Solution: The force per unit length on a wire carrying a current of I amperes can be determined by multiplying Eq. (3.4) by cross-sectional area of the wire. As the product of the current density and the cross-sectional area of the wire is simply the current I, the force per unit length acting on the wire is given as
Thus, for wire 1 carrying a current of I amperes into the paper, the θ-directed force is given as
And for wire 2 located 180° away from wire 1 and carrying a current of I amperes out of paper is given as
where l is the length of rotor in metres. The torque acting on the rotor is given by the sum of the force –moment-arm products for each wire i.e.
3.3 ENERGY BALANCE
Since, for all practical purposes, the mass of the materials used in the construction of an electrical machine remains constant under the conditions of operation, the principle of conservation of energy can be applied in the analysis of the energy conversion. The input energy must, therefore, be equal to the summation of the useful output energy, the energy converted into heat, and the change in the energy stored in the magnetic field. Thus energy balance equation may be written as
Equation (3.7) is applicable to all conversion devices; it is written so that the electrical and mechanical energy terms have positive values for motor action. The equation equally well to the generator action; these terms then simply have negative values. For generator action
Irreversible conversion of energy to heat arises due to three reasons; part of electrical energy is converted directly to heat in the resistance of the current paths, part of the mechanical energy developed within the device is absorbed in the friction and windage and converted into heat, and part of the energy absorbed by the coupling field is converted into heat in magnetic core loss (for magnetic coupling) or dielectric loss (for electric coupling). If the energy losses in the electrical system, the mechanical system, and the coupling field are grouped with the corresponding terms in Eq. (3.7), the energy balance equation may be rewritten as
The left-hand side of Eq. (3.8) can be expressed in terms of the currents and voltages in the electric circuits of the coupling device. Consider, for example, the energy-conversion device shown schematically in