Guide to Industrial Power Transformers--Theory

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The invention of the power transformer towards the end of the nineteenth century made possible the development of the modern constant voltage AC supply system, with power stations often located many miles from centers of electrical load. Before that, in the early days of public electricity supplies, these were DC systems with the source of generation, of necessity, close to the point of loading.

Pioneers of the electricity supply industry were quick to recognize the benefits of a device which could take the high current relatively low voltage out put of an electrical generator and transform this to a voltage level which would enable it to be transmitted in a cable of practical dimensions to consumers who, at that time, might be a mile or more away and could do this with an efficiency which, by the standards of the time, was nothing less than phenomenal.

Today’s transmission and distribution systems are, of course, vastly more extensive and greatly dependent on transformers which themselves are very much more efficient than those of a century ago; from the enormous generator transformers, stepping up the output of up to 19,000 A at 23.5 kV, of a large generating unit in the UK, to 400 kV, thereby reducing the current to a more manageable 1200 A or so, to the thou sands of small distribution units which operate almost continuously day in day out, with little or no attention, to provide supplies to industrial and domestic consumers.

The main purpose of this guide is to examine the current state of transformer technology, but in the rapidly shrinking and ever more competitive world of technology it is not possible to retain one's place in it without a knowledge of all that is going on the other side of the globe, so the viewpoint will, hopefully, not be an entirely parochial one.

For a reasonable understanding of the subject it is necessary to make a brief review of transformer theory together with the basic formulae and simple phasor diagrams.


A power transformer normally consists of a pair of windings, primary and secondary, linked by a magnetic circuit or core. When an alternating voltage is applied to one of these windings, generally by definition the primary, a current will flow which sets up an alternating m.m.f. and hence an alternating flux in the core. This alternating flux in linking both windings induces an e.m.f. in each of them. In the primary winding this is the 'back-e.m.f' and, if the transformer were perfect, it would oppose the primary applied voltage to the extent that no current would flow. In reality, the current which flows is the transformer magnetizing current. In the secondary winding the induced e.m.f. is the secondary open-circuit voltage. If a load is connected to the secondary winding which permits the flow of secondary current, then this current creates a demagnetizing m.m.f. thus destroying the balance between primary applied voltage and back-e.m.f. To restore the balance an increased primary current must be drawn from the supply to provide an exactly equivalent m.m.f. so that equilibrium is once more established when this additional primary current creates ampere-turns balance with those of the secondary.

Since there is no difference between the voltage induced in a single turn whether it is part of either the primary or the secondary winding, then the total voltage induced in each of the windings by the common flux must be proportional to the number of turns. Thus the well-known relationship is established that:

E1/E2 = N1/N2 (eqn. 1)

and, in view of the need for ampere-turns balance:

I1/N2 = I2/N2 (eqn. 2)

where E, I and N are the induced voltages, the currents and number of turns respectively in the windings identified by the appropriate subscripts. Hence, the voltage is transformed in proportion to the number of turns in the respective windings and the currents are in inverse proportion (and the relationship holds true for both instantaneous and r.m.s. quantities).

The relationship between the induced voltage and the flux is given by reference to Faraday's law which states that its magnitude is proportional to the rate of change of flux linkage and Lenz's law which states that its polarity such as to oppose that flux linkage change if current were allowed to flow. This is normally expressed in the form:

e =-N(dΦ/dt)

but, for the practical transformer, it can be shown that the voltage induced per turn is

E/N = KΦm f (eqn. 3)

...where K is a constant, Fm is the maximum value of total flux in Webers linking that turn and f is the supply frequency in Hertz.

The above expression holds good for the voltage induced in either primary or secondary windings, and it is only a matter of inserting the correct value of N for the winding under consideration. Figure 1 shows the simple phasor diagram corresponding to a transformer on no-load (neglecting for the moment the fact that the transformer has reactance) and the symbols have the significance shown on the diagram. Usually in the practical design of transformer, the small drop in voltage due to the flow of the no-load current in the primary winding is neglected.

FIG. 1 Phasor diagram for a single-phase transformer on open circuit. Assumed turns ratio 1:1

If the voltage is sinusoidal, which, of course, is always assumed, K is 4.44 and Eq. (eqn. 3) becomes

E = 4.44f phi N

For design calculations the designer is more interested in volts per turn and flux density in the core rather than total flux, so the expression can be rewritten in terms of these quantities thus:

E/N = 4.44BmAf x 10^-6 (eqn. 4)

where E/N= volts per turn, which is the same in both windings

Bm = maximum value of flux density in the core, Tesla

A = net cross-sectional area of the core, mm^2

f = frequency of supply, Hz.

For practical designs Bm will be set by the core material which the designer selects and the operating conditions for the transformer, A will be selected from a range of cross-sections relating to the standard range of core sizes produced by the manufacturer, whilst f is dictated by the customer's system, so that the volts per turn are simply derived. It is then an easy matter to determine the number of turns in each winding from the specified voltage of the winding.


