Pulse width modulation technique with harmonic injection and frequency modulated carrier:...

Pulse width modulation technique with harmonicinjection and frequency modulated carrier:formulation and application to an induction motor

M.J. Meco-Gutierrez, F. Perez-Hidalgo, F. Vargas-Merino and J.R. Heredia-Larrubia

Abstract: A new generated pulse width modulation (PWM) technique is presented, making it poss-ible to significantly reduce harmonics in comparison to currently used PWMs operating in realtime. This improvement means that a motor connected to an inverter that is controlled with thistechnique undergoes less overheating and vibrations, thereby improving its performance.

1 Introduction

Inverters are devices that convert a continuous input voltageinto an alternating output voltage with a specific magnitudeand frequency. They are very frequently used in industriesfor such applications as induction heating, auxiliary powersources, uninterrupted power sources and, increasingly, insystems for controlling the speed of alternating motors[1]. Fig. 1 shows a complete diagram of an AC/AC conver-ter. The last stage in this converter consists of a powerinverter that transforms the continuous voltage, obtainedat the filter’s output, into an alternating voltage with specificvalues of frequency and amplitude as required by the load,which is usually a three-phase motor.Fig. 2 shows a diagram of a three-phase inverter. The

inverter is fed by a continuous voltage E of the intermediatecircuit on the assumption that it has negligible curl [2]. Thispower source is connected to the load via three semi-bridgesmade up of electronic switches. The subscript in the name ofthe switch indicates the input sequence in conduction that isprolonged 1808 in every cycle. Source E is assumed to havean average-sized contact point to facilitate the formulationof waveforms, although this is unnecessary for practicalpurposes.After performing the Fourier decomposition, one can see

that the harmonics that are even numbered or multiples ofthree disappear in line-to-line voltages, leaving just the1st, 5th, 7th, 11th and so on. These voltage harmonicshave decreasing amplitudes (although they may be signifi-cant in lower orders) and they lead to a series of flow har-monics that superimpose themselves on top of the normalflow, thereby increasing loss in the machine and leadingto the appearance of permanent pulsating pairs that jeopar-dise its proper functioning. For their part, the current harmo-nics are filtered by the load’s own reactive effect, inproportion to the order of the harmonic in question.

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-epa:20060110

Paper first received 16th March and in revised form 29th May 2006

M.J. Meco-Gutierrez, F. Perez-Hidalgo and F. Vargas-Merino are with theElectric Engineering Department, Universidad de Malaga, Plaza de El Ejidos/n, Malaga 29013, Spain

J.R. Heredia-Larrubia is with the Electronic Technology Department,Universidad de Malaga, Plaza de El Ejido s/n, Malaga 29013, Spain

E-mail: [email protected]

714

It is also necessary to reduce the harmonic content of theoutput voltage, and to do so, the switches’ firing controlsignals are modulated [3]. Such modulation is carried outby comparing a modulator signal and a carrier signal,thereby yielding the modulated signal. This is a periodicpulse train of variable widths that contains information(frequency and amplitude) from the modulator signal. Thepulses are applied to the switches’ control electrodes,(the base in the case of power transistors or the gate in thecase of thyristors). This leads to pulse width modulation(PWM).Several different PWM techniques share the common

objective: first, to reduce the rate of harmonics in rectangu-lar waves in general or selectively eliminate some of them.Additionally, these techniques seek to control resulting fun-damental output voltage and reduce the effects of thermaland mechanical overcharging in the motor caused by thepresence of any remaining harmonics.The most advanced techniques, which are based on genetic

algorithms [4], computational procedures [5], hysteresis bandmodulation [6–9] and so on are calculated off-line and arenot convenient when attempting to control systems in realtime using analogue electronic circuits. In such cases, it ismore convenient to use generated techniques.The most classic generated PWM technique involves

using a sinusoidal modulator with fundamental frequencyvm and a high-frequency triangular signal vp as a carrier.Comparing the two in real time gives the sinusoidal pulsewidth modulated signal (SPWM), which constitutes theinverter’s firing pulses (Fig. 3a). Given an ideal inverter,this will be the waveform that has the output signal to beapplied to the motor. The main disadvantage of this tech-nique is that it generates a fundamental term with smallamplitude and a significant amount of undesired harmonics.There exists a series of more advanced techniques thatdecrease harmonics and increase the value of the fundamen-tal term of the resulting modulated signal in comparison tothe SPWM. The best of these techniques is called harmonicinjection pulse width modulation (HIPWM) and consistsof overmodulating the sinusoid – whose fundamentalfrequency is vm, – by 15%, adding 27% to the third harmo-nic in phase and 2.9% to the ninth in counterphase [10],resulting in a wave that can be expressed as follows(Fig. 3b)

y ¼ 1:15 sin(vmt)þ 0:27 sin(3vmt)� 0:029 sin(9vmt) (1)

