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Transcript
Introduction: DC Machine basics like d c. Motor, d c Generator, modes of operation of DC Motors and DC Generators are discussed in this chapter. DC Machines are part of Electrical Engineering. If you are using any kind of stereo, Walkman, DVD player or VCR you must observed a very small rotating motor inside the appliance or instrument you are using. This is no doubt a DC Motor. There are lots of application which can be completed only with DC Machines, Traction is one among these. 1.1 Modes of Operation: DC Machines are not outdated and basically a DC Machine can work in three modes: 1.2 • • • Generator Mode Motor Mode Break Mode Fig 1.1 Basic DC Electrical Motor construction 1.2.1 Generator Mode: A generator is a device that is used to produce electrical energy or electricity. The main function of a generator is to alter the form of energy. It converts mechanical energy into electrical energy. 18 In dc. generators, the machine is driven by a prime mover with the mechanical power and produce electrical energy. A dc. generator produce dc current as output by using commutator and brushes. 1.2.2 Motor Mode: A motor is just opposite in working and principle as compare to DC Generator. A motor or dc Motor converts electrical energy [dc Voltage] into mechanical energy or force. 1.2.3 Break Mode: Break mode is something different from the above said mode. In break mode, the machine ( which functions as a motor before the application of braking action ) works as a generator and the electrical power developed is either pumped back to the supply as in regenerative braking or is dissipated in the machine system as in dynamic braking. Hence in the braking mode the machine deadlerat on account of power supplied or dissipated by it and, therefore, produce a mechanical braking action. 1.3 Classification of DC Machine Based Upon Power Intake: Based intake power type DC Machine is of two types viz. DC Motor and DC Generator. Type Power Input Power Output DC Generator Mechanical power Electrical power DC Motor Electrical power Mechanical power 18 2.1 Introduction DC motors have been used in industrial applications for years. Coupled with a DC drive, DC motors provide very precise control. DC motors can be used with conveyors, elevators, extruders, marine applications, material handling, paper, plastics, rubber, steel, and textile applications to name a few. 2.2 Construction Fig.2.1 DC Motor construction DC motors are made up of several major components which include the following: • Frame • Shaft • Bearings • Main Field Windings (Stator) • Armature (Rotor) • Commutator • Brush Assembly 18 • For servo applications DC drives is still popular because of good dynamic response and ease of control. 2.7 Speed control DC motor methods: The speed control of a separately - excited DC motor is investigated in details. The steady state equivalent circuit of separately-excited DC motor is shown in Fig. 2.3. The armature circuit voltage equation and mechanical behavior of separately -excited DC motor at steady state are: (2.22) Va = Ra la + E E =ce<j>n (2.23) Tei = Cm d> Ia (2.24) It is more convenient to normalize the above equations with suitable base value to obtain dimensionless equation. The rated armature voltage (Van) is taken as the base value of the voltage, while the short circuit current at rated Voltage (Isc) is taken as the base value of the current. The corresponding short circuit torque is the value of the torque. (2.25) The base field flux is the rated field flux (<J)n) corresponding to rated filed current (Ifm) at rated field voltage (Vfm). The base motor speed is chosen to be the no load motor speed (no). Neglecting the armature voltage drop at load, the no load speed is obtained from (2.26) Vam — Ce (j) r no The motor normalized speed and current are obtained to n/n0 - Va/Van (cp / cpn f (2.27) - Ra/Ran (<p / <Pn )" (Te/Tsc) la/ Isc 2 (2.28) = ( 9 / <Pn )"’ (Tel/Tsc) 18 The motor speed given by eq. (2.27) depends on; the armature voltage (Va), the armature circuit resistance (Ra) including both armature and additional series resistance circuit and the motor field excitation ($). In practice, for speed less than the base speed, the armature current and field currents are maintained constant to meet the torque demand, and the armature voltage (Va) is varied to control the speed. For speed higher than the base speed, the armature voltage is maintained at the rated value and the field current is varied to control the speed. Figure.2.8 shows the characteristics of torque, power, armature current, and field current against the speed. The speed control of a DC motor is achieved by varying one of these