1. Non-Linear Saturation: The transformer core's B-H relationship is highly non-linear. When flux enters saturation, small changes in flux require enormous magnetizing current.
2. Asymmetric Operation: During inrush, if the transformer is switched on with residual flux in the core, the flux waveform becomes asymmetric, saturating heavily on one side.
3. DC Offset & 2nd Harmonic: The asymmetric current creates a DC offset component. In frequency domain, this asymmetry manifests as even harmonics, especially the 2nd harmonic. This is why it's called "quasi-DC" - it's a low-frequency component riding on the DC offset.
4. Current Pulses: The excitation current appears as sharp pulses each half-cycle when the core saturates, rich in harmonics extending to high orders.
5. Resistive Damping: The winding resistance causes the DC flux offset to decay exponentially with time constant τ = L/R. As the DC component decays, the flux becomes more symmetric.
6. Harmonic Decay: As flux symmetry is restored, even harmonics (especially 2nd) decay toward zero. The fundamental component remains as the transformer reaches steady-state normal magnetization.
7. Time Constant: Shorter τ means faster decay. Larger transformers have higher L/R ratios and thus longer decay times (can be several seconds in practice).
Real/Imaginary FFT Components: The real part represents in-phase components, imaginary represents quadrature. The 2nd harmonic typically has large real component indicating the DC-like offset behavior.
🎯 Interactive B-H Curve: Drag any of the 8 control points to reshape the magnetization curve! Try making it sharper (harder saturation), flatter (softer material), or asymmetric to see different inrush behaviors and harmonic content.
Voltage Waveform:
v(t) = Vexcitation × sin(ωt + θswitch)
Where θswitch is the switching angle in radians
Flux Calculation:
φ(t) = φAC(t) + φDC(t)
φAC(t) = -Vexcitation × cos(ωt + θswitch)
φDC(t) = φDC,initial × e-t/τ
φDC,initial = φresidual + Vexcitation × cos(θswitch)
Time constant: τ = L/R (in cycles, default 5.0)
B-H Magnetization Curve:
Default: B = tanh(H / ksat)
Where ksat is the saturation factor (controls curve sharpness)
Interactive: Piecewise linear interpolation between 8 control points
Inverse function H = f-1(B) computed via linear interpolation
Current Calculation:
i(t) = H(t) = f-1(φ(t))
Current is obtained by inverting the B-H curve at each flux value
FFT Analysis:
Recursive Cooley-Tukey FFT algorithm (radix-2)
DC Component: X[0] / N
Harmonic Magnitude: |X[k]| = √(Re[k]² + Im[k]²) × (2/N) for k ≥ 1
Analysis performed per cycle (500 samples/cycle)
Zero-padding applied for incomplete cycles
Phasor Representation:
Each harmonic: Hk = Re[k] + j·Im[k]
Head-to-tail addition shows vector sum of all components
Phase angle: θk = atan2(Im[k], Re[k])
1. Single-Phase Model: This simulation represents a single-phase transformer. Three-phase effects (phase shifts, zero-sequence) are not modeled.
2. Ideal Voltage Source: The supply voltage is assumed constant and unaffected by the inrush current. In reality, source impedance would cause voltage dips.
3. Linear Resistance: Winding resistance R is constant. In reality, resistance increases with temperature and frequency (skin effect).
4. No Core Losses: Hysteresis and eddy current losses are neglected. The B-H curve is single-valued (no hysteresis loop width).
5. Uniform Core: The entire core is assumed to have identical magnetic properties. Air gaps and construction details are ignored.
6. No Load Conditions: The transformer is unloaded (no secondary current). Load would affect the inrush behavior.
7. Instantaneous Switching: The circuit breaker closes instantaneously at the specified angle. Real breakers have finite closing times.
8. No Remanence Variation: Residual flux is uniform. In practice, it varies with core geometry and previous operating history.
9. Frequency Domain Analysis: FFT assumes periodic signals. During transients, the signal is not truly periodic, so spectral leakage occurs.
10. Per-Unit System: All values are normalized (per-unit). Actual magnitudes depend on transformer rating, voltage level, and core design.
Typical Real-World Values:
• Inrush current: 8-12× rated current (can reach 20× for large transformers)
• 2nd harmonic content: 15-60% of fundamental during peak inrush
• Decay time constant: 0.1-10 seconds (larger for bigger transformers)
• Saturation flux: ~1.4-1.8 pu (varies by core material)