METHODS FOR MEASURING THE POWER FACTOR UNDER OPERATING CONDITIONS OF ELECTRICAL INSTALLATIONS

R. A. Voronov.

I.

In the operation of modern electrical installations, great attention is paid to the value of the power factor, on which depends not only the degree of utilization of generators, transformers, and electrical networks, but also the cost of the electrical energy generated. Power stations are interested in having their energy consumed at as high a power factor as possible; for this purpose consumers must have good equipment, suited to the place of its operation.

The main influence on the magnitude of the power factor is exerted by electric motors operating at partial load or at increased voltage. Transformers and other induction apparatus have a somewhat smaller influence. Proper selection of equipment and rational choice of compensating devices, as well as systematic observation of their operation, can very greatly increase the value of the overall power factor of the entire network.

In order that consumers should strive to carry out measures to improve the power factor of their installations, a system of penalty tariffs was introduced, under which, in addition to payment for electrical energy, a penalty was imposed for a poor power factor, depending on the magnitude of this factor. In this connection, alongside active-energy meters, consumers acquired reactive-energy meters, from which the value of the average power factor is also calculated.

Rational selection of equipment can be made only on the basis of systematic observations of all items of equipment and the identification of those among them that have the greatest influence on the deterioration of the power factor. In carrying out such surveys and observations, great difficulties were encountered because of the absence of suitable instruments and the impossibility of having them connected in advance at all points of the networks where they might be required. Having sets of portable instruments and moving them from one place to another also proved inconvenient; moreover, in some cases such connection of instruments is altogether impossible. It was necessary to find methods of measurement by which one could get by with a minimal number of the simplest instruments, either already available on site or easily connected into the network, while the accuracy of the measurements had to be quite sufficient.

Such a method was the one-ammeter method, proposed back in 1933 by the author of this work, suitable for symmetrical three-phase circuits¹. This method, applied at several installations, showed

fully demonstrated its suitability and proved to be quite convenient, but in its application it was limited by the power of the circuit under investigation. Further development of this method led to minor alterations of the ammeter or current transformer and made it possible to apply it at any power and in any circuits.

The present work is devoted to the theory of this method and to a description of methods for its practical use.

II.

The power factor is customarily defined as the multiplier by which the product of voltage and current must be multiplied in order to obtain the power of the circuit. In ordinary circuits with sinusoidal current this factor is equal to the cosine of the phase-displacement angle of the current and voltage and is determined from the relation

\[ \cos \varphi = \frac{P}{U I} \]

for single-phase current, or

\[ \cos \varphi = \frac{P}{\sqrt{3}\,U\,I} \]

for balanced three-phase current. The most widespread method is to calculate the power factor from these equations, for which an accurate measurement of power, voltage, and current is necessary.

For such measurements it is necessary to have three instruments in the circuit: a voltmeter, an ammeter, and a wattmeter, whose measuring ranges must correspond to the operating conditions of the given circuit. In three-phase circuits, in the absence of a neutral point, a special wattmeter may be required, or else Aron’s circuit with two wattmeters must be used. Since the errors of the instruments may be summed, in order to obtain good results all the instruments must be sufficiently accurate. If ordinary technical instruments are used, the total error of the result may reach 6% (counting up to 2% per instrument), while its probable value may be about 4%. If the instruments operate not at the end of the scale, but in the middle or at the beginning of it, then this error may increase still more. Only by using precision instruments of class L or K can results be obtained with sufficient accuracy, of the order of 1–1.5%.

Since the objects of measurement may have the most varied powers, these instruments will be required with correspondingly different measuring ranges. It does not appear possible to install them in advance at all locations, while switching in an ammeter and wattmeter during operation is impossible and usually can be carried out only with the circuit disconnected, which in many cases also cannot be done.

The use of a phase meter reduces the number of necessary instruments and simplifies observation, but its connection during operation of the circuit encounters the same difficulties as the connection of a wattmeter. The readings of phase meters have sufficient accuracy under normal conditions of their operation, but under all deviations from these conditions (small current or low voltage, non-sinusoidal current, incorrect frequency, phase unbalance, etc.) often lead to large errors. In addition, good phase meters are very expensive and are rarely available on site, while simple electromagnetic ones are very unreliable. Finally, a phase meter gives incomplete observations and requires the additional connection of ammeters or wattmeters.

If two identical wattmeters are available (or if it is possible to switch the windings of one of them), then in a three-phase

current, the Aron circuit may be used. In this case, for a symmetrical system the phase-shift angle is found from the equation

\[ \tan \varphi=\sqrt{3}\,\frac{P_1-P_2}{P_1+P_2}, \]

where \(P_1\) and \(P_2\) are the instrument readings. From the phase-shift angle the power factor can also easily be found. Instead of wattmeters, meters may be used; in that case, in the equation, instead of the wattmeter readings, one must substitute the number of revolutions of the disks over the same interval of time.

