Magnetizing and Exciting Currents Waveshapes in Transformers

The graphical methods to derive the magnetizing and exciting currents in transformers are illustrative since they allow us to visualize simultaneously the elements that originate these waveforms, such as the flux waveform and the typical B-H curves of the magnetic materials. 

The resulting waveforms show that they are not sine waves, and, in the case of the excitation current, it is not symmetrical either. 

An analysis of the results using the Fourier theorem allows us to decompose the waveforms into a fundamental component and a series of odd harmonics in which the third harmonic predominates.The suppression of the third harmonic in the currents will induce a peaked EMF in the windings.

 

Graphical Derivation of the Magnetizing Current Wave from the Magnetization Curve

The relative permeability of ferromagnetic materials varies with the magnetic field’s intensity. This characteristic limits our ability to analytically derive the magnetizing current, but magnetizing current can be calculated from the normal magnetization curve for a given material, or curves of exciting volt-amperes per weight of material versus the flux density, supplied from manufacturers.

If the voltage supplied to the transformer is sinusoidal, the flux is almost sinusoidal. Assuming a sinusoidal flux wave, Figure 1 shows the derived magnetizing current-time curve.

 

Figure 1. Derivation of the magnetizing current wave from the magnetization curve

 

If the transformer iron core did not saturate, the magnetizing current (Im), generated by the flux, would be sinusoidal in shape and in phase with the flux. However, for economic reasons, transformers are operated near the knee of the magnetization curve, where some degree of saturation exists. The result is that the magnetizing current is symmetrical, but cannot be sinusoidal due to the magnetization curve nonlinearity. 

On resolving the magnetizing current wave into a series of component sine curves, it is equivalent to a sine wave of a fundamental frequency and a series of odd harmonics (3rd, 5th, 7th, etc.) with the third harmonic being the most dominant. The third harmonic ranges from 30% to 40% of the fundamental frequency, depending on the degree of saturation of the iron core. The higher-order harmonics may be neglected. The third harmonic current and its multiples (triple-harmonics) are in phase.

Figure 2 shows the fundamental and third harmonic constituents of the magnetizing current. 

 

Figure 2. Magnetizing current wave made up of a fundamental and third harmonic

 

Graphical Derivation of the Exciting Current Wave from the Hysteresis Loop

Considering hysteresis, eddy current losses and sinusoidally varying flux, the current-time curve of the exciting current (Ie) will form what is shown in Figure 3.

 

Figure 3. Exciting current wave, derived from the hysteresis loop

 

The exciting current is severely distorted, indicating significant 3rd, 5th, 7th, and higher-order harmonics, and a substantial fundamental frequency component responsible for the unsymmetrical waveshape. The second quarter cycle is not a mirror image of the first quarter cycle. The sinusoidal component Ih+e is accountable for the degree of dissymmetry observed in the exciting current.

Figure 4 shows the instantaneous values of Ie, adding the Im and Ih+e waves

 

Figure 4. Instantaneous addition of exciting current component waves

 

The exciting current may be resolved into a fundamental frequency and harmonics, as before, with the third harmonic again being dominant.

 

Suppression of the Triple-Harmonics in the Magnetizing Current

Suppressing the triple-harmonics in the magnetizing component forces a sinusoidal current to flow through the winding. As the iron core nonlinearity requires the harmonics, the flux supplies them. These harmonics, predominantly the third one, distort the flux, exhibiting a double top, as seen in Figure 5.

With a flux wave (ɸm), composed of the fundamental and the third harmonic, the resultant wave of the EMF induced in the winding also consists of a fundamental and a third harmonic. Figure 5 shows this with dashed lines.

 

Figure 5. Distorted flux and EMF with third harmonics

 

Further, the relative phase positions of the fundamental and the third harmonic of the EMF reverse the corresponding phase relations of the components of the flux wave.

The outcome is a double top flux wave and a peaked EMF wave. The induced voltage is also peaked, ignoring the voltage drop in the winding.

The scale of the third harmonic of the EMF is three times greater than that of the flux wave. If the third harmonic suppression in the magnetizing current produces a 35% third harmonic in the flux wave, this flux harmonic will generate a 3 x 35 = 105% third harmonic in voltage.

The net effect is the induction of large third harmonic voltages in the transformer windings that may impose excessive stress on the insulation.

 

Effect of the Load Upon Current Distortion

When the secondary winding delivers a load current (I2), the exciting current combines with it as seen at the primary side (N2·I2/N1) to give the total primary current.

Assuming the primary voltage and the flux are sinusoidal, which induces a sinusoidal secondary voltage, the secondary current will also generally be sinusoidal unless the nature of the secondary load introduces disturbances. When the secondary load consists of devices with magnetic cores working at high flux density, the effect would bring about the same type of distortion of secondary current as described for the primary circuit.

If the secondary current is sinusoidal and of full-load value, the resultant primary current will be slightly distorted, since the harmonics of the exciting current constitute a small percentage of the total. If the load is passive, the effect of increasing the load is to smooth out the primary current.

 

Summarizing Magnetizing and Exciting Currents Waveshapes 

With an iron core subjected to a magnetizing force, the resultant magnetization tends to approach a limit as the magnetizing force increases, i.e., the iron approaches a condition of saturation.

Assuming no hysteresis loss in the core, the coordinates of any point on the magnetization curve represent simultaneous values of the flux in the core and the magnetizing current producing the flux.

A graphical method proves that the waveform of the magnetizing current is not sinusoidal, but tends to become more peaked as the iron approaches saturation. The magnetizing current is, however, symmetrical.

The result of resolving this waveform into a series of component sine curves, is a curve of a fundamental frequency and a series of odd harmonics, with the third harmonic prevailing.

The exciting current is obtained, in a similar way, considering the hysteresis losses. The resultant waveform is neither sinusoidal nor symmetrical, and, as before, the third harmonic is dominant.

The suppression of the third harmonic in the magnetizing current will distort the flux, and its waveshape will exhibit a double top containing third harmonics, which, in turn, induces a peaked EMF in the windings.