Mechanical Engineer''''s Reference Book 2011 Part 2 doc

Basic electrical technology 211 3 Therefore R R + ~(wL - l/&) - VO,, __- V,, Using the complex conjugate and calculating the modulus of the voltage ratio gives R (2.59) [R2 + (WL - l/WC)2]”2 The phase angle 6 = -tan-’ (2.60) The voltage ratio will have a maximum value of unity when the frequency (2.61) Equation (2.61) defines the ‘resonance’ condition at which the inductive and capacitive reactances are equal and self- cancelling. The resonant frequency is usually denoted w0 and it is the frequency at which the power transferred through the circuit 11s maximum. At any other frequency above or below w0 the power transferred is reduced. Z R + j(XL - Xc) (2.62) At the resonant frequency the total reactance is zero and the circuit behaves as if only the resistive element were present. The general variation of the voltage ratio (or amplitude ratio) and phase angle with frequency is illustrated in Figure 2.18. A.lso shown in the figure are the two frequencies, w1 and w2, at which the amplitude ratio is -3 dB. The -3 dB amplitude ratio is chosen because it corresponds to a halving in the power transmitted. The ‘,bandwidth’ is the frequency range between o1 and w2. A quality parameter, used with respect to resonant circuits, is the so-called ‘Q factor’, which is defined as the ratio of the resonant frequency to the bandwidth. The impedance of the circuit is given by - 2.1.30 Semiconductors The materials commonly used for semiconductors are germa- nium and silicon. In recent times silicon has all but replaced germanium as a semiconductor material. These materials have a crystalline structure such that each atom is surrounded by equally spaced neighbours. The basic structure can be visua- lized as a two-dimensional grid where the node points repre- sent the central nucleus and the inner shell electrons, while the connecting lines of the grid represent the four valence elec- trons associated with each nucleus. This grid concept is adequate to describe an intrinsic (or ‘pure’) semiconductor. At absolute zero temperature the crystalline structure is perfect and the electrons are all held in valence bonds. Since there are no current carriers available, the crystal behaves as a perfect insulator. As the temperature rises above absolute zero an increasing number of valence bonds are broken, releasing pairs of free electrons and their associated ‘holes’. In the absence of an applied fieid the free electrons move randomly in all directions. When an electric field is applied the electrons drift in a preferential direction to oppose the field and a net flow of current is established. The covalent bond, with a missing electron, has a large affinity for electrons such that an electron from a neighbouring bond may easily be captured. This will leave the neighbouring 0.1 wo 10 wo Angular frequency (rad/s) Figure 2.18 Voltage ratio and phase angle versus frequency (series RLC) atom depleted of electrons and the flow of electrons is generally associated with a counterflow of so-called holes. The mobile hole, to all intents and purposes, is essentially a simple positive charge. 2.1.31 Doped semiconductors Doped semiconductors are those in which an impurity has been introduced into a very pure intrinsic silicon. The nature of the impurity depends on the type of semiconductor re- quired: 1. n-type: Impurities with five valence electrons can be added to produce a negative type of semiconductor. These impurities are referred to as ‘donors’, since the additional electron is very easily freed within the matrix. In the n-type semiconductor the free electrons are the dominant current carriers. 2. p-type: the p-type semiconductor is one in which the added impurities have only three valence electrons. Such impurities are called ‘acceptors’ and they produce a positive type of semiconductor within which hole conduc- tion is the dominant current carrier. 