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1048 H. A. Toliyat et al.

Flux

Controller

PWM

3-phase

Inverter

Induction

Motor

abc

Current

Model

−

Vdc

Clarke

Transformation

Park

Transformation

Inv. Park

Transformation

Mechanical Speed of Rotor

ref

FIGURE 37.23 Vector-control algorithm for induction motor.

37.6.2 DSP Controller Requirements

The controller of the induction motor control system is used to

read the feedback current and position signals, to implement

the speed or torque control algorithm, and to generate the

gate signals based on the control signal. Analog controllers or

digital signal processors, such as LF2407 can perform these

tasks.

The interface of the LF2407 is illustrated in Fig. 37.24.

Two quadrature counters detect the rising and falling edges

of the encoder signals. Two input channels related to the 10-

bit ADC are selected to read the two-phase currents. The pins

PWM1 to PWM6 output the gating signals to the gate drive

circuitry.

QEP-1

QEP-2

PWM-1 & PWM-6

Gate

Drive

TMS320F2407

En A

ADCIN-1

ADCIN-0I

En B

FIGURE 37.24 The interface of LF2407.

37.6.3 Implementation of Field-oriented Speed

Control of Induction Motor

Some practical aspects of implementing the block diagram

of Fig. 37.24 are discussed in this section and subsections.

The software organization, the utilization of different vari-

ables, and the handling of the DSP controller resources are

described. In addition, the control structure for the per-unit

model is presented. Next, some numerical considerations have

been made in order to address the problems inherent within

the ﬁxed-point calculation. As described, current model is one

of the most important blocks in the block diagram depicted

in Fig. 37.23. The inputs of this block are the currents and

mechanical speed of rotor. In the next sections, technical

37 DSP-based Control of Variable Speed Drives 1049

points that should be considered during current and speed

measurement, as well as their scaling are discussed.

37.6.3.1 Software Organization

The body of the software consists of two main modules: the

initialization module and the PWM interrupt service routine

(ISR) module. The initialization model is executed only once at

start-up. The PWM ISR module interrupts the waiting inﬁnite

loop when the timer underﬂows. When the underﬂow inter-

rupt ﬂag is set, the corresponding ISR is served. Figure 37.25

shows the general structure of the software. The complete FOC

algorithm is executed within the PWM ISR so that it runs at

the same frequency as the switching frequency or at a frac-

tion of it. The wait loop could be easily replaced with a user

interface.

Hardware

Initialization

Start

Variable

Initialization

Inf. Loop

PWM

I.S.R

FIGURE 37.25 General structure of software.

37.6.3.2 Base Values and Per-unit Model

It is often convenient to express machine parameters and

variables of per-unit quantities. Moreover, the LF2407 is a

ﬁxed point DSP, so using a normalized per-unit model of the

induction motor is easier than using real parameters. In this

model, all quantities refer to the base values. Base power and

base voltage are selected, and all parameters and variables are

normalized using these base quantities. Although one might

violate this convention from time to time when dealing with

instantaneous quantities, the rms values of the rated phase

voltage and current are generally selected as the base voltage

for the a–b–c variables while the peak value is generally selected

as the base voltage for d–q variables. The base values are deter-

mined from the nominal values by using Eq. (37.27), where

, V

, f

are the nominal phase current, the nominal phase to

neutral voltage, and the nominal frequency in a star-connected

induction motor, respectively. The base value deﬁnitions are

as follows:

√

= 2πf

(37.27)

and V

are the maximum values of the nominal phase cur-

rent and voltage, ω

is the electrical nominal rotor ﬂux speed,

and ψ

is the base ﬂux.

37.6.3.3 Speed Estimation during High-speed Region

As previously mentioned, this method is based on counting

the number of encoder pulses in a speciﬁed time interval. The

QEP assigned timer counts the number of pulses and records it

in the timer counter register (TxCNT). As the mechanical time

constant is much slower than the electrical one, the speed reg-

ulation loop frequency might be lower than the current loop

frequency. The speed regulation loop frequency is obtained in

this algorithm by means of a software counter. This counter

accepts the PWM interrupt as input clock and its period is

the software variable called SPEEDSTEP. The counter variable

is named speedstep. When speedstep is equal to SPEEDSTEP,

the number of pulses counted is stored in another variable

called n

and thus the speed can be calculated. The scheme

depicted in Fig. 37.26 shows the structure of the speed feedback

generator.

