WO2011123440A1

WO2011123440A1 - Pm brushless motor drive circuit topology and control

Info

Publication number: WO2011123440A1
Application number: PCT/US2011/030318
Authority: WO
Inventors: Krishnan Ramu
Original assignee: Ramu, Inc.
Priority date: 2010-03-29
Filing date: 2011-03-29
Publication date: 2011-10-06
Also published as: US20110234134A1; US8541920B2; US20110234027A1; CN102844971A; EP2553795A1; EP2553807A1; EP2553795A4; WO2011126836A4; US8981705B2; WO2011126836A1; WO2011123443A1; CN102844971B

Abstract

An inverter for a permanent magnet brushless dc machine, having a permanent magnet rotor and a set of stator windings, applies the full dc voltage provided to the inverter to each phase of the machine.

Description

PM Brushless Motor Drive Circuit Topology and Control

This application claims priority to U.S. provisional application 61/318,506 filed March 29, 2010, the content of which is incorporated herein by reference.

Field of the Invention

The invention relates to an inverter controlled permanent magnet brushless direct current (PMBDC) motor drive system. More specifically, the invention relates to the realization of a new inverter and its control in conjunction with a PMBDC machine to improve the operation of the entire motor drive system.

Background of the Related Art

Permanent magnet brushless direct current (PMBDC) machines are well known in literature. Consider a three phase machine having three phase windings and a rotor with permanent magnets. The phase windings are balanced, in that they have the same number of turns per phase winding, and they are spatially phase shifted by 120 electrical degrees. The theory and operation of such machines are described in chapters 1 and 9 of the book, R.

Krishnan, "Permanent Magnet Synchronous and Brushless DC Motor Drives", CRC Press, 2009. The theory and operation of a three phase H- bridge inverter is common knowledge and described in the same book in chapters 2 and 10.

Consider the magnitude of a direct current (dc) input voltage to an inverter as V_sl, which can be supplied from either a battery source or a rectified alternating current (ac) source.

Assuming two phase windings of a machine are series connected, the instantaneous maximum voltage that is applied across a machine winding phase through a three phase inverter is 0,5 V_sl. If the current in a phase is I, then the input power per phase is 0.5V_slI. Since two phases conduct at any given time in a PMBDC motor drive, the maximum input power to the PMBDC machine is 2*0.5V_slI=V_slI.

Because the two windings are in series, they carry the same current I. From the dc link or input side, the instantaneous power supplied to the inverter and machine is the product of voltage V_sl and current I, expressed as V_slI . The instantaneous power is equal to the inverter input power and the machine input power, if losses in the inverter are ignored.

The inverter supplies each phase with 120 electrical-degrees-wide current in both positive and negative cycles; that is, the inverter supplies a bipolar or alternating current to the machine phase windings. Furthermore, an inverter phase leg has an ideal duty cycle of 2/3, meaning that it is active for that part of the time in the machine's ac cycle. For the remaining 1/3 of the cycle time, the inverter is in the off condition (i.e., it is not active). This is true for all three phases in the inverter.

The machine phase windings carry the alternating currents generated through the inverter, though the current of each winding is phase shifted from the others by 120 electrical degrees. The phase shift of the current in each winding is the same amount as the spatial phase shift between the winding phases, so as to produce a uniform and constant air gap torque and power.

Consider the winding resistive losses in the PMBDC machine and let the resistance per phase be R_s in units of Ohms. The total instantaneous winding power loss is equal to I²R_S per phase. Since two winding phases conduct in a PMBDC machine at any given time, the total resistive loss is 2I²R_S.

Summary of the Invention

Because the resistive loss of a permanent magnet brushless direct current (PMBDC) machine is 2I² R_s, the voltage applied to a phase may be increased without affecting the resistive loss. As the voltage applied to a phase is increased, the phase current may be correspondingly decreased without affecting the input power to the machine. However, the resistive loss decreases as the phase current decreases, resulting in higher operational efficiency of the motor. An improved inverter, for driving a PMBDC machine, that applies higher voltage and lower current to a phase, so as to reduce resistive loss, is an object of this invention.

Other objects of the invention include increasing the applied voltage across a machine phase to greater than half a direct current (dc) input voltage, decreasing conductive losses in inverter transistors, decreasing combined power losses, achieving high efficiency of a motor drive system, achieving higher speed of operation for constant torque operation, achieving alternating current and voltage control with higher phase voltages in the machine, achieving four quadrant operation, and increasing reliability and fault tolerance for the drive system. A control system to coordinate the operation of the motor drive system is a further objective of the invention. To achieve these objects, the inverter disclosed hereinafter produces an input voltage that is greater than or comparable to that of a dc input supply voltage.

These and other objects of the invention may be achieved by a motor drive system having: (1) a first capacitive element that stores energy so as to provide a first dc voltage supply and (2) a second capacitive element that stores energy so as to provide a second dc voltage supply. A first electrically conductive switch electrically connects in series with a winding of a single motor phase and the first capacitive element to form a first series circuit when the first switch conducts current. A second electrically conductive switch electrically connects in series with the motor phase winding and the second capacitive element to form a second series circuit when the second switch conducts current. The entire voltage of the first capacitive element is applied across the motor phase winding, less the voltage drop across the first switch and the voltage drop resulting from parasitic resistance in the first series circuit, when: (1) the second switch does not conduct current and (2) the second capacitive element is neither being charged nor discharged. The entire voltage of the second capacitive element is applied across the motor phase winding, less the voltage drop across the second switch and the voltage drop resulting from parasitic resistance in the second series circuit, when: (I) the first switch does not conduct current and (2) the first capacitive element is neither being charged nor discharged.

Moreover, the above-mentioned and other objects of the invention may be achieved by a motor drive system having: (1) a phase winding of a motor that applies electromotive force to a rotor of the motor; (2) a first capacitive element that stores energy so as to provide a first dc voltage supply; and (3) a second capacitive element that stores energy so as to provide a second dc voltage supply. A first electrically conductive switch electrically connects in series with the motor phase winding and the first capacitive element to form a first series circuit when the first switch conducts current. A second electrically conductive switch electrically connects in series with the motor phase winding and the second capacitive element to form a second series circuit when the second switch conducts current. The entire voltage of the first capacitive element is applied across the motor phase winding, less the voltage drop across the first switch and the voltage drop resulting from parasitic resistance in the first series circuit, when: (1) the second switch does not conduct current and (2) the second capacitive element is neither being charged nor discharged. The entire voltage of the second capacitive element is applied across the motor phase winding, less the voltage drop across the second switch and the voltage drop resulting from parasitic resistance in the second series circuit, when: (1) the first switch does not conduct current and (2) the first capacitive element is neither being charged nor discharged.