Because this method requires two identical instruments, it is usually used only under laboratory test conditions, or else in cases where the instruments are installed at the given location for continuous operation. In addition, to obtain results with sufficient accuracy it is necessary to have fully symmetrical phases both in voltage and in load.

The use of these methods of measuring the power factor is connected with the need to have precise instruments, which must be moved from one place to another as required. All this would be greatly simplified if the current windings of the instruments could be connected through current transformers with a movable core (of the Dietze clamp type). Unfortunately, the accuracy of such transformers, especially as regards angular error, is so low that there is no possibility of trusting the readings of a wattmeter or phase meter connected through them; moreover, it is not even possible to introduce any corrections into their readings.

From the foregoing it follows that the commonly used methods are not very convenient and cannot be used very widely.

III.

To determine the power factor in sinusoidal current, it is necessary, in one way or another, to establish the value of the phase-shift angle between current and voltage and to take the cosine of this angle. Of greatest interest, because of the simplicity of the instruments used, is the three-ammeter method proposed by Fleming, in which the property of the vectors of three currents is used, one of which is the sum of the other two.

According to this method, in parallel with the load whose power factor is to be determined, a resistance is connected that has a quite definite phase shift. Usually this resistance is chosen to be active, so that its current coincides in phase with the network voltage. With three ammeters, the current in the load \(I_1\), the current in the additional resistance \(I_2\), and the total current \(I_3\) are measured. From the vector diagram it is easy to obtain the value of the load power factor

\[ \cos \varphi=\frac{I_3^2-I_1^2-I_2^2}{2I_1I_2}. \]

To obtain sufficient accuracy by this method, it is necessary for the current in the added resistance to be of the same order as that in the load for which the measurements are made; moreover, the instruments themselves must introduce as small errors as possible. Because differences appear in the equation, even small deviations in the readings can give significant errors in the result.

Because of these inaccuracies and because of the need to have three ammeters in the circuit, this method has received almost no practical application. Nevertheless, several years ago a proposal appeared to apply this

method not only in a single-phase circuit, but also in a three-phase one¹). Since in this method, in addition to the inconveniences indicated above, it is necessary to connect into the network, in star, three identical watt resistances consuming the same power as the load being examined, its application can hardly be justified.

The three-voltmeter method, which is essentially a modification of the three-ammeter method, was proposed by Ayrton, Sumpner, and Swinburne simultaneously with Fleming (1891). Replacing the ammeters with voltmeters makes it possible to manage with only one instrument, switched to three positions, which somewhat increases accuracy. The need to connect the resistance in series requires almost doubling the network voltage, which is completely impossible under operating conditions. Therefore this method is applicable even more rarely than the three-ammeter method.

The method described below is also essentially a modification of Fleming’s method, but it requires only one measuring instrument—an ammeter; moreover, the accuracy of this instrument has a significantly smaller effect on the value of the power factor obtained as a result of the measurement.

IV.

In its simplest form, the one-ammeter method was designed for determining the power factor of three-phase asynchronous motors of small power, or of other corresponding loads with symmetrical phases. Using the ordinary technical ammeter already present in the circuit, or connecting such an ammeter into one of the phases if it was absent²), three readings are taken from this instrument in accordance with the diagram in Figure 1. The first reading \(I_1\) is taken during normal operation of the circuit; the second reading \(I_2\), with the additional connection of a resistance \(R\) between this phase and the second; and the third \(I_3\), with this same resistance switched into the third phase.

Fig. 1.

Fig. 2.

As the resistance, incandescent lamps in the appropriate number, or some wire rheostat with low inductance, may be used.

These three readings are sufficient to find the phase displacement angle, and consequently also the power factor. The three values \(I_1\), \(I_2\), and \(I_3\)

¹) Gorokhov. Determination of the power factor of small-power asynchronous motors in the absence of a wattmeter. Elektrichestvo, 1933.

²) For this purpose Dietze clamps may be used if it is not possible to connect an ordinary ammeter.

correspond to the lengths of the vectors in the diagram of Figure 2, with the vectors \(I_0'\) and \(I_0''\) corresponding to the currents in the resistance \(R\) for its two connections. The latter two vectors are in phase with the line voltages, i.e., they are shifted by \(30^\circ\) from the phase voltage.