Assuming that n

is the number of encoder pulses in one

SPEEDSTEP period when the rotor turns at the nominal speed,

a software constant K

speed

should be chosen as follows:

01000h = K

speed

The speed feedback can then be transformed into a Q4.12

format, which can be used in the control software. In the

proposed control system, the nominal speed is 1800 rpm and

SPEEDSTEP is set to 125. The n

can be calculated as follows:

1800 ×64 ×4

×SPEEDSTEP ×T

= 288 (37.28)

Counter

QEP

speedstep is equal

to SPEEDSTEP?

speed

yes

No chan

From

Encoder

FIGURE 37.26 Block diagram of speed feedback calculator.

1050 H. A. Toliyat et al.

w = K

speed

speedstep = speedstep-1

speedstep = 0

speedstep = SPEEDSTEP

= TxCNT

Yes

TxCNT=0

FIGURE 37.27 Complete ﬂowchart of speed measurement block during

high-speed region.

where T

pwm

= 3 × 10

−4

(PWM frequency is 10 kHz

but the program is running at 3333 Hz) and hence K

speed

given by:

speed

4096

288

= 14.22 ⇔ 0E38h Q8.8

Note that K

speed

is out of the Q4.12 format range. The most

appropriate format to handle this constant is the Q8.8 format.

The speed feedback in Q4.12 format is then obtained from the

encoder by multiplying n

by K

speed

. The ﬂowchart of speed

measurement is presented in Fig. 37.27.

37.6.3.4 Speed Measurement during Low-speed

Region

To detect the edges of two successive encoder pulses, the devel-

oped program can use either the QEP counter or the capture

unit input pins. The program has to measure the time between

two successive pulses, therefore it must utilize another GP

timer. In this program, Timer 3 has been dedicated to the

time measurement. During the interrupt service routine of

the capture unit or counter QEP, speed can be calculated. To

obtain the actual speed of the motor, the appropriate number

is divided by the value in the count register of Timer 3.

As it can be inferred, at very low speeds an overﬂow may

occur in Timer 3. The counter would then reset itself to zero

and start counting up again. This event results in a large error

T3CNT=FFFFh

Timer 3

Disable

Start of Timer 3

Overflow ISR

End of Timer 3

Overflow ISR

Temp=T3CNT

T3CNT=0

ACC=31238*32

w=n

Enable Timer3

=ACC / Temp

Start of Capture unit or

Counter QEP ISR

End of Capture unit or

Counter QEP ISR

FIGURE 37.28 Flowchart of speed measurement at low speed.

in speed measurement. To avoid this event, Timer 3 will be dis-

abled in the overﬂow interrupt service routine. However, this

timer is enabled in the capture unit (counter QEP) interrupt.

The ﬂowchart of this implementation is presented in

Fig. 37.28

37.6.3.5 The Current Model

The Current Model is used to ﬁnd the rotor ﬂux position.

This module takes i

and i

as inputs plus the rotor electrical

speed and then calculates the rotor ﬂux position. The current

model is based on Eqs. (37.23) and (37.24). Equation (37.23)

in transient form can be written as:

dλ

= i

(37.29)

Assume λ

= i

where i

is the magnetizing current,

therefore Eq. (37.29) can be written as follows:

= i

(37.30)

Rotor ﬂux speed in a per-unit system can be shown by:

dθ

= ω

(37.31)

37 DSP-based Control of Variable Speed Drives 1051

where θ is the rotor ﬂux position and T

= L

and ω

are

the rotor time constant and rotor electrical speed, respectively.

The rotor time constant is critical to the correct functionality

of the Current Model. This system outputs the rotor ﬂux speed,

which in turn will be integrated to get the rotor ﬂux position.

Assuming that i

(k+1)

≈ i

(k)

, Eqs (37.30) and (37.31) can be

discretized as follows:

(k+1)

= i

(k)



(k)

−i

(k)



(k+1)

= n

(K +1)

(k)

(k+1)

(37.32)

Measure Ia and Ib

Regulate

speed?

Calculate speed

Start

Iabc Idqs

Clark Trans.

Speed PI regulator,

Calculate Iqs

High speed?