2.1.32 pn junction diode A pn junction is formed by doping a crystal in such a w2y that the semiconductor changes from p- to n-type over a very short 2/14 Electrical and electronics principles length (typically m). The transition zone from p- to n-type is called the ‘carrier depletion layer’ and, due to the high concentration of holes on one side and electrons on the other, a potential difference exists across this layer. The diffusion of holes from p to n and electrons from n to p is the majority carrier movement, called the ‘diffusion current’. The drift of electrons from p to n and holes from n to p is the minority carrier movement. referred to as the ‘drift current’. When there is no externally applied potential difference, the diffusion current and the drift current are balanced in equili- brium. If an electric field is applied across the device then two situations can exist, as illustrated in Figure 2.19. Figure 2.19(a) shows the reverse-bias mode in which the potential barrier is increased. The diffusion current is reduced while the drift current is barely altered. Overall, the current is negative and very small. When forward bias is applied, as in Figure 2.19(b), the potential barrier is reduced and a large diffusion current flows. Overall, the current is positive and large. These general characteristics are the basis of a semiconductor diode which displays the typical currentholtage relationship de- picted in Figure 2.20. This figure shows clearly that a very high impedance is presented by the diode to an applied voltage of reverse polarity. A low impedance is presented to a forward polarity (a) Reverse bias (b) Forward bias Figure 2.19 pn junction with applied potential difference Forward current ( mA) t Reverse saturation current Is Reverse current I (PA) Reverse voltage Forward voltage Figure 2.20 Currentivoltage relationship for a pn semiconductor diode voltage. In simple terms, the diode accommodates a forward flow of current but greatly inhibits a reverse flow. The diode may be likened therefore to a switch which is activated ‘on’ for forward voltages and ‘off‘ for reverse voltages. The reverse saturation current, Is, is typically of the order of a few nano-amperes and can sensibly be regarded as zero. The general characteristic also shows that the reverse volt- age has a critical limiting value at which a ‘breakdown’ occurs. Depending upon the diode construction, the breakdown (or ‘Zener’ voltage) may range from as low as one volt to as much as several thousand volts. Up to the breakdown voltage, the reverse saturation current is independent of the reverse volt- age. Since the currentholtage relationship for a diode is a non-linear exponential function, the analysis of circuits involv- ing diodes can become complicated. A simple awareness of the diode’s practical function as a rectifier is perhaps more important than a proficiency in analysing circuits involving diode elements. 2.1.33 A.C. rectification Figure 2.21 shows an a.c. circuit with a diode in series with a load resistor. When the diode is forward biased a current will flow in the direction indicated by the arrowhead. No current can flow when the diode is reverse biased, provided that the applied voltage does not exceed the breakdown value. The resultant current waveform through the resistor, for a sinu- soidal voltage input, will therefore consist of positive only half sine waves. Since the output waveform is positive only, then it is, by definition, a d.c. voltage. It can be shown that the r.m.s. voltage across the resistor is (2.63) where RL is the load resistance, RF is the diode forward resistance and V, is the peak input voltage. Determination of RF is problematic, however, and models of varying complexity are used to simulate the diode in the circuit. The single-diode circuit results in half-wave rectification. To obtain full-wave rectification a diode bridge circuit can be used. The diode bridge is shown in Figure 2.22. When A is positive with respect to B then diodes D1 and D3 are conduct- ing. When B is positive with respect to A then diodes D2 and 04 are conducting. The circuit arrangement ensures that the current, which consists of a continuous series of positive half sine waves, is always in the same direction through the load RL. With full-wave rectification there are twice as many half sine pulses through the load than there are with half-wave rectifica- tion. In addition, there are always two diodes effectively in series with the load. The resultant r.m.s. voltage across the load resistor for the full-wave diode bridge rectification circuit is (2.64) The ‘peak inverse voltage’ (PIV) is defined as the maximum reverse-biased voltage appearing across a diode. When used as a rectifier the diodes must have a sufficiently high reverse voltage rating in excess to the peak inverse voltage that the circuit can generate. For both the half- and the full-wave rectification circuits considered, the peak inverse voltage is equivalent to the maximum supply voltage, V,. Additional manufacturers’ diode specifications would normally include the maximum power rating and the