Brushless DC Motor Drive Schemes

While elaborate and expensive drive schemes are possible, in many applications simplifying assumptions are made that lead to readily implemented drive schemes that perform reasonably well. This section illustrates these simple drive schemes for two-and three-phase motors. The fundamental task for a motor drive is to apply current to the correct windings, in the correct direction, at the correct time. This process is called commutation, since it describes the task performed by the commutator (and brushes) in a conventional brush dc motor. The goal here is to develop an intuitive understanding, rather than discuss every nuance of every possible motor drive scheme. More detailed information can be found in references such as Leonhard (1985) and Murphy and Turnbull (1988). With this intuitive understanding, more complex drive schemes are readily understood.

.1 Two-Phase Motors

ILL. 71 Two-phase motor schematic.

Previously, torque and back-emf expressions are developed considering just one motor phase. When there is more than one phase, each individual phase acts independently to produce torque. Consider the two-phase motor illustrated in Ill. 71.

Power dissipated in the phase resistances produces heat; the phase inductances store energy but dissipate no power; and power absorbed by the back-emf sources EA and

EB is converted to mechanical power  . (Think about it: where else could it go?) Writing this last relationship mathematically gives

EAiA  EBiB  (10.1)

Here the back-emf sources are determined by the motor design, and the currents are determined by the motor drive. Because of the BLv law, the back-emf sources are linear functions of speed; that is, E =  , where k, the emf waveform, is a function of motor parameters and position. Substituting this relationship into Eq. (10.1) gives

kAiA  kBiB  T (10.2)

Thus, the mutual torque produced is a function of the back-emf waveforms and the applied currents. Most important, Eq. (10.2) applies instantaneously. Any instantaneous variation in the back-emf waveforms, or the phase currents will produce an instantaneous torque variation.

Equation (10.2) provides all the information necessary to design drive schemes for the two-phase motor. Since the back-emf waveforms are a function of position, it is convenient to consider Eq. (10.2) graphically. Making the simplifying assumption that the emf is an ideal square wave, Ill. 72 shows the back-emf waveforms, with that from phase B delayed by  /2 rad electrical with respect to phase A.

One-Phase-On Operation. Given the waveforms shown in Ill. 72, several drive schemes become apparent. The first, shown in Ill. 73, is one-phase-on operation where only one phase is conducting current at any one time. In this figure, the phase currents are superimposed over the back-emf waveforms and Eq. (10.2) is applied instantaneously to show the resulting motor torque on the lower axes. The overbars are used to signify current flowing in the reverse direction. Some important aspects of this drive scheme include the following:

Ideally, constant ripple-free torque is produced.

The shape of the back-emf of the phase not conducting at any given time (e.g., phase A over  /4  /4) has no influence on the torque production since the associated current is zero. The back-emf need only be flat when the current is applied. The smoothing of the transitions in the back-emf that exist in a real motor do not add torque ripple.

Neither phase is required to produce torque in regions where its associated back-emf is changing sign.

Each phase contributes an equal amount to the total torque produced. Thus each phase experiences equal losses and the drive electronics are identical for each phase.

Copper utilization is said to be 50 percent, since at any time only half of the windings are being used to produce torque; the other half have no current flowing in them.

The amount of torque produced can be varied by changing the amplitude of the current flowing in them.

ILL. 72 Square-wave back-emf waveforms for a two-phase motor.

Square pulses of current are required but not achievable in the real world, since the inductive phase windings limit the current slope to di/dt = v/L, where v is the applied voltage and L is the inductance. Using  t, this relationship can be stated in terms of position as di/  v  L). With either interpretation, the rate of change in current is finite, whereas Ill. 73 assumes that it is periodically infinite.

Two-Phase-On Operation. Following the same procedure used to construct Ills. 10.73 and 10.74 shows two-phase-on operation, where both phases are conducting at all times. Important aspects of this drive scheme include the following:

Ideally, constant ripple-free torque is produced.

