Industrial Power Transformers--Design fundamentals (part 2)

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It has been assumed thus far that a transformer has only two windings per phase, an LV and an HV winding. In fact, although this is by far the most frequent arrangement, there is no reason why the number of windings should be limited to 2. The most common reason for the addition of a third winding to a three-phase transformer is the provision of a delta-connected tertiary winding.

Other reasons for doing so could be as follows:

• To limit the fault level on the LV system by subdividing the infeed: that is, double secondary transformers.

• The interconnection of several power systems operating at different supply voltages.

• The regulation of system voltage and of reactive power by means of a synchronous capacitor connected to the terminals of one winding.

Tertiary windings

As indicated in Section 1, it is desirable that a three-phase transformer should have one set of three-phase windings connected in delta thus providing a low impedance path for third harmonic currents. The presence of a delta-connected winding also allows current to circulate around the delta in the event of unbalance in the loading between phases, so that this unbalance is reduced and not so greatly fed back through the system. Although system designers will aim to avoid the use of star/star-connected transformers, there are occasions when the phase shift produced by a star/delta or delta/star transformer is not acceptable as, for example, in the case of the power station auxiliary system described above. For many years it was standard practice in this situation to provide a delta-connected tertiary winding on the transformer.

Because the B-H curve of the magnetic material forming the transformer core is not linear, if a sinusoidal voltage is being applied for a sinusoidal flux (and hence a sinusoidal secondary voltage), the magnetizing current is not sinusoidal. Thus, the magnetizing current of a transformer having an applied sinusoidal voltage will comprise a fundamental component and various harmonics. The magnitude and composition of these harmonics will depend on the magnetizing characteristic of the core material and the value of the peak flux density. It is usual for third harmonic to predominate along with other higher third-order harmonics.

Since the third-order harmonic components in each phase of a three-phase system are in phase, there can be no third-order harmonic voltages between lines. The third-order harmonic component of the magnetizing current must thus flow through the neutral of a star-connected winding, where the neutral of the supply and the star-connected winding are both earthed, or around any delta-connected winding. If there is no delta winding on a star/star trans former, or the neutral of the transformer and the supply are not both connected to earth, then line to earth capacitance currents in the supply system lines can supply the necessary harmonic component. If the harmonics cannot flow in any of these paths then the output voltage will contain the harmonic distortion.

Even if the neutral of the supply and the star-connected winding are both earthed, as described above, then although the transformer output waveform will be undistorted, the circulating third-order harmonic currents flowing in the neutral can cause interference with telecommunications circuits and other electronic equipment as well as unacceptable heating in any liquid neutral earthing resistors, so this provides an added reason for the use of a delta connected tertiary winding.

If the neutral of the star-connected winding is unearthed then, without the use of a delta tertiary, this neutral point can oscillate above and below earth at a voltage equal in magnitude to the third-order harmonic component. Because the use of a delta tertiary prevents this it is sometimes referred to as a stabilizing winding.

The number of turns, and hence rated voltage, of any tertiary winding may be selected for any convenient value. Thus, the tertiary terminals may be brought out for supplying any substation auxiliary load, dispensing with the need for any separate auxiliary transformer. In the case of large transmission autotransformers, which must of necessity be star/star connected, a common use of the tertiary winding is for connection of system compensation equipment.

Although any auxiliary load may be quite small in relation to the rating of the main transformer, the rating of the tertiary must be such as to carry the maximum circulating current which can flow as a result of the worst system unbalance. Generally, this worst unbalance is that condition resulting from a line to earth short circuit of the secondary winding with the secondary neutral point earthed, see below.

Assuming a one-to-one turns ratio for all windings, the load currents in the primary phases corresponding to a single-phase load on the secondary of a star/star transformer with delta tertiary are typically as shown in FIG. 10.

This leads to an ampere-turns rating of the tertiary approximately equal to one-third that of the primary and secondary windings, and provides a common method for rating the tertiary in the absence of any more specific rating basis.

The full range of possible fault conditions are shown in FIG. 11. The magnitude of the fault current in each case is given by the following expressions. For case:

(eqns. 2-5)


IS is the fault current shown in FIG. 11(a)-(c)

ISP is the fault current due to the primary supply in FIG. 11(d)

ISS is the fault current due to the secondary supply in FIG. 11(d)

I is the normal full-load current of the transformer

IZPS is the percentage normal full-load impedance per phase between primary and secondary windings

IZPT is the percentage normal full-load impedance per phase between primary and tertiary windings

IZTS is the percentage normal full-load impedance per phase between tertiary and secondary windings

FIG. 10 Single-phase load to neutral

FIG. 11 Fault currents due to short-circuits to neutral

Expressions (2 )-( 5) apply strictly to one-to-one turns ratio of all windings, and the true currents in each case can easily be found by taking due account of the respective turns ratios.

