DETERMINATION OF THE MAGNITUDE OF THE SECONDARY CURRENT OF A MEASURING DIRECT-CURRENT TRANSFORMER WITH CORES MADE OF ELECTRICAL STEEL IN STEADY-STATE OPERATION

K. K. RYAZANTSEV

In the transmission of electric power and in the operation of industrial direct-current installations, it is necessary to carry out measurements, automatic regulation, and protection. To accomplish this, accurate, reliable, and, as far as possible, simple equipment is needed; its creation is complicated either by very large currents or by high voltages, and sometimes by both together.

An additional difficulty is introduced by the fact that the magnetic field formed around current-carrying parts is, in steady-state operation, unchanging. This rules out the possibility of transforming direct current and of separating the power circuits from the measuring circuits, as is done relatively easily with alternating current.

The measurement of direct currents is most simply carried out by means of a magnetoelectric voltmeter connected to a shunt, which is connected in series in the circuit where the current to be measured flows. But this method has major disadvantages (complexity of shunt construction, difficulty of their calibration, large energy consumption, difficulty of insulating the measuring set at high voltage, etc.), which become more noticeable the higher the installation voltage and the greater the current. Therefore, various measuring devices have been proposed whose operation is based on the use of phenomena accompanying the flow of direct current in the working circuit [1].

Most of these devices have at least one of the following disadvantages:

1) the need to place the measuring instrument in close proximity to the current-carrying part (devices of I. K. Kikoin, I. A. Cherdantsev, Keinath);

2) the presence of rotating parts, which reduces reliability and requires special maintenance (devices of K. B. Karandeev, S. S. and N. S. Dobrotvorsky, Hartmann, Nelke, Pestarini);

3) the inclusion of magnetrons and thyratrons in the circuit (devices of R. E. Albrandt, Fitz-Gerald);

4) nonautomatic operation (Ditsch’s device).

Among the measuring devices known at present, the so-called choke-type measuring direct-current transformers, or transductors, stand out for their advantages. When they are used, measuring instruments and protective relays are not under the voltage of the working circuit, since the primary and secondary windings of the transformer are insulated from one another. Direct-current transformers make it possible to connect several instruments to themselves simultaneously (amper-

meters, wattmeters, counters, relays, etc.) and, when well made, operate in accuracy class 0.5. At the same time the transformers operate automatically, have no rotating parts, and do not require continuous supervision. An advantage of the transformer is also the absence of power consumption from the direct-current network, whereas a shunt may dissipate several kilowatts.

The disadvantages of transformers include the use of auxiliary alternating current, the complexity of the device, and also the high cost in comparison with an ordinary measuring alternating-current transformer.

The measuring direct-current transformer under consideration is an electromagnetic apparatus that makes it possible to determine the magnitude of a direct current by measuring the strength of an auxiliary current. The existence of a relation between the magnitudes of the two currents is explained by the fact that the reactance of the alternating-current windings of the saturating chokes included in the transformer depends on the magnetization by the direct current.

The apparatus received its name only because of its similarity in construction and purpose to a measuring alternating-current transformer, and not because a transformation of direct current actually takes place in it.

The transformer may have one four-limb magnetic core or two separate cores of rectangular or ring-shaped form.

Fig. 1

Figure 1 shows the circuit of a transformer in which the direct-current winding (the “primary” winding) is a bus passing inside toroidal cores carrying alternating-current windings \(w_2\) (the “secondary” windings). The direct-current windings are connected in series and aiding, while the alternating-current windings are also connected in series, but opposing. Each secondary winding is divided into parallel-connected sections, which are distributed uniformly over the core in order to reduce the effect on the measurement accuracy of asymmetry in the position of each core relative to the bus.

The secondary current of the transformer is rectified by means of dry rectifiers \(B\) and is measured by the magnetoelectric instrument \(A_2\). In addition, an ammeter \(A_1\) of the electromagnetic system is additionally connected into the circuit; if the rectifiers fail, it remains the main monitoring instrument.

The circuit of a transformer intended for measuring currents above 10 kA is supplemented by a special compensating device that serves to reduce the error. It consists of a small direct-current transformer through whose primary winding the secondary current of the main transformer flows. The secondary currents of both transformers, after rectification, are measured by the instrument \(A_2\) of the circuit in Fig. 1. The compensating device also includes two additional resistances, one of which is constant, while the magnitude of the other depends on the applied voltage [5].