The shape of the back-emf is critical at all times, since torque is produced in each phase at all times.

If either current does not change sign at exactly the same point that the emf does, negative phase torque is produced, which leads to torque ripple.

Both phases are required to produce torque in regions where their associated back-emf is changing sign.

Each phase contributes an equal amount to the total torque produced. Thus, each phase experiences equal losses, and the drive electronics are identical for each phase.

Copper utilization is 100 percent.

The amount of torque produced can be varied by changing the amplitude of the square-wave currents.

Impossible-to-produce square-wave currents are required.

For a constant torque output, the peak phase current is reduced by half compared with the one-phase-on scheme.

ILL. 73 One-phase-on torque production.

ILL. 74 Two-phase-on torque production.

The Sine-Wave Motor. A square-wave back-emf motor driven by square current pulses in either one- or two-phase-on operation as previously described represents what is usually called a brushless dc (BLDC) motor. On the other hand, if the back-emf is sinusoidal, the motor is commonly called a synchronous motor. Operation of this motor follows Eq. (10.2) also. However, in this case it is easier to illustrate torque production analytically. The key to understanding the two-phase synchronous motor is by recalling the trigonometric identity sin 2

 cos 2

 1.

Let phase A have a back-emf shape of kA  K cos  and be driven by a current iA  I cos  . If as before the back-emf of phase B is delayed by  /2 rad electrical from phase A, kB  K sin  , and the associated phase current is iB  I sin  . Applying these expressions to Eq. (10.2) gives

kAiA  kBiB  T

KI (cos 2  sin^2  )  KT = T (10.3)

Thus, once again the torque produced is constant and ripple-free. In addition, the currents are continuous, and only finite di/  is required to produce them. Just as in the square-wave case considered earlier, the currents must be synchronized with the motor back-emf. To summarize, important aspects of this driven scheme include the following:

Ideally, constant ripple-free torque is produced.

The shape of the back-emf and drive currents must be sinusoidal.

If both phase currents are out of phase an equal amount with their respective back-emf the torque will have a reduced amplitude but will remain ripple-free.

Each phase contributes an equal amount to the total torque produced. Thus, each phase experiences equal losses, and the drive electronics are identical for each phase.

Copper utilization is 100 percent.

The amount of torque produced can be varied by changing the amplitude of the sinusoidal currents.

The phase currents have finite di/  .

Based on the three examples just considered, it is clear that there are an infinite number of ways to produce constant ripple-free torque. All that is required is that the left-hand side of Eq. (10.2) instantaneously sum to a constant. The trouble with the square-wave back-emf schemes is that infinite d i/  is required. The torque ripple that results from not being able to generate the required square pulses is called commutation torque ripple. The trouble with the sinusoidal emf case is that pure sinusoidal currents must be generated. In all cases, the back-emf and currents must be very precise whenever the current is nonzero; any deviation from ideal produces torque ripple. For the square-wave back-emf schemes, position information is required only at the commutation points (i.e., 4 points per electrical period). On the other hand, for the sinusoidal back-emf case, much higher resolution is required if the phase currents are to closely follow the back-emf waveforms. Thus, simple and inexpensive Hall-effect sensors are sufficient for the brushless dc motor, whereas an absolute position sensor (e.g., an absolute encoder or resolver) is required in the sinusoidal current drive case.

Despite the fact that the square-wave back-emf schemes inevitably produce torque ripple, they are commonly implemented because they are simple and inexpensive.

In many applications, the cost of higher performance cannot be justified.

ILL. 75 H-bridge circuit.

H-Bridge Circuitry. Based on Figs. 10.73 and 10.74, it is necessary to spend positive and negative current pulses through each motor winding. The most common circuit topology used to accomplish this is the full-bridge or H-bridge circuit, as shown in Ill. 75. In the figure, Vcc is a dc supply, switches S1 through S4 are commonly implemented with MOSFETs or IGBTs (though some still use bipolar transistors because they are cheap), diodes D1 through D4, called freewheeling diodes, protect the switches by providing a reverse current path for the inductive phase current, and

R, L, and Eb represent one motor phase winding.