It will be appreciated that from the point of view of continuous and short-time loads the impedances between tertiary windings and the two main windings are of considerable importance. The tertiary winding must be designed to be strong enough mechanically, to have the requisite thermal capacity, and to have sufficient impedance with respect to the two main windings to be able to with stand the effect of short circuits across the phases of the main windings and so as not to produce abnormal voltage drops when supplying unbalanced loads continuously.

When specifying a transformer which is to have a tertiary the intending purchaser should ideally provide sufficient information to enable the transformer designer to determine the worst possible external fault currents that may flow in service. This information (which should include the system characteristics and details of the earthing arrangements) together with a knowledge of the impedance values between the various windings will permit an accurate assessment to be made of the fault currents and of the magnitude of currents that will flow in the tertiary winding. This is far preferable to the purchaser arbitrarily specifying a rating of, say, 33.3 percent of that of the main windings, although the reason for use of this rule-of-thumb method of establishing a rating in the absence of any more precise information will be apparent from the example of FIG. 10. A truly satisfactory value of the rating of the tertiary winding can only be derived with a full knowledge of the impedances between windings of the transformer and of the other factors identified above.

As indicated at the start of this section, the above philosophy with regard to the provision of tertiary windings was adopted for many years and developed when the cores of transformers were built from hot-rolled steel. These might have a magnetizing current of up to 5 percent of full-load current. Modern cold-rolled steel cores have a much lower order of magnetizing current, possibly as low as 0.5 percent of full-load current. In these circumstances the effect of any harmonic distortion of the magnetizing current is much less significant. It now becomes, therefore, much more a matter of system requirements as to whether a star/star transformer is provided with a delta tertiary or not.

In the case of a star/star-connected transformer with the primary neutral unearthed and with the neutral of the secondary connected to earth, a secondary phase to earth fault may not cause sufficient fault current to flow to cause operation of the protection on account of the high impedance offered to the flow of single-phase currents by this configuration. Generally, the presence of a delta tertiary remedies this by permitting the flow of circulating currents which lead to balancing currents in the other two phases. The problem can be illustrated by considering as an example the design of the 60 MVA star/star connected 132/11 kV station transformer for the CEGB's Littlebrook 'D' Power Station in the mid-1970s. This was one of the first of the CEGB's power stations to have a station transformer as large as 60 MVA, and there was concern that if this were to follow the usual practice of having a delta-connected tertiary winding, the fault level for single phase to earth faults on the 11 kV system when operating in parallel with the unit transformer might become excessive.

Since, at this time, the practice of omitting the tertiaries of star/star-connected 33/11 kV transformers was becoming relatively common, the proposal was made to leave off the tertiary. Discussions were then initiated with transformer manufacturers as to whether there would be a problem of too little fault current in the case of 11 kV earth faults. Manufacturers were able to provide reassurance that this would not be the case and when the transformer was built and tested, this proved to be so. Analysis of problems of this type is best carried out using the concept of zero-sequence impedance and this is described below.


It is usual in performing system design calculations, particularly those involving unbalanced loadings and for system earth fault conditions, to use the principle of symmetrical components. This system is described and ascribes positive-, negative- and zero-sequence impedance values to the components of the electrical system.

For a three-phase transformer, the positive- and negative-sequence impedance values are identical to that value described above, but the zero-sequence impedance, varies considerably according to the construction of the trans former and the presence, or otherwise, of a delta winding.

The zero-sequence impedance of a star winding will be very high if no delta winding is present. The actual value will depend on whether there is a low reluctance return path for the third harmonic flux.

For three-limb designs without a delta, where the return-flux path is through the air, the determining feature is usually the tank, and possibly the core sup port framework, where this flux creates a circulating current around the tank and/or core framework. The impedance of such winding arrangements is likely to be in the order of 75-200 percent of the positive-sequence impedance between primary and secondary windings. For five-limb cores and three-phase banks of single-phase units, the zero-sequence impedance will be the magnetizing impedance for the core configuration.

Should a delta winding exist, then the third harmonic flux will create a circulating current around the delta, and the zero-sequence impedance is deter mined by the leakage field between the star and the delta windings. Again the type of core will influence the magnitude of the impedance because of the effect it has on the leakage field between the windings. Typical values for three-limb transformers having a winding configuration of core/tertiary/star LV/star HV are:

[Z0] LV approximately equal to 80-90% of positive-sequence impedance LV/tertiary

[Z0]HV approximately equal to 85-95% of positive-sequence impedance HV/tertiary where Z0 is zero-sequence impedance.

Five-limb transformers have their zero-sequence impedances substantially equal to their positive-sequence impedance between the relative star and delta windings.


