Tài liệu HIGH PERFORMANCE DRIVES_Chapter4 pdf

HIGH PERFORMANCE DRIVES  E Levi, 2001 1 1. WHAT IS A HIGH PERFORMANCE DRIVE? 1.1 GENERAL CONSIDERATIONS Equation of mechanical motion that describes rotor rotation is the same, irrespective of the type of the electric machine under consideration. It is given with the well-known expression TT J d dt eL −= ω (1.1) Variation of speed of rotation is then given with: ω () () =−1/ JTTdt eL (1.2) Let us assume that load torque is zero, that the machine is at rest and there is now application of step speed command. The fastest possible response of the actual speed to its command will be obtained if it is possible to produce instantaneous step in the machine torque from zero to the highest allowed value T emax , and to keep torque at this value until the actual speed becomes equal to its command. In such a case actual speed of the machine will be given with ω = T J t emax (1.3) Thus if a step increase in speed is required, the torque command is a step function as well and the fastest response of the machine speed (delayed with respect to reference because of inertia J) is obtained if the actual torque in the machine can be stepped instantaneously to its commanded value, which is equal to maximum permissible value. Hence the essential requirement in any drive which is to be used for highly demanding applications, where precision and fastness of speed (or position, or torque) control are of ultimate importance, is possibility of instantaneous torque control. In other words, not only steady-state but transient behaviour of the machine is controlled as well. A high performance drive is therefore a drive that provides an instantaneous torque response to the change in torque command, with, ideally, actual torque equal to the reference torque without any delay. Typical areas that ask for high performance electric drives are robotics, machine tools, rolling mills, paper industry, electric vehicles, traction, elevators, cranes, etc. Some of these ask for torque controlled drives (electric vehicles, traction), while in others the ultimate control input may be the position ( robotics, machine tools, elevators, etc.). In order to further explore behaviour and properties of a high performance drive, the simplest case of a separately excited DC motor is considered next. 1.2 HIGH PERFORMANCE SEPARATELY EXCITED DC MACHINE What is generally meant by a ‘high performance drive’ may most easily be explained by looking at a separately excited DC machine, whose schematic representation is shown in Fig. 1.1. The equivalent circuit is composed of two electric circuits. The first one represents the excitation winding, while the second one describes armature winding. Both are supplied with DC voltage and hence both currents are DC as well. Separately excited DC machine is described with a well-known set of dynamic first-order differential equations (subscript f denotes field or excitation or stator winding while subscript a denotes rotor or armature winding.): HIGH PERFORMANCE DRIVES  E Levi, 2001 2 vRiL di dt fi vRiL di dt eeK TT J d dt TK i ffff f ff aaaa a mf eL emfa =+ = =+ + = −= = Φ Φ Φ () ω ω (1.4) i f v f LR ff LR aa i v a a e Fig. 1.1 - Equivalent circuit of a separately excited DC machine As there is no relative motion between stator winding and flux density wave, there is no rotational electromotive force induced in the stator winding. Furthermore, as the rotor flux density axis is aligned with brush axis, this being perpendicular to stator magnetic axis, there are no transformer induced voltages in either of the windings due to mutual inductance. Both voltage equations contains terms which describe transformer induced voltages due to winding self-inductance (L a and L f ). Rotor winding equation contains rotational induced emf due to rotation of the rotor in stationary stator flux density. If the non-linearity of the iron core is neglected, excitation flux is directly proportional to excitation winding current. Excitation flux Φ f in the machine is dependent purely on excitation winding current and therefore it can be set to desired value regardless of the armature current value. If the excitation voltage is constant, excitation winding current and excitation flux will be constant as well, so that for constant flux operation dynamic model of a separately excited DC motor reduces to a very simple set of equations. vRiL di dt eeC TT J d dt TCi aaaa a eL ea =+ + = −= = ω ω (1.5) where C = K m Φ f . Equations (1.5) show that, if excitation flux is constant, the torque of the machine is governed purely by armature current. Therefore if the armature current can be controlled in an instantaneous manner, torque will instantaneously follow the change in armature current, without any unwanted transients. On the other hand, armature voltage equilibrium equation shows that due to the inductance of the armature winding armature current response has to be delayed with respect to applied voltage. As a consequence, closed loop armature current control is necessary in order to enable instantaneous torque and consequently instantaneous speed control. By utilising this principle, torque, speed and position of the machine can be controlled precisely not only in steady-state but during transients as well. In other words controlled variable (torque, speed, position) is fully controllable at any time. Separately excited machine enables by virtue of its construction so called decoupled (independent) flux and torque control. Very fast transition from one steady state to the other is thus possible. Moreover, the