maximum allowable forward current. Electrical machines 211 5 Diode Figure 2.21 Half-wave rectification circuit A Voltage output I Voltage across R, Figure 2.22 Full-wave rectification with a diode bridge 2.1.34 The Zener diode The diode breakdown effect is also used in a variety of circuits to provide a stabilized reference voltage. Special diodes which are designed to operate continuously in the reverse bias mode are called ‘Zener diodes’. These diodes are manufactured with a range of breakdown voltages from between 3 to 20 V. Figure 2.23 shows a Zener diode being used in a circuit to give a stable voltage which is essentially independent of the current flowing through the device. The series resistor in the circuit is included to limit the reverse current through the diode to a safe value. voltage I Figure 223 Zener diode as a reference voltage source Stabilized voltage 2.2 Electrical machines The function of a rotating electrical machine is to convert mechanical power into electrical power, or vice versa. The conversion from mechanical to electrical power is made with a ‘a generator’ and the conversion of electrical to mechanical power with a ‘motor’. Electrical machines may be further sub-divided into a.c. or d.c. machines. The major part of all electrical energy generated in the world today is produced by a particular type of a.c. machine called an ‘alternator’. The applications of electric motors are no less substantial and they are used in a great variety of industrial drives. It is muaily the mechanical features of a particular application which deter- mines the type of electric motor to be employed, and the torquespeed characteristics of the machine are therefore very important. 2.2.1 The d.c. generator All conventional electrical machines consist of a stationary element and a rotating element which are separated by a air gap. In d.c. machines - generator or motor - the stationary element consists of salient ‘poles’ which are constructed as laminated assemblies with coils wound round them to produce a magnetic field. The function of the laminations is to reduce the losses incurred by eddy currents. The rotating element is traditionally called the ‘armature’, and this consists of a series of coils located between slots around the periphery of the armature. The armature is a150 fabricated in laminations which are usually keyed onto a locating shaft. A very simple form of d.c. generator is illustrited in Figure 2.24. 2/16 Electrical and electronics principles Figure 2.24 Single-coil, two-pole d.c. generator In the figure the single coil is rotated at constant speed between the opposite poles, north and south, of a simple magnet. From Faraday's law (equation (2.25)) the voltage generated in the coil is equal to the rate of change of flux linkages. When the coil lies in the horizontal plane there is maximum flux linking the coil but a minimum rate of change of flux linkages. On the other hand, when the coil lies in the vertical plane there is zero flux linking the coil but the rate of change of flux linkages is a maximum. The resultant variation in generated voltage in the coil, as it moves through one revolution, is shown in Figure 2.24(b). It is apparent that the generated voltage is alternating with positive and negative half-cycles. To change the a.c. output voltage into a d.c. voltage, a simple yet effective mechanical device called a 'commutator' is used. The commutator (Figure 2.25) incor- porates brass segments separated by insultating mica strips. External connection to the armature coil is made by stationary carbon 'brushes' which make sliding contact with the commu- tator. Referring to Figures 2.24(a) and 2.25(a), as the coil rotates from the horizontal plane through 180" the right-hand side of the coil is under the north pole and is connected via the commutator to the upper brush. Meanwhile, the left-hand side of the coil is under the south pole and is connected to the lower brush. A further 180" of rotation effectively switches the coil sides to the opposite brushes. In this manner the coil side passing the north pole is always connected to the positive upper brush, while the coil side passing the south pole is always connected to the negative lower brush. The resultant output voltage waveform is shown in Figure 2.25(b). If two coils, physically displaced by 90°, are now used, the output brush voltage becomes virtually constant, as shown in Figure 2.26. With the introduction of a second coil, the commutator must have four separate segments. In