Theory & Practice of Electromagnetic Design of DC Motors & Actuators
George P. Gogue & Joseph J. Stupak, Jr.
6.1 Force production:
In the most basic form, two current carrying circuits can experience mutual forces. A current carrying conductor in a magnetic field experiences a field too. Torque (T) is determined from force (F) with the following relationship:
where r is the length of arm or distance from the center of action of the force.
Force F on a closed conducting surface can be calculated from the following:
where B is the flux-density, l is the wire length and I is the current flowing in that wire. Refer to Section 4.1 for further discussion.
6.1.1 Forces between a conductor and steel:
For a current carrying conductor embedded in iron, as shown in Figure 6.1, it is forced further into the iron, away from the surface. If that conductor is in the air, it is then attracted to the iron surface. The force per unit length acting on the conductor is:
where h is the distance between conductor and surface. The force will be zero if that conductor is halfway between two iron surfaces, (Reference 10 & 17).
For a conductor in a deep parallel-sided slot, the force attracting the conductor to the bottom of the slot is:
where l is the width of the slot and d, the distance to the bottom of the slot, as shown in Figure 6.2.
The forces on the sides of a deep parallel-sided slot and on the sides of an air-gap can be determined using Maxwell stresses, as shown below.
Along DA and D'A' in Figure 6.3, the force of attraction per unit area towards surface FF' is:
Along AE and A'E', the force is:
Where and are the values of the magnetic field at those surfaces. The resultant of these 2 forces is:
which when integrated along the whole surface simply results in:
This simplified analysis shows that the force is actually developed on the sides of the slot rather than directly on the conductor. This explains the paradox that the conductor being in the slot can only see a very small amount of flux to explain the forces (and torque) generated by motors, (Reference 10).
An alternative physical interpretation can also be used in the generation of force. The main magnetic field causes and at the iron surfaces. The current carrying conductors, however, while carrying current, weaken the field on one side of the slot and strengthen it on the other. The net result is a pull in one direction resulting in rotational torque, (Reference 18). Refer to Section 1.8 for a similar discussion.
6.2 Energy considerations:
A current carrying coil can be broken into a number of small loops of square areas. Each loop is rotated until its plane is parallel to the total field and no flux is linked. The potential energy of a current carrying circuit in a magnetic field can then be determined from the work required to reassemble the original circuit.
If a current loop is moved, the flux linked with it is reduced by d and the potential energy is increased by Id. The emf E is, therefore:
The energy drawn from the power source is then reduced by EIt. To maintain the current from the source at a constant value, extra energy must be supplied in time t. Integrating the increments of current over the whole time interval results in:
This indicates that half the energy withdrawn from the source is used for mechanical work and half is stored in the field.
6.2.1 The force equation:
The change in stored magnetic energy is, (Reference 19):
where is the electromagnetic force and the flux linkage.
If the variables i and X are considered independent variables, then
6.3 Torque balance equation:
The general equation for torque balance at the motor shaft is (Reference 20)
The stability or instability of the operating points depends on the motor and load speed torque characteristics. Figure 6.4 shows an example of a brushless dc motor and a fan load.
6.3.1 Dynamic determination of torque:
From Equation 6.14 and from the acceleration/deceleration curves, the torque versus current (or excitation) of the motor can be determined, (Reference 17).
and A can be determined at any value of speed S after determining the rate of deceleration S/t from the test.
During deceleration, S/t is determined and the torque at a particular speed can be determined from the above, Equation (6.16).
6.3.2 Torque development by 2 fields:
Torque can be developed as a mutual reaction between two fields under one or more of the following conditions, (Reference 21):
b. The angle between the 2 field vectors must be more than zero.
c. The two fields must change sign simultaneously if they are alternating fields.
d. At least one field must exist independently. The other can also be independent, e.g. a permanent magnet, or dependent, e.g. the rotor field in an induction motor.