From the diagram we have

\[ \begin{aligned} I_2^2 &= I_1^2 + I_0^2 + 2 I_1 I_0 \cos(\varphi + 30^\circ) = I_1^2 + I_0^2 + I_1 I_0(\sqrt{3}\cos\varphi - \sin\varphi),\\ I_3^2 &= I_1^2 + I_0^2 + 2 I_1 I_0 \cos(\varphi - 30^\circ) = I_1^2 + I_0^2 + I_1 I_0(\sqrt{3}\cos\varphi + \sin\varphi). \end{aligned} \tag{a} \]

Since the values \(I_1\), \(I_2\), and \(I_3\) are known from the instrument readings, the only unknowns are the current \(I_0\) and the angle \(\varphi\). By a series of transformations one can find the value of the current

\[ I_0 = \sqrt{A - \sqrt{A^2 - B^2}}, \]

where \(A\) and \(B\) replace the expressions

\[ 2A = I_1^2 + I_2^2 + I_3^2 \]

\[ B^2 = (I_1^4 + I_2^4 + I_3^4) - (I_1^2 I_2^2 + I_1^2 I_3^2 + I_2^2 I_3^2); \]

substituting this value of \(I_0\) into the equation, we find the desired value of the power factor

\[ \cos\varphi = \frac{I_3^2 + I_2^2 - 2I_1^2 - 2I_0^2}{2\sqrt{3} I_1 I_0} = \frac{2\sqrt{A^2 - B^2} - 3I_1^2} {2\sqrt{3} I_1 \sqrt{A - \sqrt{A^2 - B^2}}}. \]

This equation proves to be very complicated for calculation; moreover, because of the series of differences entering into it, it may give large inaccuracies as a result of the calculation. Its use turns out to be very laborious and requires much time.

Fig. 3.

In connection with this, it is much more convenient, instead of calculation, to use a previously constructed nomogram, shown in Figure 3. Introducing the ratios of the currents

\[ \frac{I_0}{I_1} = a;\quad \frac{I_2}{I_1} = b;\quad \frac{I_3}{I_1} = c, \]

we obtain equations (a) in the following form

\[ b^2 = 1 + a^2 + 2a \cos(\varphi + 30^\circ), \]

\[ c^2 = 1 + a^2 + 2a \cos(\varphi - 30^\circ). \]

By assigning various values to \(a\) and \(\varphi\), from these equations one can calculate the corresponding values of \(b\) and \(c\). Using the latter quantities as coordinate axes, it is easy to construct a family of curves for \(\cos \varphi\) and for \(a\).

Use of the nomogram is very simple. For this purpose, from the ammeter readings the values of the ratios \(b\) and \(c\) are calculated, and from them the point on the nomogram is found. From this point it is easy to determine not only the value of the power factor, but also the corresponding ratio

\[ a = \frac{I_0}{I_1}, \]

and consequently also the current \(I_0\) passing through the added resistance.

Investigation of the question of possible errors showed that their sources may be the nonproportionality of the ammeter scale to the current passing through it, the low sensitivity of the instrument, and asymmetry of the network voltages. The accuracy of the instrument in this case plays no role, since the final result is affected not by the currents themselves, but by their ratios.

In order to obtain better sensitivity of the instrument, it is necessary that the normal current of the circuit correspond to 50–75% of the nominal value of the scale. When the added resistance is connected, the readings must not go beyond the limits of the scale (25% and 100%), and at the same time at least one of them should differ as much as possible from the reading at normal current.

Considering the nomogram (Fig. 3), one may come to the conclusion that the most convenient added resistance is one for which the ratio \(a\) will lie within the limits 0.4–0.6, which corresponds to a current \(I_0\) averaging about 50% of the load current. Taking into account the possibility of obtaining such currents in portable watt resistors, it was assumed that the limit of applicability of this method is motors with powers up to 30–40 KW at network voltages up to 380 volts.

Use of this method in some installations showed that such a power is not yet the limit. According to the statement of the workers of these installations¹, they used this method to make measurements for motors with powers up to 100 KW, though with the current \(I_0\) reduced to 25–30% of the load current. Since the results fully satisfied them, they assumed that this method could be extended to still more powerful installations.

The asymmetry of voltages in a normal three-phase current circuit is very small and its influence is almost imperceptible; but when this method is applied, owing to the connection of an added resistance into one phase, additional voltage drops of appreciable magnitude may be obtained. Calculations show that, despite this, the values of the power factor will differ from the true values very little. Even in networks with a large resistance of the conductors, the error will amount only to hundredths, and even then only at large phase-shift angles.

Some assistance in correcting this influence of voltage asymmetry may be provided by metallic incandescent lamps, if they are used instead of the resistance \(R\). Since the resistance of lamps increases with incandescence, their current will change considerably more slowly than the voltage changes; consequently, the difference in the currents in the two connections will be smaller.

As can be seen from the above description of the one-ammeter method, this method has advantages over other ways of measuring the power factor, since it does not require the additional connection of precision instruments and makes do with a simple technical ammeter or Dietze clamps. The accuracy of the results obtained in this way proves quite good, differing hardly at all from the results of measurements by control instruments (the error in the results usually does not exceed 0.02–0.03).

At the same time, at high powers this method proves rather cumbersome, since it requires a large rheostat for the resistance \(R\), and also creates, for the duration of the measurement, an additional load on the network with the consumption of considerable power. In addition, this method has proved inapplicable to single-phase circuits.