Another special type of multi-winding transformer is the double secondary transformer. These transformers are sometimes used when it is required to split the number of supplies from an HV feeder to economize on the quantity of HV switchgear and at the same time limit the fault level of the feeds to the LV switchgear. This can be particularly convenient when it is required to omit an intermediate level of voltage transformation. For example a 60 MVA 132 kV feeder to a distribution network would normally step down to 33 kV. If, in order to meet the requirements of the distribution network, it is required to transform down to 11 kV, this equates to an LV current of about 3000 A and, even if the transformer had an impedance of around 20 percent, an LV fault level from the single infeed of around 15 kA, both figures which are consider ably higher than those for equipment normally used on a distribution network.

The alternative is to provide two separate secondary windings on the 60 MVA transformer, each rated at 30 MVA, with impedances between HV and each LV of, say 16 percent. Two sets of LV switchgear are thus required but these can be rated 1500 A and the fault level from the single infeed would be less than 10 kA.

In designing the double secondary transformer it is necessary that both LV windings are disposed symmetrically with respect to the HV winding so that both have identical impedances to the HV. This can be done with either of the arrangements shown in FIG. 12. In both arrangements there is a cross over between the two LV windings half-way up the limb. However, in the configuration shown in FIG. 12(a) the inner LV upper-half crosses to the outer upper-half and the inner lower-half crosses to the outer lower-half, whilst in the configuration of FIG. 12(b) upper inner crosses to lower outer and upper outer to lower inner. The LV windings of FIG. 12(a) are thus loosely coupled, whilst those of FIG. 12(b) are closely coupled, so that the leakage reactance LV1 to LV2 of FIG. 12(a) is high and that of FIG. 12(b) is low. It is thus possible to produce equivalent circuits for each of these arrangements as shown in FIG. 13(a) and (b) in which the transformer is represented by a three terminal network and typical values of impedance (leakage reactance) are marked on the networks. For both arrangements the HV/LV impedance is 16 percent, but for the transformer represented by FIG. 13(a) the LV1 to LV2 impedance is around 28 percent whilst for that represented by FIG. 13(b) it is only 4 percent. Which of the two arrangements is used depends on the constraints imposed by the LV systems. It should be noted that the same equivalent circuits apply for calculation of regulation, so that for the arrangement shown in FIG. 13(a), load on LV1 has little effect on the voltage on LV2 whereas for FIG. 13(b), load on LV1 will considerably reduce the voltage on LV2.

FIG. 12 Transformers with two secondary windings: (a) Loosely coupled; (b) Closely coupled

FIG. 13 Equivalent circuits for loosely coupled and closely coupled double secondary transformers: (a) Loosely coupled LV windings; 2 (b) Closely coupled LV windings


The voltage regulation of a winding on a three-winding transformer is expressed with reference to its no-load open circuit terminal voltage when only one of the other windings is excited and the third winding is on no-load, that is the basic voltage for each winding and any combination of loading is the no-load voltage obtained from its turns ratio.

For the case of two output windings W2 and W3, and one input winding W1, shown diagrammatically in FIG. 14, the voltage regulation is usually required for three loading conditions:

(1) W2 only loaded

(2) W3 only loaded

(3) W2 and W3 both loaded

For each condition two separate values would be calculated, namely the regulation of each output winding W2 and W3 (whether carrying current or not) for constant voltage applied to winding W1.

The voltage regulation between W2 and W3 relative to each other, for this simple and frequent case is implicit in the values (W1 to W2) and (W1 to W3) and nothing is gained by expressing it separately.

The data required to obtain the voltage regulation are the impedance voltage and load losses derived by testing the three windings in pairs and expressing the results on a basic kVA, which can conveniently be the rated kVA of the lowest rated winding.

From these data an equivalent circuit is derived as shown in FIG. 15. It should be noted that this circuit is a mathematical conception and is not an indication of the winding arrangement or connections. It should, if possible, be determined from the transformer as built.

FIG. 14 Diagram of a three-winding transformer

FIG. 15 Equivalent circuit of a three-winding transformer

The equivalent circuit is derived as follows:

Let a12 and b12 be, respectively, the percentage resistance and reactance volt age referred to the basic kVA and obtained from test, short circuiting either winding W1 or W2 and supplying the other with winding W3 on open circuit.

a23 and b23 similarly apply to a test on the windings W2 and W3 with W1 on open circuit.

a31 and b31 similarly apply to a test on the windings W3 and W1 with W2 on open circuit.

d is the sum (a12 = a23 = a31).

g is the sum (b12 = b23 = b31).

