1 Introduction and review of basic theory 1.1 Aim of the book On entering the world of electrical machines, the student meets many conceptual difficulties not experienced for example in the early studies of digital systems, with their simple and precise 2-state operation. More assistance is required to permit the new-comer to gain confidence in dealing with non-linear, 3-dimensional, rotating electromagnetic devices. The purpose of this book is to provide this aid to understanding by showing how, with a limited number of equations derived from basic considerations of power flow and elementary circuit and electromagnetic theory, the electromechanical performance can be explained and pre- dicted with reasonable accuracy. Such an aim, which will permit the calculation of power-input/output c haracteristics almost close enough in engineering terms to those of the device itself, can be achieved by representing the machine as a simple electrical circuit - the equivalent-circuit model. This concept is explained in many books, for example in the author's companion volume Electrical Machines and Their Applications. Though more detailed theoretical treat- ment is given there, substantial portions of the present text may be regarded as suitable revision material. This expanded 3rd edition can, as a whole, be considered as a textbook with particular, but not exclusive, emphasis on Electrical Drives, taught through worked examples, for a reader having some familiarity with basic machine theory. Perhaps it is appropriate to point out that complete and exact analysis of machine performance is so complex as to be virtually impossible. The additional accuracy achieved by attempts to approach such methods is primarily of interest to the specialist designer who must ensure that his 2 Electrical Machines and Drive Systems (1.1) product will meet the user's needs without breakdown and he must judge when the analytical complication is justified. For the user, and for the engineering student who is not yet a specialist, the simpler methods are adequate for general understanding and provide a lead-in if necessary for later specialisation. There are many features of all machine types which are common, the obvious example being the mechanical shaft equations. But apart from these and the fundamental electromagnetic laws, the input/output relationships and modes of operation have many similarities. These are brought together where possible and also in this first chapter, some elementary mechanical, magnetic and circuit theory is discussed briefly, as a reminder of the basic knowledge required. Students should beware of underestimating the vital importance of this material, since experience shows that it is these very points, improperly understood, which hold back progress in coming to feel at ease with machines problems. However familiar one may become with theory, as a student, the true test of an engineer is his ability to make things work. First steps to this goal of confidence are reached when a student is prepared to commit himself to selecting equations and inserting values in the algebraic expressions, producing answers to a specific problem. Hence the importance of practice with numerical examples. Understanding grows in proportion to one's ability to realise that the equations developed really can be used in a systematic fashion to solve such problems, since they describe the physical behaviour in mathematical terms. Appreciation of this last statement is the key to successful problem-solving. The chapters are planned to sequence the examples at increasing levels of difficulty. Much theoretical support is given, in that the equations are discussed either at the beginning of each chapter, or as the need arises. Solution programmes indicate the kind of problems which can be formulated for the three basic types of rotating machine: d.c., induction, and synchronous. Readers are encouraged to adopt an ordered approach to the solution; for example it is a good idea to incorporate the question data on a diagram. One of the difficulties of machines problems often lies in the amount of data given. By putting the values on a simple diagram, assimilation is easier and it helps to avoid mistakes of interpretation, especially when working with 3-phase circuits. In following this recom- mended pattern, it is hoped that the text will help to remove the mystery with which some students feel the machines area is shrouded. The emphasis is on machine terminal-characteristics, rather than on the internal electromagnetic design. In other words, the electrical-drives aspect is uppermost since this is the area in which most engineering students need to have some good knowledge. It is worth noting that about 60-70% of all electrical power is consumed by motors driving mechanical shafts and virtually all this power is produced by generators driven through (1.2) Introduction and review of basic theory 3 mechanical shafts, so that the subject is of considerable importance to engineers. The problems and solutions are discussed where appropriate, to draw out the engineering implications. Electromechanical transients, stability and control are not neglected and opportunity is also taken to consider the effects introduced by die impact of power-electronic circuits, so often intimately associated with machine control. In general, the usual methods of analysis are still reasonably effective in predicting machine performance. Full account of the influence of this important environment, in which harmonics proliferate, is a somewhat specialised topic but some indication is given of the means used to deal with the machines problems which arise. Detailed study of machine/semiconductor systems requires the use of madiematical and computer simulation procedures, which have tended to become the province of those who market commercial computer software packages. However, Chapter 8 considers diis topic in sufficient depth to provide a better understanding of such investigations. Finally, in Appendix D, some tutorial examples are given along with the answers. Some of the worked examples in the text have been taken from Appendix E of Electrical Machines and Their Applications, but many of these remain, as further exercises for the determined student. 