On the Experimental Determination of the Inductive Parameters of Commutating Sections in DC Machines

V. L. Gromok, V. V. Ivashin, I. A. Miloradov

(Presented by the scientific seminar of the Department of Electrical Machines)

At present, the commutation of electrical machines is calculated, with certain assumptions, on the basis of a system of ordinary differential equations compiled for the commutating sections of the armature winding, having the parameters \(L\), \(R\), \(M\), that is, taking into account the mutual-inductive coupling between the sections being commutated simultaneously.

The accuracy of such calculations depends primarily on the accuracy with which the parameters \(L\) and \(M\) are determined. The existing analytical formulas for calculating these parameters are of a very approximate nature and often give results that differ by a factor of 2–3. In addition, with their aid only static quantities are calculated; consequently, for large machines, in which eddy currents in the iron and copper, in the body of the conductors, play a significant role, these formulas give an even greater error. To determine equivalent parameters that take into account the influence of eddy currents, there exist a number of experimental methods [1–4], which have substantial shortcomings, such as high error and complexity.

The purpose of the present work is to search for a new, simpler and more accurate experimental method for determining the inductive parameters of commutating sections of DC machines. The availability of such a method will subsequently make it possible not only to improve the accuracy of commutation calculations, but also to refine the analytical formulas for calculating these parameters.

An electrical circuit, depending on the conditions in which it is found, may be characterized by static, dynamic, and equivalent inductance [5].

Static inductance may be defined as the inductance of a circuit in which equal increments of flux linkage correspond to equal increments of current. If, for example, under a linear change of current the flux linkage of the circuit changes nonlinearly, i.e., equal increments of one of the quantities correspond to unequal increments of the other, then such a circuit is characterized by a dynamic inductance, equal to the limit of the ratio of the increment of the self-induction flux to the increment of current when the latter tends to zero.

The influence of eddy currents on the inductance of a commutating section of an electrical machine is clarified by considering transient processes in a separately taken section when a constant ...

voltage, which, under certain assumptions, corresponds to the commutation conditions in this section.

Let us turn to the calculation of the eddy currents induced in the copper conductors of the section. A number of works devoted to such calculations are known [6, 7, 8]. Most authors come to the conclusion that the law of variation of eddy currents can be approximately approximated by a simple

Fig. 1, a, b. Equivalent circuit and current curves of the section

exponential function \(i_{\mathrm{в}} = I_{\mathrm{в}m}(1 - e^{-t/\tau_{\mathrm{в}}})\), where \(I_{\mathrm{в}m}\) is the maximum value of the current, and \(\tau_{\mathrm{в}}\) is the time constant for the establishment of these currents. This function is calculated by means of equivalent circuits in the form of a number of loops with parameters \(R_i\) and \(L_i\). Let us consider this on the simplest equivalent circuit, shown in Fig. 1, a, where

\(L\) is the inductance of the armature-winding section, assuming its active resistance to be zero; \(R_{\mathrm{в}}\) and \(L_{\mathrm{в}}\) are the parameters of the eddy-current loop. Calculation of the transient processes in such a circuit when it is connected to a constant voltage gives the following expressions:

\[ i_1 = \frac{U}{L}\cdot t;\qquad i_2 = \frac{U}{R_{\mathrm{в}}}\left(1 - e^{-t/\tau_{\mathrm{в}}}\right);\qquad \tau_{\mathrm{в}} = \frac{L_{\mathrm{в}}}{R_{\mathrm{в}}}; \]

\[ i = i_1 + i_2 = \frac{U}{L}\cdot t + \frac{U}{R_{\mathrm{в}}}\left(1 - e^{-t/\tau_{\mathrm{в}}}\right). \tag{1} \]

The curves of these currents are shown in Fig. 1, b.

It follows from Fig. 1, b that, owing to the presence of eddy currents in the slot copper, the current \(i\) in the section has the greatest rate at the beginning of the commutation period \(T_k\). As the eddy currents become established, this rate decreases, approaching that which would occur in the absence of eddy currents (curve \(i_1\)).

Now let us consider the nature of the field variation in the slot (Fig. 2). The current \(i\) flowing through the conductor creates a field in the slot with intensity \(H\), which, according to the law of total current, is determined as

\[ H = \frac{\sum I}{l} = \frac{i}{l}. \]

The current \(i_2\) is induced by this field in the closed loops of the conductor. This current, in turn, creates a field with intensity \(H_{\mathrm{в}}\), which is proportional to the magnitude of the current

\[ i_2 = \frac{U}{R_{\mathrm{в}}}\left(1 - e^{-t/\tau_{\mathrm{в}}}\right). \]

The indicated fields act in opposition; therefore the resulting field intensity in the slot is determined by: \(H_{\mathrm{p}} = H - H_{\mathrm{в}}\).

