DC link current in PWM inverters

P.D. Evans, B.Sc.(Eng.), Ph.D., D.I.C., A.C.G.I., and

R.J. Hill-Cottingham, B.Sc.
Indexing terms:

Inverters, Modulation, Power electronics

Abstract: A general method is described for calculating the current drawn by an inverter from its DC source. It
has special relevance to pulse-width modulated inverters and allows the ripple current rating of the filter capacitor to be specified with some precision. The method is for an inverter of any number of phases and uses
switching functions derived from the harmonic coefficients of the inverter waveform. Experimental results from
single- and 3-phase inverters demonstrate the validity of the method. Some generalised results are presented for
a 3-phase inverter for the special case where the output current can be assumed to be the fundamental component only. Variation of the DC link current harmonics with power factor of the load and depth of modulation
under these conditions is presented and discussed.

List of principal symbols

= peak amplitude of mth harmonic of switching function

Bt = peak amplitude of /th harmonic of DC link current
Fk = switching function for kth inverter leg
[dc = instantaneous DC link current
= instantaneous DC link current due to the kth leg
= instantaneous current in the kth leg
= peak amplitude of the nth harmonic of load current
M = depth of modulation
P = number of modulating pulses per cycle
a. = number of inverter 'legs'
V = average DC link voltage
KG, = peak amplitude of nth harmonic voltage between
the output terminal of the kth inverter leg and the
midpoint of the DC supply
Zkn = impedance to the nth harmonic between the output
terminal of the kth inverter leg and the midpoint of
the DC supply
9 = general variable for angle within the fundamental
waveform period

+ V/2

Am

Introduction

It is of interest to know the current waveform drawn from

the DC line of a pulse-width modulated (PWM) inverter.
This is partly because it improves the general understanding of inverter operation, but also because it allows the
filter capacitors to be specified satisfactorily. This becomes
more important as the inverter rating increases. The
present work relates to the current drawn by the inverter
from the DC link. The input currents to the DC link via
the mains filter are not considered here as they are considered to be well known.
A physical picture of the problem is illustrated in Fig. 1
for a 3-phase inverter. Fig. la shows the 'phase' voltage of
one leg of an inverter, i.e. the potential of the output terminal with respect to the midpoint of the DC voltage,
when there are six modulating pulses per cycle and the
depth of modulation is 0.8. A low number of pulses has
been used here for clarity. The fundamental voltage component is also shown in Fig. la together with the leg
current which is assumed to be sinusoidal. The current lags
the voltage by an arbitrary phase angle (j>, where </> = 30
has been selected as it corresponds closely to a power
factor of 0.8. When the PWM voltage is positive the top
Paper 4633B (P6), first received 1st November 1985 and in revised form 13th February 1986
The authors are with the School of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

IEE PROCEEDINGS,

Vol. 133, Pt. B, No. 4, JULY 1986

-V/2

1
0

HVIIMIJlfflriJl

Fig. 1

0
x
2ir
Derivation of DC link current in a 3-phase PWM inverter

a (i) inverter leg voltage

(ii) fundamental component of leg voltage
(iii) sinusoidal current
b Switching function
c DC link current drawn by red leg
d DC link current drawn by yellow leg
e DC link current drawn by blue leg
/Total DC link current

transistor of the leg is on and, during these periods, the leg

current is drawn from the DC rail through the active
device if it is positive and feeds back into the DC rail via
the diode if it is negative. The resulting pulses of DC link
current are shown shaded in Fig. la. Fig. \b shows a
switching function which has an amplitude of either 0 or 1
and transition edges between these states which correspond identically with the timing of the PWM voltage
waveform in Fig. la. The use of switching functions is a
well established technique for handling this type of
problem, for example, Pelly [1]. Clearly, multiplying the
sinusoidal current waveform of Fig. la by the switching
function of Fig. \b gives the current drawn from the DC
rail as shown in Fig. \c. Figs. Id and le are the current
waveforms which would be drawn from the other two legs
of a 3-phase inverter. They therefore have the same shape
as Fig. \c but are displaced by 2n/2> radians and An/3>
radians, respectively. Fig. I/is the sum of Fig. lc, d and e
and is therefore the total DC link current. For the chosen
power factor the DC link current appears to be always
217

positive. In general, and especially with lower power

factors, the current can be both positive and negative.
For comparison, the DC link current for the more familiar case of the quasisquare-wave inverter is shown in Fig.
2. It has been derived by the same switching function technique and can be seen to be of the well known form.

as bipolar power transistors, but the analysis also applies

for other devices.) If the current drawn from the output
Ieg1

Fig. 3

leg k

legq

General inverter with q legs

terminal of the /cth leg is denoted by ik, the current drawn

from the positive line of the DC supply by the /cth leg, idck,
may be expressed as
where Fk is the switching function for the top semiconductor device in the leg. It is equal to + 1 when the device
is on and zero when it is off.
Clearly, the total DC current for the inverter idc is given
by
(2)