transition itself is fully controllable enabling high performance of HIGH PERFORMANCE DRIVES  E Levi, 2001 3 the drive to be achieved. A schematic diagram of a high-performance DC drive aimed for operation in base speed region (constant flux region) is shown in Fig. 1.2. Speed controller is of PI structure while configuration of the current controller may greatly vary depending on the type of the current control method applied and on the topology of the power electronic converter. However, closed armature current control loop provides application of such an armature voltage which enables that armature current closely tracks its reference value. Hence the machine is said to be current-fed and in idealised analysis it is possible to assume that reference armature current is equal to actual armature current. This simplification is valid for very fast power electronic converters, which operate at high switching frequencies and hence allow for very fast change of the armature voltage. The idealisation introduced is equivalent to stating that the input voltage of the power electronic converter is infinite, as discussed later. From the point of view of the machine modelling, if the machine is treated as current-fed, then the armature voltage equation may be omitted from analysis. This is so since in such a case armature current equals its reference, and the reference current is known from the control system. Speed controller ω ω - * T e * 1 K m Φ f i a * Current controller firing signals electronics converter Power i a - i a DCa v ω Fig. 1.2 - Closed-loop speed control of a separately excited DC motor One possible set-up of the power electronic converter with associated current control is illustrated in Fig. 1.3. Step-down chopper circuit is shown in an idealised form (the switch S can be any fully controllable semiconductor, while the flywheel diode is not shown - it will of course carry the armature current when the switch is off). Current control is of the hysteresis type (it is often termed bang-bang control or two-point control). The main idea of this current control method is to keep the actual current inside the pre-defined error band around reference current, which is called hysteresis band. Operation of the current controller is illustrated in Fig. 1.3 too, where trajectory of actual current and appropriate armature voltage delivered by the chopper are shown as well. As can be seen from figure 1.3c, change of armature voltage will take place whenever armature current falls outside prescribed limits ii aa * ±∆ ,where∆i a is called hysteresis band. If hysteresis band is taken very small then changes of voltage will be very frequent, which means operation with high switching frequency of the chopper. In this case switching frequency of the chopper is not constant and is predominantlyinfluencedbyselectedhysteresisband(andbyinductanceinthecircuit). Itisassumedin Fig. 1.3 that the drive operates with constant excitation flux. Figure 1.3c shows that the actual armature current oscillates around reference current within prescribed limits. Hence the torque, which is directly proportional to the current, will not be constant; in order to keep fluctuations in torque as low as possible, it is an imperative to keep the hysteresis band at the smallest possible value (i.e. to operate the chopper with the highest possible switching frequency). Mechanical equation indicates that the mechanical subsystem of the machine behaves as a first order delay system. In other words, inertia of the drive filters out variations in torque and these are not reflected in speed, which in steady-state operation remains essentially constant (provided that inertia is sufficiently high for the given torque pulsation). HIGH PERFORMANCE DRIVES  E Levi, 2001 4 An example of the dynamic response of the separately excited DC machine drive of Fig. 1.2 is shown in Fig. 1.4. Inner current control loop is assumed to be infinitely fast, so that commanded (reference) armature current is taken as equal to the actual armature current. At first excitation voltage (taken as constant DC) is applied across excitation winding. During this interval reference speed is kept at zero, so that reference and actual armature currents equal zero as well. Once when the flux in the machine has been established and steady state reached, a ramp speed command is applied. As the machine is modelled as current-fed, the model of the DC machine, used in simulation for operation in the constant flux region, reduces to TT J d dt TCi eL e a −= = ω (1.6) As can be seen from Fig. 1.4, the initial excitation of the machine follows exponential law. When the flux settles and speed command is applied, actual speed follows reference speed with a very small delay, caused by inertia of the drive. Torque rises almost instantaneously and reaches the maximum value determined by the imposed limit and hence the armature current is in the limit as well. When the reference speed reaches steady-state value the actual speed overshoots (due to action of the speed controller, which is PI) and hence torque rapidly goes out of the limit, reduces and becomes negative, which means that electric braking operation takes place. Once when the speed settles at the value equal to the reference value, torque falls to zero, because the case shown in Fig. 1.4 is acceleration with zero load torque. LR aa i v a a e V DC S semi- conductor switch V DC a. v on off off offonon a v on off off offonon a i a t t t t a i aa i+∆ i a i+∆ i a t on T t v= a on T V DC b. c. f switch = 1 T Fig. 1.3 - Separately excited DC machine