IET Electr. Power Appl., 2007, 1, (5), pp. 714–726

2 Brief description of the proposed technique

This article proposes an alternative technique, whichinvolves the same number of commutations per unit timeand therefore causes the same amount of heating in the inver-ter’s transistors, while generating an output signal with anappreciable increase in the fundamental term and a signifi-cant reduction in lower order harmonics, which are most dif-ficult to filter. To achieve this, the modulating wave iscompared to a triangular carrier with variable frequencyover the period of the modulator. It is therefore convenientfor the modulating signal to have a lot of sinusoidal‘information’ in the areas of greater sampling. Hence themodulating signal defined by (1) is used, which presents a

Fig. 1 Converter structure

IET Electr. Power Appl., Vol. 1, No. 5, September 2007

greater slope in the segments where it rises and falls, aswell as a practically flat area at its peaks, yielding morerelaxed pulses.The above-mentioned situation is featured in Fig. 4,

which shows a sine wave with a period of 20 ms andanother wave with harmonic injection (HI) and the sameperiod. A triangular signal with a frequency that variesover the PWMs modulator period is used as carrier, for thepurpose of obtaining more samples in the area with the great-est slope. To achieve this, the carrier signal is modulated infrequency.The general expression for a frequency-modulated signal

is vi(t) ¼ vcþ kf f (t), where vi is the instantaneous fre-quency that varies directly in relation to the modulator fre-quency f (t) around a central frequency vc and kf is amodulator constant. This general expression is transformedinto the final expression by the following steps:

1. For the frequency of the triangular signal to be higher atthe intervals where the PWM modulator signal has a higherslope (that is, where the ‘information’ varies more quicklyand it is therefore desirable to define smaller samplingperiods), the expression of the triangular signal’s

Fig. 2 Three-phase inverter and output waves

715

Fig. 3 Principles of SPWM and HIPWM techniques

a Principle of SPWMb Principle of HIPWM

instantaneous frequency should be as follows:

vp ¼ vi ¼ vc � kf f (t) with f (0) ¼ 0

Thus, for t ¼ 0, the instantaneous frequency will bemaximum and equal to vc.2. Likewise, for the signal to be symmetrical, it is desir-able for the frequency to be lower at (T/4) and for thesame thing to happen in the negative semiperiod as inthe positive semiperiod. To accomplish this, aftertesting several functions, a squared sine function whosepulse is vm is chosen as the frequency-modulatingsignal of the triangular signal f(t). vm is the pulse ofthe desired fundamental term at the inverter output,which coincides with the modulator’s base pulse withHI for the PWM (1).

Finally, the carrier’s instantaneous frequency variesaccording to the law

vi ¼ vc � kf ( sin(vmt))2

vi ¼ vc if sin(vmt) ¼ 0

vi ¼ vc � kf if sin(vmt) ¼

þ1 or�1

8<:

(2)

where vc is a given central pulsation, kf is a given modu-lation constant and vm is the pulse of the desired fundamen-tal term at the inverter’s output, which coincides with themodulator’s base pulse (1).We call this technique the harmonic injection pulse width

modulation and frequency-modulated triangular carrier

Fig. 4 Sinusoidal wave and sinusoidal wave with harmonicinjection (HI)

716

Fig. 5 Modulator and carrier signals and the three-phase andthe three line-to-line voltages obtained with the SPWM andHIPWM-FMTC strategies

a Principle of HIPWM-FMTCb SPWM: phases A, B, C and line-to-line voltagesc HIPWM-FMTC: phases A, B, C and line-to-line voltages


(HIPWM–FMTC). Both the modulator and carrier signalscan be observed in Fig. 5a. By comparing these twosignals we can obtain the modulated signal, shown in thelower part of Fig. 5a, which has an average order (ornumber of pulses per cycle) M. The following paragraphshows that given a central frequency vc and modulationconstant kf, the modulated signal’s average order obeysthe following expression

�M ¼2vc � kf2vm

(3)

This is confirmed in Fig. 5a, where vc ¼ 25vm, kf ¼ 20vm

and M ¼ 15.Fig. 5b and c shows the three-phase and the three line-

to-line voltages obtained by using the SPWM andHIPWM-FMTC strategies.

3 Mathematical expressions of the outputsignal: discussion and results

This section presents the complete expression of the modu-lated voltage signal obtained by using the proposed tech-nique in Fourier series. First, from Fig. 6, the fundamentalterm of the output wave is

a02¼

1

2p

ð2p0

f (t) d(vpt) ¼1

2p

ð�a

�p

�E

2d(vpt)

�

þ

ða�a

E

2d(vpt)þ

ðpa

�E

2d(vpt)

�

¼ � � � ¼E

22a

p� 1

� �(4)

where E is the inverter’s continuous voltage and vp is the

Fig. 6 HIPWM-FMTC modulation


carrier pulse. Likewise, the harmonic terms are

an ¼1

p

ð2p0

f (t) cos(nvpt) d(vpt)

¼1

p

ð�a

�p

�E

2cos(nvpt) d(vpt)