Basic operation of the H bridge is fairly straightforward. As shown in Ill. 10.7 6a, if switches S1 and S4 are closed, current flows in the positive direction through the phase winding. On the other hand, when switches S2 and S3 are closed, current flows in the negative direction through the phase winding, as shown in Fig.

10.7 6b. In either case, the current climbs exponentially according to the L/ R time constant and reaches the value of ( Vcc  E b)/ R if the switches are left closed long enough.

ILL. 76 H-bridge circuit: ( a) positive current conduction, and ( b) negative current conduction.

ILL. 77 Current decay in an H-bridge circuit: ( a) switches S1 and S4 open, and ( b) only switch S4 open.

Turn-Off Behavior. What takes more work to understand is the turn-off behavior of the H bridge and how phase current is controlled to limit its magnitude. Cur-rent control is accomplished by chopping, that is, by employing PWM techniques.

Because of its fundamental nature, PWM is discussed at length later. For the time being, consider the turn-off behavior of the H bridge. This behavior is guided by the fundamental behavior of inductors. That is, that current cannot change instantaneously but must be continuous, and the larger the voltage across an inductor, the faster the current through it will change.

To start, let the phase current be a constant Im with switches S1 and S4 closed, as shown in Ill. 7 6a. Given these initial conditions, consider what happens when both switches are opened to bring the current back to zero. Now, since current no longer flows through S1 and S4, a negative voltage appears across the inductor because d i/ dt is negative. At the same time, the phase current continues to flow in the same direction because it cannot change instantaneously. The only path for current flow is through diodes D2 and D3, as shown in Ill. 7 7a. No current can flow through diodes D1 or D4. During this time, the voltage across the phase inductance is

L di / dt  Ri . Vcc  Eb (10.4)

which is clearly large and negative when i  0, Vcc  0, and Eb  0. As time progresses, the current decreases exponentially toward the negative value  (Vcc  E b)/ R. Upon reaching zero current, the diodes turn off, the energy in the inductor 0. 5Li 2 is returned to the supply, and the circuit rests. If the circuit lacks freewheeling diodes, the switches are destroyed in an attempt to provide a current path for the inductor current.

In some situations, just one of the two switches is opened. To illustrate this action, assume the conditions shown in Ill. 7 6a and open only switch S4; let S1 remain closed. The path for decaying current flow in this case is through D3 and S1, as shown in Ill. 7 7b, giving an inductor voltage of

L di / dt =. Ri - Eb (10.5)

which is much smaller in magnitude than that given in Eq. (10.4) because  Vcc is missing. Hence, the inductor current decays much more slowly in this situation.

Later, this turn-off mode will prove helpful in implementing PWM current control.

Switch Current. A major task in drive-circuit design is to size the switches, that is, to determine their rms currents. In the H bridge, switches S1 and S4 carry the positive portion of the phase current, whereas switches S2 and S3 carry the negative portion of the phase current. Because of this division, the rms switch current is less than the RMS phase current. As illustrated for the two-phase-on scheme in Ill. 78, the RMS value of the switch current is easily shown to be 100/2 _  70.7 percent of the RMS phase current. Though not shown, the same ratio applies to the one-phase-on scheme.

Summary. Important aspects of the H-bridge circuit include the following:

Bidirectional current flow is easily achieved.

Given that the back-emf and current have the same sign in Figs. 10.73 and 10.74, the back-emf acts to fight the increase in phase current amplitude during turn-on.

In the one-phase-on drive scheme in Ill. 73, the back-emf and current have the same sign at the turn-off points. Thus, by Eq. (10.4), the back-emf acts to assist the decrease in phase current during turn-off.