Further development of this method made it possible to eliminate all the shortcomings and to extend its application to any powers and any alternating-current circuits.

V.

Every ammeter of the electromagnetic system gives a deflection that depends on the total ampere-turns produced by its winding. The form of the scale in this case depends on very many factors: the dimensions of the coil and its position, the shape of the iron core, the presence and arrangement of stationary iron parts, etc. Among other things, the arrangement of the winding on the coil also influences the scale.

The scale of the instrument may be graduated either directly for the current passing through the instrument, or for a multiple of it, as is the case when current transformers are used.

If two completely identical windings, with the same arrangement of turns, are placed on the coil of such an ammeter, then the readings of the instrument must correspond to the current of either of the coils; i.e., the scales for them must coincide with one another. It is very easy to achieve such an arrangement of the windings, since for this it is sufficient to wind both windings simultaneously with two wires at once.

When currents pass simultaneously through both windings of the instrument, the readings of the latter will depend on the geometric sum of the ampere-turns

\[ I_1 W_1 + I_0 W_0 = (I_1 + I_0) W_1 \]

or, since the turns are identical, on the sum of the currents. In other words, the ammeter will indicate the geometric sum of the currents even in the case when, in reality, there is no such current in the circuit.

The correctness of this relation for any phase shifts between the currents can be checked by means of the circuit shown in Fig. 4.

Fig. 4.

Such an ammeter may be used instead of an ordinary one for finding the power factor of a symmetrical three-phase circuit, the circuit then taking the form shown in Fig. 5. One of the windings, designed for prolonged passage of current, is connected into the main circuit and works as an ordinary ammeter; the other, which may be made of considerably finer wire, serves for passing the current of the added resistance. The whole procedure of measurement and the subsequent determination of \(\cos \varphi\) remains the same as with an ordinary ammeter.

It is easy to see that in this case no advantages over the simple instrument are obtained, while there are disadvantages, since the instrument is made somewhat more complicated and additional errors may arise because the scales are not fully identical.

This instrument must be somewhat modified in such a way that the current \(I_0\) can be taken many times smaller than the load current, while the ratio \(a\) must remain the same. This can be achieved in two ways—either by an unequal number of turns in the windings, or by connecting a current transformer into the main winding.

Two windings can be made on one instrument with different numbers of turns but with the same appearance of the scales (i.e., with proportional readings) in various ways, the simplest being the following. When both windings are wound simultaneously, the main one is wound with normal wire designed for continuous operation at full load. The second winding is wound at once in \(m\) conductors of fine wire designed for the short-time passage of a current equal approximately to \((0.3—0.4)\dfrac{I}{m}\), where \(I\) is the nominal current of the instrument.

Fig. 5.

After all the necessary turns have been wound, the separate conductors of the second winding are connected in series, i.e., one winding is obtained with a number of turns \(m\) times greater than in the main one. In view of the identical arrangement of the turns of both windings, the scales will be completely identical; only the value of a division for the second winding will accordingly be \(m\) times smaller.

When current passes simultaneously through both windings, the readings of the instrument will give not simply the sum of the currents, but the value

\[ i_1 + m i'_0 = i_1 + i_0', \]

i.e., the sum of the currents reduced to the turns of the first winding. This makes it possible to produce the same changes in the readings of the instrument by means of a current \(I_0\) that is \(m\) times less than 20 in magnitude than in the case of a simple ammeter.

When such an instrument is connected according to the circuit of Figure 5, the current in the resistance \(R\) will be required \(m\) times smaller, and consequently the resistance may be taken \(m\) times larger. The power consumed and the resulting additional voltage drop in the circuit will also decrease \(m\) times.

Theoretically, with such an instrument measurements can be made for any powers, since the value \(m\) can be chosen as large as desired. In reality, because of the difficulty of making such an instrument for a large current, at high powers it is necessary to use the second circuit with a current transformer (see below).

VI.

The construction of a two-winding instrument usually presents no difficulties, since in existing designs there is always a large reserve of space on the coil, while the additional winding, being fine, occupies very little space. Even a makeshift alteration of the instrument is very inexpensive, and in factory manufacture there should be almost no difference in cost between simple and two-winding ammeters. The entire complication of the instrument, apart from this winding, will consist in one addi—

additional terminal for three-phase current, or to two for single-phase current (see below).

Instead of an auxiliary resistance, incandescent lamps may successfully be used; as indicated above, they somewhat reduce the inaccuracies arising from voltage asymmetry. The whole setup then has the appearance shown in the photograph in Figure 6. A two-winding ammeter is connected in the ordinary three-phase-current circuit. When measuring the power factor, an auxiliary resistance is connected to its additional terminal, consisting of several metal-filament incandescent lamps of different wattages (so that the current \(I_0\) can be selected), which are connected through a switch to the two remaining phases.

When the need for them has passed, the lamps together with the switch are moved to another place, while the instrument continues to operate as an ordinary ammeter.