�

þ

ða�a

E

2cos(nvpt) d(vpt)

þ

ðpa

�E

2cos(nvpt) d(vpt)

�

¼ � � � ¼4

np

E

2sin(na) (5)

bn ¼1

p

ð2p0

f (t) sin(nvpt) d(vpt)

¼1

p

ð�a

�p

�E

2sin(nvpt) d(vpt)

�

þ

ða�a

E

2sin(nvpt) d(vpt)

þ

ðpa

�E

2sin(nvpt) d(vpt)

�¼ � � � ¼ 0 (6)

a is the angle at which switching occurs, which is calculatedfrom the intersection of curves y1 and y2

y1 ¼ 1:15cos(vmt)� 0:27cos(3vmt)� 0:029cos(9vmt)

y2 ¼2a

p� 1

8<:

9=;

)a¼p

2þp

2(1:15cos(vmt)�0:27cos(3vmt)

� 0:029cos(9vmt)) (7)

Substituting the values, the voltage thus becomes

VA(t) ¼a02þX1n¼1

an cos(nvpt)þX1n¼1

bn sin(nvpt)

¼E

2

� �[1:15 cos(vmt)� 0:27 cos(3vmt)

� 0:029 cos(9vmt)]

þ4

p

E

2

� �X1n¼1

1

nsin

np

2þnp

2(1:15 cos(vmt)

��"

�0:27 cos(3vmt)� 0:029 cos(9vmt))�cos(nvpt)

�#

(8)

The first set of brackets is the fundamental term to thecarrier frequency and the second set of brackets representsthe harmonic term to the carrier frequency vp.

717

Our next objective is to lay out this harmonic term

2 sinnp

2þnp

2(1:15 cos(vmt)� 0:27 cos(3vmt)

�� 0:029 cos(9vmt))

�¼ sin

np

2

� �cos

np

2(1:15 cos(vmt)� 0:27 cos(3vmt)

h� 0:029 cos(9vmt))

iþ cos

np

2

� �� sin

np

2(1:15 cos(vmt)

h� 0:27 cos(3vmt)� 0:029 cos(9vmt))

i¼

using the following names below

A ¼ 1:15np

2

� �B ¼ �0:27

np

2

� �C ¼ �0:029

np

2

� �

8>>>><>>>>:

(9)

The lay-out expression is as follows

¼ sinnp

2

� �[ cos(A cosvmtþ B cos 3vmt) cos(C cos 9vmt)

� sin(A cosvmtþ B cos 3vmt) sin(C cos 9vmt)]

þ cosnp

2

� �[ sin(A cosvmtþ B cos 3vmt) cos(C cos 9vmt)

þ cos(A cosvmtþ B cos 3vmt) sin(C cos 9vmt)]

¼ sinnp

2

� �( cos(A cosvmt) cos(B cos 3vt)

� sin(A cosvmt) sin(B cos 3vt))

� cos(C cos 9vmt)

�( sin(A cosvmt) cos(B cos 3vt)

þ cos(A cosvmt) sin(B cos 3vt))

� sin(C cos 9vmt)

266666664

377777775

þ cosnp

2

� �( sin(A cosvmt) cos(B cos 3vt)

þ cos(A cosvmt) sin(B cos 3vt))

� cos(C cos 9vmt)

þ ( cos(A cosvmt) cos(B cos 3vt)

� sin(A cosvmt) sin(B cos 3vt))

� sin(C cos 9vmt)

266666664

377777775¼

Keeping in mind the following general mathematicalrelationships [11, 12]

sin(x cos y) ¼X1z¼0

sinzp

2

� �2�

0

z

� �� Jz(x) cos(zy) (10)

cos(x cos y) ¼X1z¼0

coszp

2

� �2�

0

z

� �� Jz(x) cos(zy) (11)

where

2�0

z

� �� ¼

1 if z ¼ 0

2 if . 0

�(12)

and Jz is the first class Bessel function of order z, the

718

following expression can be laid out

¼ sinnp

2

� �X1s¼0

cossp

2

� �2�

0

s

� �� "

� Js(C) cos(s � 9vmt)þ cosnp

2

� �X1s¼0

sinsp

2

� �

� 2�0

s

� �� Js(C) cos(s � 9vmt)

�

�X1p¼0

cospp

2

� �2�

0

p

� �� Jp(A) cos(p �vmt)

�X1r¼0

cosrp

2

� �2�

0

r

� �� Jr(B) cos(r � 3vmt)

þ cosnp

2

� �X1s¼0

cossp

2

� �2�

0

s

� �� "

� Js(C) cos(s � 9vmt)� sinnp

2

� �X1s¼0

sinsp

2

� �

� 2�0

s


�

�X1p¼0

cospp

2

� �2�

0

p


�X1r¼0

sinrp

2

� �2�

0

r


� sinnp

2

� �X1s¼0

cossp

2

� �2�

0

s


"