1.2 Foundation theory Excitation calculations Virtually all machines have iron in the magnetic circuit to enhance the flux value and/or to reduce the excitation requirements. The price to pay for tiiese advantages is reflected in iron loss and non-linearity. Figure l.la shows a typical iron magnetisation-characteristic. The economic operating point is beyond the linear region and well clear of full saturation, at about B--1 tesla, though certain short parts of the magnetic circuit, like armature teeth, may exceed this by 50% or more. Under transient conditions too, this limit can be exceeded. The equation governing the excitation requirements follows from: Multiplying by area A: In words: Flux = Magnetomotive force X Permeance (or I/Reluctance) <t> (= BA) = F(= IN) X A(=pA/Q 4 Electrical Machines and Drive Systems (1.2) Figure 1.1 Magnetic excitation. The m.m.f. is shown in ampere turns (At) (turns N being dimensionless) but is effectively the current enclosing the magnetic circuit. The calculation of excitation m.m.f. (F) is often required for a given flux and magnetic geometry, to determine the design of the coils. Frequently there are two (or more) such coils so that the resultant excitation F is the r combination of FI and F which produces <J> , see Figure l.lb. The two 2 m m.m.f.s may be produced on opposite sides of a machine air gap; /\ say, due to several stator coils, while F similarly may be due to several rotor 2 coils. Often, sinusoidal distribution of m.m.f. is assumed and the coils can be designed to approach this closely. 'Vector' techniques can then be used to combine these two 'sinusoidal quantities' giving F! + F = F and 0 , the 2 r m mutual flux = function (F). However, m.m.f. is not a vector but a scalar, so r a different term, space phasor, is becoming accepted as an appropriate designation for such representations of sinusoidal space variations. It is sometimes convenient to take the positive magnetising senses of Fj and F 2 to be in the same direction, though in practice, the one is usually magnetising in the opposite sense to the other and would then be negative with respect to this. Electromagnetic theory The most important equations for present purposes are: e = Nd<j>/dt; e = Blv; and Force = Bli; most practical machines having the directions of B, v and f at right angles to one another. (1.2) Introduction and review of basic theory 5 For a fixed magnetic geometry: where: and will fall with the onset of saturation, so the inductance L is flux/ current dependent. For a sinusoidally time-varying current: then: and in r.m.s. and complex-number expressions: and I lags V by 90°. These quantities are scalars but their sinusoidal variation can be represented by time phasors, see Figure 1.2. The word phasor alone will often be used in the text as an abbreviation for time Figure 1.2 Induced voltage (back e.m.f.). phasor. The use of the back e.m.f. expression (+L di/dt) instead of the forward e.m.f. expression (—L di/dt) is seen to be preferable, since the current I comes out direcdy as lagging V by 90° for the inductive circuit, instead of having to deal with the concept of two identical but phase- opposed voltages. For the general case with varying geometry, e = d(Li)/dt - L di/dt (transformer voltage) + i dL/dt (motional voltage). Circuit-theory conventions Figure 1.3a shows a representation of a machine with its instantaneous e.m.f. and resistive and inductive voltage-drops. The voltage arrowheads 6 Electrical Machines and Drive Systems (1.2) Figure 1.3 Motor conventions. are the assumed +ve ends. The directions of the arrows for the instantaneous terminal voltage v and for e may be assigned arbitrarily but Ri and L di/dt must oppose z, since the voltage arrowheads must be positive for +ve i and +ve di/dt respectively. The direction of i may also be assigned arbitrarily but the decision has consequences when related to the v and/or e arrows. As shown, and with all quantities assumed to be +ve, then the machine is a power sink; i.e. in a MOTORING mode; the vi and ei products are both positive. For GENERATING, when the machine becomes a power source, ei will then be negative; e or i reversed. The above is called the MOTORING convention and it is often convenient in electrical-drives studies to use this throughout and let a negative ei product indicate a generating condition. Alternatively, a GENERATING convention could be used, as sometimes preferred in power-systems studies. By reversing the i arrow say, ei would then be positive for generating and the circuit equation would have a sign reversed. It would be a good check to complete the following short exercise to see if the above statements are properly understood. Write down the MOTOR with MOTOR V = E RI equation; conventions: Write down the GENERATOR with MOTOR V = E RI equation; conventions: Write down the GENERATOR With GENERATOR V = E RI equation; conventions: Write down the MOTOR With GENERATOR V= E RI equation; conventions: (1.2) Introduction and review of basic theory 7 The mechanical equation can be expressed as a simple extension of the above. The motor (as a mechanical power source) produces (generates) an electromagnetic torque 7^, and in equilibrium at steady speed, this is balanced by the total mechanical torque T , part of which is due to the m internal mechanical resistance TJ and the remainder is the load torque OSS at the coupling 'terminals', 7^. . oupling So: T = T = r + r (cf. electrical source, E = V+ RT). e m coupling loss This is also a MOTORING convention. For a generator, with rotation unchanged, both T and T would be negative using this e