Obviously, in the absence of eddy currents, the flux linkage of the slot is characterized by a function proportional to the current \(i_1\):

\[ \Psi=\omega\Phi=\omega\mu S\cdot\frac{i_1\omega}{l}, \tag{2} \]

where \(\omega\) is the number of turns, in this case \(\omega=1\);
\(S\) is the area penetrated by the flux;
\(l\) is the length of the section over which the field strength is calculated.

In the presence of eddy currents, the flux linkage is proportional to the difference of the currents:

Fig. 2. Field pattern of a slot in the presence of eddy currents in the winding conductors

\[ \Psi_{\mathrm{v}}=\omega\mu S\cdot H_p =\omega\mu S\cdot\frac{(i-i_2)\omega}{l} =\omega\mu S\cdot\frac{i_1\omega}{l}. \tag{3} \]

From the definition of dynamic inductance it follows that

\[ L_{\mathrm{d}}=\lim_{\Delta i\to 0}\frac{\Delta\Psi}{\Delta i} =\frac{d\Psi}{di}. \tag{4} \]

It is thus clear that, in the absence of eddy currents, flux linkage and current are expressed by linear functions, and \(L_g\) assumes a constant value corresponding to the magnitude of the static inductance, which depends only on geometric factors and on the properties of the material. In an alternating magnetic field, when eddy currents occur, one of the functions becomes nonlinear, and the quantity \(L_g\), obviously, is variable and depends on \(\dfrac{d\Psi}{dt}\) and \(\dfrac{di}{dt}\).

It is not difficult to see that if dynamic inductance is regarded as the inductance of a circuit at each instant of time, then its value is variable only while the eddy currents are being established, that is, over the time interval \(T_{\mathrm{v}}\). Subsequently the ratio \(\dfrac{\Delta\Psi}{\Delta i}\) becomes constant and equal to the corresponding ratio in the case of linear functions; i.e., the dynamic inductance of the circuit approaches the static one in magnitude. In general, over the time interval \(T_{\mathrm{k}}\), the inductance of the circuit can be characterized by an equivalent quantity, which also

it is expedient to use it in calculations of electromagnetic processes in commutating sections.

The concept of the equivalent inductance of a section can also be introduced on the basis of considering the mechanism of brush operation during the commutation of a single section. Suppose that, under the action of a constant voltage, the current in the section varies in accordance with straight line 1 in Fig. 3a. In this case the current density at all points of the brush contact arc is the same, and consequently the contact voltage drops are also the same for any form of the brush volt-ampere characteristics, i.e. \(e'_k=\Delta U_c-\Delta U_n=0\) (Fig. 3b).

Fig. 3. On the concept of the equivalent inductance of a section

Under the influence of eddy currents arising when the magnetic flux changes, the current in the section will take the form of curve 2. The current density at the leading edge of the brush will increase in comparison with the current density at the trailing edge, the equality of the transition resistances will be violated, and \(e_k\ne0\) will appear; with the aid of the brush this is introduced into the commutating section, slowing down commutation.

Now suppose that there are no eddy currents, but that the excitation of the commutating poles has increased so much that the current in the section begins to vary along straight line 3. In this case the brush must likewise introduce into the commutated section an emf \(e'_k\). And if we now proceed from the assumption that the brush introduces this emf not over the entire commutation period \(t_k\), but only over the last small interval \(t_{\mathrm{sh}}\), then it can be seen that the operation of the brush will differ little in the case of curve 2 and in the case of straight line 3, since over this interval these curves almost merge. Hence, from the standpoint of the brush’s action in correcting the commutation cycle and, consequently, of the commutation quality of the given section, this section can be characterized by the inductance

\[ L_e=\frac{e_k}{(2i_a+\Delta i)/t_k}, \tag{5} \]

where \(e_k\) is the commutating emf of the commutating poles, necessary to produce the current rate

\[ \frac{2i_a}{t_k}, \]

\(\Delta i\) is the magnitude of the overcommutation current that arises owing to the presence of eddy currents.

On the basis of the propositions set forth, a method was developed for measuring the static, dynamic, and equivalent inductances of the commutating sections of DC machines.

To measure the static inductance, the section is connected to terminals 1 and 2 of the circuit in Fig. 4a. In series with it there is connected a known adjustable static inductance \(L_b\).

Fig. 4. Circuit for measuring static inductance. Curves of the inductive voltages across \(L_x L_b\)

When thyristor \(T\) is opened, a DC voltage from the special generator of power GM, which provides compensation for the active resistance of the circuit, is applied to the series-connected \(L_x\) and \(L_b\). The current in this circuit varies in accordance with the expression

\[ i=\frac{U_0}{L_{x\mathrm{st}}+L_b}\cdot t+\frac{U_0}{R_v}\left(1-e^{-t/\tau_v}\right), \tag{6} \]

where

\(U_0\) is the DC voltage at the output of GM;
\(L_{x\mathrm{st}}\) is the value of the static inductance of the section, corresponding to the inductance \(L\) in Fig. 1a;
\(r_v\) and \(L_v\) are the parameters of the eddy-current circuits in the copper of the slot conductors.