Furthermore, the above-mentioned and other objects of the invention may be achieved by a motor drive system having a first capacitive element that stores energy so as to provide a first dc voltage supply, a second capacitive element that stores energy so as to provide a second dc voltage supply, and first, second, third, and fourth electrically conductive switches each having an input current terminal and an output current terminal. A first terminal of the first capacitive element is electrically connected directly to the input current terminals of the first and second switches. A second terminal of the first capacitive element is electrically connected directly to a first terminal of the second capacitive element. A second terminal of the second capacitive element is electrically connected directly to the output current terminals of the third and fourth switches. The output current terminal of the first switch is electrically connected directly to the input current terminal of the third switch. The output current terminal of the second switch is electrically connected directly to the input current terminal of the fourth switch. A first terminal of a winding of a first motor phase is electrically connected directly to the second terminal of the first capacitive element and the first terminal of the second capacitive element. A second terminal of the first motor phase winding is electrically connected directly to the output current terminal of the first switch and the input current terminal of the third switch. A first terminal of a winding of a second motor phase is electrically connected directly to the second terminal of the first capacitive element and the first terminal of the second capacitive element. A second terminal of the second motor phase winding is electrically connected directly to the output current terminal of the second switch and the input current terminal of the fourth switch.

Still further, the above-mentioned and other objects of the invention may be achieved by a motor drive method that discharges energy stored by a first capacitive element into a single electrical phase of a motor, via a first electrically conductive switch, such that the entire voltage across the first capacitive element is applied across the motor phase winding, less the voltage drop across the first switch and the voltage drop resulting from parasitic resistance in a first series circuit comprising the first capacitive element, the first switch, and the motor phase winding. Additionally, the energy stored by the motor phase winding is discharged into a second capacitive element, via a second electrically conductive switch, such that the entire voltage across the motor phase winding is applied across the second capacitive element, less the voltage drop across the second switch and the voltage drop resulting from parasitic resistance in a second series circuit comprising the second capacitive element, the second switch, and the motor phase winding.

In accordance with the exemplary embodiment of the invention, a PMBDC machine drive system has a machine with permanent magnet (PM) rotor and a stator with three phase windings. The three phase stator windings are spatially shifted from each other by 120 electrical degrees and have a neutral connection amongst them. The induced electromotive forces (emfs) across machine phases are, ideally, trapezoidal waveforms with a constant region over 120 electrical degrees in both half cycles; a deviation from such ideality has no impact on the invention and objectives.

The PMBDC system has an H-bridge inverter to supply currents to the stator phase windings. The inverter's input dc voltage positive polarity is connected to the top rail of the inverter, and the input voltage's negative polarity is connected to the neutral of the machine stator phase windings. A dc link capacitor C₁ is connected between the positive and negative polarities of the input dc voltage. An additional dc capacitor C₂ is connected between the neutral and bottom rail of the three phase inverter subsystem.

A supply dc input voltage V_sl, operating in conjunction with the machine windings and diodes in the bottom half of the inverter bridge, charges capacitor C₂, thereby generating a voltage across it. The charging of capacitor C₂ happens whenever conducting transistors in the top half of the inverter bridge are turned off and the current in the machine phase operates to forward bias the diodes in the bottom inverter half. The terminal of capacitor C₂ tied to the neutral side of the machine side is charged to a positive polarity and the terminal of capacitor C₂ tied to the bottom rail side of the inverter is charged to negative polarity.

With this arrangement, the machine phase windings have available, on their positive cycle operation, a voltage equal to the dc input supply voltage V_sl, through the control of the transistors in the top half of the inverter bridge. On the negative cycle operation of the machine, phase windings are supplied a voltage equal to the voltage available across C₂, i.e., V_S2, through the operation of transistors in the bottom half of the inverter bridge.

When the conducting top transistor of a phase leg is turned off, the current in the phase corresponding to this transistor is taken over by the diode in the bottom of the phase leg, completing the current path via capacitor C₂. Likewise, when the conducting bottom transistor of a phase leg is turned off, the current in the phase winding is routed through the top diode of the phase leg, completing the current path via capacitor C₁ and charging it. This operation partially transfers the energy from the machine phase winding to a capacitor, and part of the energy is converted to power input for the machine. The voltages across capacitors C₁ and C₂ may be the same or different.

The drive system may use a standard H bridge inverter and a neutral connection of a three-phase machine, so as not to add additional cost beyond that of a conventional system. However, the drive system requires an additional capacitor C₂ and higher voltage ratings for transistors and diodes in the inverter, which do add an incremental cost to the total PMBDC motor drive system.

The drive system inverter and PMBDC motor disclosed herein reduce the conduction losses in the inverter transistors and diodes. The lower losses in the inverter cause lower total losses, resulting in higher efficiency. Further, the drive system provides higher peak power operation and higher speeds of operation at full load torque.

Brief Description of the Drawings

Figure 1 illustrates a related art inverter permanent magnet brushless direct current (PMBDC) motor drive system;

Figure 2 illustrates the operation of the PMBDC motor drive system of Figure 1 ;

Figure 3 illustrates an inverter PMBDC motor drive system; Figure 4 illustrates the operation of the PMBDC motor drive system of Figure 3;

Figure 5 illustrates the four quadrants of torque vs. speed and the operational domain available to the PMBDC motor drive system of Figure 3;

Figure 6 illustrates the unidirectional control operation of the PMBDC motor drive system of Figure 3;

Figure 7 illustrates the duty cycle of a positive-current conducting phase vs. current magnitude as a function of normalized stator phase resistance; and

Figure 8 illustrates an uninterruptable-supply power converter.

Detailed Description of the Invention

Figure 1 illustrates an inverter permanent magnet brushless direct current (PMBDC) motor drive system of the related art. A direct current (dc) supply voltage 100, obtained either from a rectifier connected to the utility supply or from a battery, is provided to an inverter 110. A filtering capacitor C₁ is connected in parallel with dc supply voltage 100, via a upper voltage rail 101 and a lower voltage rail 102, to smooth the voltage variations of dc supply voltage 100. The dc voltage existing across capacitor C₁ and provided to inverter 110 is expressed as V_sl.