In transformers of high accuracy classes, iron-nickel alloys of the permalloy type are used as the core material. These alloys have a high initial magnetic permeability and a small-

hysteresis losses. Therefore, transformers with magnetic cores made of permalloy-type alloys have the following properties:

1. There is a direct proportionality between the direct and alternating currents.

2. The magnitude of the secondary current depends very weakly on fluctuations of the supply-network voltage.

In [2] it is indicated that, with good construction, the transformer error does not exceed 0.5% for voltage fluctuations in the alternating-current network within ±30% of the nominal value.

1. The secondary-current curve over a half-period is close in shape to a rectangle, as a result of which, after rectification, a direct current is obtained that is proportional to the measured direct current.

2. All changes in the primary current are reflected with great accuracy on the secondary-current curve, including those whose duration is less than half a period.

3. The calculation of the transformer and the analysis of the phenomena occurring in the circuit are considerably simplified by the possibility of replacing the magnetization characteristic curve by segments of two straight lines.

However, the use of iron-nickel alloys is made difficult by their scarcity, high cost, and sensitivity to work hardening and mechanical stresses. Therefore it would be highly desirable to replace these alloys with electrical steel, especially in those cases where transformers are not required to operate in a high accuracy class. Indications of the possibility of such a replacement are given in [3, 6].

Most researchers have studied the properties of a direct-current transformer with magnetic cores made of iron-nickel alloys with high initial magnetic permeability, whereas the processes occurring in a transformer with electrical-steel cores have not been sufficiently covered.

In the case of using electrical steel as the material of the cores, the analysis of the phenomena in the transformer is complicated by the fact that the magnetization characteristic of the steel is far from an ideal rectangular characteristic. Therefore, first, it is necessary to take into account the change in induction simultaneously in both reactors during each half-period and, second, to take into consideration the phenomenon of hysteresis, which noticeably affects the magnitude of the secondary current and the shape of its curve.

The first circumstance leads to the resulting equations and their solutions being cumbersome, while the second makes it necessary to use in calculations not the main magnetization curve, but the mean magnetization curve with abscissas equal to the half-sums of the corresponding abscissas of the branches of the hysteresis loop. The mean magnetization curve may be replaced by a hysteresis-free magnetization curve, which is obtained experimentally. As is known, when it is measured, an alternating field with amplitude smoothly decreasing to zero is superposed on a constant field. The ordinates of this curve are a single-valued function of the strength of the constant magnetic field, since the superposition of the alternating field eliminates hysteresis phenomena. The hysteresis-free curve (No. 2 in Fig. 2) is closer to the mean magnetization curve than the main curve (No. 1 in Fig. 2) and gives a certain refinement of the results, especially in the region of small currents.

As already mentioned, taking into account the change in induction in both cores leads to a complication of the equations; therefore one has to resort to simplifications that facilitate obtaining solutions. Thus, for example, in the presence of a significant active resistance, the magnetization characteristic curve is replaced by segments of straight lines. If, however, the active resistance has such a value that it may be neglected, then the equations by which the instantaneous values of the secondary current are found may be co-

compiled using the analytical expression for the magnetization characteristic. In addition, under the same condition \((r = 0)\), it is possible, by a simple

Fig. 2

graphical method, to determine the instantaneous values of the secondary current and from them to construct the curve of variation of this current.

Determination of the magnitude of the secondary current taking active resistance into account

Let us turn to Fig. 3, which schematically shows the arrangement of a direct-current transformer.

It was already indicated above that the direct-current windings are regarded as being connected in agreement, and the alternating-current windings in opposition. Therefore the inductions \(B_0\) existing in both cores at the instant when the secondary current passes through zero will be identical in sign and magnitude.

Fig. 3

Fig. 4

The points \(B_0\) on the magnetization characteristics can be easily found if the value of the direct current \(I_1\) is known (Fig. 4). At subsequent instants the magnitudes of the inductions and, consequently, the positions of the points \(B'_t\) and \(B''_t\) on the characteristics will be determined by the algebraic sum of the current \(I\) and the secondary current \(i\), reduced to one and the same number of turns, for example \(w_2\). If, in the half-period under consideration, both currents create fluxes of the same direction in core I, then the point \(B'_t\) on the characteristic will move to the right as long as the current \(i\) increases, while

then, as it decreases, shifts to the left and, finally, again occupies the position \(B_0\). In this same half-period, in the core of choke II the fluxes will be directed oppositely. Therefore the point \(B_t^{II}\) moves along the characteristic in the region lying to the left of the point \(B_0\). In the next half-period the sign of the secondary current will change to the opposite one, in accordance with which both magnetic fluxes in the core of choke I will have opposite directions, while in core II they will have the same directions.