þ cosnp

2

� �X1s¼0

sinsp

2

� �2�

0

s


#

�X1p¼0

sinpp

2

� �2�

0

p


�X1r¼0

sinrp

2

� �2�

0

r


þ cosnp

2

� �X1s¼0

cossp

2

� �2�

0

s


"

� sinnp

2

� �X1s¼0

sinsp

2

� �2�

0

s


#

�X1p¼0

sinpp

2

� �2�

0

p


�X1r¼0

cosrp

2

� �2�

0

r

� �� Jr(B) cos(r � 3vmt) ¼

¼X1s¼0

2�0

s

� �� Js(C) cos(s � 9vmt) sin

np

2

� �cos

sp

2

� ��"

þ cosnp

2

� �sin

sp

2

� ��#X1p¼0

X1r¼0

cospp

2

� �cos

rp

2

� �


IE

� 2�0

p

� �� 2�

0

r

� �� Jp(A)Jr(B)cos(p �vmt)

� cos(r �3vmt)

þX1s¼0

2�0

s

� �� Js(C)cos(s �9vmt) cos

np

2

� �cos

sp

2

� ��"

�sinnp

2

� �sin

sp

2

� ��#X1p¼0

X1r¼0

cospp

2

� �sin

rp

2

� �

� 2�0

p

� �� 2�

0

r


� cos(r �3vmt)

�X1s¼0

2�0

s

� �� Js(C)cos(s �9vmt) sin

np

2

� �cos

sp

2

� ��"

þcosnp

2

� �sin

sp

2

� ��#X1p¼0

X1r¼0

sinpp

2

� �sin

rp

2

� �

� 2�0

p

� �� 2�

0

r


� cos(r �3vmt)

þX1s¼0

2�0

s

� �� Js(C)cos(s �9vmt) cos

np

2

� �cos

sp

2

� ��"

�sinnp

2

� �sin

sp

2

� ��iX1p¼0

X1r¼0

sinpp

2

� �cos

rp

2

� �

� 2�0

p

� �� 2�

0

r


� cos(r �3vmt)

¼X1p¼0

X1r¼0

X1s¼0

sinp

2(nþ s)

� �cos

pp

2

� �cos

rp

2

� ��sin

p

2(nþ s)

� �sin

pp

2

� �sin

rp

2

� �þcos

p

2(nþ s)

� �cos

pp

2

� �sin

rp

2

� �þcos

p

2(nþ s)

� �sin

pp

2

� �cos

rp

2

� �

2666666664

3777777775

� 2�0

p

� �� 2�

0

r

� �� 2�

0

s

� �� Jp(A)Jr(B)Js(C)cos(p �vmt)

�cos(r �3vmt)cos(s �9vmt)

266666666666666664

377777777777777775

¼X1p¼0

X1r¼0

X1s¼0

sinp

2(nþ s)

� �cos

pp

2

� �cos

rp

2

� ��sin

pp

2

� �sin

rp

2

� ��þ

þcosp

2(nþ s)

� �cos

pp

2

� �sin

rp

2

� ��þsin

pp

2

� �cos

rp

2

� ��

2666666664

3777777775

� 2�0

p

� �� 2�

0

r

� �� 2�

0

s



266666666666666664

377777777777777775

T Electr. Power Appl., Vol. 1, No. 5, September 2007

¼X1p¼0

X1r¼0

X1s¼0

sinp

2(nþ s)

� �cos

p

2(pþ r)

� �hþcos

p

2(nþ s)

� �sin

p

2(pþ r)

� �i� 2�

0

p

� �� 2�

0

r

� �� 2�

0

s



26666666664

37777777775¼

¼X1p¼0

X1r¼0

X1s¼0

sinp

2(nþpþ rþ s)

� �h i2�

0

p

� ��

� 2�0

r

� �� 2�

0

s



266666664

377777775(13)

After substituting this expression in (8) and undoing thechanges made to (9), we can obtain the complete expressionfor the simple voltage wave

VA0(t)¼E

2

� �[1:15cos(vmt)� 0:27cos(3vmt)

� 0:029cos(9vmt)]

þ4

p

E

2

� �X1n¼1

1

n

� �X1p¼0

X1r¼0

X1s¼0

�

sinp

2(nþ pþ rþ s)

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

d

� 2�0

p

� �� 2�

0

r

� �� 2�

0

s

� �� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

c

�Jp 1:15np

2

� �Jr �0:27

np

2

� �Js �0:029

np

2

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

c

�cos(p �vmt)cos(r � 3vmt)cos(s � 9vmt)|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}b

266666666666666664

377777777777777775

� cos(nvpt)|fflfflfflfflffl{zfflfflfflfflffl}a

with

2�0

p

� �� ¼

1 if p ¼ 0

2 if p . 0

�;

2�0

r

� �� ¼

1 if r ¼ 0

2 if r . 0

�;

2�0

s

� �� ¼

1 if s ¼ 0

2 if s . 0

�(14)

where Jp,r,s are the first class Bessel functions with orders p,r and s, respectively.The carrier frequency, vp is assumed to be constant.