variables [1]. The speed control methods are discussed in the following item. 2.7.1 armature voltage control The main field flux maintained constant at its rated value, the armature voltage is varied in the range [ -1.0 < (Va/Van)< 1.0]. When rated armature voltage is applied and the field current is fixed to its max. value , the motor speed is called the base speed (nt,). The armature applied voltage is controlled by using rotating machine converter as in ward-leonard system or static converters or a combination of them. Static converter for the armature voltage control is achieved either by a controlled rectifier or by a DC chopper control. The controlled rectifier is fed form an AC source, and the voltage is changed by adjusting the value of the firing angle. The controlled rectifier is associated with AC voltage harmonics and additional power loss. The DC chopper is fed on the contrary from a DC source, which may be the output of an uncontrolled rectified fed from the AC supply. The value of the output voltage is controlled by adjusting the chopper circuit onoff ratio or the chopping frequency. The DC chopper output voltage is associated with lower output voltage harmonics when compared with the controlled rectifier. Armature voltage control gives the ability to load the motor by its allowable maximum torque without overloading of the motor. The motor normalized speed and current are obtained to the speed-torque and current-torque characteristics of the separately -excited DC motor with armature voltage control are shown in Fig.2.9 a and b respectively. a- Speed-torque characteristics b- Current-torque characteristics Fig. 2.9 : Speed-torque (a) and current-torque (b) characteristics with armature voltage control. 21 2.7.2 Field weakening control The field weakening control is used when it is required to obtain motor speeds above the base speed or when the armature voltage cannot be changed. The armature voltage is fixed at the rated value and the field current and consequently the main flux is reduced. Maximum speed obtained by field weakening is limited by the mechanical design of the motor and the motor stability at weak field flux due to the magnetizing effect of armature reaction. The motor normalized speed and current are obtained to the speed-torque and currenttorque characteristics of the separately-excited DC motor with field weakening control are shown in fig.2.10 a and b respectively. Fig. 2.10 : Speed-torque (a) and current-torque (b) characteristics with fieid weakening control. 2.7.3 Armature resistance control A variable additional resistance is inserted in the armature circuit. This method is accompanied by additional copper losses especially when the motor speed is greatly decreased resulting in very poor motor efficiency. So, this method is economically relevant for small motors and in drives with narrow speed control ranges operating for short periods in the reduced speed range. The motor normalized speed and current are obtained to speed-torque characteristics of the separately- excited DC motor with armature resistance control is shown in Fig. 3.5. Fig. 2.11 : Speed-torque characteristics with armature resistance control. 3.1 Introduction In many industrial applications, it is required to convert a fixed voltage DC source into a variable voltage dc source of controllable average value. A dc chopper converts directly from dc to dc and is also known as a dc converter. Dc chopper are used usually where the source is a battery as in industrial electrically powered trucks and where it is a third rail or over heat trolley wire as in rapid transit cars. They provide smooth acceleration control, high efficiency and fast dynamic responses. Chopper are used to replace switched, series armature circuit resistors, the advantage is higher efficiency continuous control, and the ability to operate the motor in regenerative braking mode and it can be used in dc voltage regulators and also used in conjunction with an inductor to generate a DC current source, especially for the current source inverter. When a dc drive is used, then there are the following possibilities: a- The use of current converter circuit with low reactive power consumption, for example, half-controlled bridge, b- Tapped-transformer with adjustable secondary voltage and an uncontrolled rectifier. 