For determining the magnitude of the secondary current, let us replace the curve of the characteristic by three straight-line segments, of which the first and second meet at the point \(B_0\) (Fig. 5). Let us superpose the characteristics at the points \(B_0\), rotating one of them through \(180^\circ\) for this purpose. Such an arrangement of the characteristics reflects, first, the fact that the inductions in the cores are equal when there is no alternating current in the secondary windings and, second, the fact that the fluxes created by the direct and alternating currents during each half-period coincide in direction in one of the cores and are opposite in the other. On the other hand, by arranging the characteristics in this way, we can easily follow the change in induction in each core, plotting the value of \(i\), depending on its sign, to the right or to the left of the vertical passing through the point \(B_0\).

Fig. 5

According to the region of the characteristic, the inductive resistance of the secondary winding of a choke may be equal to \(x_1\), \(x_2\), or \(x_3\). Thus, for example, at the instant shown in Fig. 5, the inductive resistance of the winding of the first choke is equal to \(x_1\), and that of the second to \(x_2\). In each half-period the resistance of the winding of the choke being magnetized by the alternating current is equal to \(x_1\), while the resistance of the winding \(w_2\) of the choke being demagnetized by the same current will be equal to \(x_2\) or \(x_3\), depending on the magnitude of the induction,

Fig. 6

In what follows we shall assume that the voltage at the terminals of the secondary circuit varies according to a sine law and that there is no transfer of energy from the alternating-current network into the direct-current circuit. The latter is close to reality, since the primary winding of the transformer of the design under consideration has only one turn; as a result, the emf arising at the terminals of this winding amounts to fractions of a volt and cannot create appreciable currents.

The equivalent circuit of the transformer secondary circuit is shown in Fig. 6. It consists of the permanently connected reactances \(x_1\), \(x_n\), \(r\), and the reactances \(x_2\), \(x_3\), of which one or another is shunted. The resistance \(r\) includes all active resistances of the secondary circuit. The inductive reactance of the load \(x_n\) is assumed constant.

Fig. 7

According to [4], the transition of the induction in the choke steel from one region of the characteristic to another should be represented as the instantaneous shunting of one of the resistances and the simultaneous introduction of another resistance into the circuit by opening its switch. Therefore, in accordance with the equivalent circuit, we may write that the secondary current \(i\), within the limits from \(-I_k\) to \(+I_k\) (Figs. 5 and 7), varies according to the law:

\[ i_1 = I_{1m}\sin(\omega t-\varphi_1)+I_{n1}e^{-\frac{t-t_1}{\tau_1}} \tag{1} \]

where

\[ I_{1m}=\frac{U_m}{\sqrt{r^2+(x_1+x_2+x_n)^2}}, \]

\[ \varphi_1=\operatorname{arctg}\frac{x_1+x_2+x_n}{r}, \]

\(I_{n1}\)—the initial value of the transient component arising when the circuit parameters change at the instant \(t_1\) of the displacement of the point \(B_t'\) from region 3 to region 2 of the characteristic,

\[ \tau_1=\frac{x_1+x_2+x_n}{\omega r} \]
—the time constant of the circuit.

Further, when the current in magnitude exceeds \(I_k\), it is equal to

\[ i_2=I_{2m}\sin(\omega t-\varphi_2)+I_{n2}e^{-\frac{t-t_2}{\tau_2}}, \tag{2} \]

where

\[ I_{2M}=\frac{\dot U_M}{\sqrt{r^2+(x_1+x_3+x_\mu)^2}}, \]

\[ \varphi_2=\operatorname{arctg}\frac{x_1+x_3+x_\mu}{r}; \]

\(I_{n2}\) is the initial value of the transient component appearing at the instant \(t_2\) when the point \(B_t^{II}\) moves from region 2 of the characteristic into region 3;

\[ \tau_2=\frac{x_1+x_3+x_\mu}{\omega r} \]

is the time constant of the circuit.

The unknown quantities \(t_1\), \(t_2\), \(I_{n1}\), and \(I_{n2}\), entering into expressions (1) and (2), are related by equations (3)—(6), which are drawn up taking into account the fact that at the instant \(t_1\) the current \(i_1\) is equal to \(-I_k\), at the instant \(t_2\) this same current has the magnitude \(+I_k\), and the current \(i_2\) at the instants \(t_2\) and \(\dfrac{T}{2}+t_1\) is equal to \(+I_k\) (Fig. 7).

\[ I_{1M}\sin(\omega t_1-\varphi_1)+I_{n1}=-I_k, \tag{3} \]

\[ I_{1M}\sin(\omega t_2-\varphi_1)+I_{n1}e^{-\frac{t_2-t_1}{\tau_1}}=I_k, \tag{4} \]

\[ I_{2M}\sin(\omega t_2-\varphi_2)+I_2=I_k \tag{5} \]

\[ I_{2M}\sin(\omega t_1+\pi-\varphi_2)+I_{n2}e^{-\frac{t_1+\frac{T}{2}-t_2}{\tau_2}}=I_k. \tag{6} \]