Current Model

Module

Ids and Iqs

to PI regulator

and calculate

ds and Vqs

dqs

) (V

dqs

)

New θ

Space Vector PWM

End

dqs

Calculate Sin and

Cos of

Yes

∗

FIGURE 37.29 Flowchart of digital implementation of FOC algorithm.

For example, let the constants T

and 1/T

be renamed

to K

andK

, respectively. Here L

= 73.8 mH, r

= 0.73 ,

and f

= 60 Hz. So for K

and K

we have:

(

10000/3

)

−1

101.09 ×10

−3

= 2.967 ×10

−3

⇔ 000Ch Q4.12

30.232 ×10

−3

×377

= 26.237 ×10

−3

⇔ 006Bh Q4.12

1052 H. A. Toliyat et al.

By knowing the rotor ﬂux speed (f

), the rotor ﬂux position

(θ

) is computed by the integration formula in the per-unit

system.

(k+1)

= θ

(k)

+ω

·f

(k)

·T (37.33)

In Eq. (37.33), let ω

T be called θ

inc

. This variable is the

rotor angle variation within one sampling period. Thus, the

Current Model Module has three input variables i

, i

, and

and one output, which is the rotor ﬂux position θ

represented as a 16-bit integer value. The ﬂowchart of the

ﬁeld-oriented speed control of induction motor is presented

in Fig. 37.29. This routine is placed inside the PWM interrupt

service routine.

Power Quality

S. Mark Halpin and

Angela Card

Department of Electrical and

Computer Engineering,

Auburn University,

Alabama, USA

38.1 Introduction ........................................................................................ 1053

38.2 Power Quality ...................................................................................... 1054

38.2.1 Steady-state Voltage Frequency and Magnitude • 38.2.2 Voltage Sags

• 38.2.3 Grounding • 38.2.4 Harmonics • 38.2.5 Voltage Fluctuations and Flicker

• 38.2.6 Transients • 38.2.7 Monitoring and Measurement

38.3 Reactive Power and Harmonic Compensation............................................ 1061

38.3.1 Typical Harmonics Produced by Equipment • 38.3.2 Resonance • 38.3.3 Harmonic

Filters

38.4 IEEE Standards..................................................................................... 1063

38.5 Conclusions ......................................................................................... 1065

Further Reading ................................................................................... 1066

38.1 Introduction

Power electronics and power quality are irrevocably linked

together as we strive to advance both broad areas. With the

dramatic increases over the last 20 years in energy conversion

systems utilizing power electronic devices, we have seen the

emergence of “power quality” as a major ﬁeld of power engi-

neering. The power electronic technology has played a major

role in creating “power quality,” and simple control algorithm

modiﬁcations to this same technology can often play an equally

dominant role in enhancing overall quality of electrical energy

available to end-users.

Power electronics has given us, as a industrial society,

a plethora of new ways to manufacture products, provide

services, and utilize energy. From a power quality impact

viewpoint, applications such as

1. Switched-mode power supplies,

2. DC arc furnaces,

3. Electronic ﬂuorescent lamp ballasts,

4. Adjustable speed drives, and

5. Flexible ac transmission components.

are often cause for concern. From the viewpoint of a utility

supply system, these converter-based systems can lead to oper-

ational and life expectancy problems for other equipment,

possibly not owned or operated by the same party. It was

from this initial perspective that the ﬁeld of power quality

emerged.

In most cases, the same devices and systems that create

power quality problems can also be used to solve power quality

problems. “Problem solving” applications such as

1. Active harmonic ﬁlters,

2. Static and adaptive var compensators, and

3. Uninterruptable power supplies.

all utilize the same switching device technology as the “prob-

lem causing” applications.

As the number of potentially problematic power electronic-

based loads has increased over time, so attention has given to

enhanced converter control to maximize power quality. Perfect

examples of these improvements include

1. Unity power factor converters,

2. Dip-proof inverters, and

3. Limited-distortion electronic lamp ballasts.

While these direct product enhancements are not mandatory

in North America, today’s global economy necessitates con-

sideration of power quality standards and limits in order to

conduct business in the European Union.