Inverter 110 is a three-phase H bridge inverter and is electrically connected in parallel with capacitor C₁ via voltage rails 101 and 102. Inverter 110 has two transistors in each of three phase legs. Phase leg A includes transistors T₁ and T₄ electrically connected in series. A diode D₁ is connected in parallel with transistor T₁, such that the cathode of diode D₁ is electrically connected to transistor T₁'s collector and the anode of diode Di is electrically connected to the emitter of transistor T₁ A diode D₄ is similarly connected in parallel with transistor T₄. Phase leg B includes transistors T₃ and T₆ electrically connected in series, and phase leg C includes transistors T₅ and T₂ electrically connected in series. Each of transistors T₂, T₃, T₅, and T₆ has a corresponding diode D₂, D₃, D₅, and D₆ electrically connected in parallel with the respective transistor, in the same way that diode D₁ is connected to transistor T₁,

The current conduction in each of transistors T₁-T₆ and diodes Di-D^ is indicated by an arrow in Figure 1. Each of transistors T₁-Τ₆ conducts current from collector to emitter, and each of diodes D₁ -D₆ conducts current from anode to cathode.

The collectors of transistors T₁, T₃, and T₅ and cathodes of diodes D₁, D3, and D₅ are electrically connected to upper voltage rail 101. The emitters of transistors T₂, T₄, and T₆ and anodes of diodes D₂, D4, and D₆ are electrically connected to lower voltage rail 102. The gates of transistors T₁-T₆ are individually controlled to govern the flow of current through the respective transistor.

A PMBDC motor 105 has three phase stator windings with phases A, B, and C. Phase leg A controls the current flow through the windings of phase A, phase leg B controls the current flow through the windings of phase B, and phase leg C controls the current flow through the windings of phase C. One end of the stator winding for phase A is electrically connected to neutral N and the other end is electrically connected to the emitter of transistor T₁, the collector of transistor T₄, the anode of diode D₁, and the cathode of diode D₄. Similarly, one end of the stator winding for phase B is electrically connected to neutral N and the other end is electrically connected to the emitter of transistor T₃, the collector of transistor T₆, the anode of diode D₃, and the cathode of diode D₆. And similarly, one end of the stator winding for phase C is electrically connected to neutral N and the other end is electrically connected to the emitter of transistor T₅, the collector of transistor T₂, the anode of diode D₅, and the cathode of diode D₂.

The theory and operation of the PMBDC machine illustrated in Figure 1 are described in chapters 1 and 9 of the book, R. rishnan, "Permanent Magnet Synchronous and Brushless DC Motor Drives", CRC Press, 2009. The theory and operation of the three phase H- bridge inverter illustrated in Figure 1 are described in the same book, in chapters 2 and 10. Figure 2 illustrates the operation of the PMBDC motor drive system of Figure 1. The induced electromotive forces (emfs) of phases A, B, and C are denoted and e_cs, respectively. The induced emf of a phase is the instantaneous voltage induced across its corresponding phase winding. Each of phases A-C of the PMBDC machine has a trapezoidal, induced emf with a constant voltage magnitude of E_p, both in the phase's positive and negative 120° cycles, for some period of a 360° cycle and a ramp voltage rising from -E_p to +Ep or falling from +E_p to -E_p during the remaining period of the 120° cycle. The induced emfs of each of phases A-C are electrically phase shifted from each other by 120°. The induced emf magnitude Ep is expressed as:

where N is the number of conductors in series per phase, v is the velocity, 1 is the length of the conductor, r is the radius of the rotor bore, ω_m is the angular velocity, and B is the flux density of the field in which the conductors are placed. This flux density is solely due to the rotor magnets. The product (Blv), also expressed as has the dimensions of flux and is directly proportional to the airgap flux φ_g expressed in equation (2):

The product of flux and number of conductors in series has the dimension of flux linkage and is denoted by λp. Since the product of flux and number of conductors in series is proportional to phase flux linkage by a factor of , the product is hereafter referred to as

modified flux linkage.

The electromagnetic torque is given by:

The instantaneous induced emfs, having units of volts, may be expressed as:

where the functions f_es(θ_r) , f_bs(θ_r) and f_cs(θ_r) have the same shape as e_as, and with a maximum magnitude of ±1.

The electromagnetic torque, having units of N-m, may be expressed, after substituting for induced emfs from equations (4, 5 and 6) into equation (3), as:

Currents are generated when the functions f_as(G,), f_bs(θ_r) and f_cs (θ_r) attain a value of ±1 in PMBDC motor drives. Only two of the three phases are excited, with current going into one (with positive function value of 1 , say for phase A) and the current coming out of the other one of the excited phases(with the negative function value of -1 , say for phase B), such that the electromagnetic torque becomes:

For an example in which phases A and B are excited, if the current in phase A is positive, then the current in phase B is negative. The currents in phases A and B are equal in magnitude because they are produced by turning on transistors T₁ and T₆, and the activation of transistors T₁ and T₆ causes phase A winding and phase B winding to be connected in series against dc supply voltage 100. Denoting the current in both phases as I_p, the torque equation may be expressed as:

The air gap power acquired from summing the products of the induced emfs of the phases, with their respective currents, at any time is expressed by the equation:

Figure 2 illustrates the phase-induced emfs, respective phase currents, respective air gap power for each phase, and total air gap power for the machine as a function of electrical rotor position.

Suppose the related art is maintained the same, as shown in Figure 2, except that the induced emf is increased by a factor of k for all phases and the applied currents from the inverter is reduced by a factor of 1/k for all phase currents. The induced emfs are increased by increasing the number of turns in the phases. The operation with such scaling is illustrated in Figure 4, which is very similar to Figure 2, and all the variables and their symbols mean the same things in both figures . The consequence of such an operation yields the torque and air gap power as:

Nothing has changed in the air gap power or in the electromagnetic torque by increasing the induced emf and decreasing by the same proportion the applied currents to the machine phases between the scaled approach and the related art system. The induced emf for the scaled approach may be increased to match that the supply voltage, and the phase currents can be decreased by the same proportion to which the induced emfs are increased.

Usually the alternating current (ac) or dc supply voltage is a fixed value and not subject to change in practical applications. However, the dc supply voltage to the inverter may be increased by boosting the rectified ac voltage or battery voltage by a boost converter placed between the dc supply and the inverter. This modification introduces an additional power converter, which decreases system efficiency and increases the number of power electronic components and system cost.