The rms value of the alternating current can be found from the expression:

\[ I=\left(\frac{2}{T}\int_{t_1}^{t_1+\frac{T}{2}} i^2\,dt\right)^{\frac12} = \left[ \frac{2}{T} \left( \int_{t_1}^{t_2} i_1^2\,dt+ \int_{t_2}^{t_1+\frac{T}{2}} i_2^2\,dt \right) \right]^{\frac12}. \tag{7} \]

If in (7) we substitute, in place of \(i_1\) and \(i_2\), the right-hand sides of expressions (1) and (2), we obtain:

\[ \begin{aligned} I=&\left(\frac{2}{T}\right)^{\frac12} \Biggl\{ \frac{1}{\omega}I_{1M}^{2} \left[ 0.5(\omega t_2-\omega t_1)+0.25\sin 2(\omega t_1-\varphi_1) \right.\\ &\left. \qquad\qquad\qquad -0.25\sin 2(\omega t_2-\varphi_1) \right] + \frac{2\tau_1}{1+\omega^2\tau_1^2}I_{1M}I_{n1} \left[ d\left(e^{-\frac{t_2-t_1}{\tau_1}}\sin\omega t_2-\sin\omega t_1\right) \right.\\ &\left. \qquad\qquad\qquad -f\left(e^{-\frac{t_2-t_1}{\tau_1}}\cos\omega t_2-\cos\omega t_1\right) \right] + \frac{1}{\omega}I_{2M}^{2} \left[ 0.5(\omega t_1-\omega t_2+\pi)-0.25\sin 2(\omega t_1-\varphi_2) \right.\\ &\left. \qquad\qquad\qquad +0.25\sin 2(\omega t_2-\varphi_2) \right] + \frac{2\tau_2}{1+\omega^2\tau_2^2}I_{2M}I_{n2} \left[ -g\left(e^{-\frac{t_1+\frac{T}{2}-t_2}{\tau_2}}\sin\omega t_1+\right. \right. \end{aligned} \]

\[ +\sin \omega t_2)+h\left(e^{-\frac{t_1+\frac{T}{2}-t_2}{\tau_2}}\cos \omega t_1+\cos \omega t_2\right)\right] \]

\[ \left. +\frac{\tau_1}{2} I_{n1}^{2}\left(1-e^{-2\frac{t_2-t_1}{\tau_1}}\right) +\frac{\tau_2}{2} I_{n2}^{2}\left(1-e^{-2\frac{t_1+\frac{T}{2}-t_2}{\tau_2}}\right) \right\}^{\frac{1}{2}}, \tag{8} \]

where

\[ d=\omega \tau_1 \sin \varphi_1-\cos \varphi_1, \]

\[ f=\sin \varphi_1+\omega \tau_1 \cos \varphi_1, \]

\[ g=\omega \tau_2 \sin \varphi_2-\cos \varphi_2, \]

\[ h=\sin \varphi_2+\omega \tau_2 \cos \varphi_2. \]

Determination of the Secondary Current Without Taking Active Resistance into Account

If the active resistance of the secondary circuit is small and is neglected, a formula for determining the instantaneous values of the alternating current can be obtained by replacing by a straight-line segment only that part of the magnetization characteristic which lies beyond the bend point, and by expressing analytically the initial part of the same characteristic.

To derive the formula indicated above, taking \(r=0\) and taking into account the opposition connection of the secondary windings, we write the fundamental voltage equation for the secondary circuit:

\[ -\left[\left(-w_2 S\frac{dB^I}{dt}\right)-\left(-w_2 S\frac{dB^{II}}{dt}\right)\right]+L_n\frac{di}{dt} =U_m\sin(\omega t+\varphi), \tag{9} \]

where \(B^I\) and \(B^{II}\) are the instantaneous values of the induction in the steel of the first and second reactors;

\(w_2\) is the number of turns of the secondary winding;

\(S\) is the cross section of the reactor core;

\(L_n\) is the inductance of the load of the secondary circuit;

\(U_m\) is the amplitude value of the voltage applied to the terminals of the transformer secondary circuit;

\(\varphi\) is the phase angle at the instant when the current \(i\) passes through zero \((t=0)\).

