Question 10.2: A permanent magnet dc motor is being used to drive a mechani...

Asimplified schematic of the dc motor is shown in Fig. 10.11. The armature winding (only one turn of the winding is shown in Fig. 10.11) is placed within a uniform magnetic field of flux density B. The resistance of the winding is R and its inductance is L .When the input voltage e_{i}    is applied, current i flows in the armature winding. Under these conditions (a current-carrying conductor within a magnetic field), force F_{e}   is exerted on both the top and the bottom sections of the armature winding as illustrated in Fig. 10.11. The force F_{e}  is given by Eq. (10.4),

where l is the length of the armature coil. Because the winding is free to rotate around its longitudinal axis, force F _{e}   produces a torque:

T_{e} = F_{e} r  . (10.19)

This torque acts on each (top and bottom) section of each turn in the armature winding.

Assuming that there are N turns in the winding and they are all within the uniform magnetic field, the total induced torque exerted on the armature winding is

T_{e} =2N F_{e} r  . (10.20)

When Eq. (10.20) is compared with the equation for a rotational–electromechanical transducer given in Fig. 10.2, the coupling coefficient for the dc motor can be identified as

\alpha_{r}=\frac{1}{2NlBr}   , (10.21)

and hence the electrically induced torque can be expressed as

T_{e} =\frac{1}{\alpha_{r} } i  . (10.22)

When the armature winding starts to rotate within a uniform magnetic field, voltage e _{m}   is induced in the winding, given by Eq. (10.6):

e_{m}= vBl  .

Replacing translational velocity v  with the rotational velocity Ω times radius r and accounting for N turns and two sections of each turn in the armature winding, one finds that the total voltage induced is

e_{m}= 2N\Omega rBl    (10.23)

or

e_{m} =\frac{1}{\alpha_{r} } \Omega       (10.24)

where \alpha_{r} is the coupling coefficient defined by Eq. (10.21). Equations (10.22) and (10.24) describe mathematically the coupling between mechanical and electrical parts of the system and are often called the coupling equations.