-offset estimate uses $K_a=(1-|r|)E_{lim}/(I_{sf}|Z_t|)$ and $t=-T_p\ln[1-(K_a-1)/(X/R)]$ between $1<K_a<1+X/R$. It returns immediate exceedance of that approximation below the range and no full-offset crossing above it. This estimate assumes resistive burden, full current DC offset and adverse remanence; it does not use the entered phase and is not an upper bound across all inception phases. SEL Eq.27 (PDF p.13) provides the published 13.3 ms comparison.

$K_{ssc}=I_f/I_{pn}$. The legacy $1+X/R$ factor, its EMF estimate, and its primary-current estimate apply only to the hypothetical full-offset case. At low X/R, a zero-DC-current phase can demand more flux because the integral of its AC current has an initial-condition offset. For the entered inception phase, the finite-window factor is $\omega\max|\lambda_{ideal}-\lambda_r|/(\sqrt{2}I_{sf}|Z_t|)$, calculated from the exact ideal-transfer flux including the inductive term. EMF demand divides by $1-|r|$ as an adverse-polarity allowance. It is a study factor, not a declared IEC class rating.

The Fourier estimate is $\sqrt{2}f\left|\int_{t-1/f}^{t}i_s(u)e^{-j\omega u}du\right|$. Only complete fault windows are shown; total RMS uses the same window.

PSRC Theory for CT SAT Calculator (2003), attached PDF p.11, limits initial remanent flux to knee flux for this approximation. Because this calculator expresses remanence against the entered limiting EMF, it requires $|r|E_{lim}\le V_k$. It also blocks initial exciting current above 1% of $I_{sf}$; that numerical limit is a screening choice to enforce the small-initial-current approximation, not a standard-specified limit.

• The default excitation points are illustrative, user-replaceable assumptions, not catalogue data. Use a test report for the connected CT tap and frequency.
• A single-valued, lossless power law approximates the excitation curve. The low-excitation region and progressive slope changes are not separately fitted. Extrapolation beyond the higher point is flagged.
• Remanence is an initial flux condition. This approximation has no hysteresis memory and can relax initial flux; it cannot model retained remanence after clearing or repeated faults. Initial remanent flux above the knee and initial exciting current above 1% of symmetrical RMS ratio current block the results. Reduce the remanence assumption or use a hysteresis model.
• Fault magnitude and system X/R stay constant over 1–12 cycles. Prefault load flux, fault clearing, breaker arcing and reclosing are omitted.
• Winding and lead resistances are constant at the temperatures represented by the entered values. Relay burden is a constant series R–L equivalent; leakage inductance and core losses are omitted.
• A Fourier window is a generic measurement model. It includes decaying-DC leakage and has no relay-specific saturation correction, analogue filter, clipping, pickup logic, or trip characteristic.

For screening and education with inductive protection CTs at 15–100 Hz, positive burden resistance, and fitted exponents from 3 to 40. A numerical waveform is not laboratory validation of a particular CT. Resolve extrapolation warnings and initial-flux errors using measured data or an appropriate hysteresis/EMT model. Time resolution is one cycle divided by the displayed numerical intervals per cycle (0.0625 ms at the defaults). Long windows have coarser resolution; threshold interpolation does not establish equivalent physical accuracy.

The two independent reference cases cover the SEL full-offset time approximation and its remanence example. Excitation-anchor recovery and RMS reconstruction are calibration consistency properties, not independent performance references. Numerical regression properties cover the reviewer's 6 Ω lead-loop case and the low-X/R zero-offset case. First-cycle step refinement checks time and Fourier convergence for the 6 Ω case. A separate SciPy Radau integration cross-checks those regressions. None of these numerical checks establishes measured transient accuracy.

IEEE C37.110-2007 methods reproduced in the 2013 SEL paper supply the legacy screening expressions; IEEE C37.110-2023 supersedes that edition. IEC 61869-2:2012 supplies the relevant current-transformer terminology and transient-performance context. This tool does not certify IEEE accuracy class, IEC P/PX/TP classification, rated ALF, or a complete IEC duty cycle. Check against the standard in force and the relay manufacturer’s CT requirements.

A qualified protection engineer must check CT selection and protection performance. Installation, secondary-circuit changes and CT testing belong to qualified personnel.