REDUCING THE ERROR OF A DC INSTRUMENT TRANSFORMER WITH CORES MADE OF ELECTRICAL STEEL

K. K. RYAZANTSEV

A DC instrument transformer with cores made of electrical steel, operating without rectifying the alternating current, has a current error of various magnitude and sign depending on the value of the component of the field strength in the cores,

Fig. 1

\[ H_1=\frac{I_1 w_1}{l_{\mathrm{cp}}}, \tag{1} \]

where
\(I_1\) — magnitude of the direct (“primary”) current,
\(w_1\) — number of turns of the direct-current winding,
\(l_{\mathrm{cp}}\) — length of the mean magnetic line of the core.

Figure 1 gives curves 1 and 2, expressing the dependence on \(H_1\) of the effective value of the secondary current \(I_2\) and of the error \(\Delta k\%\) of the transformer.

direct current, made from parts of an a.c. instrument transformer of type T. P. The error was calculated from the known expression

\[ \Delta k\%=\frac{k_{\mathrm{nom}} I_2-I_1}{I_1}\,100 . \tag{2} \]

Here

\[ k_{\mathrm{nom}}=\frac{w_2}{w_1}=\frac{225}{45}=5 \]

is the nominal transformation ratio, and \(w_2\) is the number of turns of the a.c. winding.

For comparison, the same figure also shows a dashed line giving the relation between the primary and secondary currents of an ideal transformer without error and having the same transformation ratio. The initial part of curve 1 lies above the straight line; then they intersect, after which the curve passes below it, bending toward the \(H_1\) axis.

Fig. 2

The reason for this behavior of the curve is as follows. At small values of \(H_1\), the external voltage \(u_2\) applied to the secondary circuit of the transformer is balanced mainly by the counter-e.m.f. \(e_2\) induced in the secondary windings of the transformer. If the active resistance of the a.c. windings and the load of the secondary circuit are neglected, then, in order to balance \(u_2\), the difference between the instantaneous values of the inductions in both cores must vary according to a sine law:

\[ B' - B'' = B_m \sin \omega t, \tag{3} \]

where

\[ B_m=\frac{U_{2m}}{\omega \cdot w_2 S}. \tag{4} \]

Here \(B_m\) is the amplitude of the difference of inductions,

\(U_{2m}\) is the voltage amplitude,

\(\omega\) is the angular frequency,

\(S\) is the cross section of the core steel \((20\ \text{cm}^2)\).

Figure 2 shows the graphical determination of the maximum value of the alternating component of the field strength of the core:

\[ H_{2m}=I_{2m}\frac{w_2}{l_{\mathrm{cp}}}, \]

based on the use of expression (3). Here \(I_{2m}\) is the greatest instantaneous value of the secondary current (Fig. 3).

For convenience in plotting, the magnetization characteristic of the second choke of the transformer is rotated by \(180^\circ\). Both characteristics are superposed at the points \(B_0\) corresponding to \(H_1\).

To the left in Fig. 2 the construction is carried out for a relatively small value of \(H_1\), and to the right—for a large one. In the first case the minimum of the induction \(B_n\) goes far into the region of negative values and \(H_{2m} > H_1\), which is the cause of an error of positive sign. The rms value of the secondary current \(I_2\) proves to be greater than the primary current \(I_1\) divided by the nominal transformation ratio.

Fig. 3

Such a relationship between the currents does not exist in all cases when the point \(B_n\) falls below the abscissa axis, since the curve of variation of the secondary current is trapezoidal (Fig. 3), as a result of which the rms value of the current is always less than the maximum. However, at small \(H_1\) (up to \(10\,\dfrac{a}{cm}\) in Fig. 1) a positive error is present. At large \(H_1\), the point \(B_n\), as is seen from Fig. 2, remains in the positive region, and then the error will clearly have a negative sign, since \(H_{2m} < H_1\).

In addition, in this case, owing to the increase of the secondary current, the voltage drop across the active resistance of the windings and the load increases, which leads to a decrease in the component of the external voltage that must be balanced by the emf \(e_2\) induced in the alternating-current windings. The difference of the inductions in the cores loses its sinusoidal character and decreases, as a result of which the proportionality between the direct and alternating currents is still more strongly disturbed. Therefore the magnitude of the secondary current found from the magnetization characteristics, proceeding from sinusoidal variation of the difference of inductions, will be overestimated; but even in this case, from Fig. 2, a relative decrease is clearly seen both in the instantaneous and in the rms values of the alternating current with increasing \(I_1\) and, consequently, \(H_1\). Thus, at small field strengths the current error of the transformer will have a positive sign, and in the region of medium and large values of \(H_1\)—a negative one.

To obtain the smallest error, the amplitude of the sinusoid \(B_m\) must be chosen so that the minimum value of the induction \(B_n\) over the entire range of variations of the primary current oscillates on the steepest initial portion of the magnetization characteristic.

Choosing the amplitude \(B_m\) according to the considerations indicated above also ensures the least sensitivity of the transformer to fluctuations of the ac mains voltage, which can be traced from Fig. 2. Indeed, if the voltage has changed, then the amplitude of the difference of inductions, for the creation of which a current \(I_{2m}\) must flow in the secondary circuit, changes simultaneously in the same proportion. The latter will differ little from the current occurring at normal voltage if the point \(B_n\) does not leave the initial steep portion of the characteristic. With a small deviation from the norm, the maximum value of the secondary current will likewise be changed little, as will the rms value of the same current, which is what is required for reducing the error.