According to (2), the carrier’s instantaneous pulse can bedefined as

vp ¼ vi ¼du

dt¼ vc � kf ( sin(vmt))

2 (15)

719

where the instantaneous phase is

u ¼

ð[vc � kf ( sin(vmt))

2] dt (16)

This is the instantaneous phase of the frequency-modulated triangular signal and the term ‘a’of (11) thusbecomes

cos(nvct) ¼ cos n

ð(vc � kf ( sin(vmt))

2) dt

� �

¼ cos(nvct) cosnkf sin(2vtmt)

4vm

� ��

� sin(nvct) sinnkf sin(2vtmt)

4vm

� ��cos

nkf t

2

� �

þ cos(nvct) sinnkf sin(2vtmt)

4vm

� ��

þ sin(nvct) cosnkf sin(2vtmt)

4vm

� ��sin

nkf t

2

� �

¼ cos(nvct) sinnkf t

2

� ��

� sin(nvct) cosnkf t

2

� ��sin

nkf4vm

sin(2vmt)

� �

þ cos(nvct) cosnkf t

2

� ��

þ sin(nvct) sinnkf t

2

� ��cos

nkf4vm

sin(2vmt)

� �

Using (10), (11) and (12) and substituting, it thenbecomes

¼ sin n vc �kf2

� �t

� �X1v¼0

sinvp

2

� �2�

0

v

� ��

� Jvnkf4vm

� �cos v 2vmt �

p

2

� ��

þ cos n vc �kf2

� �t

� �X1v¼0

cosvp

2

� �2�

0

v

� ��

� Jvnkf4vm


p

2

� �� ¼

which finally yields

¼X1v¼0

cos n vm �kf2

� �t �

vp

2

� �2�

0

v

� ��

� Jvnkf4vm


p

2

� �� (17)

This expression, when substituted in (14) yields the generalexpression of the simple output voltage of the PWM inver-ter with harmonic injection and frequency-modulated tri-angular carrier

720

VA0(t) ¼E

2

� �[1:15 cos(vmt)� 0:27 cos(3vmt)

� 0:029 cos(9vmt)]þ4

p

E

2

� �X1n¼1

1

n

� �X1p¼0

X1r¼0

X1s¼0

X1v¼0

�

sinp

2(nþ pþ r þ s)

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

d

�

2�0

p

� �� 2�

0

r

� ��

2�0

s

� �� 2�

0

v

� �� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

c

� Jp 1:15np

2

� �Jr �0:27

np

2

� ��Js �0:029

np

2

� �Jv

nkf4vm

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}c

� cos(p � vmt) cos(r � 3vmt)

� cos(s � 9vmt) cos v 2vmt �p

2

� �� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

b

� cos n vc �kf2

� �t �

vp

2

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

a

2666666666666666666666666666666666664

3777777777777777777777777777777777775

2�0

p

� �� ¼

1 if p ¼ 0

2 if p . 0

�;

2�0

r

� �� ¼

1 if r ¼ 0

2 if r . 0

�;

2�0

s

� �� ¼

1 if s ¼ 0

2 if s . 0

�;

2�0

v

� �� ¼

1 if v ¼ 0

2 if v . 0

�(18)

where E is the inverter’s level of continuous voltage, n is theharmonic and Jp,r,s,v are first class Bessel functions withorders p, r, s and v, respectively [11].In the above expression, there are two distinct terms: the

first generates the voltage’s fundamental term, while thesecond expresses the total content of harmonics.Importantly, the first term includes a third and ninth harmo-nic, but they disappear by subtraction in the line-to-linevoltage.The following considerations can be made with regard to

the second term:

† For every n, when p ¼ r ¼ s ¼ v ¼ 0: a harmonic will orwill not exist, depending on whether n is odd and therefore‘d’ is non-zero; ‘c’ is the amplitude of the harmonic and ‘b’is equal to 1; and ‘a’ is the pulse at which the harmonicoccurs: n(vc2 (kf/2))† For every n, and for any value of p, r, s and v (exceptp ¼ r ¼ s ¼ v ¼ 0): (18) defines the side bands and term‘d’ determines whether or not such a side band exists,depending on the value of the resulting sine. The term ‘c’gives the amplitude of the side band’s components, andlastly, terms ‘b’ and ‘a’ give the frequencies at which theside band occurs.