3.2 Principle of operation of DC chopper Fig.3.1 shows the equivalent circuit and the corresponding output voltage of a DC chopper. In this circuit the chopper is represented by on-off switch. When the switch CH is closed for a certain time (ton), the input voltage appears across the load. When this switch is opened for another certain time (toff), the voltage across the load drops to zero. During a specified time period T, the onoff ratio of the chopper is controlled, this results in controlling the average output voltage across the load. Fig. 3.1: chopper circuit configuration and output voltage wave form The average output voltage is given by: V a v = ^j;° n V s d t = t - f V s = K V s = t o n FV s = (l-toffF)Vs (3.1) The average voltage can be adjusted either by controlling the duty cycle of the chopper while keeping the frequency constant (pulse width modulation) or controlling the chopper frequency (pulse frequency modulation) while keeping certain (on-off) ratio. 3.3 methods of control of output chopper voltage 3.3.1 Pulse width modulation In this technique the chopping frequency f is kept constant and the on-time and the off- time of the chopping switch is varied as shown in Fig.3.2 (a), this method has many advantages such as, 1. 2. Easier isolation. Signal filtering due to constant frequency operation. (a) Pulse width modulation (b) pulse frequency modulation Fig. 3.2 :(a) principles of PWM & (b) pulse frequency modulation (PFM) 3.3.2 Pulse Frequency Modulation (PFM) The on-time or the off-time of the chopper is kept constant and the frequency is varied as shown in Fig.3.2 (b). To obtain a full output voltage range using the frequency modulation control, the frequency is varied over a wide range, consequently, harmonics at new frequencies would be generated and the filter design became difficult. 3.4 Classification of DC chopper circuit DC chopper circuits are classified according to the power flow between the source and the load. The direction of the power flow depends on the forced path for the power and the type of the load. 3.4.1 First-Quadrant chopper (Class A chopper) The circuit arrangement of the first-quadrant chopper is shown in Fig.3.3(a). The current flows from the source to the load and the chopper circuit operates as a rectifier. This configuration is used for motoring operation of a DC motor. (a)Chopper circuit configuration (b) Voltage-current characteristics Fig. 3.3: Configuration (a) and voltage-current characteristics (b) of first-quadrant chopper Both load current and voltage are positive, i.e. operation is in the first quadrant as shown in Fig.3.3 (b). During the on-period the chopper CHi is switched on for a time interval ton. During the off-period the switch CHi is switched off for a time interval Wf, the armature current will flow in the closed loop via the freewheeling diode Di. 3.4.2 Second-Quadrant chopper (Class B chopper) The circuit arrangement of the first-quadrant chopper is shown in Fig.3.4(a). The current flows from the machine to the source. The machine voltage is positive, i.e. operation is in the second quadrant. The chopper circuit operates as an inverter and is used for regenerative braking of DC motor. (a)Chopper circuit configuration (b) Voltage-current characteristics Fig. 3.4: Configuration (a) and voltage-current characteristics (b) of secondquadrant chopper During the on-period the chopper CH2 is switched on for a time interval ton , and the load voltage vanishes Vo=0.0. During the off-period in which the chopper CH2 is switched off for a time interval t0ff, the energy stored in the magnetic circuit is returned to the supply through the diode D2. 3.4.3 Two-Quadrant Type-A chopper (Class C chopper) The circuit arrangement of this type is shown in Fig.3.5(a). the load voltage is positive and the load current is either positive or negative. (a) Chopper circuit configuration (b) Voltage-current characteristics Fig. 3.5: Configuration (a) and voltage-current characteristics (b) of Two-quadrant type-A chopper The load current state depends on the switching case of the choppers CHi and CH2. When chopper CHi is operated and diode Di is used, the circuit operates in the first quadrant (motoring). When chopper CH2 is operated and diode Do is used, the second quadrant operation is obtained (braking). It must be avoided to switch both CHi and CH2 simultaneously to prevent short circuit on the source. The voltage-current characteristics are shown in Fig3.5 (b). 3.4.4 Two-Quadrant Type-B chopper (Class D chopper) The circuit arrangement of this type is shown in Fig.3.6 (a). The load current is positive and the machine voltage is either positive or negative. This configuration allows to operate either as a rectifier as an inverter. (a)Chopper circuit configuration (b) Voltage-current characteristics Fig. 3.6: Configuration (a) and voltage-current characteristics (b) of Two-quadrant type-B chopper When the choppers CHi and CH2 are turned on, the power flows from the DC source to machine in the shown current direction (motoring). But when chopper CHi and CH2 are switched off, the load current will continue to flow in that path created by the diodes D1 and D2. The power flows from the machine to the DC source in serves direction to the current path shown (regeneration). 