Fig. 2

Derivation of DC current in DC link of a quasisquare-wave

inverter
a (i) inverter leg voltage
(ii) fundamental component of leg voltage
(iii) sinusoidal current
b Switching function

c DC link current drawn by red leg

d DC link current drawn by yellow leg
e DC link current drawn by blue leg
/ T o t a l DC link current

These complex DC link current waveforms are most

conveniently defined in terms of their harmonic components. To calculate them therefore the method described
pictorially in the time domain in Figs. 1 and 2 above can
be repeated mathematically by representing the switching
functions and voltage waveforms harmonically in terms of
their Fourier series. In the resulting definition of the DC
current, the zero order harmonic or DC term of the series
corresponds to the average flow of real power into the
inverter and all the AC harmonic components define the
ripple current which circulates in the main filter capacitor.
A general method of solution for this problem is presented for an inverter with q legs. The two most common
cases are the 3-phase bridge inverter, q = 3, and the singlephase inverter, q = 2. The expressions for DC link current
are worked in terms of the DC voltage, the load impedance and the harmonic coefficients of the PWM waveform.
Although sinusoidal PWM is probably the most common
form, and a method of calculating these harmonic coefficients is suggested in Appendix 8, the method applies for
other switching strategies, provided that the appropriate
harmonic coefficients are used. The basis of the method is
to calculate the current drawn from each leg in terms of
voltage harmonics and impedance and to compute the DC
current by multiplying the leg current by a switching function for the leg which is based on the same set of harmonic
coefficients as the voltage. The total DC link current is the
sum of the contributions from all the inverter legs.
2

Analysis of DC current waveform

Consider the kth leg of inverter with q legs, as shown in

Fig. 3. (The power semiconductor devices are represented
218

The general problem of defining ik and Fk (eqn. 2) can be

solved for the general q leg inverter if each inverter leg is
treated in isolation. The method of doing this is described
below.
2.1 Calculation of leg current ik

The 'phase' voltage of the inverter leg, i.e. the voltage

between the output terminal of the leg and the notional
midpoint of the DC supply G, is used together with the
impedance between these two points to calculate ik. A connection is not made to this midpoint and no current flows
into it. It is simply used as a convenient concept for the
development of the mathematical method. In a 3-phase
inverter, q = 3, the impedance existing between the output
terminal and the notional ground is the impedance of one
phase of a star load, or of the equivalent star load. For a
single-phase inverter, q = 2, it is, in effect, one-half of the
single-phase load.
If the amplitude of the nth harmonic of the phase
voltage of the /cth leg is denoted by VkGn, and the method
of calculating this is discussed in Appendix 8, the corresponding harmonic leg current ikn is given by
i = V JZ

(3)

where Zkn is the impedance to the nth harmonic frequency

between terminal and notional DC midpoint.
Consequently, the total leg current is given in the usual
way by
(4)

It has been assumed that leg one is the voltage reference

phase so that, for an inverter giving a balanced set of voltages, the /cth leg is displaced in phase by 2n(k l)/q. This
being the case, the sum of all the leg currents i0 is given by
(5)

'o=

= Z
k=

(vkGjzkny"ee-j"2k-

(6)

(_n= xi

1EE PROCEEDINGS,

Vol. 133, Pt. B, No. 4, JULY 1986

If we assume that the impedances in all legs are the same,

i.e. Zkn = Zn, eqn. 6 can be rearranged to
i - V

(7)

The k summation is a geometric series which becomes

-jn?n(k-l)iq _
fc=

e-jn{q-l)lq

S1

" \nVl>

sin (nn/q)

(8)

From the physical model adopted it is clear that we

require i0 to equal zero as no net current flows into the
notional midpoint G. Eqn. 8 is zero for all harmonics
except when
n = sq, s = 0, 1 , 2, ...
These harmonics cannot exist, however, and so the expression for ik must be restated as

where n # sq, s = 0, + 1 , 2, ..., and m + n = s'q, s' = 0,

1, 2 , . . . .
Eqn. 14 is therefore the final general expression for the
total DC current in the time domain. Performing the
double summation of n and m and observing the constraints on the values of n and m + n, allows the DC
current waveshape to be calculated over a cycle, as 9 varies
from 0 to 2n.
The harmonic content of this waveform can be deduced
in the usual way. That is to say, if Bt is the /th harmonic of
the DC current, then
,-M

(15)

Substituting the expression for idc (eqn. 14) into this and
simplifying, produces the final result:
n m

