With the advent of high speed electronic trip devices in circuit breakers, low impedance sources, and safety consciousness, engineers designing power systems have become increasingly aware of the phenomenon of transformer inrush current.

Although inrush current is unavoidable, Quality Transformer & Electronics has developed design techniques that address geometries and flux density that can be employed to limit this initial current to desirable levels.

The initial rush of magnetizing current or inrush current in a transformer is caused by two basic factors:

  1. Initial angle of sinusoidal voltage impressed across the primary
  2. Magnitude and direction of the residual flux in the core.

If we assume that the voltage impressed across the primary has the for

Equation 1

Where θ0 is the phase angle, so that the value of the voltage at t=0 is

Equation 2

And if we assume that the value of the residual flux in the core at the time of switching is Φr, we can derive the basic equations of the inrush phenomena by using Faraday's Law:

Equation 3
Equation 4

Solving for dΦ/dt:

Equation 5

Integrating with respect to time, we get:

Equation 6

where K is the constant of integration. Since we assume that Φ=Φr at t=0, substitution of these values in equation (6) gives:

Equation 7

Solving for K we get:

Equation 8

If we substitute the above value for K in equation (6) we obtain the general equation of flux as a function of time:

Equation 9

Equation (9) gives the value of the flux during the first few cycles of operation. The current corresponding to this value of flux is readily obtained from the magnetization curve. If we divide both sides of equation (9) by the core’s cross-section area A, we get:

Equation 10


Equation 11


Equation 12

If we refer to an appropriate magnetization curve we can find a value of the current for eave value of B in equation (10).

If Br=0 just before switching, and if θ0=90° (or π/2 radians), then equation (1) will become:

Equation 13


Equation 14

and equation (10) will become:

Equation 15


Equation 16

Equations (12) and (14) show that no noticeable inrush current will be drawn from the line if the input voltage at the time of switching has the form as expressed by equation (12).

If Br=0 just before switching, and if θ0=0°, then equation (1) will be of the following form:

Equation 17

Equation (10) will be:

Equation 18

Equations (15) and (16) tell us that the voltage and flux density will have the following wave forms:

Figure 1

As will be understood from Figure 1, the peak value of the flux density in the first cycle reaches two times the peak value of the flux density in equation (14). The value of the flux density in equation (14) is the value of the steady state of flux density. So the case represented by Figure 1 is one where the flux density reaches such a high value that it drives the core material far into saturation and, as can be judged from the magnetization curve, will cause the primary winding to draw a very high current form the line.

An even worse case is still possible if, at the instance of switching, there is an appreciable residual flux density Br in the core, and if this flux density has a direction such that it will add to the flux density produced by the impressed voltage, the core will go even further into saturation with a corresponding increase in inrush current.

So if Br has a finite value and θ0=0°, the voltage and flux density equations will have the form:

Equation 19


Equation 20
Figure 2