3.4.5 Four-Quadrant chopper (Class E chopper) The circuit arrangement of this type is shown in Fig.3.7 (a). It is the general case in which the load current or the machine voltage may be positive or negative according to the switching choppers. The voltage - current characteristic are shown in Fig.3.7 (b) which shows different modes of operation (a) Chopper circuit configuration (b) Voltage-current characteristics Fig. 3.7: Configuration (a) and voltage-current characteristics (b) of Four-quadrant chopper. By controlling the sequence of switching of the choppers (CHi, CH2, CH3&CH4), one can obtain the desired case of operation, i.e. operation is possible with reversible regenerative. This chopper is the basis for the single phase bridge inverter and it is characterized by circuit complexity rather than its high cost. .4 Types of DC Motors The dynamic behavior of the DC machine is mainly determined by the type of the connection between the excitation winding and the armature winding. 2.4.1 separately-excited DC motor The equivalent circuit of separately-excited DC motor is shown in Fig. 2.3 O------------------->— + iI a ! ;P ii 12 B Fig. 23: equivalent circuit of separately-excited Dc motor When a separately excited dc motor is excited by a field current and armature current flows in the armature circuit, the motor develops a back electromotive force (emf) and a torque to balance the load torque at a particular speed. The field current of a separately excited dc motor is independent of the armature current and any change in the armature current has no effect on the field current. The field current is normally much less than armature current. The equations describing the characteristics of separately excited dc motor can be determined from figure 2.3. 12 (2.1) (2.2) £ v f = Rf if + L f (% ) 05 II The instantaneous equations of the separately excited DC motor are Va = R a ia + La ( % ) + e (2.3) T'd ~if i- a (2.4) T a = J {% ) + Bo + T L (2.5) • Under steady state condition, the time derivative in the above equations are zero and hence, the steady-state average quantities are: (2.6) (2.7) E = Kv a) If Va — I a Ra Eg = Ra ^I f Td = f y if ia = + T L (2.8) (2.9) The developed power is (2.10 ) The speed of separately excited DC motor can be found from V a -R a Ia 60 — KvIf (2.11 ) 2.4.2 Self excited DC motor 2.4.2.1 Series DC motors In the Series motors, the field windings are connected in series with the armature circuit. Figure 2.4 presents the equivalent circuit of the series DC motor. The field circuit is designed to carry the armature current. Fig. 2.4: The equivalent circuit of the series DC motor The speed of the DC series motor is determined and can be varied by controlling the armature voltage or armature current meanwhile, the demand torque is determined as the following. ^a— I a ( Ra~/o K-v *a Td= KfIfla=B(o+ TL 0) = — — (2.12) (2.13) The series motor can provide a high torque especially at starting and for this reason DC series motors are commonly used in traction applications. For speed up to the base speed the armature voltage is varied and the torque is maintained constant. 2A.2.2 Shunt DC motors The field circuit is independent of the armature circuit because both circuits are fed from voltage source. Armature current and speed depend on the mechanical load. Fig. 2.5: The equivalent circuit of the shunt DC motor 2A.2.3 Compound DC motors Compound- wound motor which has two field winding, one connected in parallel with the armature and the other in series with it. There are two types of compound motor connections. When the shunt field winding is directly connected across the armature terminals [Fig. 2.6 a], it is called short-shunt connection. When the shunt winding is so connected that it shunts the series combination of armature and series field [Fig. 2.6 b], it is called long-shunt connection. 2.5 Speed /Torque characteristics The following chart compares speed/torque characteristics of DC motors. At the point of equilibrium, the torque produced by the motor is equal to the amount of torque required to turn the load at a constant speed. At lower speeds, such as might happen when load is added, motor torque is higher than load torque and the motor will accelerate back to the point of equilibrium. At speeds above the point of equilibrium, such as might happen when load is removed, the motor’s driving torque is less than required load torque and the motor will decelerate back to the point of equilibrium. The DC drive system is designed to control and to vary the speed and torque of the DC motors and they span wide range from a very simple designed hard ware to a very complicated programmable software microcontroller kits. DC motor drives have many advantages such as: The DC drive is relatively simple and cheap compared to induction motor drives nut DC motor itself is more expensive. For low power applications the cost of DC motor plus drives is still economical. 