Bearing in mind that the point \(B_0\) lies beyond the bend point (Fig. 4) and that the portion of the characteristic located beyond this point can be replaced by a straight-line segment, we represent the first term of expression (9) in the form

\[ -w_2 S\frac{dB^I}{dt}=-L^1\frac{di}{dt}, \tag{10} \]

where

\[ L^1=\frac{d\Psi^1}{di}\simeq \text{const}. \]

We express the initial portion of the magnetization characteristic as follows (Fig. 4):

\[ I_1-i=\alpha \operatorname{sh}\beta B^{II}. \tag{11} \]

Here \(I_1\) is the value of the direct current referred to the number of turns of the secondary winding.

Then

\[ -\frac{di}{dt}=-\alpha \beta \operatorname{ch}\beta B^{II}\frac{dB^{II}}{dt}. \tag{12} \]

After substituting (10) and (12) into expression (9), we have:

\[ (L^1+L_n)\alpha\beta\,\operatorname{ch}\beta B^{II}\frac{dB^{II}}{dt} +w_2S\frac{dB^{II}}{dt} +U_m\sin(\omega t+\varphi)=0. \]

Separating the variables and integrating, and also taking into account that, in the absence of eddy currents, hysteresis, and active resistance, \(\varphi\) may be assumed equal to \(90^\circ\), we obtain

\[ \alpha(L^1+L_n)\operatorname{sh}\beta B^{II} \left|_{B_0}^{B_t^{II}}\right. + w_2SB^{II}\left|_{B_0}^{B_t^{II}}\right. + \frac{U_m}{\omega}\sin\omega t \left|_{0}^{t}\right. =0 \]

or

\[ \alpha\operatorname{sh}\beta B_t^{II} +\frac{w_2S}{L^1+L_n}B_t^{II} -\alpha\operatorname{sh}\beta B_0 -\frac{w_2S}{L^1+L_n}B_0 + \frac{U_m}{\omega(L^1+L_n)}\sin\omega t =0. \tag{13} \]

Here the only unknown is \(B_t^{II}\)—the instantaneous value of the induction at the time \(t\), while the remaining quantities, including \(B_0\), are given. \(B_0\) is easily found from the magnetization characteristic, since at the instant \(t=0\), when the secondary current is absent, the magnitude of the induction is determined only by the specific ampere-turns of the direct current.

Finding from equation (13) the values \(\alpha\operatorname{sh}\beta B_t^{II}\) corresponding to different instants of time, and substituting them into expression (11), we obtain the instantaneous values of the secondary current \(i\), knowing which we construct the curve of variation of this current over half a period. The associated operations can be simplified; for this purpose we rewrite expression (13) in the following form:

\[ \alpha\operatorname{sh}\beta B_t^{II} + \frac{w_2S}{L^1+L_n}B_t^{II} = \alpha\operatorname{sh}\beta B_0 + \frac{w_2S}{L^1+L_n}B_0 - \frac{U_m}{\omega(L^1+L_n)}\sin\omega t. \]

Let us denote

\[ F(B^{II})=\alpha\operatorname{sh}\beta B_t^{II} + \frac{w_2S}{L^1+L_n}B_t^{II}. \]

\[ F_1(t)= \alpha\operatorname{sh}\beta B_0 + \frac{w_2S}{L^1+L_n}B_0 - \frac{U_m}{\omega(L^1+L_n)}\sin\omega t. \]

Assigning values to the instants of time \(t\) within the range from \(0\) to \(T/2\), and to the inductions \(B^{II}\), starting from \(B_0\), we construct the curves \(F_1(t)\), \(F(B^{II})\), and \(\alpha\operatorname{sh}\beta B^{II}\) (Fig. 8). In addition, a dashed horizontal line should be drawn at a distance \(I_1\) from the induction axis.

After this, the determination of the instantaneous values of the current is very simple, namely: having selected the instant of time \(t\) (see the right-hand part of Fig. 8), we move upward to the curve \(F_1(t)\), then proceed horizontally to the left to the curve \(F(B^{II})\). The vertical \(AB\), drawn through the point of intersection of the horizontal with the curve \(F(B^{II})\), is, on the scale, equal to the instantaneous value of the current, since \(i=I_1-\alpha\operatorname{sh}\beta B^{II}\). The value \(i\) must be laid off on the right-hand part of Fig. 8 from the \(t\)-axis (segment \(A^1B^1\)).

Repeating these operations the required number of times, we obtain a series of points belonging to the curve \(i=f(t)\).

In exactly the same way, an analytical expression for the instantaneous values of the secondary current can be obtained if the magnetization characteristic is described by the formula \(i=aB+bB^3\). In this case one can dispense with constructing auxiliary curves, calculating the values of \(i\) directly.

