Transformer Mutual Induction (2D)

Laminated-core transformer with live voltage and current waveforms, a saturating B-H curve and efficiency against load.

Switch to 3D

Presets

Playback

Windings

Turns set the voltage ratio and the core flux.

Source and load

Switch-on

Inrush depends on the voltage phase when the switch closes.

Layers

Model checks

The science

Faraday induction and the turns ratio

A changing flux Φ through N turns induces an electromotive force e = N dΦ/dt (Faraday). In a transformer the same core flux links both windings, so each turn sees the same voltage and

vs / vp = Ns / Np,    is / ip = Np / Ns (ideal, load connected)

For a sine drive the peak flux is Φm = Vp√2 / (2π f Np). That is why a transformer designed for 50 Hz overheats at 25 Hz: the flux doubles and the core saturates.

The model in this page

The page integrates, with fourth-order Runge-Kutta, the equivalent circuit referred to the primary: winding resistances Rp, Rs, a combined leakage inductance Llk on the secondary side, a core-loss resistance Rc and a non-linear magnetising branch with flux linkage λ:

dλ/dt = (vp − Rp(im + is')) / (1 + Rp/Rc),    Llk dis'/dt = dλ/dt − (Rs' + RL') is' im(λ) = (λ / Lm0) (1 + (|λ| / λk)6),    Lm0 = μ0μr Np2 A / l,    λk = 0.8 Np A Bsat

The core is 40 cm², magnetic path 0.4 m, μr = 5000; ratings 230 V, 1 kVA. The sixth-power term is a smooth stand-in for saturation: below the knee the core is linear, above it the magnetising current rises steeply. The B-H graph shows that curve and the trace actually followed.

Inrush and saturation

Switching on at the voltage zero crossing needs a flux that starts from 0 and swings up to twice Φm (the DC offset decays slowly with Rp). The core is driven into the knee and the magnetising current spikes. Switching on at the voltage peak starts the flux at its natural value and nothing happens. Try both buttons with the Overvoltage preset.

Losses and efficiency

Copper loss is I2R in both windings and grows with the square of the load; core loss v2/Rc is nearly constant. The efficiency peaks where the two are equal. The green curve is computed with the steady-state phasor solution of the linear circuit.

Honest simplifications

  • Eddy and hysteresis loss are lumped into one resistor Rc (1 % of rating at rated voltage); real hysteresis loss depends on the loop area.
  • The magnetisation curve is single-valued (no remanence), so inrush is not sensitive to the previous switch-off.
  • Leakage is lumped on the secondary side; the real split between windings affects only the details of a short-circuit transient.
  • Winding capacitance and skin effect are ignored (valid to a few kHz). The load is purely resistive.
  • Time is slowed by a factor of 20 so the waveforms can be seen.

Not a model of any specific product.

What to try
  • Run Open circuit: Vs/Vp equals Ns/Np and the primary draws only the small magnetising current (red).
  • Run Overload 200 % and watch the secondary voltage sag below the ideal ratio (winding and leakage drop).
  • Run Overvoltage: core saturates and press the two switch-on buttons: at the zero crossing the magnetising current spikes, at the peak it does not.
  • Run Low frequency 25 Hz: the flux doubles and the B-H trace reaches the knee.
  • Sweep the load slider and read the efficiency maximum on the green curve (where copper loss equals core loss).
Primary and secondary voltage (V)
Currents: primary, secondary, magnetising (A)
Core magnetisation: flux density against magnetising current (dashed = model curve)
Efficiency and secondary voltage against load (phasor model, dot = now)