Then the mathematical values to be inserted in the equivalent circuit are:

Arm W1: a1 _ d/2 _ a23 b1 _ g/2 _ b23

Arm W2: a2 _ d/2 _ a31 b2 _ g/2 _ b31

Arm W3: a3 _ d/2 _ a12 b3 _ g/2 _ b12

It should be noted that some of these quantities will be negative or may even be zero, depending on the actual physical relative arrangement of the windings on the core.

For the desired loading conditions the kVA operative in each arm of the net work is determined and the regulation of each arm is calculated separately. The regulation with respect to the terminals of any pair of windings is the algebraic sum of the regulations of the corresponding two arms of the equivalent circuit.

The detailed procedure to be followed subsequently for the case of two output windings and one supply winding is as follows:

(1) Determine the load kVA in each winding corresponding to the loading being considered.

(2) For the output windings, W2 and W3, this is the specified loading under consideration; evaluate n2 and n3 for windings W2 and W3, being the ratio of the actual loading to the basic kVA used in the equivalent circuit.

(3) The loading of the input winding W1 in kVA should be taken as the phasor sum of the outputs from the W2 and W3 windings, and the corresponding power factor cos f and quadrature factor sin f deduced from the in-phase and quadrature components.

Where greater accuracy is required, an addition should be made to the phasor sum of the outputs and they should be added to the quadrature component to obtain the effective input kVA to the winding W1, (the output kVA from winding W2)

bn 22 100

_ (the output kVA from winding W2)

bn 33 100

n for each arm is the ratio of the actual kVA loading of the winding to the basic kVA employed in determining the equivalent circuit.

A more rigorous solution is obtained by adding the corresponding quantities (a, n, output kVA) to the in-phase component of the phasor sums of the out puts, but this has rarely an appreciable effect on the voltage regulation.

Equations (eqn. 7) and (eqn. 8) may now be applied separately to each arm of the equivalent circuit, taking separate values of n for each arm as defined earlier.

To obtain the voltage regulation between the supply winding and either of the loaded windings, add algebraically the separate voltage regulations deter mined for the two arms, noting that one of these may be negative. A positive value for the sum determined indicates a voltage drop from no-load to the loading considered, while a negative value for the sum indicates a voltage rise.

Repeat the calculation described in the preceding paragraph for the other loaded winding. This procedure is applicable to autotransformers if the equivalent circuit is based on the effective impedances measured at the terminals of the autotransformers.

In the case of a supply to two windings and output from one winding, the method can be applied if the division of loading between the two supplies is known.

An example of the calculation of voltage regulation of a three-winding transformer is given in the following.

Assume that:

W1 is a 66 000 V primary winding.

W2 is a 33 000 V output winding loaded at 2000 kVA and having a power factor cos Phi 2 = 0.8 lagging.

W3 is a 11 000 V output winding loaded at 1000 kVA and having a power factor cos Phi 3 = 0.6 lagging.

The following information is available, having been calculated from test data, and is related to a basic loading of 1000 kVA.

a12 = 0.26 b12 = 3.12

a23 = 0.33 b23 = 1.59

a12 = 0.32 b31 = 5.08


d _ 0.91 and g _ 9.79

Then for

W1: a1 = 0.125 and b1 = +3.305

W2: a2 = 0.135 and b2 = -0.185

W3: a3 = 0.195 and b3 = +1.775

The effective full-load kVA input to winding W1 is:

(i) With only the output winding W2 loaded, 2000 kVA at a power factor of 0.8 lagging.

(ii) With only the output winding W3 loaded, 1000 kVA at a power factor of 0.6 lagging.

(iii) With both the output windings W2 and W3 loaded, 2980 kVA at a power factor of 0.74 lagging.

Applying expression (eqn. 7) or (eqn. 8) separately to each arm of the equivalent circuit, the individual regulations have, in W1 under condition

(i) where n1 = 2.0, the value of 4.23% W1 under condition

(ii) where n1 = 1.0, the value of 2.72% W1 under condition

(iii) where n1 = 2.98, the value of 7.15%

W2 where n2 =2.0, the value of -0.02%

W3 where n3 = 1.0, the value of 1.53%

Summarizing these calculations therefore the total transformer voltage regulation has:

(i) With output winding W2 fully loaded and W3 unloaded,

at the terminals of winding W2, the value of 4.23 - 0.02= 4.21%

at the terminals of winding W3, the value of 4.23 + 0 = 4.23%

(ii) With output winding W2 unloaded and W3 fully loaded,

at the terminals of winding W2, the value of 2.72 + 0 = 2.72%

at the terminals of winding W3, the value of 2.72 + 1.53 = 4.25%

(iii) With both output windings W2 and W3 fully loaded,

at the terminals of winding W2, the value of 7.15 - 0.02 = 7.13%

at the terminals of winding W3, the value of 7.15 + 1.53= 8.68%

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