Method for Determining the Resultant Armature Reaction of Electric DC Commutator Machines
Abstract Generated abstract
The paper addresses the calculation of resultant armature reaction in DC commutator machines, a quantity important for circuit and transient process analysis but difficult to obtain by existing componentwise or experimental methods. It proposes a computational procedure that derives the resultant armature-reaction flux from catalog characteristics, chiefly the speed versus armature current curve and, where needed, the magnetization characteristic, by comparing the useful flux under load with the flux absent armature reaction. The method also gives corresponding resultant ampere-turns and an equivalent number of armature turns, with a simplified variant for shunt and separately excited motors using ideal no-load speed. The contribution is a more direct way to tabulate the dependence of armature reaction on armature current and to support transient calculations using magnetization-curve partial cycles.
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PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV
Volume 183 1968
METHOD FOR DETERMINING THE RESULTANT ARMATURE REACTION OF DC COMMUTATOR ELECTRICAL MACHINES
A. E. Alekhin, V. A. Beinarovich
(Presented by the scientific seminar of the Department of Electric Drive)
In calculations of electrical circuits with direct-current machines, and especially in calculations of transient processes in them, it is necessary to take into account the resultant armature reaction. Existing methods for determining armature reaction by individual components—longitudinal, transverse, and commutating—are very labor-intensive and cumbersome, or else require the prior acquisition of experimental characteristics [1–9].
In the present work a method is proposed for determining the resultant armature reaction, which consists in the following.
The resultant flux of the armature reaction \(\Phi_{r a}\) is determined as the difference between the flux \(\Phi_0\) in the absence of armature reaction (from the magnetization curve or the ideal no-load condition) and the useful flux \(\Phi_{\mathrm{res}}\), determined for a given operating condition from the characteristic of the dependence of speed on armature current \(n=f(I_a)\).
To determine the dependence of the magnetic flux of the resultant armature reaction \(\Phi_{r a}\) on the armature current \(I_a\), two catalog motor characteristics are used:
1) the dependence of rotational speed on armature current \(n=f(I_a)\), (Fig. 1);
Fig. 1
Fig. 2
2) the machine magnetization characteristic \(\Phi_{\mathrm{res}} = f(Iw_{\mathrm{res}})\) (Fig. 2)—the dependence of the magnetic flux on the magnetizing ampere-turns; moreover, for shunt- and separately excited motors its availability is not obligatory.
For the selected value of the current \(I_{\mathrm{a}}\) in the armature circuit of the motor, the corresponding value of the rotational speed \(n\) is determined from the characteristic (Fig. 1). The counter-e.m.f. of rotation \(E\), corresponding to the obtained value of the speed, is determined from expression (1):
\[ E = U - I_{\mathrm{a}} R, \tag{1} \]
where \(U\) is the voltage applied to the armature circuit of the motor;
\(R\) is the active resistance of the armature circuit;
\(I_{\mathrm{a}}\) is the current in the armature circuit.
The resultant magnetic flux of the motor (taking the armature reaction into account) is determined as
\[ \Phi_{\mathrm{res}} = \frac{E}{c n}, \tag{2} \]
where \(c\) is a constant coefficient, determined from the catalog data of the motor,
\[ c = \frac{U_{\mathrm{n}} - I_{\mathrm{an}} R}{n_{\mathrm{n}} \Phi_{\mathrm{n}}} = \frac{pN}{60a}. \tag{3} \]
In expression (3), \(p\) is the number of pole pairs;
\(N\) is the number of active conductors of the armature winding;
\(a\) is the number of pairs of parallel branches of the armature winding;
\(U_{\mathrm{n}}, I_{\mathrm{an}}, n_{\mathrm{n}}\) are the nominal values of the armature-circuit voltage, armature current, and speed;
\(\Phi_{\mathrm{n}}\) is the magnetic flux of the machine under nominal operating conditions.
The magnetic flux \(\Phi_0\) in the absence of armature reaction is determined from the magnetization curve (Fig. 2), with known excitation ampere-turns, in the general case for a compound-wound motor,
\[ I w_{\mathrm{v}} = I_{\mathrm{v}} w_{\mathrm{sh}} + I_{\mathrm{a}} w_{\mathrm{s}}, \tag{4} \]
where \(I w_{\mathrm{v}}\) is the total excitation ampere-turns;
\(I_{\mathrm{v}}, w_{\mathrm{sh}}\) are the current and turns of the shunt or separately excited field winding;
\(I_{\mathrm{a}}, w_{\mathrm{s}}\) are the current and turns of the series field winding.
From the found values \(\Phi_{\mathrm{res}}\) and \(\Phi_0\), the magnetic flux of the resultant armature reaction \(\Phi_{\mathrm{ra}}\) at the selected value of the armature current \(I_{\mathrm{a}}\) is
\[ \Phi_{\mathrm{ra}} = \Phi_{\mathrm{res}} - \Phi_0. \tag{5} \]
If it is necessary to obtain a graphical or tabular dependence \(\Phi_{\mathrm{ra}} = f(I_{\mathrm{a}})\), the above procedure is carried out for a series of values of the armature current \(I_{\mathrm{a}}\).
By the indicated method, the resultant ampere-turns of the armature reaction \(I w_{\mathrm{a}}^{*}\), or the calculated number of turns of the armature winding \(w_{\mathrm{a}}^{*}\) creating the flux of the resultant armature reaction \(\Phi_{\mathrm{ra}}\), are also very simply determined. The ampere-turns \(I w_{\mathrm{res}}\) creating the magnetic flux \(\Phi_{\mathrm{res}}\) of the machine are determined from the magnetization characteristic by the value of \(\Phi_{\mathrm{res}}\) calculated from expression (2). The ampere-turns of the windings
...of excitation \(I w_{\mathrm{v}}\), for the selected value of the armature current, are determined from catalog data by expression (4).
Then
\[ I w_{\mathrm{a}}^{*}=I w_{\mathrm{res}}-I w_{\mathrm{v}}, \tag{6} \]
and the number of turns is
\[ w_{\mathrm{a}}^{*}=\left|\frac{I w_{\mathrm{res}}-I w_{\mathrm{v}}}{I_{\mathrm{a}}}\right|. \tag{7} \]
Determination of the dependence of the magnetic flux of the resultant armature reaction on the armature current for a motor with separate and shunt excitation can be carried out using only the motor characteristic \(n=f(I_{\mathrm{a}})\) (Fig. 1, b) and catalog data. The simplification consists in determining \(\Phi_0\), whose value for a motor with shunt or separate excitation can be determined as
\[ \Phi_0=\frac{u}{c n_0}. \tag{8} \]
where \(n_0\) is the speed of the ideal no-load operation of the motor.
Use of the proposed method for determining the resultant armature reaction makes it possible, by a very simple computational procedure, to determine the dependence on armature current of one of the components of the armature reaction from the known remaining components. In addition, this method, in comparison with others, considerably simplifies the determination of the armature reaction or its components when calculating transient processes by partial cycles of the magnetization curve.
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