Thus, it would be desirable to make the same battery voltage or rectified ac output voltage available to one phase instead of two phases in series, such that the available voltage to the machine phase becomes doubled compared to the related art system of Figure 1. In the system of Figure 1 , two phase windings are in series and their currents are equal but the applied voltage to each phase winding from the dc supply voltage is half the dc supply voltage. The subject matter disclosed herein provides a unique way of applying the entire dc supply voltage to each phase of the machine, without an additional power converter, such as a boost converter.

Figure 3 illustrates an inverter controlled P BDC motor drive system. Components of the drive system illustrated in Figure 3 that are common to those of the drive system of Figure 1 are identified with the same reference characters in each drawing.

A dc supply voltage 100, obtained either from a rectifier that is connected to a utility supply or from a battery, is provided to an inverter 110. A filtering capacitor C₁ is connected in parallel with dc supply voltage 100, via an upper voltage rail 101 and an intermediate voltage rail 104, to smooth the voltage variations of dc supply voltage 100. A capacitor C₂ is electrically connected between intermediate voltage rail 104 and a lower voltage rail 103. The dc voltage existing across capacitor C₁ is expressed as V_sl, and that existing across capacitor C₂ is expressed as V_s2. Ideally, V_sl is the same voltage as supply dc voltage 100.

Inverter 110 is a three-phase H bridge inverter and is electrically connected in parallel with the series connection of capacitors C₁ and C₂, via voltage rails 101 and 103. Inverter 110 has two transistors in each of three phase legs. Phase leg A includes transistors T₁ and T₄ that are electrically connected in series. A diode Di is connected in parallel with transistor T₁, such that the cathode of diode Di is electrically connected to transistor T₁ 's collector and the anode of diode D₁ is electrically connected to the emitter of transistor T₁. A diode D₄ is similarly connected in parallel with transistor T₄. Phase leg B includes transistors T₃ and T₆ that are electrically connected in series, and phase leg C includes transistors T₅ and T₂ that are electrically connected in series. Each of transistors T₂, T₃, T₅, and T₆ has a corresponding diode D₂, D3, D₅, and D₆ electrically connected in parallel with the respective transistor, in the same way that diode Dj is connected to transistor T₁

The current conduction in each of transistors Tj-T₆ and diodes D₁-D₆ is indicated by an arrow in Figure 1. Each of transistors T₁-T₆ conducts current from collector to emitter, and each of diodes Di-D₆ conducts current from anode to cathode.

The collectors of transistors T₁, T₃, and T₅ and cathodes of diodes D₁, D₃, and D₅ are electrically connected to upper voltage rail 101. The emitters of transistors T₂, T₄, and T₆ and anodes of diodes D₂, D₄, and D₆ are electrically connected to lower voltage rail 103. The gates of transistors T₁- T₆ are individually controlled to govern the flow of current through the respective transistor.

A PMBDC motor 105 has three phase stator windings with phases A, B, and C. Phase leg A controls the current flow through the windings of phase A, phase leg B controls the current flow through the windings of phase B, and phase leg C controls the current flow through the windings of phase C. One end of the stator winding for phase A is electrically connected to neutral N and the other end is electrically connected to the emitter of transistor T₁, the collector of transistor T₄, the anode of diode Di, and the cathode of diode D4. Similarly, one end of the stator winding for phase B is electrically connected to neutral N and the other end is electrically connected to the emitter of transistor T₃, the collector of transistor Τ₆, the anode of diode D3, and the cathode of diode D₆. And similarly, one end of the stator winding for phase C is electrically connected to neutral N and the other end is electrically connected to the emitter of transistor T₅, the collector of transistor T₂, the anode of diode D₅, and the cathode of diode D₂. Neutral N is electrically connected to intermediate voltage rail 104. Consider the operation of phase A as characteristic of that for each of phases A-C.

Turning on transistor T₁ applies voltage V_sl to phase A winding, causing a positive current to flow from upper voltage rail 101 through transistor Tl and phase winding A to intermediate voltage rail 104. If the induced electromotive force (emf) across phase winding A is positive, positive power is supplied to phase winding A, resulting in positive torque and positive speed (CW). As positive power is applied to phase winding A, energy is transferred from dc supply voltage 00 to motor 105. Motor 105's forward motoring operation is defined by a first quadrant I of the torque versus speed characteristics illustrated in Figure 5.

The four quadrant operation of the PMBDC motor drive is shown in Figure 5. The x axis of the illustrated coordinate system denotes rotor speed, and the y axis denotes the applied torque. Particularly, positive x axis 200 corresponds to positive, forward, or clockwise rotation in speed while the negative x axis 210 corresponds to negative, reverse, or counterclockwise rotation in speed. Positive torque corresponds to positive y axis 205, and negative torque corresponds to negative y axis 215.

When transistor Tj is turned off to control the current in phase A, energy stored by phase winding A is conveyed by diode D₄ to capacitor C₂, which is charged to a voltage of V_s2.

During this time, energy is transferred from motor 105 to capacitor C₂. The applied voltage across phase winding A is -V_s2 at this time and the torque is positive, as the current and speed are still positive; therefore, the power output is positive. This operation also corresponds to quadrant I in Figure 5.

Suppose the speed of motor 105 is positive and transistor T₄ is turned on to circulate a current from C₂ to phase winding A in a negative direction while the induced emf is positive (i.e., with the positive polarity of the induced emf existing at the collector of transistor T₄ and the negative polarity existing at neutral N). This action will build a large current in phase winding A and, in order to keep within safe current limits, transistor T₄ will have to be turned off. Then, the current in phase winding A will find a path through diode Dj to charge capacitor C₁, thereby transferring energy from motor 105 to dc supply source 100, During this time, the induced emf is positive and current is negative in phase winding A, with the result that motor 105 produces a negative torque (i.e., generative torque and hence negative power). This operation corresponds to quadrant IV of Figure 5. Hence, for positive or forward rotation of the rotor, quadrants I and IV correspond to forward motoring and forward regeneration.

Likewise quadrant I is replaced by quadrant III and quadrant IV replaced by quadrant II for reverse motoring and reverse regeneration, respectively. For brevity, those operations are not elaborated here as they can be deduced and derived from the foregoing understanding of quadrant I and IV operations using the inverter. This description of operation proves that the motor drive system of Figure 3 provides full dc bus voltage V_sl to each phase winding during its positive cycles. Also the description proves that the motor drive system of Figure 3 lends itself to four quadrant operation.