a typical d.c. machine there may be as many as 36 coils, which would require a 72-segment commutator. The simple d.c. generator of Figure 2.24 can be improved in perhaps three obvious ways. First, the number of coils can be increased, second, the number of turns on each coil can be increased and third, there is no reason why another pair of Coil voltage output (b) -ve T output voltage wavefo r rn 0 180 360 (b) Figure 2.25 Commutator connections to armature Electrical machines 2/17 2.2.1.2 Armature torque I The force on a current-carrying conductor is given by equation (2.27). i.e. Outpui: voltage 0 180 360 F = BlI The torque on one armature conductor is therefore T = Fr = BavlIar (2.68) where r is the radius of the armature conductor about the centre of rotation, I, is the current flowing in the armature conductor I is the axial length of the conductor, and B,, is the average flux density under a pole. Note that dl Figure 2.26 Two-coil. twopole d.c. generator output voltage poles cannot be introduced. A typical d.c. machine would therefore normally incorporate four poles, wired in such a way that each consecutive pole has the opposite magnetic polarity to each of its neighbouring poles. If the e.m.f.’s generated in the armature coils are to assist each other then while one side of the coil is moving under a north pole, the other side must be moving under a south pole. With a two-pole machine the armature coils must be wound such that one side of the coil is diametrically opposite the other. With a four-pole machine the armature coils can be wound with one side of the coil physically displaced 90” from the other. The size of the machine will generally dictate how many coils and the number of turns on each coil that can be used. 2.2.1.1 Armature e.m.f. If a coiiductor cuts flux then a voltage of 1 V will be induced in the conductor if the flux is cut at the rate of 1 Wbis. Denoting the flux per pole as @ and the speed in revolutions per second as N, for the single-turn coil and two-pole generator of Figure 2.24(al the e.m.f. indcced in the coil is Flux per pole aj EmI = - - 2N@ Time for half revolution 1/(2N) For a machine having Z, armature conductors connected in series, i.e. 242 turns, and 2p magnetic poles, the total induced e.m.f. is 2 2 E = 2!V@ 2p = 2N@Z, p volts (2.65) Zs depends on the type of armature winding, and the two main types are ‘lap-wound’ and wave-wound’. The lap winding is characterized by the fact that the number of para.lle1 paths through the winding is equal to the number of poles. In the alternative wave winding the number of parallel paths through the winding is always equal to two. If 2 denotes the total number of armature conductors then for the lap winding Z I Number of parallel paths Number of poles 2p -_ - Z - - Z =- (2.66) and for the wave winding Z Number of parallel paths 2 -_ - Z z, = (2.67) Lap windings are generally used in low-voltage, heavy-current machines and wave winding in all other cases. B,, = ~ (27rr1)/2p The resultant torque per conductor is T=L @2plI r @pia 2nd 7~ For Z, armature conductors connected in series the total torque on the armature is - Newton-metres T=- @PIaZs 7r 2.2.1.3 Terminal voltage (2.69) Denoting the terminal voltage hy V, the induced e.m.f. by E and the armature resistance by R,, V = E - IaRa (for a generator) (2.70) V = E + I,R, (for a motor) (2.71) For the motor, the induced e.m.f. is often called the ‘back e.m.f.’. 2.2.2 Methods of connection The methods of connecting the field and armature windings may be grouped as follows: 1. Separately excited - where the field winding is connected to a source of supply independently of the armature Self-excited - which may be further sub-divided into: (a) across the armature terminals; (b) in series with the armature winding; (c) and series windings. supply; 2. Shunt-wound - where the field winding is connected Series-wound - where the field winding is connected Compound-wound - which is a combination of shut The four alternative methods of connection are illustrated in Figure 2.27. 2.2.3 The separately excited generator Consider the separately excited generator, shown in Figure 2.27(a), running at a constant rated speed with no load across the output. It is assumed that initially the poles were comp- letely de-magnetized. If the field current, and hence the magnetic field, is gradually increased then a plot of terminal voltage against field current takes the form shown in Figure 2.28. As the field current increases, the iron poles begin to saturate and the proportionality between the flux and the field current no longer exists. If the field current is ?hen reduced. 