Fig. 8

Without repeating all the operations, let us indicate that, proceeding by the same path, we obtain a cubic equation; solving it, we find that

\[ B_t^{\prime\prime}=v-\frac{p}{3v}, \tag{14} \]

where

\[ v=\sqrt[3]{-q+\sqrt{q^2+p^3}}, \]

\[ p=\frac{a(L_1+L_n)+w_2S}{3b(L_1+L_n)}, \]

\[ q=\frac{U_m\sin\omega t}{2\omega b(L_1+L_n)} -\frac{a(L_1+L_n)+w_2S}{2b(L_1+L_n)}\,B_0 -\frac{1}{2}B_0^3 . \]

After substituting (14) into the expression

\[ I_1-i=aB_t^{\prime\prime}+b(B_t^{\prime\prime})^3 \]

we have

\[ i=I_1-a\left(v-\frac{p}{3v}\right) -b\left(v-\frac{p}{3v}\right)^3 . \tag{15} \]

The disadvantages of both methods discussed in this paragraph are the complexity of the calculations and the impossibility of directly calculating the rms value of the secondary current. At the same time, it should be noted that when the analytical expression for the magnetization characteristic \(i=aB+bB^3\) is used, less accurate results are obtained; the reason for this is a certain discrepancy between the calculated curve constructed from this expression and the actual characteristic curve.

Determination of the Secondary Current without Taking into Account the Active Resistance and the Load of the Secondary Circuit

Assuming that the active resistance and the load of the secondary circuit are equal to zero, on the basis of equation (9) it can be shown that the difference of the instantaneous values of the inductions in the cores of the direct-current transformer varies according to a sine law, i.e.,

\[ B_t^I - B_t^{II} = B_M \sin \omega t, \tag{16} \]

where

\[ B_M = \frac{U_M}{\omega w_2 S}. \]

Substitute into (16) the values of \(B_t^I\) and \(B_t^{II}\) obtained from the expression

\[ H = \alpha \operatorname{sh} \beta B, \]

which describes the magnetization characteristic. In doing so, we take into account that during each half-period, only for one of the cores do the constant and variable components of the field intensity have the same signs. Therefore it should be written that

\[ B_t^I = \frac{1}{\beta}\operatorname{Arsh}\frac{H_1 + H_{2t}}{\alpha}, \tag{17} \]

\[ B_t^{II} = \frac{1}{\beta}\operatorname{Arsh}\frac{H_1 - H_{2t}}{\alpha}. \tag{18} \]

Here \(H_1\) is the constant component of the magnetic-field intensity;
\(H_{2t}\) is the instantaneous value of the field intensity produced by the alternating current.

Substituting (17) and (18) into expression (16), we obtain

\[ \frac{1}{\beta}\left( \operatorname{Arsh}\frac{H_1 + H_{2t}}{\alpha} - \operatorname{Arsh}\frac{H_1 - H_{2t}}{\alpha} \right) = B_M \sin \omega t, \]

or, after transformations,

\[ (H_1 + H_{2t}) \left[ 1+\left(\frac{H_1-H_{2t}}{\alpha}\right)^2 \right]^{\frac12} - (H_1 - H_{2t}) \left[ 1+\left(\frac{H_1+H_{2t}}{\alpha}\right)^2 \right]^{\frac12} = \alpha \operatorname{sh}(\beta B_M \sin \omega t). \]

Squaring this expression twice, we obtain the biquadratic equation

\[ H_{2t}^4 + p H_{2t}^2 + q = 0, \]

in which

\[ p = \alpha^2 - 2H_1^2 \left\{ 1+\frac{2}{\operatorname{sh}^2(\beta B_M \sin \omega t)} \right\}, \]

\[ q = \alpha H_1^2 + H_1^4 - \left[ \frac{1}{2}\alpha^2 \operatorname{sh}(\beta B_M \sin \omega t) \right]^2. \]

The solution of this equation is

\[ H_{2t} = \left[ -\frac{p}{2} + \left( \frac{p^2}{4}-q \right)^{\frac12} \right]^{\frac12}. \]

The instantaneous value of the secondary current at time \(t\) will be equal to:

\[ i_{2t}=\frac{l_{cp}}{w_2}\,H_{2t}, \tag{20} \]

where \(l_{cp}\) is the length of the mean magnetic line.

Owing to the complexity of the integral \(\int H_{2t}^{2}\,dt\), it was not possible to obtain an expression for determining the effective value of the secondary current, which, however, can be calculated by the formula

\[ I_2=\frac{l_{cp}}{w_2}\sqrt{(H_{2t}^{2})_{cp}}. \]

For this it is necessary to construct the curve \(H_{2t}^{2}=f(t)\) and find the mean value of the squares of the variable component of the field strength \((H_{2t}^{2})_{cp}\) over the time of one half-period.

