1
Introductory Concepts

1.1 Introduction

This chapter is a review for most since the material is covered in undergraduate courses in the analysis of electromechanical devices [1]. The material is presented to start everyone with the same background. The chapter begins with coupled circuits (transformers) where the phasor equivalent circuit is established. Since phasors are not always taught the same, they are covered briefly in Appendix B to make sure everyone understands the concept of phasors as used in this text. Although we will give several approaches for the calculation of torque of electric machines; Section 1.1-3 sets forth a method of calculating force and torque that is generally taught at the undergraduate level.

Some instructors may choose to skip some material and/or select topics that were not covered in undergraduate courses at their school. As mentioned, the material will be a review for most and can be covered rather fast. On the other hand, Chapter 2 dives into machine analysis that contains new material and can be taught at a much slower pace.

1.2 Stationary Magnetically Coupled Circuits

Magnetically coupled electric circuits are central to the operation of transformers and electromechanical motion devices. In the case of transformers, stationary circuits are magnetically coupled for the purpose of changing the ac voltage and current levels. The two windings shown in Fig. 1.2-1 consist of turns N1 and N2, and they are wound on a common core, which is a ferromagnetic material with a permeability large relative to that of air. The magnetic core is illustrated in two dimensions.

Figure 1.2-1 Magnetically coupled circuits.

The flux produced by each winding can be separated into two components: a leakage component denoted by the subscript l and a magnetizing component denoted by the subscript m. Each of these components is depicted by a single streamline with the positive direction determined by applying the right-hand rule to the directions of current flow in the winding. The leakage flux associated with a given winding links only that winding, whereas the magnetizing flux, whether it is due to current in winding 1 or winding 2, links both windings.

The flux linking of each winding may be expressed as

The leakage flux Fl1 is produced by current flowing in winding 1, and it links only the turns of winding 1. Likewise, the leakage flux Fl2 is produced by current flowing in winding 2, and it links only the turns of winding 2. The flux Fm1 is produced by current flowing in winding 1, and it links all turns of windings 1 and 2. Similarly, the magnetizing flux Fm2 is produced by current flowing in winding 2, and it also links all turns of windings 1 and 2. Both Fm1 and Fm2 are called magnetizing fluxes. With the selected positive directions of current flow and the manner in which the windings are wound, the magnetizing flux produced by positive current flowing in one winding can add to or subtract from the magnetizing flux produced by positive current flowing in the other winding. Thus, the mutual inductance can be positive or negative. In Fig. 1.2-1, it is positive.

It is appropriate to point out that this is an idealization of the actual magnetic system. It seems logical that all of the leakage flux will not link all the turns of the winding producing it; hence, Fl1 and Fl2 are "equivalent" leakage fluxes. Similarly, all of the magnetizing fluxes of one winding may not link all of the turns of the other winding.

The voltage equations may be expressed as

In matrix form,

The resistances r1 and r2 and the flux linkages ?1 and ?2 are related to windings 1 and 2, respectively. Since it is assumed that F1 links the equivalent turns of winding 1 (N1) and F2 links the equivalent turns of winding 2 (N2), the flux linkages may be written as

where F1 and F2 are given by (1.2-1) and (1.2-2), respectively.

If we assume that the magnetic system is magnetically linear (i.e., core losses and saturation are neglected), we may apply Ohm's law for magnetic circuits to express the fluxes. Thus, the fluxes may be written as

where k = 1 or 2 and and are the reluctances of the leakage paths, and is the reluctance of the path of magnetizing fluxes. Typically, the reluctances associated with leakage paths are much larger than the reluctance of the magnetizing path. The reluctance associated with an individual leakage path is difficult to determine exactly, and it is usually approximated from test data or by using the computer to solve the field equations numerically. On the other hand, the reluctance of the magnetizing path of the core shown in Fig. 1.2-1 may be computed with sufficient accuracy.

For the iron

where li is the length of the path in iron, µr is the relative permeability of iron, µ0 is the permeability of free space, and Ai is the cross-sectional area of the flux in the iron. In electromechanical devices, we will find that the magnetizing flux must transverse air gaps and

Substituting (1.2-8) and (1.2-9) into (1.2-1) and (1.2-2) yields

Substituting (1.2-12) and (1.2-13) into (1.2-6) and (1.2-7) yields

When the magnetic system is linear, the flux linkages are generally expressed in terms of inductances and currents. We see that the coefficients of the first two terms on the right-hand side of (1.2-14) depend on N1 and the reluctance of the magnetic system, independent of the existence of winding 2. An analogous statement may be made regarding (1.2-15) with the roles of winding 1 and winding 2 reversed. Hence, the self-inductances are defined as

where Ll1 and Ll2 are the leakage inductances and Lm1 and Lm2 are the magnetizing inductances of windings 1 and 2, respectively. From (1.2-16) and (1.2-17), it follows that the magnetizing inductances may be related as

which is .

The mutual inductances are defined as the coefficient of the third term on the right-hand side of (1.2-14) and (1.2-15). In particular,

We see that L12 = L21 and, with the assumed positive direction of current flow and the manner in which the windings are wound as shown in Fig. 1.2-1, the mutual inductances are positive. If, however, the assumed positive directions of the current or the direction of the windings were such that Fm1 opposed Fm2, then the mutual inductances would be negative.

The mutual inductances may be related to the magnetizing inductances. Comparing (1.2-16) and (1.2-17) with (1.2-19) and (1.2-20), we see that

The flux linkages may now be written as

where L11 and L22 are defined by (1.2-16) and (1.2-17), respectively, and L12 and L21 by (1.2-19) and (1.2-20), respectively. The self-inductances L11 and L22 are always positive; however, the mutual inductances L12(L21) may be positive or negative, as previously mentioned.

Although the voltage equations given by (1.2-3) and (1.2-4) may be used for purposes of analysis, it is customary to perform a change of variables that yields the well-known equivalent T circuit of two windings coupled by a linear magnetic circuit. To set the stage for this derivation, let us express the flux linkages from (1.2-22) and (1.2-23) as

With ?1 in terms of Lm1 and...