coupling convention. To illustrate how these conventions affect the machine considered as a system, with electrical-power terminals and mechanical-power 'terminals' - excluding for the moment the control-power terminals - consider Figure 1.3b. In general, either or both terminal powers can be negative and here, a motoring convention is being considered. The three practicable conditions are: Electrical power Mechanical power MOTORING (A) positive; positive; Electrical power; Mechanical power GENERATING (B) negative; negative; Electrical power Mechanical power BRAKING (C) positive; negative; In the last mode, it will be noticed that both mechanical and electrical 'terminals' are accepting power into the machine system. All the power is ir fact being dissipated within this system, which may include resistance L external to the machine itself. The mechanical power is usually coming from energy stored in the moving parts, and since this cannot be released without a fall of speed, the action is one of braking. The machine is generating; not feeding power into the electrical supply, but assisting this to provide the power dissipated; see Section 3.5. To understand how the mechanical 'terminals' respond to these three modes, assume that Ti is 1 unit and T is 10 units. Let the speed be oss e positive and remembering that power is (torque X speed), use the mechanical balance equation to find: T ^ T — T ^coupling *e •'loss Mode A; Motoring r = 10 - 1 = +9. (O T +ve. coupling m coupUnK 8 Electrical Machines and Drive Systems (1.2) Mode B; Generating T = -10-1 = - 11. <y r -ve. coup]ing m coupling [7^. will be -ve for +ve o> ] m Mode C; Braking (i) T = -10 - 1 = -11. <w r -ve. coupling m coupling [o) +ve. /. T = +1 m Joss T will be -ve] e Mode C; Braking (ii) r = +10- (-1) = +11. co T -ve. coupiing m coupling [o) -ve. /. T = -1 m loss 7^ will be +ve] Note that if rotation reverses, 7"i will reverse because it always opposes oss rotation. In mode C, the sign of T is opposite to that of co because the c m machine itself is generating, so for either rotation, the mechanical 'terminal' power, w 7; , is negative. m oupling Sinusoidal a.c. theory Most a.c. sources are of nominally constant r.m.s. voltage so the voltage phasor is taken as the reference phasor. It need not be horizontal and can be drawn in any angular position. A lagging power factor cos <p means current lagging the voltage as shown on Figure 1.4a. The instantaneous power vi, which pulsates at double frequency, is also shown and has a mean value of V7 cos <p. If <p were to be greater than 90°, the power flow would have reversed since /cos (p would be negative as seen on the phasor diagram for a current /'. Note that the phasor diagrams have been drawn at a time <vt = IT/2 for a voltage expressed as v= Vsin cat. For the reverse power-flow condition, if the opposite convention had been chosen (with v or i reversed), then VT cos <p would have been positive. This is shown on Figure 1.4b where it will be noted that the current is at a leading power factor. Taking Figure 1.4a as a motoring condition, it shows electrical power being absorbed at lagging power factor Figure 1.4 Power flow in single-phase a.c. circuit. (1.2) Introduction and review of basic theory 9 whereas Figure 1.4b shows electrical power being delivered at a leading power factor. Phasor diagram including machine e.m.f.; motoring condition The equation, allowing for inductive impedance, is: V = E + W+jXl, arid is shown as a phasor diagram on Figure 1.5 for two different values of E. Note that on a.c., the e.m.f. may be greater than the terminal voltage V arid yet the machine may still operate as a motor. The power factor is Figure 1.5 Phasor diagrams. affected but the power flow is determined by the phase of E with respect to V. Frequently, the current is the unknown and this is found by rearranging the equation as: N.B. <p will normally be taken as -ve for lagging power factor. The appropriate exercise to check that these phasor diagrams are understood is to draw the corresponding diagrams for a generator using (a) motor conventions and (b) generator conventions. 10 Electrical Machines and Drive Systems (1.2) Meaning ofVxl components Multiplying the /, /cos (f> and /sin <p current phasors by Vgives: VI (Svoltamperes, VA) VI cos <p (P watts) and V7sin^> (Qvoltamperes reactive, VAr) and a 'power phasor diagram' can be drawn as shown on Figure 1.5c. Power devices are frequently very large and the units kVA, kW and kVAr, (X 103), and MVA, MW and MVAr, (X 106), are in common use. The largest single-unit steam-turbine generators for power stations are now over 1000MW= 1GW, (109W). 3-phase circuit theory For many reasons, including efficiency of generation and transmission, quite apart from the ease of producing a rotating field as in any polyphase system, the 3-phase system has become virtually universal though there are occasions when other m-phase systems are used. For low powers of course, as in the domestic situation for example, single-phase supplies are satisfactory. For present purposes consideration will only be given to balanced 3-phase circuits, i.e. where the phase voltages and also the phase currents are mutually displaced by 120 electrical degrees (2ir/3) radians). Electrical angles are given by a) - 2ir/£ radians. t On the assumption of balanced conditions, the power in a 3-phase system can be considered as available in three equal power 'packages', each handling 1/3 of the total power, i.e. where <p is the same for each phase. The pulsating components of power cancel, giving steady power flow. There are two symmetrical ways of connecting the three phases as shown on Figure 1.6: in STAR (or wye); for which it is obvious that the current through the line terminals is the same as the current in the phase itself, or: in DELTA (or mesh); for which it is obvious that the voltage across the line terminals is the same as the voltage across the phase itself.