The simplest rotating dc machine is shown below:
It consists of a single loop of wire rotating about a fixed axis. The rotating part is called rotor, and the stationary part is the stator.
The magnetic field for the machine is supplied by the magnetic north and south poles. Since the air gap is of uniform width, the reluctance is the same everywhere under the pole faces.
If the rotor is rotated, a voltage will be induced in the wire loop. To determine the magnitude and shape of the voltage, examine the figure below:
To determine the total voltage etot on the loop, examine each segment of the loop separately and sum all the resulting voltages. The voltage on each segment is given by eind = (vxB)l
Thus, the total induced voltage on the loop is: eind = 2vBl
When the loop rotates through 180o, segment ab is under the north pole face instead of the south pole face. At that time, the direction of the voltage on the segment reverses, but its magnitude remains constant. The resulting voltage etot is shown below:
There is an alternative way to express the eind equation, which clearly relates the behaviour of the single loop to the behaviour of larger, real dc machines. Examine the figure below:
The tangential velocity v of the edges of the loop can be expressed as v = r. Substituting this expressing into the eind equation before gives:
eind = 2rBl.
The rotor surface is a cylinder, so the area of the rotor surface A is equal to 2rl.
Since there are 2 poles, the area under each pole is Ap = rl. Thus,
eind = (2/).Ap.B.
Since the flux density B is constant everywhere in the air gap under the pole faces, the total flux under each pole is = APB. Thus, the final form of the voltage equation is:
eind = (2/)..
The voltage out of the loop is alternately a constant positive and a constant negative value. How can this machine be made to produce a dc voltage instead of the ac voltage? This can be done by using a mechanism called commutator and brushes, as shown below:
Here 2 semicircular conducting segments are added to the end of the loop, and 2 fixed contacts are set up at an angle such that at the instant when the voltage in the loop is zero, the contacts short-circuit the two segments.
Thus, every time the voltage of the loop switches direction, the contacts also switches connections, and the output of the contacts is always built up in the same way. This connection-switching process is known as commutation. The rotating semicircular segments are called commutator segments, and the fixed contacts are called brushes.
Suppose a battery is now connected to the machine as shown here, together with the resulting configuration:
How much torque will be produced in the loop when the switch is closed? The approach to take is to examine one segment of the loop at a time and then sum the effects of all the individual segments.
The force on a segment of the loop is given by : F = i (l x B) , and the torque on the segment is = r.F sin.
The resulting total induced torque in the loop is: ind = 2rilB.
By using the fact that AP = rl and = APB,
the torque expression can be reduced to: ind = (2/)i.
In this figure:
the armature circuit is represented by an ideal voltage source EA and a resistor RA. This representation is really the Thevenin equivalent of
the entire rotor structure, including rotor coils, interpoles and compensating windings, if present.
The brush voltage drop is represented by a small battery Vbrush opposing the direction of current flow in the machine.
The field coils, which produce the magnetic flux in the motor are represented by inductor LF and resistor RF. The separate resistor Radj represents an external variable resistor used to control the amount of current in the field circuit.
Some of the few variations and simplifications:
The brush drop voltage is often only a very tiny fraction of the generated voltage in the machine. Thus, in cases where it is not too critical, the brush drop voltage may be left out or included in the RA.
The internal resistance of the field coils is sometimes lumped together with the variable resistor and the total is called RF.
Some generators have more than one field coil, all of which appear on the equivalent circuit.
The internal generated voltage is given by: EA = K.. and the torque induced is
ind = K..IA.