In addition, the motor drive system of Figure 3 makes possible full dc bus voltage application to the machine phases while creating an additional voltage source for higher degree of freedom in operation. This is made possible by cycling energy, from the turn off of currents in the positive half cycles of the phases, to charge capacitor C₂, resulting in dc voltage source V_s2. DC voltage source V_s2 is applied during the negative half cycle operation of the machine phases with the full magnitude of V^ being applied to the phases of motor 105. The magnitude of V_s2 is comparable to V_sl .

The voltage available for operation in quadrants I and IV is V_s1 and in quadrants of II and III is V_s2. The fact that V_s2 is derived from V_sl through the motor drive system of Figure 3 during the positive half-cycle turn off of the phases makes it a somewhat dependent source. Source voltage V_s2 can be controlled by the inverter operation and its application to machine phase windings increases the phase root-mean-square( rms) (or effective) voltage with the result that the torque and power harvested out of the machine is increased. The phase voltage of the motor drive system of Figure 3 is twice that of the motor drive system of Figure 1.

Only one dc input supply voltage, V_sl, is available for the motor drive system of Figure 3. With that, another voltage source, V_s2, for operating motor 105 is generated using capacitor C₂ and the bottom diodes of inverter 110. In the process, a. separate buck or boost or buck-boost circuit is not required to create the additional voltage source. Therefore, the motor drive system of Figure 3 does not require additional devices to achieve higher phase voltages, but uses only the devices that exist in inverter 110.

Further, no additional charging of capacitor C₂ takes place without affecting, by controlling the magnitude of the current in the machine, the current control using pulse width modulation and resulting torque control. Two voltage sources, V_sl and V_s2, being available to control and generate torque in PMBDC motor 105 endows many choices in the control and maximization of efficiency in motor 105. The motor drive system of Figure 3 supports the operation of the PMBDC machine with mostly unidirectional current, using only source voltage V_s1 in the positive cycle, or partly bidirectional current control using both sources, V_sl for positive half cycle currents and V_s2 for negative half cycle currents, as illustrated in Figure 4.

With the drive control system of Figure 3, the phase voltage (i.e., the voltage applied across each phase winding) is the entire supply voltage V_sl. By contrast, the motor drive system of Figure 1 has two phase windings in series such that each phase is supplied with half the input voltage (i.e., 0.5V_sl). Implications of this feature are described below.

Consider the PMBDC machine drive system of Fig. 1 in relation to that of Fig. 3. The phase current requirement for an input power of V_slI is calculated as follows and given in Table 1.

For only unidirectional current in the machine phases of the system of Figure 3, the current requirement is the same as for the system of Figure 1. For the unidirectional current operation of the system illustrated in Fig. 3, which operation is illustrated in Figure 6, only positive half-cycle currents flow in the phase windings of motor 105. The positive-cycle current operation of any phase lasts only for 120°, which is one third of the full cycle of a phase voltage or current having 360 electrical degrees. Therefore, the phase currents exist for only 1/3 of the time and have a duty cycle d of 1/3. The rms value of a constant current I_p having a duty cycle d is 1/(d)^1/2. Therefore, the rms current through a phase winding during a duty cycle of 1/3 I_p/(3)^1/2.

The motor drive system of Figure 1 has phase conduction for both positive and negative half cycles for electrical 120° in each of these cycles. Therefore, the phase currents are present for electrical 240° which constitutes 2/3 of the entire phase cycle . Thus, the duty cycle of the phase currents is 2/3 and the rms current is calculated from current magnitude divided by the square root of the duty cycle, which is expressed as I_p (2/3)^{1 2}.

The difference in the rms phase winding currents of the systems of Figures 1 and 3 are also reflected in the average phase winding currents for these two systems. Thus, the current rating of the motor drive system illustrated in Figure 3 is lower than that for the system of Figure 1.

The motor drive system of Figure 1 cannot provide unidirectional current in the phase windings of motor 105. The reason for this is that the phase windings are connected in series and for a positive current to flow in one phase winding the flowing current must come from the series-connected phase winding. But the current flowing into the one phase winding is positive while the current flowing out of the other phase is negative. Because the current in one phase winding must be equal and opposite to the current in the other phase winding of the series connected phase windings, it is impossible to have unidirectional currents. Therefore, unidirectional currents cannot be applied to the phase winding of the motor drive system illustrated in Figure 1.

As illustrated in Figure 6, the induced emf of a phase is 2E_P for the system of Figure 3, whereas the induced emf for a phase of the system illustrated by Figure 1 is E_p, as illustrated in Fig. 2. Since two phase windings are in series in the system of Figure 1 , the total induced emf facing the supply input voltage is twice the induced emf per phase, which then is equal to 2Ep. For the system illustrated in Figure 3, the supply dc input voltage faces only one induced emf per phase at any time. Assuming the supply voltage is the same for the systems of Figures 1 and 3, the same dc input voltage supply should be able to support the same induced emf facing it, which is 2E_P. Thus, the number of turns for a phase of the system illustrated by Figure 3 may be doubled with respect to those of the system of Figure 1, so as to achieve the same amount of induced emf facing the voltage supply. With equal currents of I_p being drawn for each phase in both the systems of Figures 1 and 3, the air gap power is the same, as illustrated in Figures 2 and 6, and there is no difference in power output or torque for the two systems

For bipolar current in the machine phases of the system illustrated in Figure 3, the current in the phases is 0.5I_P, which is 50% less than that within the system of Figure 1. This advantage in current rating for the system of Figure 3 is due to the doubling of the phase voltage available to the phase windings. The doubling of the available phase voltage can also be used to extend the operational speed range of a motor, as described later.

Assume that unidirectional current plays a major role in torque generation for the system of Figure 3 and that there are no bidirectional currents. In this case, the number of winding turns in the system of Figure 3 has to be twice that of the system of Figure 1 to achieve the same induced emf. Further assume that the copper volume and cost is matched in both systems. From these assumptions, the information within Table 2 may be derived for the systems of Figures 1 and 3.

For a bipolar drive, the phase currents are 1/2 for the system illustrated by Figure 3. The subscript p is deleted hereafter, for simplicity, from current I. The total stator resistive loss for any two conducting phase windings, for example phases A and B, with phase A current I_a and phase B current I_b , is:

And this resistive loss is the same as that for the system illustrated by Figure 1. Thus, for bipolar operation, the systems of Figures 1 and 3 have the same resistive loss. However, the inverter losses are also an important consideration for improving the efficiency of a machine.