211 8 Electrical and electronics principles If Field Armature (a) Separately excited (b) Shunt-wound (c) Series-wound Figure 2.27 Methods of field connection (d) Compound-wound the magnetic hysteresis causes the terminal voltage to have a slightly greater value than that obtained when the field current was being increased. When the field current is reduced to zero, a ‘residual voltage’ remains. On increasing the field current once more, the curve follows the broken line to merge with the original lower curve. These curves are termed the ‘open-circuit characteristics’ of the machine. and driven at constant speed with a constant field current; I,, the terminal voltage variation with armature current is as shown in Figure 2.29. The decrease in terminal voltage with increase in load is due mainly to the voltage drop across the armature resistance, R,. Additionally, the decrease in ter- minal voltage is attributed to a decrease in flux caused both by the de-magnetizing ampere-turns of the armature and the magnetic saturation in the armature teeth. These effects are collectively known as ’armature reaction’. Figure 2.29 is referred to as the ‘load characteristic’ of the generator. The separately excited generator has the disadvantage inhe- rent with a separate source of direct current required for the field coils. They are, however. used in cases where a wide range in terminal voltage is required. Saturation 0) +- - !2 K E & + - m If the generator is now connected to a variable external load Field current Figure 2.28 Open-circuit characteristics of a separately excited generator Electrical machines 211 9 The shunt-wound machine is the most common type of d.c. generator employed. The load current, however, must be limited to a value well below rhe maximum value to avoid excessive variation in terminal voltage. Open-circuit voltage a, OI 0 + I m Separately ._ excited E generator - a, OI 0 + I m Separately - excited generator \ \ Shunt-wound I generator I Armature current, I, Figure 2.29 Load characteristic of a separately excited generator 2.2.4 The s~~nt-wound generator The field winding in the shunt-wound generator is connected across the armature terminals as shown in Figure 2.27(b) and is therefore in parallel (or ’shunt’) with the load. A shunt generator will excite only if the poles have some residual magnetism and the resistance of the shunt circuit is less than some critical value. If, when running at constant speed, the field is disconnected from the armature, the voltage generated across the armature brushes is very small and entirely due to the residual magnet- ism in the iron. When the field is connected, the small residual voltage generates a flow of current in the field winding. The total flux in the field winding will gradually build up and the final terminal voltage will depend on the resistance of the field winding and the magnetization curve of the machine. The general characteristic is shown in Figure 2.30. When connected to an external load the shunt-wound generator exhibits a drop in terminal voltage as the armature current is increased (see Figure 2.29). The drop in voltage in the shunt-wound generator is much greater than that in the separately excited generator. This stems from the fact that, as the termiiial voltage drops, the field current also reduces, which causes a further drop in terminal voltage. Final no-load voltage Field current, 1, Figure 2.30 No-load characteristic of a shunt-wound generator 2.2.5 The series-wound generator For the series-wound generator the field winding is connected in series with the armature terminals as shown in Figure 2.27(c). The armature current therefore determines the flux. The constant speed load characteristic (Figure 2.31) exhibits an increase in terminal voltage as the armature (or load) current increases. At large values of load current the armature resistance and reactance effects cause the terminal voltage to decrease. It is apparent from Figure 2.31 that the series-wound generator is totally unsuitable if the terminal voltage is required to be reasonably constant over a wide range of load current. 