Energy in Terms of Electrical Parameters
In the preceding article, the energy and force were related in terms of magnetic-system parameters, namely flux and mmf, through the third parameter, the permeance.
It is at times convenient to relate these things in terms of electrical-system-parameters, namely, the inductances and currents. That is being dealt with here only for linear systems. Let be the permeance of the magnetic circuit and L be the coil-inductance.
Thus, a force exists if the coil-inductance is dependent on x. Such analysis is more suitable when the system has more than one coils coupled through the magnetic circuit. If two such coils are considered, following data should be known for evaluation of the force, in case of linear displacement:
From the right-hand side of this equation, it is noted that the inductance-term which is dependent on x contributes to the force.
Rotary Motion
Most popular systems for electro- mechanical energy are Generators and Motors. The preceding discussion dealt with the Linear motions, wherein x represented the displacement parameter, and force was being calculated.
Now we shall deal with the rotary systems, wherein angular displacement parameters (such as q) and corresponding torque developed by the system will be correlated, through a systematic procedure for a typical rotary macRhointaer.y motion
Description of Simple System
A simple rotary system has a ‘stator’ and a ‘rotor’. Air-gap separates these two. Stator has two similar coils ‘a’ and ‘b’ located at 900 electrical, with respect to each other. Inner surface of the stator is cylindrical. Outer surface of the rotor is also cylindrical resulting into uniform air-gap length for the machine.
The diagram represents a two- pole machine. Axis of coil ‘a’ may be taken as reference, with respect to which the rotor-coil axis makes an angle of q, at a particular instant of time. For a continuous rotation of the rotor at w radians / sec, q = wt. Coil- ‘b’-axis is perpendicular to the reference, as shown. Due to the uniform air-gap length, and due to the perpendi-cularity between coils ‘a’ and ‘b’, inductance-parameters exhibit the following patterns:
Let x represents the inductance parameter as a function of q. The subscripts indicate the particular parameter. xaa = self-inductance of coil ‘a’ , xab = mutual inductance between coils a and b and so on. x represents value of the particular inductance parameter, which will help in knowing the variation of inductance with q.
Energy stored in the coils
Energy stored in the magnetic field can either be expressed in terms of mmf and flux or be expressed in terms of inductance-terms and coil currents. If ia, ib and ir are the coil-currents, stored- energy-terms are as given below:
If ia, ib, ir are assumed to be constant currents, for simplicity, so that their derivatives with respect to q (and hence with respect to time t) are zero, the energy-terms which include constant inductances do not contribute to torque. W1, W2, and W3 thus cannot contribute to torque. W4 and W5 contribute to torque related by:
Different Categories
From the torque expressions above, it is clear that the torque exists only when stator and rotor-coils carry currents. When only stator-coils (or only rotor coil) carry current, torque cannot be produced.
(a) One coil each on Stator and on Rotor
In the above mentioned case, let us excite only one stator-coil. Let ia = is, ib = 01 and ir maintained as before.
Magnitudes of these terms are maximum for q = 900. If q can be set at 900, at all instants of time, torque obtained is maximum. Such a situation does exist in a d. c. machine in which rotor carries an armature winding which is a lap- or wave-connected commutator winding. The brushes are so placed on the commutator that rotor-coil-axis satisfies the above- mentioned condition of q = 900, irrespective of the rotor-position or rotor speed. Such an equivalence of a rotating armature coil with such an effectively stationary coil is referred to as a quasi-stationary coil. It means that a rotating coil is being analyzed as a stationary coil due to its typical behaviour for electro-mechanical energy conversion purposes.
(b) Two stator coils carrying two-phase currents and rotor-coil carrying d. c.: When two stator coils carry two-phase alternating currents, a synchronously Permanent magnet synchronous motor for washing machine rotating mmf is established. If the rotor-coil carries direct current, and the rotor is run at same synchronous speed, a unidirectional constant torque is developed. Mathematically, similar picture can be visualized, with a difference that the total system is imagined to rotate at synchronous speed. Such a machine is Synchronous machine, (to be discussed in Later chapters). It can be understood through the simple system described here.
(c) Machines with Permanent Magnets.
With suitable interpretation, the field side of the simple system can be imagined to be with permanent magnets in place of coil-excited electromagnets. All the interpretations made above are
valid, except for the difference that in this case there is no scope for controlling the rotor-coil- current-magnitude.
(d ) Machines with no rotor coil, but with premeance variation.
Smooth cylindrical rotor surfaces do not exist in such cases. There are no rotor-coils. Due to geometry of the rotor surface, stator-coil- self inductances vary with rotor position. Thinking on lines of relating energy terms and their derivatives for torque-calculations, the working principles can be understood. With simple construction, Reluctance motors belong to this category.
(e) Switched currents in Stator Coils.
In yet another type, stator coils aredistributed and properly grouped. One group carries currents during certain time interval. Then, this current is switched off. Another group carries current in the next time interval and so on. The rotor surface is so shaped that it responds to this current switching and torque is produced. Even though stator-coil- inductances are complicated functions of rotor position, the method of analysis for such machines is same. Prominent types of Current in stator coils machines of this type are: switched reluctance motors, stepper motors, etc.
Vital Role of Air-gap
Magnetic circuit of an electrical machine has a flux established due to coil-mmfs. This flux is associated with stator core, rotor core and air-gap. An important point for understanding is to know which out of these three stores major portion of the field energy. Through an illustrative case, it will be clear below, in example 25.1.
Example 25.1. Let a machine with following data be considered.
Solution.
(c) Repeat (b) above based on force-calculations and mechanical displacement.
(d) What will be change in above results of mechanical work done, if the mechanical movement is fast, keeping the flux initially constant ?
Solution.
It indicates that with fast movement, the electrical energy-input and the field-stored energy have decreased by 0.157 J each but the mechanical-energy-term remains unaffected by fast or slow movements of armature.