Consider the transistor conduction loss in the inverter. The switching losses are usually smaller compared to the conduction loss and, hence, are not considered.

Let V_t designate the conduction voltage drop in each transistor of the system of Figure 3, and let each phase transistor carry only half the rated current to produce the equivalent power produced by the system of Figure 1. Then, the conduction loss P_t for the two conducting phases at any given time in the system of Figure 3 is:

In the system illustrated by Figure 1 , two transistors conduct to carry the same load current I at any given time, resulting in two transistor conduction losses P_to given by:

Therefore, the ratio of conduction loss between the systems of Figures 1 and 3 is:

Thus, the system of Figure 3 reduces the conduction power loss by 50% with respect to that of the system of Figure 1. This is on the order of 0.5 kW to 1kW for an electric vehicle hybrid drive. And due to the lower conduction loss, the cooling requirement for the inverter is reduced, resulting in higher thermal robustness and reliability for the inverter.

The above description assumes that the phase voltage for the system of Figure 3 is doubled with respect to that of the system of Figure 1 and that each system provides the same base speed and base torque. An attractive parametric study is that in which the phase voltage at base speed for the system of Figure 3 is something other than twice, such as less than twice, that of the system of Figure 1 .

Let the applied phase voltage for the system of Figure 3 be expressed as:

where V_a is the applied phase voltage for the system of Figure 1. At base speed, the applied phase voltage is equal to Vdc/2, where Vd_C is the dc supply voltage input to the inverter. Factor k varies the applied voltage from 2V_a to 2kV_a or kvdc through the inverter. Accordingly, the system of Figure 3 has to have a base voltage (i.e., at base speed) of 2kV_a and, hence, the number of winding turns has to be multiplied by 2k as seen from emf equation (18) .

= (2k)N_{s (] 8)}

Keeping the copper volume constant, for achieving a constant copper cost between the systems of Figures 1 and 3, results in a conductor cross section a<s of:

where a is the cross section of the conductor in the system of Fig. 1. The resistance of the machine of the system of Figure 3 becomes: 2 (20) where R_s is the per phase resistance of the system of Figure 1. The copper loss in the system of Figure 1 , in which a factor of I is introduced, is:

The ratio of copper losses between the systems of Figures 1 and 3 is expressed as;

But 7_ac = I_bc = I (in system of Figure 1 ) (23) And

Letting

which gives way to:

The resistive loss in the system of Figure 3 may be expressed as:

And that in the system of Figure 1 may be expressed as:

Thus, the systems of Figures 1 and 3 have the same copper loss under the above- described circumstance.

The ratio of inverter conduction power losses is:

where P_tn is the inverter conduction losses with the system of Figure 3 and Pi_c is the inverter conduction losses with the system of Figure 1. Some sample calculations are supplied in Table 3 to provide a feel for the implications of varying k, with respect to the number of turns per phase winding, resistance per phase, phase current, and ratio of conduction losses for the systems of Figures 1 and 3. N_s is the number of turns per phase, R_s is the resistance per phase, and I is the per phase current in the system of Figure 1.

Based on these numerical examples, the following inferences can be made:

1. As k increases, decreases inversely.

2. Reserve voltage available to a machine phase is (v_rfe - kv_dc ) = (l - k)v_dc, , and this increases with decreasing k.

3. The reserve voltage can be used to obtain a speed of operation that is greater than that of the base speed (at which machine delivers rated or base torque).

4. The new base speed for the system of Figure 3 is 1 k in per unit. For example:

Table 4 exemplifies that a very large extension of base speed, at which base torque is obtained, may be achieved by changing the value of k in a design. In Table 4, p.u. is the normalized speed, which is non dimensional and limited to low values, such as 1 or 2 or 3. For example, speed in p.u. is equal to the actual speed in revolutions per minute (rpm) divided by base, which is usually the rated speed. Suppose that the actual speed is 1000 rpm and base speed is 2000 rpm, then the speed in p.u. is equal to 1000/2000= 0.5 p.u. This allows scaling to be done regardless of the variations in many variables such as in voltages, currents, torque, and power output by using base values for each of these variables to obtain manageable numbers of p.u. values. 5. Base speed extension also allows for higher range of flux-weakening operation, with the result that the system of Figure 3 provides an excellent fit for electric vehicle or hybrid motor drive.

The system of Figure 3 enables the operation of each phase independently of the other phases. For example, if the phases are given only unidirectional currents, then the total power and torque can be harvested from only one phase at any time, which has consequences on resistive losses, transistor conduction losses, and total losses. For example:

Resistive losses (31)

Transistor conduction loss = V_tI (32) P_ct = total losses using only unidirectional currents = V_tI + 4R_SI² Likewise:

P_ct = total drive losses of the system of Figure 1 = 2V_tI + 2R_SI (34) Equating losses for the systems of Figures 1 and 3 leads to:

2V_tI = 2R_sI²

which, in turn leads to the following condition:

or

V_t = 2R_sI (37)

Two scenarios arise in regard to power losses:

(i) If V₍ > 2R_SI , then the system of Figure 3 provides lower power losses.

(ii) If V_t < 2R_SI , then the system of Figure 1 provides lower losses. For example, in electric vehicle drives, V_t≥ 2R_SI and, thus, the system of Figure 3 will be better than that of Figure 1. Even if V_i < 2R_SI , the deficit in losses for the system of Figure

3 can be wiped out by using the energy in C₂ to make a bidirectional drive, which will reduce the positive current to provide high efficiency operation. Further, the commutation of positive current is faster in the system of Figure 3 because it charges C₂ directly from the phase current winding without involving another phase during its commutation. In addition, the system of Figure 3 provides greater fault tolerance for a failed machine phase or inverter transistor.

It may be stated that mostly single-phase operation with a small amount of energy harvested with another phase excitation is an important mode of operation for the system of Figure 3. That does not mean other modes of operation are excluded; other modes will play a role depending on the application of the motor drive system. The control principle for this is derived here where bipolar currents are used in machine phases but only small negative currents, as opposed to large positive currents, are employed. For illustration of this control strategy, consider the motor drive having a phase sequence of a,b,c. The conventional operation with bipolar currents is shown in Ref [R Krishnan, "Electric Motor Drives", Prentice Hall, 2001, pp. 523]. Consider a time instant when phase A current is positive and phase B current has to be negative. Let the phase currents be scaled up or down by a fact or of a and b for phases A and B, respectively, from their rated or base value of I and be denoted as:

A phase current I_a = aI, and

B phase current I = bI, where I is the rated current.