While many studies suggest increases in power electronic-

based energy utilization as high as 70–80% (of all energy

consumed), it is equally clear that we are beginning to real-

ize the total beneﬁt of such end-use technologies. Power

quality problems associated with grounding, sags, harmonics,

and transients will continue to increase because of the sheer

number of sensitive electronic loads expected to be placed

1053

1054 S. M. Halpin and A. Card

in service. At the same time, we are only now beginning to

realize the total beneﬁts that such loads can offer.

38.2 Power Quality

The term “power quality” means different things to differ-

ent people. To utility suppliers, power quality initially referred

to the quality of the service delivered as “measured” by

the consumer’s ability to use the energy delivered in the

desired manner. This conceptual deﬁnition included such

conventional utility planning topics as voltage and frequency

regulation and reliability. The end-user’s deﬁnition of power

quality also centers around their ability to use the delivered

energy in the desired manner, but the topics considered can

be much more speciﬁc and include magnitude and duration of

different events as well as waveshape concerns. Fortunately, a

good working deﬁnition of power quality has not been a point

of contention, and most parties involved consider “power

quality” to be that, which allows the user to meet their end-use

goals. The working deﬁnition is not complicated by particular

issues; engineers are well aware that topics from many aspects

of power engineering may be important.

Power quality can be roughly broken into categories as

follows:

1. Steady-state voltage magnitude and frequency,

2. Voltage sags,

3. Grounding,

4. Harmonics,

5. Voltage ﬂuctuations and ﬂicker,

6. Transients, and

7. Monitoring and measurement.

The remainder of this section discusses each of the major

categories in turn.

38.2.1 Steady-state Voltage Frequency

and Magnitude

In most areas of North America, steady-state frequency regu-

lation is not a signiﬁcant issue due to the sufﬁcient levels of

generating capacity and the strong interconnections among

generating companies and control areas. In other parts of

the world, and North America under extreme conditions,

frequency can deviate from 1/4 to 1/2 Hz during periods of

insufﬁcient generating capacity. Under transient conditions,

frequency can deviate up to 1–2 Hz.

Frequency deviations can affect power electronic equipment

that use controlled switching devices unless the control sig-

nals are derived from a signal that is phase-locked with the

applied voltage. In most cases, phase locks are used, or the

converters consist of uncontrolled rectiﬁers. In either case, fre-

quency deviations are not a major cause of problems. In most

TABLE 38.1 ANSI C84.1 Voltage ranges

Service voltage (%) Utilization voltage (%)

Range A 114–125 108–125

Range B 110–127 104–127

Range A is for normal conditions and Range B is for emergency or short-time

conditions.

cases, frequency deviations have more impacts on conven-

tional equipment that does not use electronics or in very

inexpensive electronic devices. Clocks can run fast (or slow),

motor speeds can drop (or rise) by a few revolutions per

minute, etc. In most cases, these effects have minimal eco-

nomic impact and are not considered a real power quality

problem.

Steady-state voltage regulation is a much more pronounced

issue that can impact a wide range of end-use equipment.

In most cases, utility supply companies do a very effective

job of providing carefully regulated voltage within permissi-

ble ranges. In North America, ANSI Standard C84.1 suggests

steady-state voltage ranges both at the utility service entrance

and at the point of connection of end-use equipment. Fur-

thermore, equipment manufacturers typically offer equipment

that is tolerant of steady-state voltage deviations in the range

of ±10%. Table 38.1 shows the voltage ranges suggested by

ANSI C84.1, with speciﬁc mention of normal (Range A)

and contingency (Range B) allowable voltages, expressed in

percent.

Virtually all equipment, especially sensitive electronic equip-

ment, can be effected by deviating voltage outside the ±10%

range. In most cases, overvoltages above +10% lead to loss of

life, usually over time; excessive overvoltages can immediately

fail equipment. Undervoltages below −10% usually lead to

excessive current demands, especially for equipment that has a

controlled output like an adjustable speed drive controlling a

motor to a constant speed/torque point. The impacts of these

prolonged excessive currents can be greater voltage drop, tem-

perature rise in conductors, etc. In the extreme, undervoltages

of greater than 15–20% can cause equipment to immediately

trip. In most cases, such extreme undervoltages are associated

with system faults and the associated protection system. These

extreme undervoltages are so important that they are classiﬁed

in a power quality category of their own called voltage sags.

