Transformer inrush current, doubling effect

Inrush current is the large transient magnetising current drawn by a transformer when it is first connected to the supply. Its peak can reach 5 to 10 times the full-load current (even more in small transformers), and it dies away over a few cycles to a few seconds (longer in large units).

Doubling effect

If the supply voltage is $v = V_m\sin(\omega t + \theta)$, the flux needed in steady state is $\phi = -\phi_m\cos(\omega t + \theta)$, because flux lags voltage by 90°. At the instant of switching the flux cannot change suddenly, so it starts from the residual flux $\phi_r$, and a decaying transient flux is added to the steady-state value.

Case: switching on at the instant when the supply voltage is zero ($\omega t = 0$).

• In steady state, the flux at this instant would be at its negative maximum ($-\phi_m$).
• But the core flux just before switching is the residual flux $\phi_r$ (often small), and it cannot change suddenly.
• So the flux starts from $\phi_r$ and follows

$$\phi = \phi_r + \phi_m(1 - \cos\omega t)$$

• Half a cycle later ($\omega t = \pi$), the flux reaches

$$\phi_{max} = 2\phi_m + \phi_r$$

The peak flux is therefore about twice the normal peak, and more if residual flux is present in the same direction. This is the doubling effect.

 flux
  2phi_m |        .-.
         |      /     \
   phi_m |    /   steady-state  
         |  /  .-"-.
      0 -+-/-"------"-------> t
         voltage zero when switched

Consequence

• A normal transformer is designed to work at a flux density near the knee of the $B$–$H$ curve (about 1.5 to 1.7 T). Twice the flux drives the core deep into saturation, where the permeability falls sharply.
• A very large magnetising current is then needed to produce this flux, and this is the inrush current. It has a high harmonic content (mainly second harmonic) and decays as the transient flux is damped by the winding resistance.

Case: switching on at voltage maximum gives no transient: the flux starts at zero, which is already its steady-state value at that instant, so the inrush is minimal.

Effects and remedies: inrush can trip overcurrent relays and stress the windings. It is reduced by switching through a series (pre-insertion) resistor and by point-on-wave closing, and relay maloperation is avoided by harmonic restraint in differential relays.