Determination of the Secondary Current Value of a DC Measuring Transformer with Electrical-Steel Cores under Transient Conditions

K. K. Ryazantsev

The presence of a transient process in a direct-current circuit into which a measuring transformer is connected by its primary winding affects the shape of the curve of its secondary current. Examining the oscillograms of both currents, it is easy to notice that there is a direct proportionality between them (Fig. 1). Thus, for example, the instantaneous values of the secondary current at the same instants, counted from the beginning of periods following one another, will be increasing or decreasing depending on the nature of the change in the primary current.

Determining the law that relates the instantaneous values of the direct and alternating currents is of undoubted interest; therefore, below an attempt is made to elucidate this question.

Fig. 1

Fig. 2

Let us turn to the circuit of a direct-current measuring transformer (Fig. 2), having four windings connected in pairs by magnetic fluxes. The primary windings are a busbar passing inside saturation reactors that form part of the transformer. The fact that the primary winding consists of only one turn makes it possible to disregard the transfer of energy from the secondary circuit into the primary, since the emf induced in the busbar is insignificant and cannot create appreciable currents.

The secondary circuit of the transformer is supplied from an alternating-current network in which the voltage varies according to a sine law.

If it is assumed that the primary windings of both reactors are connected in agreement, then the secondary windings should be regarded as connected in opposition. Therefore, during each half-period the magnetic fluxes created by the direct and alternating currents in one of the cores coincide in direction, while in the other at the same time they are opposite.

Taking into account the minus sign in parentheses due to the opposing connection of the windings, we can write the following equation for the secondary circuit:

\[ u_2=i_2 r-\left(e_L^{\mathrm{I}}-e_L^{\mathrm{II}}\right)-\left(e_M^{\mathrm{I}}-e_M^{\mathrm{II}}\right)-e_{Lн}, \tag{1} \]

in which

\(e_L^{\mathrm{I}},\ e_L^{\mathrm{II}}\) are the self-induction e.m.f.’s induced in the secondary windings of the chokes by the current \(i_2\);

\(e_M^{\mathrm{I}},\ e_M^{\mathrm{II}}\) are the mutual-induction e.m.f.’s arising in the secondary windings of the chokes as a result of the change in the primary current \(i_1\);

\[ e_{Lн}=-L_{н}\frac{di_2}{dt} \]

is the self-induction e.m.f. of the load;

\(r\) is the sum of the active resistances of the secondary circuit.

After a slight transformation we obtain:

\[ u_2=i_2 r-\left(e_L^{\mathrm{I}}+e_M^{\mathrm{I}}\right)+\left(e_L^{\mathrm{II}}+e_M^{\mathrm{II}}\right)+L_{н}\frac{di_2}{dt}. \tag{2} \]

The resultant e.m.f.’s of each winding

\[ e^{\mathrm{I}}=e_L^{\mathrm{I}}+e_M^{\mathrm{I}};\qquad e^{\mathrm{II}}=e_L^{\mathrm{II}}+e_M^{\mathrm{II}} \]

are a consequence of changes in the flux linkages of the secondary windings under the action of the primary and secondary currents, and therefore one may write

\[ e^{\mathrm{I}}=-\frac{d\Psi^{\mathrm{I}}}{dt}=-w_2 S\frac{dB^{\mathrm{I}}}{dt}, \tag{3} \]

\[ e^{\mathrm{II}}=-\frac{d\Psi^{\mathrm{II}}}{dt}=-w_2 S\frac{dB^{\mathrm{II}}}{dt}. \tag{4} \]

Here

\(w_2\) is the number of turns of the secondary winding;
\(S\) is the cross-section of the core;
\(B^{\mathrm{I}}\) and \(B^{\mathrm{II}}\) are the instantaneous values of induction in the cores.