In the two-phase-on drive scheme in Ill. 74, the back-emf and current have opposite signs immediately after the turn-off points. Thus, the back-emf acts to fight the decrease in phase current during turn-off. Thus, the back-emf hinders commutation at both turn-on and turn-off in the two-phase-on drive scheme.

At no time can vertical pairs of switches (i.e., S1 and S2 or S3 and S4) be closed simultaneously. If this happens, a shoot-through fault occurs where the motor sup-ply is shorted. In implementation, a short delay is often added between commutations to guarantee that no shoot-through condition occurs.

For the square-wave back-emf schemes, the rms switch current is equal to 70.7 per-cent of the RMS phase current.

For two-phase motors, H bridges are required, giving a total of eight switches to be implemented by power electronic devices.

ILL. 78 Phase and switch currents for two-phase-on operation.

.2 Three-Phase Motors

Three-phase motors overwhelmingly dominate all others. The exact reasons for this dominance are not known, but the historical dominance of three-phase induction and synchronous motors and the minimal number of power electronic devices required are likely contributing factors. The addition of a third phase provides an additional degree of freedom over the two-phase motor, which manifests itself in more drive schemes and terminology. For example, wye (Y) and delta  ) connections are possible.

In three-phase motors, the power balance equation leads to

kAiA  kBiB  kCiC  T (10.6) where kC and iC are the back-emf shape and current, respectively, of the third phase.

By construction, the back-emfs of each phase have the same shape but are offset from each other by  /3 rad electrical, or 12  deg electrical. The back-emf shapes for the ideal square-wave back-emf motor are shown in Ill. 79.

ILL. 79 Square-wave back-emf waveforms for a three-phase motor.

Three-Phase-On Operation. The most obvious drive scheme for the three-phase motor is to extend the two-phase-on operation of the two-phase motor, as shown in Ill. 80. Here each phase conducts current at all times and contributes equally to the torque at all times. At each commutation point, one phase current changes sign and the others remain unchanged. The important aspects previously listed for the two-phase-on two-phase motor apply here as well.

Despite the conceptual simplicity of this drive scheme, it is hardly ever implemented in practice because three H bridges as shown in Ill. 75 are required, one for each phase winding. The resulting 12 power electronic devices make the drive expensive compared with other drive schemes.

ILL. 80 Three-phase-on operation.

ILL. 81 Y-connected three-phase motor and drive circuitry.

ILL. 82 Torque production in a Y-connected three-phase motor.

Y Connection. Just as the Y connection is a popular configuration in three-phase power systems, it is also the most common configuration in three-phase brushless PM motors. As shown in Ill. 81, the center or neutral of the Y is not brought out, each external terminal or line is connected to a half-bridge circuit, and the collection of three half bridges is called a three-phase bridge. In this way, an H bridge appears between each set of terminals. Only six power electronic devices are needed for the switches in the three-phase bridge, as opposed to eight for a two-phase motor. The supply voltage is applied from line to line through the switches rather than from line to neutral. Compared with the three-phase-on case, the supply voltage works against two back-emf sources to force current into the motor. Furthermore, independent control of the phase currents is not possible since Kirchhoff's current law, IA  IB  IC  0, must be satisfied.

Torque production follows the idea that current should flow in only two of the three phases at a time, and that there should be no torque production near the back-emf sign crossings. Ill. 82 shows the phase currents superimposed on the back-emfs. Each phase conducts currents over the central  /3 rad electrical of each half cycle. The resulting torque is shown at the bottom of the Ill. with the letter designating the current polarities contributing to the torque. At each com-mutation point, one switch remains closed, one opens, another closes, and the rest remain open. There are six commutations per electrical period, and thus this drive scheme is often called a six-step drive (Murphy and Turnbull, 1988). The six numbered arrows shown in Ill. 81 illustrate these steps, as do the respective circled step numbers in Ill. 82.