Then:

Desired torque equals the rated torque, which may be defined as l .u. (where p.u. is the per unit used as normalized non-dimensional unit and 1 p.u. indicates 100% or rated value of a variable, be it current, voltage, torque, power, etc.).

Equate this to conventional operation P_cu loss as

Include device conduction loss; then the equations become for the system of Figure 3:

Where v_t is the transistor conduction voltage drop in the inverter. And for the system of Figure 1 , the total loss is:

If a and b are fractions and do not add up to 1 , different results are possible. Both cases will be examined (i.e., a+b=l, and a+b<l) in the following. Consider:

The ratio of the total loss P_r between the systems of Figures 3 and 1 is given

where the normalized stator resistance R_sn is defined as:

Case (i): a + b = l and P_r = 1

leading to:

Solving for a:

Phase current for the positive conducting phase, identified as the first phase in Figure 4, versus I for various Rgn, ranging from 0.5 to 3 p.u., is shown in Figure 7. Figure 7 illustrates the current in positive phase, normalized units. From the value of a, the value of b can be extracted as b=l -a, and this is the normalized current in negative phase. The latter indicates the current to be supplied by C₂ (i.e., auxiliary capacitor). Figure 6 shows the operating points with the constraints that the sum of two phase currents equal one p.u. and the loss in each of systems 1 and 3 is the same. As may be determined from Figure 7, for R_sn=2.1 or higher values and for a<0.7, the losses in the system of Figure 3 are attractive compared to those of the system of Figure 1.

Case(ii): a + b < 1 means less than rated current being supplied, say for partial loads.

In this case:

Given the value of m, the value of a can be found and then the value of b for a given P_r (assume it is equal to 1, i.e., P_r=l). Note that the torque is assumed to be equal for both systems of Figures 1 and 3.

Case(iii): a = b , but α + £ < 1

Under the constraints described above, the system of Figure 3 has lower loss than that of the system of Figure 1, as the numerator of the above equation is smaller than the denominator. Such is advantageous for a partial load condition, which is what most drive applications encounter in practice. Consider the system of Figure 1 and assume that one of the machine phase windings has an open-circuit fault. In such a case, current in the other two phase windings cannot be controlled as required with electrical 120° in both their positive and negative half cycles.

Contrast this with the system of Figure 3 in which only the phase winding having an open- circuit fault is deprived of current. The remaining phases will get their respective currents, as shown in Figure 4, without any change as compared to a pre- fault condition. Similar reasoning can be applied to short-circuit fault cases in the machines, with the same advantage being maintained in the system of Figure 3. Similar reasoning is extendable to inverter transistor failures and the availability of currents to phases that are unaffected by the fault.

Although the currents in the above discussion have been considered as ideal currents, ideal currents are not possible to realize, but may be closely realized by resorting to pulse width modulation (PWM) control of the inverter transistors. The control of transistor inverters to shape currents is well known in text books, one of which is R. Krishnan, "Electric Motor Drives", Prentice Hall, 2001. Those knowledgeable in the art are familiar with PWM control of a transistor inverter.

Variations of the invention exist, such as provisioning the dc supply voltage from a battery or connecting an ac-rectified dc supply across C₂ instead of C[ within the system of Figure 3. The change in the supply input points supports negative half-cycle based

unidirectional current control, without affecting bidirectional current control. The change in the supply input points does not affect the torque or power output.

Consider a case in which the top capacitor C₁ in the system of Figure 3 is connected to a dc supply voltage, obtained from a rectified ac supply, and a battery is connected across C₂ with the illustrated polarity. If the ac supply fails, the dc input voltage to the capacitor C₁ is lost, but

C₁ can be charged from the battery across C₂ by using the bottom transistors, machine phase windings, and respective top diodes of the inverter. The battery can be kept charged when the ac supply is available, which is then rectified and fed to charge C₁. Then, the charge stored by C₁ can be transferred to C₂ using the machine phase windings, top transistors, and bottom diodes of the inverter. Therefore, this system provides an uninterrupted power supply to run the PMBDC motor drive when one of the power sources, either from a rectified ac supply or from a battery is absent. This arrangement of having the battery across C₂ allows the battery to be charged through the controlled flow of current in the machine windings. Therefore, battery charging is accomplished without additional components, such as a transistor, an inductor, and a circuit breaker that usually are required in a separate charger such as buck or boost charger.

Figure 8 illustrates such an uninterruptable power supply for a power converter. Figure 8 illustrates the same system as Figure 3, with the exception that a battery or dc supply is connected in parallel with capacitor C₂. With this arrangement, the uninterruptable operation of the system illustrated in Figure 3 may be achieved as described in the preceding paragraph.

By contrast to the system of Figure 1, the system of Figure 3 provides:

1. higher reliability and fault tolerance;

2. higher speed of operation;

3. lower conduction losses in an inverter;

4. higher efficiency in the machine; and

5. higher efficiency operation.

Moreover, a split phase dc is created without an additional buck or boost or buck-boost converter at the front-end. And four quadrant operation with a higher torque-speed envelope for transient operation and steady-state operation is achieved. Other benefits of the system of Figure 3 include:

1. full dc input voltage to the phase winding;

2. unidirectional current operation;

3. four quadrant operation; 4. bidirectional or bipolar current operation;

5. flexible operation to vary the magnitude of currents and voltages across the machine windings using pulse width modulation or other known means;

6. minimization of total losses that include stator resistive losses and transistor conduction losses;

7. extension of base speed by as much as two times with full rated or base torque;

8. extension of base speed can be varied from 1 to 2 at least;

9. fault tolerance is high in both the machine as well as in the inverter;

10. uninterrupted operation of a PMBDC motor drive when a power source fails; and

11. controlled current charging of a battery from a rectified ac supply without additional components,

The foregoing has been a detailed description of possible embodiments of the invention. Other embodiments of the invention will be apparent to those skilled in the art from

consideration of the specification and practice of the invention. Accordingly, it is intended that this specification and its disclosed embodiments be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.