Fig. 6.

The entire connection of the lamps and the making of the measurement take one or two minutes.

Practice has shown that making such an instrument is convenient only for currents up to 100–150 amperes, with a current in the auxiliary winding of about 5 amperes and a multiplier \(m\) up to 10. This makes it possible, with a rheostat consuming about 5 amperes, to measure the power factor of motors up to 70–100 kilowatts at voltages up to 380 volts.

VII.

At present, instead of a multi-ampere instrument, an ammeter for only 5 amperes is often used, connected into the network through a measuring current transformer. Since such a connection considerably simplifies and reduces the cost of the measuring instrument and the circuit for bringing current to it, while the cost of the transformer is very low, these circuits have become widespread even at low voltage.

When such a circuit is used for a two-winding ammeter, both windings of the latter may be chosen for 5 amperes, differing only

thickness of the wire. The manufacture of such an ammeter is, of course, much simpler than that of a multi-ampere one with different numbers of winding turns.

The connection diagram for such an instrument takes the form shown in Figure 7; moreover, in order to avoid breakdown of the insulation between the windings of the instrument, these must necessarily be electrically connected with one another, which also entails connecting both windings of the transformer. It follows from this that, with such a circuit, the secondary winding of the transformer must not be grounded.

Fig. 7.

All calculations and the use of the nomogram remain as before, with the role of the number \(m\) being played by the transformation ratio of the current transformer. In the error term there is added the angular error of the current transformer, which enters into the final result by making the calculated phase-shift angle smaller than it actually is. Since under normal conditions the angular error of current transformers is very small (less than \(1^\circ\)), the error in the obtained value of the power factor will be insignificant and may be entirely neglected.

When this circuit is used, the ammeters for all circuits are the same, with the current in the main winding up to 5 amperes and in the additional winding up to 2–3 amperes. In all cases, metal-filament incandescent lamps may serve as the additional resistance. At 220 volts it is convenient to have three lamps: one of 100 watts, a second of 200, and a third of 300 watts. With the aid of these lamps, the required additional current can always be selected.

VIII.

When using a current transformer, instead of placing a second winding on the ammeter coil, one may wind a third winding on the current transformer itself, and take the ammeter to be the most ordinary type. Such a circuit, while not differing essentially from the preceding ones, has a number of very substantial advantages.

For a three-winding current transformer there must hold the equality of ampere-turns

\[ \dot I_1 w_1 + \dot I_\delta w_\delta + \dot I_2 w_2 = \dot I_0 w_1, \]

where \(I_\delta\) is the current in the additional winding, and \(I_0\) is the magnetizing component of the current of the total ampere-turns, referred to the main primary winding. If all currents are recalculated to the number of turns of this winding, then the equation for the reduced currents takes the form

\[ \dot I_1 + \dot I'_\delta + \dot I'_2 = \dot I_0, \]

whence the current passing through the ammeter will be

\[ -\dot I'_2 = (\dot I_1 + \dot I'_\delta) - \dot I_0; \]

the arrangement of these currents on the vector diagram is shown in Figure 8.

In current transformers of the normal type the magnetizing component \(I_0\) is very small and may be neglected, especially since the angular

the error for $I_2$ in this case is of no significance. Moreover, this component, being proportional to the current $I_2$¹), will affect the instrument readings in the same way in all cases, so that the current ratios will not change. This makes it possible to regard the ammeter readings as equal to the sum of the currents

\[ \dot I_1=\dot I'_0=\dot I_1+m\dot I_0=\dot I_1+\frac{w_0}{w_1}\dot I_0 \]

without any errors.

The circuit for measuring the power factor in a three-phase-current circuit when such a three-winding transformer is used has the form shown in Fig. 9. By choosing the number of turns of the auxiliary winding equal to the number of turns of the secondary winding, the same lamps as in the preceding cases may be used as the resistance.

This circuit is of interest because in it the angular errors of the current transformer are eliminated and there is no need for a connection between the windings. This makes it possible to use such a circuit at any power levels, and the voltage may be quite considerable. Under laboratory conditions, with certain precautions taken (switching the lead by means of a long insulated rod, good insulation of the windings, grounding of the ammeter, and a small current in the auxiliary winding, of the order of 0.5 ampere), it proved possible to use this circuit even at voltages of 2–3 thousand volts. In production, the voltage limit will probably have to be considered 500 volts.

Fig. 9.

Fig. 8.

IX.

As was indicated above, portable current transformers of the Dietz clamp type cannot be used for measurements with wattmeters and phase meters, both because of large angular errors and because of the incomplete proportionality between the currents of the primary and secondary circuits. The reason for this is the small number of ampere-turns in the primary circuit (a feed-through single-turn transformer with a small primary current), as a result of which a large magnetizing component of the current is obtained. In addition, owing to the split core and the nonrigid position of the primary conductor with respect to this core, considerable scatter of the magnetic fluxes is obtained, differing under different operating conditions of the clamps.