38.2.2 Voltage Sags

Other than improper grounding, voltage sags are probably the

most problematic of all power quality problems. At this time,

a number of standards-making bodies, including IEEE, ANSI,

and IEC, are working on standards related to sags. In most

cases, sags are generally agreed to be more severe and outside

of the scope of ANSI C84.1 and they are temporary in nature

38 Power Quality 1055

due to the operation of system protection elements. Because

the electrical system is a continuous electrical circuit, faults

in any location will have some impact on voltages through-

out the network. Of course, areas closer to the faulted area

will see a greater voltage sag due to the fault than other, more

(electrically) remote areas. Sags can originate anywhere in a

system, but are more pronounced in utility distribution sys-

tems because of the greater exposure of low-voltage systems to

the causes of short circuits.

Most utility companies implement distribution system pro-

tection in what is known as a “fuse saving” methodology.

Figure 38.1 shows a typical overhead distribution system with

two feeders being supplied from the same substation trans-

former. Each primary circuit has its own automatic circuit

recloser (ACR) and shows one fused tap.

With the protection system set up based on fuse-saving

methodology, any fault downstream of a fault will be cleared

ﬁrst by the substation recloser followed by a reclosing oper-

ation (re-energization of the circuit) 1/2–2 later. If the fault

is still present, the closest fuse should blow to permanently

isolate the fault. (Note that in some cases, multiple reclosing

attempts are made prior to the clearing of the fuse.)

For a fault on the load side of fused tap #2 in Fig. 38.1, cus-

tomers on feeder #1 will see a voltage sag determined by the

system and transformer impedance at the substation. Because

this impedance is typically on the same order (or larger) as

the feeder circuit impedance, a sag in substation bus voltage of

50% is common. This sag will persist until feeder #2 is cleared

by the recloser opening. When the recloser re-energizes the

circuit, a permanent fault will still be present and the substa-

tion bus will again experience a voltage sag. Of course, any sag

in substation bus voltage will be delivered directly to all cus-

tomers on feeder #1, even though there is no electrical problem

on that feeder. Figure 38.2 shows a possible rms voltage pro-

ﬁle that might be supplied to the customers on feeder #1 for

100

120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.1

2.2

2.3

2.4

2.5

Time (s)

Voltage (%)

FIGURE 38.2 The rms voltage supplied to feeder #1 customers.

Utility

equivalent

Substation

transformer

Feeder #1

Feeder #2

Fused Tap #2

Fused Tap #1

FIGURE 38.1 Overhead distribution system.

a permanent fault on the load side of fused tap #2. Only one

recloser operation is shown prior to fuse clearing.

Just based on the voltage information shown in Fig. 38.2, it

is impossible to tell if the end-use loads on feeder #1 will expe-

rience a problem. Equipment tolerance curves are required to

assess the vulnerability of equipment to voltage deviations,

including sags, and all equipment is different. Figure 38.3

shows the lower portions of two equipment tolerance curves,

the (older) CBEMA and the (newer) ITIC curves for com-

puter equipment. Most, but not all, power electronic-based

equipment has a similar shape. Voltage sags with a duration

that correspond to a point that is “below and to the right” of

the tolerance curve will result in loss of equipment function,

while sags of duration that plot “above and to the left” of the

tolerance curve will not effect equipment performance. Note

that only the lower portion of the curve has been shown; an

upper tolerance curve also exists that is often used in transient

(overvoltage) studies.

Voltage sags are probably the most common power quality

problem that is “given” to the end-user by the supplying utility.

However, improper equipment grounding is responsible for

1056 S. M. Halpin and A. Card

100

1.00 10.00 100.00 1000.00 10000.00 100000.00

Duration of Sag (milliseconds)

[0 to 6,000 Cycles]

% of Nominal Voltage

FIGURE 38.3 CBEMA (curved line) and ITIC (square shape) tolerance curves.

the vast majority of power quality problems on the customer’s

side of the meter.

38.2.3 Grounding

Grounding of equipment was originally conceived as a per-

sonnel safety issue. However, the presence of an electrical

conductor that is at zero potential has been widely used in

many power electronic and microprocessor-controlled loads.

In the United States, electrical systems in residential, com-

mercial, and industrial facilities fall under the purview of the

National Electric Code (NEC) which establishes speciﬁc cri-

teria for grounding of equipment. While it was once thought

that proper grounding according to the NEC was detrimental

to power quality concerns, these opinions have gradually faded

over time.