Since the eddy-current field affects only the flux linkage of the slot and does not affect the flux linkage of the coil \(L_b\), while the total voltage applied to them is constant, it is evident that the inductive voltages across \(L_x\) and \(L_b\) are represented by curves 1 and 2, respectively, in Fig. 4b, provided that \(L_{x\mathrm{st}}=L_b\), which can be achieved by adjusting \(L_b\). Straight line 3 corresponds to the voltage on each of the inductances in the absence of eddy currents. After the eddy currents have become established, the voltage \(U_0\) will be divided between \(L_x\) and \(L_b\) and will be maintained constant.

Since the section under investigation is located in the slot and is inaccessible for measuring the inductive voltage drop across it, using the measuring coil \(L_{\mathrm{meas}}\) one can observe curve 2 on the oscilloscope screen. If \(U_0\) is the voltage at the output of GM, \(U_{\mathrm{meas}}\) is the amplitude of the horizontal portion of curve 2, and \(k_w\) is the transformation coefficient between \(L_b\) and \(L_{\mathrm{meas}}\), then, by adjusting \(L_b\), it is possible to achieve

\[ U_{\mathrm{meas}}\cdot k_w=\frac{1}{2}U_0. \]

Then the static inductance of the section \(L_{x\mathrm{st}}\) is equal to the known inductance \(L_b\), which can be calibrated in advance. For these measurements the magnitude of the voltage \(U_0\) is immaterial, since the measurements are made in the steady-state regime.

Using curve 2, the dynamic inductance of the section \(L_{xd}\) can be expressed by the following function of time over the interval \(T_v\):

\[ L_{xd}=L_b\cdot \frac{U_0-U_b(t)}{U_b(t)}, \tag{7} \]

where \(U_b(t)\) is the instantaneous value of the inductive voltage across \(L_b\).

For calculating \(L_{xd}\), the voltage \(U_0\) plays an essential role, since the value of the eddy-current component of the circuit current (formula 6), and consequently the flux linkage of the slot and the voltage \(U_b(t)\), depend on this voltage. Obviously, its value must be specified on the basis that the ratio

\[ \frac{U_0}{L_{x\mathrm{st}}+L_b} \]

must be equal to the rate of change of the section current that it would have in the absence of eddy currents, i.e.

\[ \frac{2i_a}{T_k}. \]

For the further measurement of currents and calculation of the equivalent inductance, \(L_x\) and \(L_b\) are connected in parallel according to the circuit in Fig. 5.

Fig. 5. Circuit for measuring section currents

To preserve the previous rate of rise of the current, \(U_0\) is reduced by a factor of two. If the shunts \(Sh_x\) and \(Sh_b\) are calibrated and the active resistances \(L_b\) and \(L_x\) are equal, then, taking into account the equality of the static inductances, from points 1--2 one can record the curve of variation of the eddy-current component of the section current, which will be equal to the difference \(i_x-i_b\) (curve \(i_2\) in Fig. 2). From shunt \(Sh_b\) the current curve analogous to \(i_1\) in Fig. 2 is recorded; from shunt \(Sh_x\), the current curve analogous to \(i\) in Fig. 2. Having such curves, using formula (5) one can calculate the equivalent inductance and its percentage decrease relative to the static value caused by eddy currents.

Thus, the developed method makes it possible:

1. To measure the static inductance of windings, for example, of a commutating section of a DC machine, even in those cases when the influence of eddy currents is significant.

2. To measure the dynamic inductance of the section at any required instant of time.

3. To measure the exponential component of the section current that appears owing to the presence of eddy currents in the copper of the slot.

4. To calculate the equivalent inductance of the section required for calculating the electromagnetic processes in the section.

References

1. V. V. Pruss-Zhukovskii, M. K. Ziesina, et al. Experimental determination of the parameters of commutated sections of DC machines. Report No. 12--62, Institute of Electromechanics, Leningrad, 1962.

1. A. F. Gritsenko. Measurement of the inductance of commutated sections, Transactions of OMIIIT, vol. 40, 1963.

2. A. V. Ivanov-Smolenskii, A. I. Dulkin. Investigation of the magnetic permeances and inductances of windings of electrical machines and apparatus by the method of modeling on conducting paper. Izvestiya vuzov, “Elektromekhanika,” 1963, No. 10.

3. V. V. Magidson. Measurement and calculation of the inductive parameters of armature sections of a DC machine. Izvestiya vuzov, “Elektromekhanika,” 1963, No. 6.

4. L. R. Neiman and K. S. Demirchyan. Theoretical Foundations of Electrical Engineering, “Energiya,” 1967.

5. Yu. A. Bakhvalov, I. B. Bityutskii. Synthesis of a circuit modeling commutation eddy currents in the armature slot. “Elektromekhanika,” 1967, No. 10.

6. V. T. Frolkhin. Pulse Devices, Moscow, 1966.

7. R. Rüdenberg. Transient Processes in Electric Power Systems. IL, Moscow, 1955.