Nevertheless, with a slight modification these clamps can be used to measure $\cos\varphi$ by the method indicated above. For this it is necessary to apply to them one more winding, having $m$ turns and placed over the secondary winding. In this case, a three-winding transformer is likewise obtained, but one somewhat different from the usual one.

¹) See the following section.

When such a transformer is operating, the total magnetic flux that produces the electromotive force of the secondary winding is composed geometrically of three parts

\[ \Phi_0=\Phi+\Phi_a+\Phi_b. \]

The flux component \(\Phi\) (Fig. 10) is common to all three windings and is proportional to the geometric sum of all three currents, taken in reduced values,

\[ \Phi=k(\dot I_a+\dot I_b+\dot I'_2), \]

where \(k\) is the coefficient of proportionality, and \(\dot I_a\) and \(\dot I_b\) are the currents in the two primary windings, the main and the auxiliary.

The components \(\Phi_a\) and \(\Phi_b\) are common only to two windings, one of the primary windings and the secondary; moreover, for them, in the general case, the multipliers \(k\) will be different and unequal:

\[ \Phi_a=k_a(\dot I_a+\dot I'_2), \]

\[ \Phi_b=k_b(\dot I'_b+\dot I'_2). \]

The flux component common only to the two primary windings, and the leakage fluxes of each of them, will not affect the relation of operation of the transformer, causing only an additional voltage drop in the primary circuit.

Fig. 10.

The total magnetic flux must produce an electromotive force directed toward overcoming the resistances of the entire secondary circuit, including the leakage reactance of the secondary winding.

\[ E'_2=-j4.44 f \omega \Phi_0=\dot I'_2 Z'_2; \]

substituting into this equation the value of \(\Phi_0\), expressed through the currents of the windings, and transforming the equation, we obtain

\[ \dot I'_2= -\frac{\dot I_a(k+k_a)+\dot I'_b(k+k_b)} {-\dfrac{z'_2}{4.44 f\omega}+(k+k_a+k_b)}. \]

From this equation it is easy to see that the current \(\dot I'_2\) may be considered equal to the sum of the currents \(\dot I_a\) and \(\dot I'_b\) only in the case when the fluxes \(\Phi_a\) and \(\Phi_b\) (partial leakages) are small in comparison with the flux embracing all three windings, and when, at the same time, the effect of the magnetizing current, i.e. the term with \(z'_2\), is small. Such a condition obtains in a normal current transformer, as was assumed above.

In the case of Dietz clamps or similar devices, the leakages acquire an appreciable magnitude, and the coefficients \(k\), \(k_a\), and \(k_b\) become comparable with one another. This destroys the equality of the currents, but nevertheless a certain proportionality between them is preserved. If we introduce the ratio

\[ \frac{k+k_b}{k+k_a}=\alpha, \]

then the equation may be brought to the following form:

\[ \dot I'_2=-\beta(\dot I_a+\alpha \dot I'_b), \]

where the coefficient of proportionality \(\beta\) combines all the remaining terms of the equation

\[ \beta=\frac{\kappa+\kappa_a}{\kappa+\kappa_a+\kappa_b-j\,\frac{z'_2}{4.44 f w}} . \]

Since the ammeter for such transformers is made quite definite and is usually not used in other places, it can already be calibrated in advance not for the magnitude of the reduced secondary current \(I'_2\), but for a somewhat larger one \((W_1=1)\)

\[ \frac{I_2}{\beta}=\frac{I_2 w_2}{\beta w_1}=I'_2\frac{w_2}{\beta}, \]

as is usually done.

The scale for the current of the additional winding will differ from the scale for the current of the primary winding (the conductor) by \(m\alpha\) times, i.e., it likewise will not correspond to reduced values. Only in the case of selecting the partial leakages of the windings to be equal \((\kappa_a=\kappa_b)\) will the multiplier \(\alpha\) be equal to unity, and the scales for the reduced values will coincide with one another.

The use of clamps for the purposes of measuring the power factor is in no way different from that indicated above for a three-winding current transformer. In Fig. 11 the first three positions give the necessary measurements, from which the ratios \(b\) and \(c\) are found, and from the latter the value of \(\cos\varphi\) is determined from the nomogram. In this case the current in the additional resistance will be equal to

\[ I_0=I_1\frac{a}{m\alpha}. \]

Fig. 11.

This current can easily be measured in the fourth position of Fig. 11, when the additional winding is connected and the clamps are removed from the conductor.

If in this case the additional resistance \(R\) is constant, then a voltage scale may be applied to the instrument

\[ U=I_0R=I_1\frac{Ra}{m\alpha}, \]

i.e., the same clamps can simultaneously serve as a voltmeter as well.