Keeping in mind that the expression for the modulationorder is defined as the relationship between the triangularcarrier frequency and the modulator frequency, then

M ¼vp

vm

If the triangular carrier is also frequency modulated, then vp

is its instantaneous frequency.This means that there exists a modulation order for every

instant, which varies around an average modulation order,given by

�M ¼((vp,max þ vp,min)=2Þ

vm

¼

(vc � kf ( sin(0))2

þvc � kf ( sin(p=2))2)=2

" #vm

¼2vc � kf2vm

Comprobation: Fig. 7 shows the frequency-modulatedsignal, following (2). In this case, a value of 25 � 50 . (2p)rad/s is chosen for the central frequency (pulse) vc and avalue of 20 � 50 . (2p) rad/s is chosen for the modulationconstant kf. In other words, 25vm and 20vm are chosen,where vm is the base pulse of the PWM modulator with har-monic injection (1), which coincides with the pulse of thesinusoidal frequency modulator according to (2).The lower graph shows the triangular wave after being

modulated in frequency, and can be seen that 15 completebilateral triangular signals fit into 20 ms, which correspondsto the period of vm; that is the system is equivalent to amodulation order of M ¼ 15.The upper graph shows how the instantaneous relation-

ship M ¼ vp/vm varies sinusoidally, from a maximum ofM ¼ 25, which corresponds to (2) at time t ¼ 0, to aminimum of M ¼ 5, which corresponds to (2) for a valueof vp ¼ 25vm – 20vm at t ¼ 5 ms (p/2 radians of fre-quency vm), only to increase once again at t ¼ 10 ms andthus continue the cycle. All these values oscillate sinusoid-ally aroundM ¼ 15, which coincides with the same order asthe modulated triangular signal.


We reach to the same conclusion by starting from (2) andneed not check against a graph that the pulse vp ranges froma maximum value of 25vm to a minimum value of 25vm –20vm ¼ 5vm. The average value is thus (25þ 5)vm/2 ¼ 15vm, which according to term ‘a’ of (18), is alsoequal to vc2 (kf/2); that is the average order of the instan-taneous orders, which is ultimately the final order of thefrequency modulated signal, obeys the following expression

�M ¼2vc � kf2vm

¼2vc � kf

2(19)

The instantaneous order varies sinusoidally around thisvalue, as seen above, where vc, kf are called the central fre-quency order and modulation constant order, respectively.This expression is of great importance, since it sets thedesired average order for the triangular signal (15 in thisexample) and also makes it possible to set the central fre-quency order and the modulation constant order.

4 Comparison of different PWM techniques

For the purpose of testing the advantages of the new tech-nique proposed herein, ‘PWMS with harmonic injectionand frequency-modulated triangular carrier’, the new tech-nique will be compared with the most commonly usedPWM techniques. First, to accurately compare differentPWM techniques, one must define a set of ‘quality’ par-ameters to measure them by. The parameters adoptedherein are the same as those used in the literature [13],and are as follows.

4.1 Total harmonic distortion (THD)

A signal’s THD measures the extent to which it approxi-mates a perfect sine wave.

THD ¼1

V1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX1n¼2,3,...

V 2n

vuut (20)

where V1 is the effective value of the fundamental term andVn represents the different harmonic components. Thus, the

Fig. 7 Instantaneous modulation order and frequency-modulated triangular

721

lower the THD value, the more the signal approximates aperfect sine wave, because the contribution from harmonicsis very weak.

4.2 Distortion factor (DF)

The THD reflects the total content of harmonics, but itdoes not indicate the significance of each harmoniccomponent’s magnitude. Lower order harmonics (with alow value of n) are the most difficult to filter. Thereforethe DF aims to minimise the contribution from veryhigher order harmonics, thereby showing the contributionof lower order harmonics, which are more problematic tofilter.

DF ¼1

V1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX1n¼2,3,...

Vn

n2

� �2

vuut (21)

where V1 is the same as in the previous case.It is likewise desirable for this value to be as low as

possible.

4.3 Lower order harmonic (LOH)

This is a measurement that seeks to select the harmonic thatmost greatly distorts the signal. The LOH is a harmonicwhose frequency is closest to the fundamental frequencyand whose amplitude is greater than or equal to 3% ofthe fundamental component. In this case, it is desirablefor this parameter to be as high as possible. In additionto these parameters, one must add the effective valueof the voltage of the fundamental term obtained in eachcase, Vef. It is also best to let this value be as high aspossible.Further, the modulation order adopted should meet a set

of requirements:

† M should be a whole value to allow for phase constancybetween the modulator and carrier signals.† It is preferable forM to be a multiple of three. This makesthe carrier, which is the highest harmonic in the simplevoltage, to disappear by subtraction in the line-to-lineoutput voltage.† M should be odd in order to guarantee that even-numbered harmonics are eliminated. It is also recommendedthat M to be at least equal to 10, in order to make thesampling that the carrier subjects on the modulator as effec-tive as possible, and allow for the gathering of sufficientinformation.

All the foregoing leads one to choose M ¼ 15 as themodulation order, since 15 is the first value that meets allthe above requirements. This order is equivalent to acarrier frequency of 15 � 50 ¼ 750 Hz.The different PWM techniques studied are:

† PWM with trapezoidal modulator, modulation orderM ¼ 15. (Fig. 8a and 8b).† Sinusoidal PWMwith sawtooth carrier, modulation orderM ¼ 15. Fig. 8c and 8d).† Modified sinusoidal PWM with modulation orderM ¼ 15 (Fig. 8e and f ).† Sinusoidal PWM with triangular carrier and modulationorder M ¼ 15 (Fig. 8g and h).† PWMS with harmonic injection and modulation orderM ¼ 15 (Fig. 8i and j).