A rather simple formula for calculating the effective value of the secondary current can be obtained in the same way, but by substituting into (16) the instantaneous values of the induction from the analytical expression of the magnetization characteristic

\[ B=\frac{H}{a+bH}. \tag{21} \]

Reasoning in the same manner as before, we have the right to write

\[ B_t^{\mathrm{I}}=\frac{H_1+H_{2t}}{a+b(H_1+H_{2t})} \]

for the core being magnetized in the given half-period, and

\[ B_t^{\mathrm{II}}=\frac{H_1-H_{2t}}{a+b(H_1-H_{2t})} \]

for the core being demagnetized by the alternating current.

Using expression (16), we obtain

\[ \frac{H_1+H_{2t}}{a+b(H_1+H_{2t})} - \frac{H_1-H_{2t}}{a+b(H_1-H_{2t})} = B_{\mathrm{m}}\sin\omega t. \]

After simplification we shall have the equation

\[ H_{2t}^{2}+2\frac{m}{\sin\omega t}\,H_{2t}-n=0, \]

in which

\[ m=\frac{a}{b^{2}B_{\mathrm{m}}} \quad\text{and}\quad n=\left(\frac{a+bH_1}{b}\right)^{2}. \]

The solution of this equation

\[ H_{2t}=-\frac{m}{\sin\omega t} + \left(\frac{m^{2}}{\sin^{2}\omega t}+n\right)^{\frac{1}{2}} \tag{22} \]

makes it possible to find the instantaneous value of the current

\[ i_{2t}=\frac{l_{cp}}{w_2}\,H_{2t}. \tag{23} \]

If all the operations necessary for finding the effective value are carried out, we obtain the following:

\[ H_{2\mathrm{eff}}=\left\{\,n+\frac{4}{\pi}m\sqrt{m^{2}+n}\,[E(\alpha)-K(\alpha)]\,\right\}^{\frac{1}{2}}, \tag{24} \]

where \(K(\alpha)\) and \(E(\alpha)\) are the normal Legendre forms of elliptic integrals of the first and second kinds, with

\[ \alpha=\arcsin\left(\frac{n}{m^{2}+n}\right)^{\frac{1}{2}}. \]

The values of \(K(\alpha)\) and \(E(\alpha)\) are easily found from tables, for example in [Lit. 7]. Therefore the determination of \(H_{2\mathrm{eff}}\), if \(m\) and \(n\) are given, is carried out without difficulty.

Knowing \(H_{2\mathrm{eff}}\), we can find the effective value of the secondary current

\[ I_{2}=\frac{l_{\mathrm{av}}}{w_{2}}H_{2\mathrm{eff}}. \tag{25} \]

For example, let it be given that

\[ m=2.43;\qquad n=500;\qquad \frac{l_{\mathrm{av}}}{w_{2}}=0.203, \]

\[ \alpha=\arcsin\left(\frac{500}{2.43^{2}+500}\right)^{\frac{1}{2}}\simeq 83^\circ. \]

From the table [7, p. 177] we find

\[ E(\alpha)=1.0223,\qquad K(\alpha)=3.5004, \]

\[ I_{2}=0.203\left[500+\frac{4}{\pi}\,2.43\sqrt{2.43^{2}+500}\,(1.0223-3.5004)\right]^{\frac{1}{2}}=3.67\,a. \]

It should be noted that expression (21) describes the magnetization characteristic not for all negative values of the field strength, but only within small limits. Therefore, in calculations it is necessary first, by some method—for example graphically (see below)—to check whether the minimum value of the induction \(B_{\min}\) extends too far downward along the characteristic.

Determination of instantaneous values of the secondary current by a graphical method

A general disadvantage of the methods described above for determining the magnitude of the transformer secondary current is the dependence of the results on the accuracy with which the magnetization characteristic is replaced by some curve or by straight-line segments. To avoid the additional errors associated with this circumstance, the magnitude of the secondary current should be found using the magnetization characteristic itself directly. To clarify the possibility of this, let us return to equation (9):

\[ w_{2}S\frac{dB'}{dt}-w_{2}S\frac{dB''}{dt}+L_{n}\frac{di}{dt}=U_{m}\sin(\omega t+\beta). \]

Separating the variables and integrating, we obtain:

\[ B'\bigg|_{B_{0}'}^{B_{t}'}-B''\bigg|_{B_{0}''}^{B_{t}''} +\frac{L_{n}}{w_{2}^{2}S}\,i\bigg|_{0}^{i_{t}} = -\frac{U_{m}}{\omega w_{2}S}\cos(\omega t+\beta)\bigg|_{0}^{t}. \]

Assuming that the inductions in both cores at the instant when the secondary current passes through zero \((t=0)\) are identical and equal to \(B_0\), and also setting \(\beta=\dfrac{\pi}{2}\), we have

\[ B_t^I - B_t^{II} + \frac{L_n}{w_2 S}\, i_t = B_m \sin \omega t, \tag{26} \]

where

\[ B_m = \frac{U_m}{\omega w_2 S}. \]

Expression (26) can be used for the graphical determination of the instantaneous values of the secondary current and for constructing the curve of variation of this current.