From all that has been said it follows that the use of Ditz clamps or current transformers similar to them considerably simplifies all measurements, making it possible to switch them on during operation in all circuits. Because of the variable value of the leakage, the accuracy of such measurements will, of course, be less than with other circuits, but this decrease is very small and will not be of substantial significance. Moreover, the constancy of the leakages can be obtained quite well by placing the conductor, as far as possible, in the center of the iron and by not moving the clamps during the taking of all three readings.

Below, the table gives data from measurements made simultaneously by four methods.

From these values it is evident that the results of measurements by the two-winding ammeter and by the tongs proved to be very close to the results of measurements by the control instruments, having errors smaller than with ordinary technical instruments.

X.

In the circuits indicated above with a two-winding ammeter or with a three-winding current transformer, the vector ratios were used in the same way as in the simplest one-ammeter method, and in all cases the result was found from one and the same nomogram of Fig. 3. Apart from this case, there may be many other circuits pursuing somewhat different aims and having other equations and nomograms. Let us consider some of them.

When an additional winding is used for measurement in a three-phase circuit, its ends may be switched in the opposite direction, i.e., differences of currents may be obtained instead of sums, as shown in the diagram of Fig. 12. In this case we have:

\[ I_2^2 = I_1^2 + I_0^2 - 2 I_1 I_0 \cos(\varphi + 30^\circ), \]

\[ I_3^2 = I_1^2 + I_0^2 - 2 I_1 I_0 \cos(\varphi - 30^\circ). \]

Introducing the current ratios, we obtain

\[ b^2 = 1 + a^2 - 2a \cos(\varphi + 30^\circ), \]

\[ c^2 = 1 + a^2 - 2a \cos(\varphi - 30^\circ). \]

From these equations it is easy to construct a nomogram analogous to the nomogram of Fig. 3, but such a nomogram proves to be much less convenient to use, and therefore there is no sense in applying it.

In some cases, especially at large phase-shift angles, it is convenient, when switching the additional circuit into another phase, to change the direction of the current in it (Fig. 13). The readings \(I_2\) and \(I_3\) would then be closer to one another, which would somewhat increase the accuracy. Complicat-

...in using such a circuit, and the confusion that may result in the connections, are by no means justified by this negligible increase in accuracy.

Similar nomograms may be constructed for other additional resistances (for example, a capacitance or an inductive coil), or for another range of measurements. Thus, for example, by using a three-winding

Fig. 12.  Fig. 13.

current transformer, the auxiliary winding may be connected not in the same phase in which the current is being measured, but in the other two.

The use of all these circuits has shown that they are more difficult to implement and less convenient than those described above with an auxiliary watt resistance.

XI.

One of the applications of three-winding current transformers, or of two-winding ammeters, is the measurement of the power factor in a single-phase circuit. In this case three readings of the instrument are used: without the auxiliary winding, the sum of the currents, and the difference of the currents. To obtain the last reading, the auxiliary winding has to be switched by its ends; for this, two additional terminals must be provided. The switching may be performed either by simply transferring the ends, or by means of a special double-pole switch.

Figure 14 gives three connection diagrams of a two-winding ammeter for such a measurement. The readings of the instrument in these connections will correspond to the vectors \(I_1\), \(I_2\), and \(I_3\) of the diagram in Figure 15. Denoting by \(I_0\) the reduced value of the current of the auxiliary winding, we have

\[ I_2^2 = I_1^2 + I_0^2 + 2 I_1 I_0 \cos(\varphi - \psi), \]

\[ I_3^2 = I_1^2 + I_0^2 - 2 I_1 I_0 \cos(\varphi - \psi), \]

Fig. 14.

where \(\psi\) is the phase-shift angle for the current in the auxiliary winding. Adding these equations, we find

\[ I_0^2=\frac{1}{2}\left(I_2^2+I_3^2-2I_1^2\right) \]

and, subtracting one from the other, we obtain

\[ \cos(\varphi-\psi)=\frac{I_2^2-I_3^2}{4I_1I_0}; \]

substituting into the latter the value of \(I_0\) obtained above, we find the equation

\[ \cos(\varphi-\psi)= \frac{I_2^2-I_3^2}{2\sqrt{2}I_1\sqrt{I_2^2+I_3^2-2I_1^2}}. \]

Fig. 15.

From this equation the phase-shift angle \(\varphi\) can be found, and from it the power factor can then be determined. Although this equation is simpler than the one derived above for three-phase current, nevertheless using it leads to lengthy calculations and may produce considerable errors.

In connection with this, for a single-phase circuit it is also better to construct a nomogram and then use it to determine the value of the power factor.

To construct such a nomogram we introduce the previous ratios of currents, after substitution of which the equations take the form

\[ b^2=1+a^2+2a\cos(\varphi-\psi), \]

\[ c^2=1+a^2-2a\cos(\varphi-\psi); \]

Fig. 16.

the values of \(b\) and \(c\), calculated for various values of \(\varphi\) and \(a\) with wattful auxiliary resistance \((\psi=0)\), are given in Table No. 2, and the corresponding nomogram is shown in Figure 16.