2.2.6 The compound-wound generator The compound-wound generator (Figure 2.27(d)) is a hybrid between the shunt- and the series-wound generators. Normally. a small series field is arranged to assist the main shunt field. This is termed ‘cumulative compounding’. The shape of the load characteristic (Figure 2.32) depends upon the number of turns on the series winding. If the series field is arranged to oppose the main shunt field (‘differentially com- pounded’) a rapidly falling load characteristic is obtained. The number of turns on the series coil can be varied to give an over-compounded, level-compounded or an under-com- pounded characteristic as shown in Figure 2.32. 2.2.7 The d.c. motor There is no difference in basic construction between a d.c. generator and a d.c. motor. The only significant distinction between the two machines is quantified by equations (2.70) and (2.71). These illustrate the fact that, for a d.c. generator, the generated e.m.f. is greater than the terminal voltage. For the d.c. motor, the generated e.m.f. is less than the terminal voltage. Equation (2.65), which gives the relationship between the induced e.m.f. and the speed of a d.c. generator, applies Armature current Figure 2.31 generator Constant speed load characteristic for the series-wound 2/20 Electrical and electronics principles Over-compounded Level under T- shunt I Differentially I compounded 1- I Cumulative compounded Full load Load current Figure 2.33 The shunt-wound motor Figure 2.34(a) shows that no torque is developed until the armature current is large enough to supply the constant losses in the machine. Since the torque increases significantly for a slight decrease in speed, the shunt-wound motor is particularly suitable for driving equipment such as pumps, compressors and machine tool elements, where the speed must remain ‘constant’ over a wide range of load. Figure 2.32 Load characteristic for the compound-wound generator 2.2.9 The series-wound motor equally well to the d.c. motor. Since the number of poles and number of armature conductors are fixed, a proportionality relationship can be derived to relate speed as a function of induced e.m.f. and flux, i.e. N = El4 or, using equation (2.71), N = (V - IaRa)/@ (2.72) (2.73) The value of I,R, is usually less than about 5% of the terminal voltage such that, to a reasonable approximation, N = VI@ (2.74) Similarly, equation (2.69), which gives the armature torque on a d.c. generator, also applies to the d.c. motor. A proportion- ality relationship for the d.c. motor torque is therefore T = Ia@ (2.75) Equation (2.74) shows that the speed of a d.c. motor is approximately proportional to the voltage applied to the armature and inversely proportional to the flux. All methods of controlling the speed of d.c. motors are based on these proportionality relationships. Equation (2.75) indicates that the torque of a given d.c. motor is directly proportional to the product of the armature current and the flux per pole. 2.2.8 The shunt-wound motor The shunt-wound motor is shown schematically in Figure 2.33. Under normal operating conditions the field current will be constant. As the armature current increases, however, the armature reaction effect will weaken the field and the speed will tend to increase. The induced voltage will decrease due to the increasing armature voltage drop, and this will tend to decrease the speed. The two effects are not self-cancelling, and, overall, the motor speed will fall slightly as the armature current increases. The motor torque increases approximately linearly with the armature current until the armature reaction starts to weaken the field. These general characteristics are shown in Figure 2.34, along with the derived torque-speed characteristic. The series-wound motor is shown in Figure 2.35. As the load current increases, the induced voltage, E, will decrease due to reductions in the armature and field resistance voltages. Because the field winding is connected in series with the armature the flux is directly proportional to the armature current. Equation (2.74) therefore suggests that the speed/ armature current characteristic will take the form of a rectan- gular hyperbola. Similarly, equation (2.75) indicates that the torquelarmature current characteristic will be approximately parabolic. These general characteristics are illustrated in Figure 2.36 along with the derived torque-speed characteristic. The general characteristics indicate that if the load falls to a particularly low value then the speed may become dangerously high. A series-wound motor should therefore never be used in situations where the load is likely to be suddenly relaxed. The main advantage of the series-wound motor is that it provides a large torque at low speeds. These motors are eminently suitable, therefore, for applications where a large starting torque is required. This includes, for example, lifts, hoists, cranes and electric trains. 