Claims

What is claimed is:

1. A motor drive system comprising:

a first capacitive element that stores energy so as to provide a first direct current (dc) voltage supply;

a second capacitive element that stores energy so as to provide a second dc voltage supply;

a first electrically conductive switch for electrically connecting in series with a winding of a single motor phase and the first capacitive element to form a first series circuit when the first switch conducts current; and

a second electrically conductive switch for electrically connecting in series with the motor phase winding and the second capacitive element to form a second series circuit when the second switch conducts current, wherein:

the entire voltage of the first capacitive element is applied across the motor phase winding, less the voltage drop across the first switch and the voltage drop resulting from parasitic resistance in the first series circuit, when: (1) the second switch does not conduct current and (2) the second capacitive element is neither being charged nor discharged, and the entire voltage of the second capacitive element is applied across the motor phase winding, less the voltage drop across the second switch and the voltage drop resulting from parasitic resistance in the second series circuit, when: (1) the first switch does not conduct current and (2) the first capacitive element is neither being charged nor discharged.

2. The motor drive system of claim 1, wherein:

the first series circuit passes current in a first polarity through the motor phase winding when the first switch conducts current, the second switch does not conduct current, and the second capacitive element is neither being charged nor discharged, and the second series circuit passes current in a second polarity, opposite to the first polarity, through the motor phase winding when the second switch conducts current, the first switch does not conduct current, and the first capacitive element is neither being charged nor discharged.

3. The motor drive system of claim 1, further comprising:

an Nth electrically conductive switch, N being an integer greater than 1 , for electrically connecting in series with an Nth motor phase winding and the first capacitive element to form an Nth series circuit when the Nth switch conducts current; and

an Nth + 1 electrically conductive switch for electrically connecting in series with the Nth motor phase winding and the second capacitive element to form an Nth + I series circuit when the Nth + 1 switch conducts current, wherein:

the entire voltage of the first capacitive element is applied across the Nth motor phase winding, less the voltage drop across the Nth switch and the voltage drop resulting from parasitic resistance in the Nth series circuit, when: (1) the Nth + 1 switch does not conduct current and (2) the second capacitive element is neither being charged nor discharged, and

the entire voltage of the second capacitive element is applied across the Nth motor phase winding, less the voltage drop across the Nth + 1 switch and the voltage drop resulting from parasitic resistance in the Nth + 1 series circuit, when: (1) the Nth switch does not conduct current and (2) the first capacitive element is neither being charged nor discharged.

4. The motor drive system of claim 3, wherein:

the Nth series circuit passes current in a first polarity through the Nth motor phase winding when the Nth switch conducts current, the Nth + 1 switch does not conduct current, and the second capacitive element is neither being charged nor discharged, and the Nth + 1 series circuit passes current in a second polarity, opposite to the first polarity, through the Nth motor phase winding when the Nth + 1 switch conducts current, the Nth switch does not conduct current, and the first capacitive element is neither being charged nor discharged.

5. The motor drive system of claim 3, wherein an H-bridge inverter comprises the first, second, Nth, and Nth + 1 switches.

6. The motor drive system of claim 1, further comprising:

a first unidirectional current element that passes current in only one direction and is electrically connected in parallel with the first switch;

a second unidirectional current element that passes current in only one direction and is electrically connected in parallel with the second switch, wherein:

the first or second unidirectional current element discharges energy stored by the motor phase winding to the first or second capacitive element, respectively, when both the first and second switches are not conducting current.

7. The motor drive system of claim 1, further comprising:

a dc supply electrically connected in parallel with one of the first and second capacitive elements; and

a battery electrically connected in parallel with other of the first and second capacitive elements.

8. A motor drive system comprising:

a phase winding of a motor that applies electromotive force to a rotor of the motor; a first capacitive element that stores energy so as to provide a first direct current (dc) voltage supply;

a first electrically conductive switch for electrically connecting in series with the motor phase winding and the first capacitive element to form a first series circuit when the first switch conducts current; and

9. The motor drive system of claim 8, wherein:

10. The motor drive system of claim 8, further comprising:

11. A motor drive system comprising:

a second capacitive element that stores energy so as to provide a second dc voltage supply; and

first, second, third, and fourth electrically conductive switches each comprising an input current terminal and an output current terminal, wherein:

a first terminal of the first capacitive element is electrically connected directly to the input current terminals of the first and second switches,

a second terminal of the first capacitive element is electrically connected directly to a first terminal of the second capacitive element,

a second terminal of the second capacitive element is electrically connected directly to the output current terminals of the third and fourth switches,

the output current terminal of the first switch is electrically connected directly to the input current terminal of the third switch, the output current terminal of the second switch is electrically connected directly to the input current terminal of the fourth switch,

a first terminal of a winding of a first motor phase is electrically connected directly to the second terminal of the first capacitive element and the first terminal of the second capacitive element,

a second terminal of the first motor phase winding is electrically connected directly to the output current terminal of the first switch and the input current terminal of the third switch, a first terminal of a winding of a second motor phase is electrically connected directly to the second terminal of the first capacitive element and the first terminal of the second capacitive element,

a second terminal of the second motor phase winding is electrically connected directly to the output current terminal of the second switch and the input current terminal of the fourth switch.

12. The motor drive system of claim 11 , further comprising:

fifth and sixth electrically conductive switches each comprising an input current terminal and an output current terminal, wherein:

the first terminal of the first capacitive element is electrically connected directly to the input current terminal of the fifth switch,

the second terminal of the second capacitive element is electrically connected directly to the output current terminal of the sixth switch,

the output current terminal of the fifth switch is electrically connected directly to the input current terminal of the sixth switch, a first terminal of a winding of a third motor phase is electrically connected directly to the second terminal of the first capacitive element and the first terminal of the second capacitive element,

a second terminal of the third motor phase winding is electrically connected directly to the output current terminal of the fifth switch and the input current terminal of the sixth switch.

13. The motor drive system of claim 11 , further comprising:

14. A motor drive method comprising:

discharging energy stored by a first capacitive element into a single electrical phase of a motor, via a first electrically conductive switch, such that the entire voltage across the first capacitive element is applied across the motor phase winding, less the voltage drop across the first switch and the voltage drop resulting from parasitic resistance in a first series circuit comprising the first capacitive element, the first switch, and the motor phase winding;

discharging energy stored by the motor phase winding into a second capacitive element, via a second electrically conductive switch, such that the entire voltage across the motor phase winding is applied across the second capacitive element, less the voltage drop across the second switch and the voltage drop resulting from parasitic resistance in a second series circuit comprising the second capacitive element, the second switch, and the motor phase winding.