Because only two phases are conducting current and contributing to torque production at any one time, the amplitude of the current must be 50 percent larger here than in the three-phase-on case, where all three phases contribute simultaneously.

When two phases are called upon to produce the same torque that three phases do, the current in each phase must be 3/2 as large, since (3/2)(2 phases)  (1)(3 phases).

As a result, if this drive scheme is implemented, the current equations must be modified to reflect the current waveforms shown in Ill. 82.

The RMS phase currents are required to produce a specified rated torque. Based on the preceding discussion, these currents must be increased in amplitude by a factor of 3/2.Moreover, the equations must reflect the RMS value of the phase currents, which is 2/3Ipeak, based on the waveforms shown in Ill. 82.Combining these factors, the phase current Iph becomes:

Iph  3/2 SQR-RT (2/3)  Is/3ns = SQR-RT (3/2) I2/ 3ns

where Is  slot current

ns  number of turns per slot for the six-step driven three-phase motor. Compared with the three-phase-on case, the RMS phase current is approximately 22 percent larger and the ohmic motor loss is 50 percent greater. Thus, while the Y connection minimizes the number of power electronic devices used, it does not minimize losses.

To summarize, important aspects of this drive scheme include the following:

Ideally, constant ripple-free torque is produced.

Only six switches are required, which is a minimum number.

Phases are not required to produce torque in regions where their associated back-emf is changing sign. Thus, the back-emf can be more trapezoidal than square.

Each phase contributes an equal amount to the total torque produced. Thus, each phase experiences equal losses, and the drive electronics are identical for each phase.

Copper utilization is 67 percent, since at any one time only two of the three phases are conducting current.

For the same output, ohmic motor losses are 50 percent greater than those in the three-phase-on drive scheme.

The amount of torque produced can be varied by changing the amplitude of the square-wave currents.

Impossible-to-produce 12  -wide square-wave currents are required. The inherent finite rise and fall time of the current creates torque ripple, commonly called commutation torque ripple.

Independent control of phase currents is not possible.

From IA  IB  IC  0, it can be shown that the phase currents cannot have any harmonics that are multiples of three, that is, tripl e-n or triplen harmonics.

Because phase windings appear in series, the supply voltage must be greater than the vector sum of the back-emfs at rated speed.

Delta Connection. The delta connection shown in Ill. 83 is the dual of the Y connection. This connection is not that popular because it has a major weakness, that being the additional ohmic motor loss and torque ripple due to circulating currents flowing around the delta. Three-phase power-system utility generators are never delta-connected for this reason. It is relatively easy to show that if the back-emf waveforms of each phase do not have exactly the same shape, are not exactly 12  out of phase with one another, or contain any triplen harmonics, circulating currents will flow around the delta. Because of this weakness, connected motors appear only in lower-performance motors at low-output power levels (e.g., in the fractional horsepower range), where their higher losses can be offset with lower material costs.

ILL. 83 Delta-connected three-phase motor and drive circuitry. Based on the preceding discussion, a motor having the ideal square-wave back-emf shape as shown in Ill. 79 cannot be connected in the delta connection because a square-wave back-emf motor has very high triplen harmonic content.

Given the nature of dual circuits, it is not surprising that swapping the current and back-emf waveforms of the Y connection gives a workable solution for the delta connection, as shown in Ill. 84. Creating a motor with 12  -wide square-wave back-emf waveforms is not difficult. Simply making the magnet arc narrower works, which results in the use of less magnet material.

To ease the explanation of the delta connection, the rising edges of the back-emfs and currents are aligned in Ill. 84. As shown, the back-emf of one phase is zero at all times. Each takes a turn at being zero for 6  . Because of this zero back-emf, the line current splits approximately equally through the remaining two phases, which conduct current in opposite directions. As before, the torque produced is given by applying Eq. (10.6). The lowercase letters under the torque curve signify the line currents during the respective commutation intervals. The line not given in each commutation interval is left floating electrically and is associated with the phase having zero back-emf. A comparison of these states with those of the Y connection in Ill. 82 show that the three-phase bridge circuit switches identically for both configurations. It is for this reason that the commutation logic in commercial driver ICs for small brushless motors works with either Y- or delta-connected motors.