722

† PWMS with harmonic injection and frequency-modulated triangular carrier, vc/vm ¼ 24.75; kf/vm ¼ 19.5, which is equivalent to M ¼ 15 (Fig. 8k and l)

The results are summarised in Table 1.It is obvious that this new technique outperforms the rest

in all of the quality parameters examined, thereby demon-strating its usefulness.

5 Comparison of the SPWM, HIPWM andHIPWM-FMTC techniques when fed on aninduction motor

The assembly was carried out following the diagram inFig. 9a; the firing pulses were obtained using a PC and aDSP, model DS1102 of Dspace (work is currently beingcarried out to implement such pulses in electronics). Thepulses govern the electronic switches of the power inverter(Skiip by Semikrom), which feeds the induction motor (M).The motor used in the test was an eAM 90SY rEx; 1 kW,380/220 V; 50 Hz; 1415’6 rpm. It was provided by theAEG Motor Factory in Martorell (Barcelona, Spain) andcontains a feature making it especially useful for heatingtests, since it has a K-type thermocouple probe.Temperature readings were taken using a C.A864 modeldigital thermometer by Chauvin Arnoux.An independent excitation dynamo (D) was attached at

the AC-motor’s output, which then fed a rheostat. Thisdevice was adjusted to load the AC-motor, keeping theresistant torque constant with the three waveforms ofstudy: SPWM, HIPWM and HIPWM-FMTC.Measurements were taken with a network analyser. Thisnetwork analyser is actually a spectrum analyser (specifi-cally TeamWares Equa model), which is capable ofcapturing and processing the first 50 voltage and currentharmonics, using them to represent waveforms andfrequency spectra, respectively; that is the waveform thatappears is obtained as a reconstruction made from thefirst 50 harmonics. It is therefore not a real-time measure-ment, such as that which could be obtained using anoscilloscope.The three approaches, SPWM, HIPWM and

HIPWM-FMTC were tested, this time using a modulationorder of M ¼ 27 in order to deal with different modulationorders in one paper. Adjustments were made in such a waythat in all three cases the same voltage of the fundamentalterm at vm was obtained. The results of temperaturemeasurements inside the stator for each of the three tech-niques is shown in Fig. 9b, where one can observe that start-ing from atmospheric temperature, the permanenttemperature reached with the proposed technique wasapproximately 2% less than that reached using the SPWMtechnique and approximately 1% less than that of theHIPWM, which means that the proposed technique makesthe machine heat up less while generating the sameamount of useful power.The results obtained with the network analyser at the

inverter’s output are shown in Fig. 10. As can be seen,the proposed technique provides an extraordinary improve-ment to the inverter’s output wave, since the firstsignificant harmonic is of the 25th order with the SPWMand the 23rd order with the HIPWM, while with theHIPWM-FMTC it is in the 47th order, making it possibleto feed the motor with a signal containing fewer lowerorder harmonics, thereby causing the motor to heat up andvibrate less.


Fig. 8 Different PWM techniques and its parameters

a Trapezoidal PWM; modulation order M ¼ 15Modulator and carrier signalSimple and line-to-line voltagesb Frequency spectrum for trapezoidal PWM: THD(%) ¼ 63.3395; DF(%) ¼ 0.2321; LOH ¼ 5; Vef ¼ 0.6427c Sinusoidal PWM with sawtooth carrier, modulation order M ¼ 15Modulator and carrier signalSimple and line-to-line voltagesd Frequency spectrum for PWM with sawtooth carrier: THD(%) ¼ 66.9394; DF(%) ¼ 0.2269; LOH ¼ 10; Vef ¼ 0.6123e Modified sinusoidal PWM, modulation order M ¼ 15Modulator and carrier signalSimple and line-to-line voltagesf Frequency spectrum for modified sinusoidal PWM: THD(%) ¼ 66.5354; DF(%) ¼ 0.1965; LOH ¼ 13; Vef ¼ 0.6200g Sinusoidal PWM with triangular carrier, modulation order M ¼ 15Modulator and carrier signalSimple and line-to-line voltagesh Frequency spectrum for sinusoidal PWM with triangular carrier: THD(%) ¼ 68.0176; DF(%) ¼ 0.1922; LOH ¼ 13; Vef ¼ 0.6122i PWMS with harmonic injection, modulation order M ¼ 15Modulator and carrier signalSimple and line-to-line voltagesj Frequency spectrum for PWMS with harmonic injection: THD(%) ¼ 50.1266; DF(%) ¼ 0.1958; LOH ¼ 11; Vef ¼ 0.7089k PWMS with harmonic injection and frequency-modulated triangular carrier, vc/vm ¼ 24.75; kf/vm ¼ 19.5, which is equivalent to M ¼ 15Modulator and carrier signalSimple and line-to-line voltagesl Frequency spectrum for PWMS with harmonic injection and frequency-modulated triangular carrier: THD(%) ¼ 47.3222; DF(%) ¼ 0.1719;LOH ¼ 19; Vef ¼ 0.7452