Mention has already been made in the introduction of the fact that the trans formation between primary and secondary is not perfect. Firstly, not all of the flux produced by the primary winding links the secondary so the transformer can be said to possess leakage reactance. Early transformer designers saw leakage reactance as a shortcoming of their transformers to be minimized to as great an extent as possible subject to the normal economic constraints. With the growth in size and complexity of power stations and transmission and distribution systems, leakage reactance -- or in practical terms since transformer windings also have resistance - impedance, gradually came to be recognized as a valuable aid in the limitation of fault currents. The normal method of expressing transformer impedance is as a percentage voltage drop in the transformer at full-load current and this reflects the way in which it is seen by system designers. For example, an impedance of 10 percent means that the voltage drop at full-load current is 10 percent of the open-circuit voltage, or, alternatively, neglecting any other impedance in the system, at 10 times full load current, the voltage drop in the transformer is equal to the total system voltage. Expressed in symbols this is:

VZ=%Z = I_FL Z / E = 100

where Z is √(R2 + X2 ), R and X being the transformer resistance and leak age reactance respectively and IFL and E are the full-load current and open circuit voltage of either primary or secondary windings. Of course, R and X may themselves be expressed as percentage voltage drops, as explained below.

The 'natural' value for percentage impedance tends to increase as the rating of the transformer increases with a typical value for a medium sized power transformer being about 9 or 10 percent. Occasionally some transformers are deliberately designed to have impedances as high as 22.5 percent. More will be said about transformer impedance in the following section.


The transformer also experiences losses. The magnetizing current is required to take the core through the alternating cycles of flux at a rate determined by system frequency. In doing so energy is dissipated. This is known variously as the core loss, no-load loss or iron loss. The core loss is present whenever the transformer is energized. On open circuit the transformer acts as a single winding of high self-inductance, and the open-circuit power factor averages about 0.15 lagging. The flow of load current in the secondary of the trans former and the m.m.f. which this produces is balanced by an equivalent primary load current and its m.m.f., which explains why the iron loss is independent of the load.

The flow of a current in any electrical system, however, also generates loss dependent upon the magnitude of that current and the resistance of the system.

Transformer windings are no exception and these give rise to the load loss or copper loss of the transformer. Load loss is present only when the transformer is loaded, since the magnitude of the no-load current is so small as to pro duce negligible resistive loss in the windings. Load loss is proportional to the square of the load current.

Reactive and resistive voltage drops and phasor diagrams

The total current in the primary circuit is the phasor sum of the primary load current and the no-load current. Ignoring for the moment the question of resistance and leakage reactance voltage drops, the condition for a transformer sup plying a non-inductive load is shown in phasor form in Fig. 2. Considering now the voltage drops due to resistance and leakage reactance of the trans former windings it should first be pointed out that, however the individual voltage drops are allocated, the sum total effect is apparent at the secondary terminals. The resistance drops in the primary and secondary windings are easily separated and determinable for the respective windings. The reactive volt age drop, which is due to the total flux leakage between the two windings, is strictly not separable into two components, as the line of demarcation between the primary and secondary leakage fluxes cannot be defined. It has therefore become a convention to allocate half the leakage flux to each winding, and similarly to dispose of the reactive voltage drops. Figure 3 shows the phasor relationship in a single-phase transformer supplying an inductive load having a lagging power factor of 0.80, the resistance and leakage reactance drops being allocated to their respective windings. In fact the sum total effect is a reduction in the secondary terminal voltage. The resistance and reactance voltage drops allocated to the primary winding appear on the diagram as additions to the e.m.f. induced in the primary windings.

FIG. 2 Phasor diagram for a single-phase transformer supplying a unity power factor load. Assumed turns ratio 1:1

FIG. 4 shows phasor conditions identical to those in Fig. 3, except that the resistance and reactance drops are all shown as occurring on the secondary side.

Of course, the drops due to primary resistance and leakage reactance are converted to terms of the secondary voltage, that is, the primary voltage drops are divided by the ratio of transformation n, in the case of both step-up and step-down transformers. In other words the percentage voltage drops considered as occurring in either winding remain the same.

To transfer primary resistance values R1 or leakage reactance values X1 to the secondary side, R1 and X1 are divided by the square of the ratio of trans formation n in the case of both step-up and step-down transformers.

FIG. 3 Phasor diagram for a single-phase transformer supplying an inductive load of lagging power factor cos Φ2. Assumed turns ratio 1:1. Voltage drops divided between primary and secondary sides

The transference of impedance from one side to another is made as follows.

Let Zs =total impedance of the secondary circuit including leakage and load characteristics

Z's = equivalent value of Zs when referred to the primary winding.


(eqn. 5)

FIG. 4 Phasor diagram for a single-phase transformer supplying an inductive load of lagging power factor cos Φ2. Assumed turns ratio 1:1. Voltage drops transferred to secondary side


(eqn. 6)

Comparing Eqs (eqn. 5) and (eqn. 6) it will be seen that Z's= Zs(N1/N2)2.

The equivalent impedance is thus obtained by multiplying the actual impedance of the secondary winding by the square of the ratio of transformation n, that is, (N1/N2) 2

This, of course, holds good for secondary winding leakage reactance and secondary winding resistance in addition to the reactance and resistance of the external load.