ILL. 84 Torque production in a delta-connected three-phase motor. To summarize, important aspects of this drive scheme include the following:

Ideally, constant ripple-free torque is produced.

Only six switches are required, which is a minimum number.

Each phase contributes an equal amount to the total torque produced. Thus, each phase experiences equal losses, and the drive electronics are identical for each phase.

Copper utilization remains 67 percent, even though all three phases conduct cur-rent simultaneously. At all times, one phase is conducting current and adding to the ohmic motor loss but is not producing torque, since the back-emf is zero in each phase one-third of the time.

The amount of torque produced can be varied by changing the amplitude of the square-wave currents.

Impossible-to-produce square-wave currents are required. The inherent finite rise and fall time of the current creates torque ripple.

With all else being equal, ohmic motor losses are 50 percent greater than those in the Y connection, but the motor requires only two-thirds of the magnetic material (Miller, 1989).

Just as in the Y-connected case, the phase current amplitude must be increased by 50 percent to make up for the fact that only two phases are producing the required torque. Since the phase currents are square waves, the current equation becomes

Iph  I s/ 2n s.

Compared with the three-phase-on case, ohmic motor losses are 125 percent greater.

Independent control of phase currents is not possible.

From IA  IB  IC  0, it can be shown that the phase currents cannot have any harmonics that are multiples of three, that is, tripl e-n or triplen harmonics.

Because phases appear in parallel, the supply voltage need only be greater than the peak phase back-emfs at rated speed.

The delta connection is traditionally found in low-power, lower-performance motors.

The Sine-Wave Motor. The sine-wave back-emf motor completes the discussion of three-phase motors. A three-phase motor with sinusoidal back-emf can be Y or delta connected because there are by definition no triplen harmonics. Excitation of a sinusoidal motor with sinusoidal current gives constant ripple-free torque just as the two-phase sinusoidal motor does. In this case, the back-emfs and currents are all off-set from each other by 12  electrical. Following the notion used earlier, the torque is found by substitution into Eq. (10.6) and is given by

kAiA  kBiB  kCiC  T KI sin 2

 KI sin 2  12  )  KI sin 2  24  )  T (10.8)

3/2 KI  T

The simple elegance of Eqs. (10.3) and (10.8) is due to the pure sinusoidal content of back-emf and phase currents. Because of this elegance, a greater deal of work goes into the design of some motors to minimize the higher harmonics in the back-emf so that a sinusoidal drive can be implemented. The sinusoidal motor commonly appears in high-performance applications where high accuracy and minimal torque ripple are required.

As shown by Eq. (10.8), each phase produces torque proportional to half the peak value of the current and back-emf as compared with a unity ratio for the square-wave back-emf motor driven three-phase-on. Therefore, in a sinusoidal motor driven by sinusoidal currents, correction of the phase current equation is necessary to establish the rms phase current required to produce a specified torque. The factor of 1/.2 is taken into account by increasing the current amplitude by a factor of 2.

Combining this information with the factor of 1/ 2 for the RMS values of a sinusoid, the phase current becomes

Iph  SQR-RT ( 2) Is / 3ns

.3 PWM Methods

Specific current waveforms are assumed in each of the motor drive schemes discussed previously. To produce these waveforms from a voltage source requires cur-rent control. For maximum efficiency, this current control cannot require sustained operation of the power electronic device in its linear operating region. Rather, devices should act as switches having two states: off, where power dissipation is zero because there is no current flow, and on, where power dissipation is low because the voltage across the device is minimized. As a result, current control is implemented as a switching strategy in which the switch duty cycle is varied according to some error criterion, and the current maintains the correct shape in an average sense only. If switching action occurs at a much higher rate than any variation in the desired current waveform, the deviation between the actual and desired current can be made small. As a whole, these switching strategies are called pulse-width modulation (PWM).