Since the scale of the instrument usually begins at 20–25% of its limiting value, when this nomogram is used it is difficult to obtain good results at small phase-displacement angles, since in this case one of the readings must be very small and the other very large. In this case

Fig. 17.

there is usually no possibility of obtaining 40; the value of $a$ is greater than $-0.5$, which does not give sufficient accuracy of the result.

On the contrary, at large phase-displacement angles it is easy to obtain readings for values of $a$ greater than 0.6–0.8, with good accuracy of the result. From such considerations one may conclude that this measurement circuit and the nomogram given are good only for those displacement angles for which the power factor is less than 0.7–0.8.

In Fig. 17 a nomogram is given, constructed for an additional resistance in the form of a capacitance without ohmic resistance, i.e., for the value of the angle $\psi = -90^\circ$. It is easy to see that this circuit and nomogram, on the contrary, are better suited for small phase-displacement angles ($\cos \varphi$ greater than 0.7–0.8) than for large ones.

Fig. 18.

The nomograms of Figs. 16 and 17 are equally suitable both for inductive loads and for capacitive loads; only in the latter case the values $b$ and $c$ of the second nomogram change places. If, instead of the additional resistance, an inductive coil without iron is taken, then because of the presence of ohmic resistance (in this case $\psi \ne 90^\circ$) the curves for inductive and capacitive loads will not coincide with one another and they will have to be constructed separately.

The circuit of Fig. 14 is easily extended also to the case of using a current transformer. Thus, Fig. 18 gives the circuit of a permanent installation with a two-

with pole switching. In the normal state the switch is in such a position that the additional winding is disconnected. When measuring the power factor, this winding is alternately connected for direct and reverse currents.

XII.

One of the causes of inaccuracy in the result obtained by these methods is the influence of the inductance of the coils of the auxiliary circuit and the mutual inductance of the main circuit upon the auxiliary one. Nothing was said above about the possibility of such causes, but the question of them can easily arise. Let us determine the magnitude of the errors that may result from these influences.

It was assumed everywhere above that a purely active resistance was connected in the auxiliary circuit and that, in consequence, the current \(I_0\) was in phase with the voltage. In reality, this circuit must also have inductive reactance in the form of an additional winding, or of an ammeter, or of a current transformer; moreover, in the latter case only the leakage inductance plays a role.

If it is assumed that in the auxiliary circuit the current should on average be equal to 3 amperes, then the additional resistance of this circuit will be equal to 70 ohms at 220 volts and 125 ohms at 380 volts. At the same time, the coils of such an ammeter (20–40 turns with an average diameter of up to 2–3 cm) have an inductance of the order of \(10^{-4}\) henry, which gives an inductive reactance in hundredths of an ohm (0.03–0.08 ohm). The phase displacement in such a circuit will be measured in a few minutes and will have no effect on the value of the power factor.

In the same way, the influence of the mutual inductance of the coils will also be negligibly small, since the electromotive forces of mutual induction must be smaller than those of self-induction. Moreover, in many cases mutual inductance acts opposite to self-induction, i.e., they partially compensate each other.

When current transformers of the ordinary type are used, the leakage of the windings is very small, and therefore its influence will also be extremely small. Only for Dietz clamp-type transformers could it be at all noticeable; but even then, according to the measurements made, the difference in the angles is not more than tenths of a degree.

With a decrease in the current in the auxiliary winding and a corresponding increase in its number of turns (both for the ammeter and for the transformer), the inductance begins to grow faster than the resistance, which increases the angle of error. It follows from this that one should not strive for very small currents in the auxiliary winding.

XIII.

Summarizing everything stated above, one may arrive at the following conclusions:

1. The one-ammeter method requires considerably fewer and simpler instruments than any other method; moreover, the accuracy of the result of determining the power factor is quite high and is not inferior to the accuracy of other methods.

2. In its simplest form, using an ordinary technical ammeter available in the network, this method can be applied only for small powers.

3. When the ordinary ammeter is replaced by a two-winding one, the convenience of applying this method increases still further; moreover, the cost of the instrument

almost does not increase. Changes may be made for any powers.

1. Instead of a two-winding ammeter, an ordinary one may be used, but with an additional third winding placed on the current transformer. This makes it possible, for measurement purposes, to use current transformers of the Dietz clamp type.

2. The use of an additional winding on the ammeter or on the transformer makes it possible to extend this method also to single-phase current circuits.

3. The value of the added resistance is immaterial and may each time be chosen so that it is convenient to take readings. This makes it possible to use metal incandescent lamps instead of a resistance.

4. In any measurement, the entire calculation may be reduced to finding two ratios of the instrument readings and determining the required power factor from the corresponding nomogram.

May 1937.

1. Anzherka CES. ↩↩