2.3 Theory of operation When voltage is applied to the motor, current begins to flow through the field coil from the negative terminal to the positive terminal. This sets up a strong magnetic field in the field winding. Current also begins to flow through the brushes into a commutator segment and then through an armature coil. The current continues to flow through the coil back to the brush that is attached to other end of the coil and returns to the DC power source. The current flowing in the armature coil sets up a strong magnetic field in the armature. Fig. 2.2: (a) Armature and commutator segments, (b) Armature prior to the coil's wire being installed, (c) Coil of wire prior to being pressed into the armature, (d) A coil pressed into the armature. The end of each coil is attached to a commutator segment. 3.5 separately excited DC motor fed from DC chopper To study the performance of the chopper circuit, its operation is divided in to two main modes. During the on mode, the chopper is switched on and the current flows from the supply to the load. 35 During the off mode, the chopper is switched off and the load current continues to flow through freewheeling diode (D). •o £>f Chopper L A D 7WD Fig.3.8: DC chopper fed separately excited DC motor 3.5.1 Mode of operation 3.5.1.1 Continuous armature current During the on mode (1) (0< t < ton) the voltage equation is then given by Ks= i?aia+ L a ^ + E (3.2) And its solution gives the load current (3.3) _ (3.4) At t= 0 — I (0) = I a so 35 iai® = *±(l-e~tba)+lsoe~t'Ta At the end of model (t=ton) the current attains the initial value for mode2 iDo = iai(t) = ^7(1 - e_t°n/T“) + Iso e During mode 2, ton < t < T or Tm The load is short circuited through the freewheeling diode and the voltage equation is then given by: 35 At the end of mode2, the chopper is turned on again for the next cycle after the time period, T = ton + toff (3-8) The current at the end of mode2, is the initial current for the next cycle h = ia(ton + toff) = IDo e /t«J (3.9) With the duty cycle (3.10) la 3.5.1.2 Discontinuous armature current u=0 (3.n) To know if the motor operates in the discontinuous armature current we calculate at first "tm" if a: tm < T we have discontinuous armature current b: tm > T we have continuous armature current At, ton < t < T or Tm 35 FI ton)/ \ (P ton)( ia(t)=j-a[ 1-e /t*) + IDO e ^ (3.12) We assume the motor operate in the discontinuous armature current At t— tm , ia(tm) =0 0 = -t(l~e /Ta) + lD0 e ^ trn = Ta ln[l + —j£- (3.13) E(e ,Ta-l 3.5.2 Motor performance 3.5.2.1 Average Armature Voltage Raia+ La^ + E (3-14) Vs dt = Raia+ Ladia + Edt (3.15) By integration from (0 to t) and dividing by T 1 rton R, rT L flDO r^DO ^ fton [T - Vsdt = — I iadt + -^{ iadt + I dia] + — { I Edt + Edt] 1 Jo ‘a Jo la Jiso Jiso I a J0 Jton 35 Vst-f=Raia + 0 + Et-f (3.16) Vst-f = Raia + Et-f+E-E (3.17) Vst-f = Va + Et-f-E (3.18) Vs = VsD + E(l-tf) (3.19) 35 3.5.2.2 Average Armature current Ia=Ya fo Wdt 3.5.2.3 Torque T= Cra . Ia for continuous flux 3.5.2.4 Torque / speed characteristic The motor used to obtain the speed - Torque characteristic is Ce=0.17, Vs = 200v, Ra = 3ohm, n=1000rpm, La= 0.06H , Fch= 500HZ The flow chart of the program used to calculate this characteristic as in figure 3.10 T Fig. 3.10: speed - Torque characteristic 35 3.5.3 Ripple Armature current Under steady- state condition, Is0=l3 and the peak -to-peak load ripple current can be determined to ton (7~~ton) A/a = ho ~ ho = T1—T Ra — (3'23> 1 -e~a The maximum peak-to-peak ripple current at duty ratio D=0.5 and is given by A Ia max = ^ ■ Tanh-|Ka (3-24) 4 Ia For the switching period t« 4 Ta this expression for the maximum excursion reduce to Vs T AIa max = TT ■ 77^ (3-25) a max Rg 4Tg The armature current excursion at steady operation condition can however be approximately obtained from the voltage -time area applied to the inductance La of the armature circuit. The average output voltage is given by Ys. = £°2l = D (3.26) vs T And the voltage across the motor inductance is given by 35 VL(t)=Va(t)-Va(0) (3.27) V L(t)=V a(t)-V a(to„) (3.28) A /« = ^ = £ (T - t„„) = f ^ U(3.29) ^a *a The voltage-time area applied the inductance La is given by VL.T0n=La. A I a (3.30) From which the current excursion is obtained to 35 For a duty ratio D=0.5, the same maximum current excursion is obtained as in above equation . This equation can be used to determine the inductance of the armature circuit necessary to limit the current excursion to a preset value usually a value of 10% of the armature current tolerated. It is also obvious that the switching chopper frequency fCh should be as high as possible to reduce the load current excursion and to reduce the size of any additional series inductors in the load circuit the chopper is also limited. 