Because PWM is applied in countless applications in addition to motor drives, there are hundreds of articles on PWM in the literature. Many of these articles pertain to voltage PWM, where one seeks to control voltage rather than current (Holtz, 1992). A smaller number pertain to current-control PWM, which is of interest here.

As before, the goal is to develop an intuitive understanding, rather than discuss every nuance of every PWM scheme. More detailed information can be found in references such as Holtz (1992),Anunciada and Silva (1991), Brod and Novotny (1985), and Murphy and Turnbull (1988).

In motor drive applications, PWM is almost always implemented by controlling the bridge switches themselves. However, switching can also be implemented external to the bridge. Moreover, because motor windings have inductance, PWM action causes the phase inductance to charge and discharge, giving a continuous current despite the presence of a discontinuous applied voltage. As discussed earlier and shown in Ill. 77, inductor discharge can be fast or slow depending on which switch or switches are controlled by PWM. Intelligent use of this capability can lead to improved performance (Freimanis, 1992).

Hysteresis PWM. Hysteresis PWM, conceptually the simplest PWM scheme, controls the on-off state of switches to keep the current within a band around the desired value, as shown in Ill. 85. In the figure, I* is the reference current wave-form (i.e., the desired current),  I is the tolerance band, I = I*  I is the lower bound, and I = I*  I is the upper bound. Whenever the current crosses the upper bound, a switch is opened, allowing current to decay or discharge. Likewise, whenever the current crosses the lower bound, a switch is closed, forcing current to climb in amplitude or change. Clearly, the rate at which the inductance involved charges and discharges influences the rate at which switching occurs. In a motor drive, where the voltage across the inductance is a function of the difference between a supply voltage and the back-emf, the switching frequency will be high at low speeds and low at high speeds. The switching frequency at low speeds can be decreased by increasing the tolerance band. However, this increases the percent-age ripple in the current.

Important aspects of this PWM scheme include the following:

Precise current control is possible, as the tolerance band width is a design parameter.

The frequency at which switches change state is not a design parameter. As a result, the switching frequency can vary by an order of magnitude or more.

Acoustic and electromagnetic noise are difficult to filter, because their respective spectral components vary with the switching frequency.

This PWM method is more commonly implemented in motor drives where motor speed and load are constant. Under these circumstances, the variation in switching frequency is small.

ILL. 85 Hysteresis PWM waveforms.

ILL. 86 Clocked turn-on PWM waveforms.

Clocked Turn-On PWM. This PWM method is the most commonly implemented scheme. Rather than control the peak-to-peak error as the hysteresis controller does, here the switching frequency is held constant. Clocked turn-on PWM is shown in Ill. 86, where the top trace is a synchronizing clock. Whenever this clock pulse appears, a switch is closed, causing the inductance to charge. At some point later when the current reaches I  , a switch opens, initiating inductor discharge, which continues until the next clock pulse appears.

Important aspects of this PWM scheme include the following:

Current control is not as precise here, since there is no fixed tolerance band that bounds the current.

The frequency at which switches change state is a fixed design parameter.

Acoustic and electromagnetic noise are relatively easy to filter, because the switching frequency is fixed.

This PWM method has ripple instability that produces sub-harmonic ripple components for duty cycles above 50 percent. While this instability does not lead to any destructive operating mode, it is a chaotic behavior that reduces performance. The pre-dominant current ripple occurs at half the switching frequency.

Ripple instability can be eliminated by adding a stabilizing ramp to the reference current.

Clock Turn-Off PWM. Clocked turn-off PWM is the complement of clocked turn-on PWM. In this method, shown in Ill. 87, the clock pulse initiates inductor discharge.