The construction procedure is as follows. Superpose the points \(B_0\) of both magnetization characteristics, rotating one of them by \(180^\circ\), which makes it easy to find the difference of the inductions (Fig. 9). Starting from point \(A\), draw the straight line \(AC\) at an angle \(\alpha\), whose tangent is numerically equal to

\[ \frac{L_n}{w_2 S}. \]

Then, assigning successively increasing values of the current \(i\) and laying them off from point \(A\) to the right, we determine the differences \((B_t^I - B_t^{II})\) enclosed between the characteristic curves, and add to them the quantities

\[ \frac{L_n}{w_2 S} i_t, \]

located on the same vertical with them.

Fig. 9

The sums

\[ (B_t^I - B_t^{II}) + \frac{L_n}{w_2 S} i_t \]

should be laid off on the same vertical from the \(Oi\) axis. By joining with a curve the upper points of the segments representing these sums to scale, we obtain the curve \(OD\), which, together with the sinusoid \(B_m \sin \omega t\), serves to determine the instantaneous values of the secondary current. For example, having chosen the instant of time \(t\), we move upward to the intersection with the sinusoid at point \(K\), then proceed horizontally to point \(M\) on the curve \(OD\). Next we find the position of point \(E\), lying at the intersection of the vertical passing through point \(M\) with the horizontal drawn through the second point \(t\) on the vertical axis. The segment \(tE\), to scale, gives the magnitude of the secondary current.

Repeating a similar construction for various instants of time, we find a series of points; by connecting them we obtain the curve of variation of the secondary current over half a period.

Verification

To verify the material set forth above, a direct-current transformer was made from parts of a measuring alternating-current transformer of type TP. The transformer has two toroidal cor...

cores, assembled from stampings 0.35 mm thick. The cross section of each core is 20 cm² with a mean-line length of 45.6 cm. The primary winding consists of 45 turns of copper wire with a cross section of 6 mm². The secondary winding of each choke is uniformly distributed over the core and has 225 turns of aluminum winding wire with a cross section of 3.2 mm². The active resistance of both windings in the heated state is 1.02 ohm. A load in the form of a coil with an air core, having an inductive reactance of 2 ohms and an active resistance of 5.35 ohms, is connected into the secondary circuit.

Figure 2 shows the main (1) and hysteresis-free (2) magnetization curves of the core, taken with a fluxmeter.

The table below gives the results of calculations and direct measurement of the rms value of the secondary current at a constant component of the field strength equal to 20 A/cm.

Table

The rms value of the secondary current was found in the first and fourth cases without graphical constructions; in the remaining cases the instantaneous values were found and squared, after which curves

$$ i^2 = f(t), $$

were plotted, which were used to determine

$$ I = \sqrt{(i^2)_{\mathrm{av}}}. $$

As is evident from the table, the deviation from experiment is in the direction of underestimation, which may be explained as follows. First, the allowance for the hysteresis phenomenon was incomplete, although corrections were introduced by using the hysteresis-free magnetization curve. Second, the current value known from experiment is overestimated, since, despite the inclusion of smoothing chokes in the direct-current circuit, there was a small transfer of energy from the secondary circuit to the primary circuit due to even-harmonic currents.

In addition, it should be noted that, in all cases except No. 5, the accuracy of the analytical expression of the magnetization curve has a noticeable influence on the calculation results.

REFERENCES

1. R. B. Popov. Control of Electric-Power Consumption, Metallurgizdat, 1945.
2. Ya. M. Chervonenkiс. Transmission of Energy by Direct Current, Chapter VIII, compiled by Candidate of Technical Sciences Yu. G. Tolstov, Publishing House of the Academy of Sciences of the USSR, 1948.
3. A. G. Nikiforov. Measuring direct-current transformer, VEP, Nos. 10–11, 1939.
4. A. A. Bessonov. Electrical Circuits with Steel, GEI, 1948.
5. W. Krämer. Prüfung des Gleichstrommesswandlers und seine Fehlerkompensation. AEG.-Mitt., H. 5, 1939.
6. W. Krämer. Ein einfacher Gleichstrommeswandler mit echten Stromwandlereigenschaften. ETZ, H. 49, 1937.
7. E. Jahnke and F. Emde. Tables of Functions, OGIZ, 1948.