4.1 DC Chopper operation 4.1.1 First-Quadrant chopper (Class A chopper) 4.1.1.1 Chopper output controlled by Pulse frequency modulation. Fig. 4.1 First-Quadrant chopper (Class A chopper) controlled by 555 timer. Fig.4.1 shows the first-Quadrant chopper (Class A chopper) controlled by 555 timer circuit. The output of 555 timer is the firing of the S19CF power transistor. Fig.4.2 is presents the firing pulse train with different frequency which can be controlled by varying the CT from 0.1 |xF to 0.01 |iF. Fig. 4.2 Pulse frequency modulation (PFM) Fig.4.3 shows the output voltage of the chopper which will be applied on the motor armature. It can be seen that the armature voltage changed by operating the chopper by PFM which tested in Fig.4.2. 35 Fig.4.3 The output voltage of the chopper(Armature voltage) with respect to the firing pulse train(PFM). The yellow pulse is related to the chopper output voltage (armature voltage) and the green pulses is related to the chopper firing signal. 35 4.1.1.2 Chopper output controlled by Pulse width modulation. Fig.4.4 is presents the firing pulse train with different width which can be controlled by varying the RA in the circuit shown in figure 3.1 from 1 kQ to 2 kQ. Fig.4.4 pulse width modulation(PWM) 35 Fig.4.5 shows the output voltage of the chopper which will be applied on the motor armature. It can be seen that the armature voltage changed by operating the chopper by PWM which tested in Fig.4.4. Fig.4.5 The output voltage of the chopper(Armature voltage) with respect to the firing pulse train(PMM). The yellow pulse is related to the chopper output voltage (armature voltage) and the green pulses is related to the chopper firing signal. 4.1.2 Second-Quadrant chopper (Class B chopper) The circuit arrangement of the second-quadrant chopper is shown in 4.6 Second-Quadrant chopper (Class B chopper) Fig.4.6. The current flows from the machine to the source. The machine voltage is positive, i.e. operation is in the second quadrant. 35 Fig.4.7 The output voltage of the chopper(Armature voltage) with respect to the firing pulse train in Second-Quadrant chopper (Class B chopper) The yellow pulse is related to the chopper output voltage 35 (armature voltage) and the green pulses is related to the chopper firing signal. The chopper circuit operates as an inverter and is used for regenerative braking of DC motor. 4.1.3 Comparison between (Class A chopper) and (Class B chopper) ig.4.8 clarify the in vers operation relation between Class A and Class B Fig.4.8 Class A and Class B output chopper(armature voltage) 4.2 Determination of Speed -torque characteristics Fig.4.9 shows the linear Speed-torque curves of separately excited DC motor fed by DC Chopper with various value of duty cycle D at continuous current mode. D _ Va _ T-Ton (4 j) Vs T Va= D Vs (4.2) Vn—E From Eq. 3.21 (Chapter-3) average current: Ia = —— a 35 At duty cycle D the Eq. will be: la = ~~R~ (43) Ka From Eq. 3.22 (Chapter-3) Torque: Td= K . Ia for continuous flux (4.4) From Eq. 4.3 and 4.4 Td = k(Dvs-E2 (4 5) Ra E = kwd (4.6) From Eq. 4.5 and 4.6 Td = (4,7) wd = £r-%r<i =| (4.9) Tan = ^ 35 From Eq. (4.8),(4.9) and (4.10): (4.11) (4.12) From Eq. 4.9 and 4 . 1 1 : Wd = Dw dn-—wdn 1 dn d — £) _ 1A. dn ^dn n n (4.13) Where: Wd=27aid ,and Wdn=27tndn We will use the Eq. 4.13 to plot the speed torque characteristics at various value of duty cycle D (0.2, 0.4, 0.6, 0.8 &1): MATLAB CODE: %Wd=2*Pi*nd; %Wdn=2*Pi*ndn; %n=nd/ndn=Wd/Wdn; %T=Td/Tdn; %Wd/Wdn=Duty cycle -Td/Tdn; %The analysis for continuous current mode; %The duty cycle is 0.2,0.4,0.6,0.8,1; clc clear all n=(0:0.01:1) ; T=l-n; n_l=(0:0.01:0.8); T_1=0 . 8-nJ; n_2=(0:0.01:0.6) ; 35 T_2=0.6-n_2; n_3=(0:0.01:0.4) ; T_3=0.4-n_3; n_4=(0:0.01:0.2) ; T_4=0.2-n_4; figure (1) plot (n, T, ' b' , n_l, T_1, 'b',n_2,T_2, 'b \n_3,T_3, 'b',n_4,T_4, 'b') xlabel('Td/Tdn') ylabel ( ' nd/’ndn ' ) axis([0 1.2 0 1.2]) ..--No load speed. Max. Power - \ x X* _ 0.6 - 0.8 | 0 T3 N. Nv ' -------------------------- \ _ * \ v. 1 I. \ \ \ ■ \ \ 1\ '■ \ 111 C 0.4 \1 0.2 0 0.2 1.2 0.4 0.6 0.8 1 Fig.4.9 Speed-torque curves of separately excited DC motor fed by DC Chopper with various value of duty cycle D at continuous current mode 35 11