Later, when the current decays to I  , a switch closes and the inductance charges until the next clock pulse appears. Once again, the switching frequency is fixed by the clock frequency.

Important aspects of this PWM scheme include the following:

Current control is not as precise here, since there is no fixed tolerance band that bounds the current.

The frequency at which switches change state is a fixed design parameter.

Acoustic and electromagnetic noise are relatively easy to filter, because the switching frequency is fixed.

This PWM method has ripple instability that produces sub-harmonic ripple components for duty cycles below 50 percent. While this instability does not lead to any destructive operating mode, it is a chaotic behavior that reduces performance. The pre-dominant current ripple occurs at half the switching frequency.

ILL. 87 Clocked turn-off PWM waveforms.

Dual Current-Mode PWM. This PWM method was developed by Anunciada and Silva (1991) to eliminate the ripple instability present in the previous two methods.

Their scheme combines the clocked turn-on and clocked turn-off methods in a clever way. For duty cycles below 50 percent, the method implements stable clocked turn-on PWM, whereas for duty cycles above 50 percent, the method implements stable clocked turn-off PWM.

As illustrated in Ill. 88, this method has two clock signals, where the turn-off clock is delayed one-half period with respect to the turn-on clock. Operation is determined by logic that initiates inductor charging when the turn-on clock pulse appears or the current reaches I  , and initiates inductor discharge when the turn-off clock appears or the current reaches I  . As shown in Ill. 88, the method smoothly moves from one mode to the other. This scheme has all the attributes of the two previous PWM schemes, except for the ripple instability. Furthermore, this scheme reduces to hysteresis PWM if the clock frequency is low compared with the rate at which the inductance charges and discharges.

Triangle PWM. Triangle PWM is a popular voltage PWM scheme that is commonly used to produce a sinusoidal PWM voltage. When used in this way, it is called sinusoidal PWM .

Application of this scheme to current control is accomplished by letting the PWM input be a function of the difference between the desired current and the actual current. As shown in Ill. 89, both the turn-on and turn-off of the switch are determined by the intersections of the triangle waveform and the processed current error. As the processed current error increases, so does the switch duty cycle. Typically, the processed current error is equal to a linear combination of the current error and the integral of the current error (i.e., PI control is used). As a result, as the steady-state error goes to zero, the switch duty cycle will go to the correct value to maintain it there. Though Ill. 89 shows a unipolar triangle waveform and error signal, both signals can also be bipolar, in which case zero current error produces a 50 percent duty cycle PWM signal (Murphy and Turnbull, 1988).

ILL. 88 Dual-current-mode PWM waveforms.

ILL. 89 Triangle PWM waveforms.

ILL. 90 Conceptual logic PWM implementation: ( a) hysteresis PWM, ( b) triangle PWM, ( c) clocked turn-on PWM, ( d) clocked turn-off PWM, and ( e) dual-current-mode PWM.

Summary. The PWM methods discussed in this section represent the most common methods implemented in practice. Each method has its own strengths and weaknesses; no one PWM scheme is the best choice for every motor drive. Implementation details for these PWM methods are not presented in order to focus attention on the fundamental switching concepts. For reference, conceptual logic diagrams for all the methods are shown in Ill. 90. These diagrams apply for positive currents only. When the reference current is bipolar, more complex logic diagrams are required.

Because of the finite switching time of power electronic devices, duty cycles near 0 and near 1 must be avoided in all PWM methods. Switching devices must remain on and off for sufficient time to reach equilibrium before being switched back to the opposite state. Sustained operation at either duty cycle extreme increases the loss experienced by the switching devices and can lead to destructive device heating. The choice of PWM frequency is a tradeoff. Generally, the higher the switching frequency, the smaller the current error will be. On the other hand, the higher the switching frequency, the greater the switching loss incurred by the switches. Furthermore, PWM schemes are only as accurate as the current sensors used. Sensor type, placement, shielding, and signal processing are all critical to accurate operation of a current-control PWM method.