IET Electr. Power Appl., Vol. 1, No. 5, September 2007 723

Fig. 8 Continued

Table 1: Results of the different PWM techniques

M ¼ 15 PWM with trapezoidal

modulator

SPWM with sawtooth

carrier

Modified SPWM SPWM HIPWM HIPWM-FMTC

Figures 8a, b 8c, d 8e, f 8g, h 8i, j 8k, l

THD (%) 63.3395 66.9394 66.5354 68.0176 50.1266 47.3222

DF (%) 0.2321 0.2269 0.1965 0.1922 0.1958 0.1719

LOH 5 10 13 13 11 19

Vef 0.6427 0.6123 0.6200 0.6122 0.7089 0.7452


Fig. 9 Experimental assembly and the results of temperature measurements inside the stator

a Diagram of the experimental assemblyb Temperature measurements in the stator with each technique

Fig. 10 Results obtained with the network analyser at the inverter’s output

a Line-to-line voltage and spectrum obtained with the analyser with SPWM for M ¼ 27b Line-to-line voltage and spectrum obtained with the analyser with HIPWM for M ¼ 27c Line-to-line voltage and spectrum obtained with the analyser with HIPWM-FMTC for vc ¼ 52.5 vm and kf ¼ 51 vm, equivalent to M ¼ 27

6 Conclusions

A new PWM technique has been presented. The mathemat-ical expression of the simple output signal has beenshown. Likewise, it has been shown that the methodproduces a very significant increase in the fundamentalterm of the output voltage as well as a substantialdecrease in harmonics, thereby causing the load which isconnected to the inverter’s output, in this case a motor, toheat up less.

7 Acknowledgments

This paper was supported by the CICYT projectDPI2002-00754 of the Spanish Ministry of Science andTechnology. We would like to express our most sincere


thanks to the AEG factory in Martorell (Barcelona,Spain), for providing the motor used in the test.

8 References

1 Yu, X., Dummingan, M.W., andWilliams, B.: ‘Phase voltage estimationof a PWM VSI and its application on vector-controlled inductionmachine. Parameter estimation’, IEEE Trans. Ind. Electron., 2000, 47,pp. 1181–1184

2 Pande, M., Joos, G., and Jin, H.: ‘Output voltage integral controltechnique for compensating non-ideal DC buses in voltages sourceinverters’, IEEE Trans. Power Electron., 1997, 12, pp. 302–310

3 Bowes, S.R., and Grewal, S.: ‘Modulation strategy for single phasePWM inverters’, Electron. Lett., 1998, 34, (5), pp. 420–422

4 Sundereswaran, K., and Kumar, A.P.: ‘Voltage harmonic eliminationin PWM AC chopper using genetic algorithm’, IEE Proc., Electr.Power Appl., 2004, 151, pp. 26–31

5 Negin, M.M.: ‘SVPWM verification of an optimal control of inductionmachines’. UPEC’2002, Standffordshire University, UK, September2002

725

6 Kukrer, O., and Komurcugil, H.: ‘Variable sampling frequency PWMwaveforms’, IEEE Power Electron. Lett., 2003, 1, pp. 14–16

7 Iwaji, Y., and Fukuda, S.: ‘A pulse frequency modulated PWM inverterfor induction motor drives’, IEEE Trans. Power Electron., 1992, 7,pp. 404–410

8 Tse, K.K., Chung, H.S.-H., Hui, S.Y.R., and So, H.C.: ‘A comparativestudy of carrier-frequency modulation techniques for conductedEMI suppression in PWM converters’, IEEE Trans. Ind. Electron.,2002, 49, pp. 618–627

9 Hashem, G.M., Mashaly, H.M., and Shams, A.: ‘Generation ofoptimum pulse width modulation based on artificial neural network’.UPEC 2002, September 2002, vol. 1

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10 Boost, M.A., and Ziogas, P.D.: ‘State-of-the-art carrier PWMtechniques: a critical evaluation’, IEEE Trans. Ind. Appl., 1998, 24,(2), pp. 271–280

11 Watson, G.N.: ‘A treatise on the theory of Bessel functions’(Cambridge University Press, Cambridge, England, 1999, 2nd edn.)

12 Stemmler, H., and Ellinger, T.: ‘Spectral analysis of the sinusoidalPWM with variable switching frequency for noise reduction ininverter-fed induction motors’. Power Electronics Specialist Conf.PESC, 1994

13 Rashid, M.H.: ‘Power electronics; circuits, devices andapplications’ (Prentice-Hall International Inc., New Jersey, USA,1993, 2nd edn.)


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