To select the induction amplitude \(B_m\), it is necessary to have at one’s disposal the magnetization characteristic of the steel of the transformer cores and to construct a family of curves (Fig. 4)

\[ H_{2m}=f(B_m)\quad \text{for}\quad H_1=\mathrm{const}. \]

To achieve stability of the secondary-current values when the voltage \(U_2\) and the direct current vary, the quantity \(B_m\) must fluctuate within the limits of the upper, slightly sloping portions of the curves in Fig. 4. For example, if one is guided by a change in \(H_1\) from 2.5 to \(50\,\frac{a}{cm}\), then for this grade of steel one should choose

\[ B_m=12000 \div 14000\ \text{gauss}. \]

The selected value of \(B_m\) requires subsequent correction, which is carried out by a small change in the number of turns of the secondary windings after testing the completed transformer, having the rated load in the secondary circuit.

Fig. 4

The magnitude of the error also depends on the transfer of energy from the alternating-current circuit to the direct-current circuit. Since the secondary windings are connected in opposition, energy is transferred only by the e.m.f.’s of the even harmonics induced in the primary windings. However, in a transformer of normal design, analogous in construction to a single-turn alternating-current transformer, these e.m.f.’s are small and cannot create appreciable currents. When tests were made on a prototype transformer, chokes with cores of large cross-section, not saturated by the measured currents, were included in the direct-current circuit in order to suppress the currents of the even harmonics.

Thus, the magnitude of the transformer error is affected by the following factors:

1) the deviation of the magnetization characteristic of the core steel from the ideal rectangular characteristic;
2) the correctness of the choice of transformer parameters;
3) the magnitude of the load in the secondary circuit of the transformer;
4) fluctuation of the voltage in the alternating-current network;
5) transfer of energy from the alternating-current circuit to the direct-current circuit.

As indicated above, after testing the completed transformer it is necessary to change the number of turns of the secondary windings so as to obtain the most favorable distribution of errors along the scale of the measuring instrument.

Figure 1 shows curve 3, expressing the dependence of \(\Delta k\%\) after decreasing \(w_2\) from 225 to 221. The latter led to an increase of the error at small \(H_1\) and to its decrease at the end of the scale. In addition, the point with zero error shifted to the right. It is obvious that, with an increase in the number of turns, this point will shift to the left. But this measure alone is not sufficient, since it makes it possible to increase the accuracy only within narrow limits of the scale.

Further improvement could be achieved by calibrating the ammeter scale, which, however, would cause difficulties when replacing it. It is therefore necessary to introduce additional elements into the transformer circuit in order to be able to use a standard electromagnetic-system ammeter without any alteration and to determine the magnitude of the direct current by multiplying its readings by the nominal transformation ratio.

Below a method is described for compensating the error of both signs, consisting in superposing on the transformer secondary current flowing through the ammeter two additional currents in mutually opposite directions.

Figure 5 shows the connection diagram of a direct-current instrument transformer into which a capacitance of \(2\ \mu\mathrm{F}\) and an auxiliary transformer with a turns ratio of the primary and secondary windings of \(5:100\) are included. The core of the auxiliary transformer is taken from an old coil-type current transformer \((S = 12.3\ \mathrm{cm}^2;\ l_{\mathrm{av}} = 47\ \mathrm{cm})\).

Fig. 5

In addition, Fig. 5 shows the curve \(I_2 = f(H_1)\) at the design AC mains voltage of \(130\ \mathrm{V}\), as well as the curves of variation of the error \(\Delta k\%\) at the design voltage (middle curve) and at deviations from it by \(\pm 5\%\) (dashed curves), constructed from test data. If the middle curve \(\Delta k\%\) is compared with curves No. 2 and 3 of Fig. 1, a considerable improvement in measurement accuracy can be noted both at the beginning of the scale and at its end.

The role of the capacitance is that, with its aid, a current shifted in phase relative to the transformer secondary current is passed through the ammeter—

approximately by \(180^\circ\), and this leads to a decrease in the ammeter readings at the beginning of the scale. The capacitance, if it has the necessary value, also performs a second function, consisting in compensating changes in the secondary current during fluctuations of the AC mains voltage (for more details see the article by A. G. Nikiforov, “DC Instrument Transformer,” VEP, 1939, Nos. 10–11).

The auxiliary transformer serves to superimpose on the secondary current a second additional current, nearly coincident with it in phase and proportional to it in magnitude. As a result of such combined action on the ammeter, the noted increase in accuracy occurs.

Fig. 6

To clarify the role of each element of the circuit, Fig. 6 gives the following current-error curves:

\(a\)—for the DC transformer \((w_2 = 223)\) without capacitance and without the auxiliary transformer;

\(b\)—the same, with a connected capacitance of \(2\ \mu\mathrm{F}\);

\(v\)—the same, with the auxiliary transformer switched in;

\(g\)—the same, with capacitance and the auxiliary transformer.

The DC transformer on which the test was carried out had a rather high sensitivity to fluctuations of the auxiliary voltage \(U_2\) (see the dashed \(\Delta k\%\) curves in Fig. 5), a reduction of which can be achieved by increasing the capacitance with a corresponding change in the number of turns of the secondary windings of the chokes and in the transformation ratio of the auxiliary transformer.

In conclusion, it should be noted that, judging from the character of curve \(g\), Fig. 6, it may be assumed that even with an increase of the field intensity \(H_1\) above \(25\ \dfrac{\mathrm{a}}{\mathrm{cm}}\), the measures considered above would improve the accuracy of the measurements.