Substituting expressions (3) and (4) into (2), and also taking into account that

\[ u_2=U_M\sin(\omega t+\varphi), \]

where \(\varphi\) is the phase angle at the instant when the current \(i_2\) passes through zero, we have:

\[ U_M\sin(\omega t+\varphi)=i_2 r+w_2 S\frac{dB^{\mathrm{I}}}{dt}-w_2 S\frac{dB^{\mathrm{II}}}{dt}+L_{н}\frac{di_2}{dt}. \tag{5} \]

To carry the solution through to the end, it is necessary to neglect the active resistance and hysteresis. Even with such a simplification, as will be seen below, the expression for calculating the instantaneous values of the secondary current will be rather complicated. In addition, we shall assume that, under the adopted assumptions,

\[ \varphi=-\frac{\pi}{2}. \]

If all the above is taken into account, equation (5) will take the form

\[ U_M \cos \omega t = w_2 S \frac{dB^{\mathrm{I}}}{dt} - w_2 S \frac{dB^{\mathrm{II}}}{dt} + L_{\mathrm{н}} \frac{di_2}{dt}. \]

After separating the variables and integrating, we obtain:

\[ B_M \sin \omega t \bigg|_{0}^{t} = B^{\mathrm{I}}\bigg|_{B_0^{\mathrm{I}}}^{B_t^{\mathrm{I}}} - B^{\mathrm{II}}\bigg|_{B_0^{\mathrm{II}}}^{B_t^{\mathrm{II}}} + k i_2 \bigg|_{0}^{i_{2t}}, \]

where

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

\[ k=\frac{L_{\mathrm{н}}}{w_2 S}, \]

\(B_0^{\mathrm{I}}, B_0^{\mathrm{II}}\) are the values of the inductions in the cores at the instant of the nearest passage of the current \(i_2\) through zero. If hysteresis is not taken into account, they should be considered equal in magnitude and sign, since they are determined only by the values of the specific ampere-turns of the direct current.

Fig. 3

\(B_t^{\mathrm{II}},\ B_t^{\mathrm{I}},\ i_t\) are the instantaneous values at time \(t\) of the inductions and the secondary current.

Substituting the limits, we find:

\[ B_M \sin \omega t = B_t^{\mathrm{I}} - B_t^{\mathrm{II}} + k i_{2t}. \tag{6} \]

The inductions \(B_t^{\mathrm{I}}\) and \(B_t^{\mathrm{II}}\) are functions of the instantaneous values of the resultant field strengths of each choke (Fig. 3):

\[ H_t^{\mathrm{I}} = H_{1t} + H_{2t}, \tag{7} \]

\[ H_t^{\mathrm{II}} = H_{1t} - H_{2t}. \tag{8} \]

The relation between the resultant field strengths and the inductions is given by analytical expressions for the magnetization characteristic, for example,

\[ B=\frac{H}{a+bH}. \tag{9} \]

or

\[ H=\alpha \, sh\, \beta B . \tag{10} \]

Replacing, in the first of them, \(H\) by the right-hand sides of expressions (7) and (8) and substituting the results into (6), we find that

\[ B_M \sin \omega t = \frac{H_{1t}+H_{2t}}{a+b(H_{1t}+H_{2t})} - \frac{H_{1t}-H_{2t}}{a+b(H_{1t}-H_{2t})} + k\,\frac{l_{cp}}{w_2}\,H_{2t}, \]

whence

\[ H_{2t}=u+\frac{1}{9u}(3e+3n+d^2)+\frac{1}{3}d, \tag{11} \]

where

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

\[ q=\frac{1}{2}dn-\frac{1}{6}(e+n)d-\frac{1}{27}d^3, \]

\[ p=-\frac{1}{9}(3e+3n+d^2), \]

\[ d=\frac{w_2^2 S}{L_{\mathrm{н}}\,l_{cp}}\,B_M \sin \omega t, \]

\[ e=\frac{2a w_2^2 S}{L_{\mathrm{н}}\,l_{cp}\,b^2}, \]

\[ n=\left(\frac{a+bH_1}{b}\right)^2 . \]

Knowing \(H_{2t}\), we determine the instantaneous value of the secondary current:

\[ i_{2t}=\frac{l_{cp}}{w_2}\,H_t, \tag{12} \]

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

If the magnetization characteristic is expressed in the form

\[ H=\alpha \, sh\, \beta B, \]

then the induction will be equal to

\[ B=\frac{1}{\beta}\, Arsh\,\frac{H}{\alpha}. \tag{13} \]