fed from step-down chopper: a. Idealised power circuit; b. Chopper input and output (armature) voltage - frequency of operation is constant, average armature voltage is varied by varying on-time of the switch; c. Current controlled operation of the chopper - voltage across armature changes when armature current reaches any of the two borders around reference current (switching frequency is variable). HIGH PERFORMANCE DRIVES  E Levi, 2001 5 Summarising the introductory considerations, related to a separately excited DC motor drive, accurate speed or position control requires that instantaneous torque of a motor can be controlled instantaneously. This yields so-called high performance. In a separately excited DC motor high performance is easily achievable, since flux in the machine is influenced solely by the excitation current. Change of armature current does not effect the value of the excitation flux. Hence, if the excitation current is constant, then the torque of the machine can be controlled instantaneously with armature current, provided that instantaneous change of armature current is possible. It is said that the torque and flux control are decoupled (or independent one of the other). Such a situation arises in a separately excited DC machine by virtue of its construction: commutator of the machine places magnetic axis of the armature winding in quadrature to the excitation winding magnetic axis, so that variation of armature current has no impact on excitation flux. Flux is therefore controllable solely by excitation current and, with constant excitation current, torque is controllable directly with armature current. Two currents of the machine are therefore used for independent control of flux and torque. Instantaneous armature current change can be approximately achieved by using a power electronic converter, whose output voltage is determined with closed loop armature control (i.e. the converter, which is inherently a voltage source, is controlled using closed loop current control, so that it behaves as a current source). torque limit T e Φ f 0 Φ f ω * ω 0time Fig. 1.4 - Dynamic behaviour of current-fed separately excited DC motor drive: Initial excitation and rapid acceleration in constant flux region. One important consideration, related to current control in general, is how close to reality the assumption that current can be changed instantaneously is. Suppose that the machine operates at a certain speed in steady-state and that the speed reference is increased in a step-wise manner, leading to instantaneous step increase in torque reference and in the armature current reference. Speed will for a short period of time remain unchanged, since time constant that governs speed change is much larger than the electrical time constant. Hence induced emf in the armature can be regarded as constant for the period of time under consideration. Since armature current needs to be increased, the chopper will be turned on so that the full chopper input DC voltage will appear across the armature. From (1.5) it follows that VERiL d idt DC a a a a −= + / (1.7) This equation shows that armature current will increase exponentially and that the rate of increase will depend on the difference between the input DC voltage of the chopper and the induced emf. Higher this difference is, faster the change of current will be. As induced emf depends on the speed of rotation, the difference between the DC input voltage and the induced emf will be large at low speeds and small at high speeds. This means that rate of change of armature current will, first of all, never be instantaneous, and secondly, that it will be significantly higher at low speed than at high speed. The assumption of ideal current control is therefore never ever fully fulfilled. However, lower the operating speed is, closer to reality this assumption will be. HIGH PERFORMANCE DRIVES  E Levi, 2001 6 In order to enable satisfactory current control at all speeds, it is necessary to always have so-called voltage reserve. This means that the input DC voltage of the chopper has to be higher than the voltage required for steady-state rated speed, rated torque operation (i.e. rated armature voltage). Note that these considerations related to current control are valid not only for a separately excited DC motor discussed here, but for all the AC motor drives that will be discussed in what follows. The considerations of this section can be summarised in the following five statements: • High performance operation requires that the electro-magnetic torque of the motor is controllable in real time; • Instantaneous torque of a separately excited DC motor is directly controllable by armature current, since flux and torque control are inherently decoupled; • Independent flux and torque control are possible in a DC machine due to its specific construction that involves commutator with brushes, whose position is fixed in space and is in quadrature to the excitation flux; • Instantaneous flux and torque control require that the machine windings are fed from current controlled DC sources; • Current and speed sensing is necessary in order to obtain the feedback signals for real time control (current and speed are controlled in closed loop manner, with current control loop embedded within the speed control loop). . controllable enabling high performance of HIGH PERFORMANCE DRIVES  E Levi, 2001 3 the drive to be achieved. A schematic diagram of a high- performance DC drive. HIGH PERFORMANCE DRIVES  E Levi, 2001 1 1. WHAT IS A HIGH PERFORMANCE DRIVE? 1.1 GENERAL CONSIDERATIONS Equation