2.2.10 The compound-wound motor Compound-wound motors, like compound generators, are produced by including both series and shunt fields. The resulting characteristics of the compound-wound motor fall somewhere in between those of the series- and the shunt- wound machines. 2.2.11 Starting d.c. motors With the armature stationary, the induced e.m.f. is zero. If, while at rest, the full voltage is applied across the armature winding, the current drawn would be massive. This current would undoubtedly blow the fuses and thereby cut off the supply to the machine. To limit the starting current, a variable external resistance is connected in series with the armature. On start-up the full resistance, is connected in series. As the machine builds up speed and increases the back e.m.f.; the external resistance can be reduced until the series resistance is disconnected at rated speed. Electrical machines 2/21 Applied voltage, V Armature current (a) Figure 2!.34 The shunt-wound motor load characteristics IL Figure :!.35 The series-wound motor E I Rated I speed Variable-resistance ’starters’ are also usually equipped with a return spring and an electromagentic ‘catch plate’. The latter keeps the starter in the zero resistance position while the machine is running at its rated speed. The electromagnet is powered by the field current and, in the event of a supply failure. the electromagnet is de-energized and the return spring pulls the starter back to the full-resistance ‘off‘ position. This ensures that the full starting resistance will always be in series .with the armature winding when the machine is re- started. An overload cut-out switch is another normal feature incor- porated into the starter mechanism. The overload cut-out is another electromagnetic switch which this time is powered by the supply current. The overload switch is normally ‘off‘. but if the supply current becomes excessive, the switch is activated and it short circuits the supply to the electromagnetic catch plate. This, in turn. de-energizes the catch plate and the return spring takes the starter back to the ‘off‘ position. Figure 2.37 illustrates the essential features of a starter device for a shunt-wound motor. 2.2.12 Speed conUrol of d.c. motors Equatimon (2’74) shows that the speed of a d.c. motor is influenced both by the applied voltage and the flux. A Speed (b) variation in either of these parameters will therefore effect a variation in the motor speed. 2.2.12.1 Field regulator For shunt- and compound-wound motors a variable resistor, called a ‘field regulator‘, can be incorporated in series with the field winding to reduce the flux. For the series-wound motor the variable resistor is connected in parallel with the field winding and is called a ‘diverter’. Figure 2.38 shows the various methods of weakening the field flux for shunt-, compound- and series-wound motors. In all the above methods of speed control the flux can only be reduced, and from equation (2.74) this implies that the speed can only be increased above the rated speed, and may, in fact, be increased to about three or four times the rated speed. The increased speed, however, is at the expense of reduced torque, since the torque is directly proportional to the flux which is reduced. 2.2.12.2 Variable armature voltage Alternatively. the speed can be increased from standstill to rated speed by varying the armature voltage from zero to rated value. Figure 2.39 illustrates one method of achieving this. The potential divider, however, carries the same current as the motor, and this limits this method of speed control to small machines. Additionally, much of the input energy is dissipated in the controller, which consequently renders the system inefficient. 2.2.12.3 Ward Leonard drive In this case the variable d.c. voltage for the speed-controlled motor is obtained from a separate d.c. generator which is itself driven by an induction motor (see Figure 2.40). The field coil for the d.c. generator is supplied from a centre-tapped poten- tial divider. When the wiper arm is moved from 0 to A the armature voltage of the d.c. motor is increased from ZCKJ and the motor speed will rise. In moving the wiper from A to 0 [...]... Coffee = (Z + 7 7 (Z + T ) + =(Z+R)+(TTT) (2. 123 ) Tea = T 0 0 OR realization using NAND gates + (2. 124 ) Equations (2. 123 ) and (2. 124 ) are identical to equations (2. 121 ) and (2. 