Taking into account expressions (7) and (8) and substituting them successively into (13), one may write that

\[ B_t^I=\frac{1}{\beta}\, Arsh\,\frac{H_{1t}+H_{2t}}{\alpha}, \tag{14} \]

\[ B_t^{II}=\frac{1}{\beta}\, Arsh\,\frac{H_{1t}-H_{2t}}{\alpha}. \tag{15} \]

Taking (6), (14), and (15) into account, we have

\[ B_M \sin \omega t = \frac{1}{\beta} \left( Arsh\,\frac{H_{1t}+H_{2t}}{\alpha} + Arsh\,\frac{H_{1t}-H_{2t}}{\alpha} \right) + k\,\frac{l_{cp}}{w_2}\,H_{2t}. \tag{16} \]

In view of the complexity of this equation, the load inductance \(L_n\) has to be disregarded. Then it is considerably simplified, and its solution will be as follows:

\[ H_{2t}=\left[-\frac{g}{2}+\left(\frac{g^2}{4}-h\right)^{1/2}\right]^{1/2}. \tag{17} \]

Here

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

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

Determination of the instantaneous values of the secondary current from expressions (11), (12), and (17) is associated with lengthy calculations. To avoid this, a simple graphical method is proposed, based on the use of equation (6). To clarify this possibility, let us first consider Fig. 3, which shows the relation existing between the instantaneous values of the field strengths and inductions without taking the load into account.

Let at the instant of time \(t\) there exist an instantaneous value of the field strength \(H_{1t}\), produced by the primary current \(i_{1t}\) and readily found from the given curve \(H_1=F(t)\). On the magnetization characteristics, \(H_{1t}\) corresponds to the inductions \(B_{1t}\). If at the same instant a current \(i_{2t}\) flows through the secondary windings, then the instantaneous values of the inductions \(B_t^{\mathrm{I}}\) and \(B_t^{\mathrm{II}}\) may be determined graphically from the resultant field strengths found from expressions (7) and (8). In the case when \(L_n=0\), the difference of the inductions must vary according to the sine law; therefore the horizontals drawn through the points \(B_t^{\mathrm{I}}\) and \(B_t^{\mathrm{II}}\) cut off on the sinusoid \(B_M\sin\omega t\) the ordinate \(B_t\) corresponding to the time \(t\). This makes it possible to find the relation between \(H_{2t}\) and \(t\).

To facilitate the construction, it is necessary to rotate the characteristic of the second core by \(180^\circ\) and to make the points \(B_{1t}\) coincide (Fig. 4). Then, in order to take into account the load inductance, a dashed straight line should be drawn at an angle to the upper part of the characteristic through the point \(B_{1t}\), with the calculation that the vertical segments of the straight lines enclosed between it and the upper characteristic would represent the last term of equation (6):

Fig. 4

\[ ki_{2t}=k\frac{l_{cp}}{w_2}H_{2t}. \]

Next, taking with dividers on the sinusoid the value \(B_t=B_M\sin\omega t\), one should find on the drawing the position in which the segment \(B_t\) fits, being arranged vertically, between the lower characteristic and the dashed straight line. The distance from the vertical \(B_t\) to the vertical passing through the point \(B_{1t}\) will give, to scale, the value of the field strength

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

Knowing $H_{2t}$ and the instant $t$, we are able to construct one point of the curve $H_{2t} = f(t)$, which, to scale, represents the curve $i_{2t} = f_1(t)$.

To construct the complete curve, the operations described above must be repeated many times in order to obtain a sufficient number of points. In doing so, it is necessary to superpose the points $B_{1t}$ of both characteristics, with $t$ serving as the running coordinate; i.e., each time one of the characteristics must be plotted in a new position.

To avoid repeated constructions, one may draw one characteristic on millimeter paper and the other on tracing paper, or make a template from celluloid. The superposition of the points is then achieved by simply shifting the tracing paper or the template.

In conclusion, we note that the curves of variation of the secondary current constructed for the experimental transformer specimen, in their shape and dimensions, closely correspond to the oscillograms recorded with a loop oscillograph. As the magnetization characteristic, a hysteresis-free curve was used, close to the mean magnetization curve, whose abscissae are equal to the half-sums of the corresponding abscissae of the limiting hysteresis loop.