DC machine commutation
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Figure1
Introduction
Recently powerful direct current (DC) machines were widely used in electric drives for vehicles, vessel propulsion systems and for steel rolling mills. Fast development of alternating current drives, including thyristor or transistor inverters gradually decreases the range of application of DC machines.
Still it seems these machines will be used for certain time period in low-powered equipment because of their ability to ensure simple control of motor speed, low cost and high reliability. It is even possible to create a very simple DC drive with the aid of a motor, power source and rheostats for excitation and armature current control.  There exist DC motors which include permanent magnet excitation instead of the electromagnetic one. It should be noted that so far direct current generators have been almost out of use so only DC motors could be of interest in the nearest future.
The most problematical part of a DC motors is its mechanical commutator.
Figure 1 shows  a simple DC machine without commutating poles which is typical for low-powered  motors. When the motor rotates clockwise its armature coils reach the commutation zone and the current in these coils changes its direction. Several coils are being short-circuited and the current arising in them creates the so-called commutating armature reaction. The corresponding magnetomotive force may significantly affect the total magnetic flux in transient modes taking place in DC machines, particularly at high armature currents.
Commutation mode
In the ideal case commutation is considered linear. Let us assume that there in no electromotive force (e.m.f.) in the commutating armature coils. This may take place if the rotor rotates very slowly. While the coil passes by the brush the current flowing through the coil changes from + Ia to -Ia. Such a commutation is also called the straight-line one. Usually several commutator segments are short-circuited by brushes and this is characterized by the so-called brush overlapping coefficient.

This coefficient is equal to
γ = b
br /bc,                                                                                                       (1)
b
br is equal to the brush width,
b
c is the distance between the centers of the adjacent commutator segments, that equals
b
к = πDк /K,                                                                                                     (2|)     Dк is the commutator diameter,
K is equal to the total number of the commutator segments.
As it has already been noted by L.Dreyfus the average speed of the current alteration related to all commutating machine coils does not depend on the brush width and is only determined by the current value and the commutator speed. This thesis is considered fundamental for the commutation theory.
The real commutation differs from this ideal case (Curve 1), as shown in Figure 2. Curve 2 represents the delayed commutation, and Curve 3 shows the case of the over commutation. It is recommended to adjust machine parameters in such a way that slightly over commutation is obtained. This way enables us to almost suppress sparks arising on the leaving edge of a brush.
Reference list
1. Modern numerical methods for ordinary differential equations. Edited by G.Hall and J.M.Watt. Clarendon Press. Oxford. 1976.
2. S.S. Abramov. Development of a method of electric machine regimes analysis working in an autonomous power system with an electromechanic converter: Thesis. Saint Petersburg Technical University. 1995. 233 pages.