122 ), respectively This, of course, must be true, since the circuits from which the expressions were deduced perform identical logical functions Similarly, the circuit in Figure 2. 92, involving one NAND and three AND gates, may... core From Faraday’s law (equation (2. 25)) the induced e.m.f in the primary coil is El = Nl(d@dt) Transformer voltage equation = @ sin(wt) (2. 91) The induced e.m.f., from Faraday’s law, is Primary side, el = N,(d+/dt) = N1@w cos(ot) The r.m.s value of the induced e.m.f is 2~ ifN1@ El = _ _ = 4.44 fN@ v 2 (2. 92) Similarly, for the secondary side, E2 = 4.44 fN2@ (2. 86) 2. 2. 32 Transformer losses Since the magnetic... required and changing the number of poles is a simple and effective method of achieving this Stardelta starter 2. 2 .24 .3 Changing the rotor resistance 2. 2 .22 .2 Auto-transformer starter The aulo-transformer represents an alternative method of reducing the starting current drawn by an induction motor 2. 2 .22 .3 Rotor resistance With slip-ring induction motors it is possible to include additional resistance in... Figure 2. 82 As point E is a virtual earth, then Figure 2. 81 shows the operational amplifier connected up for a non-inverting output Assuming that the currents through resistors R1 and R2 are equal and that point E is a virtual earth, Therefore vi Rl - v, - v i R2 Figure 2. 80 Unity gain amplifier -id = il + iz + i3 VO R4 Figure 2. 82 Summing amplifier Analogue and digital electronics theory 2/ 41 or (2. 1 12) ... amplifier in Figure 2. 80 gives Figure 2. 81 Non-inverting amplifier Hence v - R2 + Rl o Vi RI Since E is a virtual earth, then Vi = V I and V, _ - - - R2 + Ri v 1 v,+ v, = vn Since the internal impedance of the amplifier is very large then V , is effectively zero and the gain is 5 Vn v 1 R1 (2. 110) R1 If, in addition, R2 9 R1, vo/vl 1 = (2. 109) (2. 111) 2. 3.14.4 Summing amplifier 2. 3.14.3 Non-inverting... NAND gates This equivalent circuit is shown in Figure 2. 100 Inspection of the circuit gives the Boolean expressions - Coffee = ( C M) (C T ) ~ ( C M) (C T ) (2. 125 ) Tea Q = ( T M ) = TM = (2. 126 ) Perhaps as expected, the Boolean expressions are identical to equations (2. 119) and (2. 120 ), which were deduced from the logic circuit of Figure 2. 92 The realization of Boolean expressions in either all... full-load position 2. 2 .22 Starting induction motors As with d.c motors, the current drawn during starting of a.c motors is very large, up to about five times full-load current A number of devices are therefore employed to limit the starting current but they all involve the use of auxiliary equipment, which is usually quite expensive 2. 2 .22 .1 Star-delta starter The star-delta switch (Figure 2. 51) is the cheapest... secondary current In this manner the main flux is generally maintained In steady state the ampere-turns in the primary and secondary windings are balanced, i.e 2* 2*33 Determination Of lransformer losses 2. 2.33.1 I1Nl = IzN2 or (2. 90) 2. 2.33 .2 I Open-circuit test The secondary coil is on open-circuit and the full-rated voltage is applied to the primary winding The transformer takes a small no-load current... depending on the load With the synchronous speed given by equation (2. 80) it is clear that the speed may be varied by changing either the frequency of the supply current or the number of poles Speed Figure 2. 52 resistances Torque-speed characteristics for various rotor 2/ 28 Electrical and electronics principles Sau irrel-caae 2. 2 .24 .4 Reduced stator voltage By reducing the applied stator voltage a... the stator windings been connected in the delta pattern on start-up ,-Full-load torque \ I1 0 Figure 2. 49 Schematic representation of an induction motor 0. 020 .06 Starting torque v I I slip 1 Figure 2. 50 Torqueslip characteristic for an induction motor Electrical machines 21 27 Three-phase supply 2. 2 .24 .1 1 I Solid state variable-frequency drives first began to appear in 1968 They were originally applied . and V2 = E2. Therefore (2. 89) 2~ ifN1@ v2 El = __ = 4.44 fN@ Similarly, for the secondary side, E2 = 4.44 fN2@ (2. 92) 2. 2. 32 Transformer losses Equations (2. 89) and (2. 90). Figure 2. 51 Stardelta starter 2. 2 .22 .2 Auto-transformer starter The aulo-transformer represents an alternative method of reducing the starting current drawn by an induction motor. 2. 2 .22 .3. armature conductors connected in series, i.e. 24 2 turns, and 2p magnetic poles, the total induced e.m.f. is 2 2 E = 2! V@ 2p = 2N@Z, p volts (2. 65) Zs depends on the type of armature