FIG. 5 Phasor diagram for a single-phase transformer supplying a capacitive load of leading power factor cos Φ2. Assumed turns ratio 1:1. Voltage drops transferred to secondary side

FIG. 5 is included as a matter of interest to show that when the load has a sufficient leading power factor, the secondary terminal voltage increases instead of decreasing. This happens when a leading power factor current passes through an inductive reactance.

Preceding diagrams have been drawn for single-phase transformers, but they are strictly applicable to polyphase transformers also so long as the conditions for all the phases are shown. For instance Fig. 6 shows the complete phasor diagram for a three-phase star/star-connected transformer, and it will be seen that this diagram is only a threefold repetition of Fig. 4, in which primary and secondary phasors correspond exactly to those in Fig. 4, but the three sets representing the three different phases are spaced 120º apart.


The output of a power transformer is generally expressed in megavolt-amperes (MVA), although for distribution transformers kilovolt-amperes (kVA) is generally more appropriate, and the fundamental expressions for determining these, assuming sine wave functions, are as follows.

Single-phase transformers

Output = 4.44 f Φm NI with the multiplier 10-3 for kVA and 10-6 for MVA.

FIG. 6 Phasor diagram for a three-phase transformer supplying an inductive load of lagging power factor cos Φ2. Assumed turns ratio 1:1. Voltage drops transferred to secondary side. Symbols have the same significance as in Fig. 4 with the addition of A, B and C subscripts to indicate primary phase phasors, and a, b and c subscripts to indicate secondary phase phasors

Three-phase transformers

Output = 4.44 f Φm NI x _/3 with the multiplier 10-3 for kVA and 10-6 for MVA.

In the expression for single-phase transformers, I is the full-load current in the transformer windings and also in the line; for three-phase transformers, I is the full-load current in each line connected to the transformer. That part of the expression representing the voltage refers to the voltage between line terminals of the transformer. The constant _3 is a multiplier for the phase voltage in the case of star-connected windings, and for the phase current in the case of delta-connected windings, and takes account of the angular displacement of the phases.

Alternatively expressed, the rated output is the product of the rated secondary (no-load) voltage E2 and the rated full-load output current I2 although these do not, in fact, occur simultaneously and, in the case of polyphase transformers, by multiplying by the appropriate phase factor and the appropriate constant depending on the magnitude of the units employed. It should be noted that rated primary and secondary voltages do occur simultaneously at no load.

Single-phase transformers

Output = E2I2 with the multiplier 10^-3 for kVA and 10^-6 for MVA.

Three-phase transformers

Output = E2I2 x _/3 with the multiplier 10^-3 for kVA and 10^-6

for MVA.

The relationships between phase and line currents and voltages for star and for delta-connected three-phase windings are as follows:

Three-phase star-connection

phase current = line current I = VA/(E x _/3)

phase voltage = E/ _/3

Three-phase delta-connection

phase current = I/ _/3 _ VA/(E x 3)

phase voltage = line voltage = E

E and I = line voltage and current respectively.


The regulation that occurs at the secondary terminals of a transformer when a load is supplied consists, as previously mentioned, of voltage drops due to the resistance of the windings and voltage drops due to the leakage reactance between the windings. These two voltage drops are in quadrature with one another, the resistance drop being in phase with the load current. The percent age regulation at unity power factor load may be calculated by means of the following expression:

copper loss x 100 /output = (percentage reactance )2/ 200

This value is always positive and indicates a voltage drop with load.

The approximate percentage regulation for a current loading of a times rated full-load current and a power factor of cos phi 2 is given by the following expression:

(eqn. 7)


VR = percentage resistance voltage at full load = copper loss x 100 / rated kVA

V=percentage reactance voltage = I2X"e/V2 x100

Equation (eqn. 7) is sufficiently accurate for most practical transformers, however for transformers having reactance values up to about 4 percent a further simplification may be made by using the expression:

percentage regulation = a(VR cos Phi 2 + Vx sin Phi 2) (eqn. 8)

and for transformers having high reactance values, say 20 percent or over, it is sometimes necessary to include an additional term as in the following expression:

(eqn. 9)

At loads of low power factor the regulation becomes of serious consequence if the reactance is at all high on account of its quadrature phase relationship.

This question is dealt with more fully in another section of this guide.

FIG. 7 Simplified equivalent circuit of leakage impedance of two-winding transformer

Copper loss in the above expressions is measured in kilowatts. The expression for regulation is derived for a simplified equivalent circuit as shown in Fig. 7, that is a single leakage reactance and a single resistance in series between the input and the output terminals. The values have been represented in the above expressions as secondary winding quantities but they could equally have been expressed in primary winding terms. Since the second term is small it is often sufficiently accurate to take the regulation as equal to the value of the first term only, particularly for values of impedance up to about 4 percent or power factors of about 0.9 or better.

VX may be obtained theoretically by calculation or actually from the tested impedance and losses of the transformer. It should be noted that the percent resistance used is that value obtained from the transformer losses, since this takes into account eddy current losses and stray losses within the transformer. This is sometimes termed the AC resistance, as distinct from the value which would be measured by passing direct current through the windings and measuring the voltage drop.

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