From a power quality perspective, improper grounding can

be considered in three broad categories

1. Ground loops,

2. Improper neutral-to-ground connections, and

3. Excessive neutral-to-ground voltage.

The ground loop problem is a signiﬁcant issue when power,

communications, and control signals all originate in different

locations, but come together at a common electrical point.

Transients induced in one location can travel through the

created ground loop, damaging equipment along the way.

Load Equipment

FIGURE 38.4 Powering and control ground loop.

Improper neutral-to-ground connections will create a “noisy”

ground reference that may interfere with low-voltage com-

munications and control devices. Excessive neutral-to-ground

voltage may damage equipment that is not properly insulated

or that has an inexpensive power supply.

Figure 38.4 shows a common wye-connected service

(assumed at the terminals of a transformer) that supplies

power to equipment that is also remotely monitored and

controlled from another location with a separate ground

reference.

38 Power Quality 1057

Load Equipment

FIGURE 38.5 Improper neutral-to-ground connections.

For any shift in ground potential for the power circuit, often

caused by lightning as shown in Fig. 38.4, potentially large cur-

rents can ﬂow through the grounding circuits and through the

sensitive electronic equipment. Such currents can easily lead

to equipment damage. Situations like these are common in

1. Residential areas, if power and CATV or telephone

grounds are not the same and

2. Commercial and industrial complexes consisting of

multiple buildings with linking communications, com-

puter, or control circuits, when each building has its

own power service (and therefore ground).

Figure 38.5 shows an example of an improper neutral-to-

ground connection, and how this connection can create power

quality problems.

Load current returning in the neutral conductor will, at

the point of improper connection to ground, divide between

neutral and ground. This current ﬂow in the ground conductor

will produce a voltage at the load equipment, which can easily

disrupt equipment operation.

Figure 38.6 shows an example of the possibility for exces-

sive neutral-to-ground voltage and how this can lead to power

quality problems.

For load equipment that produces signiﬁcant voltage drop

in the neutral, such as laser printers and copying machines

when the thermal heating elements are on, the voltage from

the neutral-to-ground reference inside the equipment can

exceed several volts. In many cases, this voltage is sufﬁcient

to damage printed circuit boards, disrupt control logic, and

fail components.

38.2.4 Harmonics

In most cases, power electronic equipment is considered to

be the “cause” of harmonics. While switching converters of

all types produce harmonics because of the non-linear rela-

tionship between the voltage and current across the switching

Load Equipment

FIGURE 38.6 Excessive neutral-to-ground voltage.

device, harmonics are also produced by a large variety of

“conventional” equipment including

1. Power generation equipment (slot harmonics),

2. Induction motors (saturated magnetics),

3. Transformers (overexcitation leading to saturation),

4. Magnetic-ballast ﬂuorescent lamps (arcing), and

5. AC electric arc furnaces (arcing).

All these devices will cause harmonic currents to ﬂow and

some devices, actually, directly produce voltage harmonics.

Any ac current ﬂow through any circuit at any frequency

will produce a voltage drop at that same frequency. Harmonic

currents, which are produced by power electronic loads, will

produce voltage drops in the power supply impedance at those

same harmonic frequencies. Because of this inter-relationship

between current ﬂow and voltage drop, harmonic currents

created at any location will distort the voltage in the entire

supply circuit.

In most cases, equipment is not overly sensitive to the

direct impacts of harmonic current ﬂow. Note, however, that

equipment heating is a function of the rms value of the cur-

rent, which can signiﬁcantly exceed the fundamental frequency

value when large harmonic components are present. It is

because harmonic currents produce harmonic voltages that

there is a real power quality concern.

Most equipment can operate satisfactorily as long as the

voltage distortion at the equipment terminals does not exceed

around 5%. Exceptions to this general rule include ripple-

control systems for converters (which are impacted by small

even-order harmonics) and small harmonics at sufﬁciently

high frequency to produce multiple zero crossing in a wave-

form. (Note that voltage notching due to simultaneous com-

mutation of switching devices can also create multiple zero

crossings.) Such a multiple crossing scenario is shown in

Fig. 38.7 and represents a 60 Hz waveform plus a 1% voltage

harmonic at 3000 Hz.

Converters that have a time-limited ﬁring signal can directly

suffer from excessive voltage distortion. For a six-pulse