where n # sq, s = 0, 1 , 2, ..., m + n = s'q, s' = 0, + 1 ,

+ 2, ..., and I = s'q are the only nonzero harmonics

present.
Eqn. 16 is the general expression for the harmonic
For a 3-phase inverter, q = 3, this is stating the well
content of the DC current waveform. The most notable
known fact that triplen currents are absent from the load.
feature of it is that the only harmonic currents present in
the DC link of a 3-phase inverter are multiples of three, i.e.
2.2 Calculation of switching function Fk
the triplens. This is, of course, because 3-phase sets of all
The method for calculating Fk, which is, of course, closely
harmonics except the triplens sum to zero, so that there
related to the phase voltage, is described in Appendix 8.
can only be triplens in the DC link. Correspondingly, only
Here, we state the following relationship:
even current harmonics occur in the DC line of a singlephase inverter, when q = 2.
(10)
Eqns. 14 and 16 are possibly unnecessarily complicated
for some applications. In the case where it is known that
the output current is substantially sinusoidal, i.e. the load
2.3 Derivation of expression for total DC current
time constant is long compared to the period of the switchIntroducing eqns. 9 and 10 into eqn. 2 gives the following
ing frequency, some simplifications can be made. This
result:
applies for most AC drive systems. In the first case, only
the fundamental component of output current is significant
i* = E { E
and, in the second case, the maximum value of this current
fc=l
and the specified range of power factor are more likely to
be the known parameters from the equipment specifix 1 A.
cation. For these conditions, in terms of the complex form
of Fourier analysis being used the following substitution
n^sq,s
= 0, 1 , 2, ... (11)
can readily be shown to be true for terms in eqns. 14 and
Also in eqn. 11, VkGn has been replaced by VAn, where V is 16.
the total DC link voltage. The harmonic coefficients of the
switching function Am and the phase voltage are the same.
(17)
vAjzn =
The only two differences between the two functions are
n=l
first that the amplitude of the phase voltage is V times the
where lx is the peak fundamental current and 0 is the
amplitude of the switching function and secondly, that the
power factor angle.
phase voltage has no DC component, a fact which is
For other applications, such as inverters in uninterruptalready taken into account by the s = 0 condition in
ible power supplies where filters are used to remove the
eqn. 9.
switching frequencies from the output waveform, the filter
Eqn. 11 rearranges to
currents modify the device currents and the DC link currents and the complete expressions, eqns. 14 and 16, may
(12)
be more appropriate.
'* = Z Z \{VAJZn)Aj(m+n) t
n oo

n^sq,s

= 0, 1 , 2 , . . . (9)

=Z

m (.

k= 1

As in eqn. 7 above, the k summation is a geometric series

which sums to

=i

sin n(m + n)/q

This is zero unless m + n = s'q, where s' = 0, 1 , 2, ...,

where the sine quotient becomes equal to q. Hence, eqn. 12
simplifies to

i* = Z Z
IEE PROCEEDINGS, Vol. 133, Pt. B, No. 4, JULY 1986

(14)

Comparison of experimental and theoretical

results

A number of measurements were taken to verify the equations developed in Section 2. In the first instance, a singlephase bridge {q = 2) was used and the spectrum of the DC
link current was measured with high and low power
factors with and without a 'trap' filter as detailed in Table
1. In all cases the switching rate was 2.1 kHz (42 pulses per
cycle). The trap filter consisted of a series inductance of
219

Table 1: Single-phase bridge test conditions

Test

Load

Load
power
factor

Depth of
modulation

DC link
voltage, V

Load
current, A

1A
1B

FURL with trap

filter
FURL with trap
filter

0.91
0.91

0.58
0.58

200
200

42.4
39.5

0.45
0.45

0.73
0.73

200
200

33.4
31.8

2A
2B

0.26 mH and a series LC resonance circuit of 0.28 raH and

6.6 //F in parallel with the load.
The calculated and experimental results for these tests
are summarised in Fig. 4. It can be seen that agreement is

80 82 84 86

162 164 166 168170 172 174

harmonic order

the amplitude of the 84th harmonic current in test IB

would rise from 20 A to around 35 A, together with a
significant increase in most other harmonic currents.
The power transistors in the single-phase inverter had
both capacitive and inductive snubbers to control switching conditions. The capacitor changing currents were
limited by the snubber inductance to around 20 A. As their
duration was only 3 /is, however, they did not appear in
spectrum measurements that were limited to a bandwidth
of 20 kHz by the spectrum analyser.
A second set of results is shown in Fig. 5 for a 3-phase
inverter (q = 3). In this case, the load power factor was 0.4
lagging, the depth of modulation 0.76, the DC link voltage
200 V and the load current 10 A RMS. The sinusoidal
pulse-width modulation was at a switching rate of 750 Hz
(15 pulses per cycle). It can be seen that agreement between

244 246 248 250 252 254 256 258 260

Fig. 4
Comparison of theoretical and
experimental results for single-phase
inverter
a Test 1
b Test 2
O # experimental results

theoretical results
O A B

generally good for all cases. Only even harmonics exist, in

accordance with eqn. 1 above. Apart from the DC and
second harmonic components, the harmonics occur in
clusters around the even multiples of the switching frequency. This is a consequence of using an even number of
modulating pulses per cycle of output frequency which, in
effect, results in a doubling of the frequency which appears
across the load. The greatest disparity between measured
and calculated results occur at the highest frequencies displayed which are in the range 12.5-13 kHz. This is thought
to be due to the departure of the actual output waveform
from the ideal caused by the effect of the 'dead time' of
10 /is between one device switching off and the other
device in the same leg being switched on. During this time
the actual output voltage is dependent upon the sense of
the current. In addition, other parasitic effects which are
not allowed for in the model may become significant at
these frequencies. The greatest effect of the increased load
time constant, test 2A compared to test 1A, is the
reduction of the predominant current at twice the switching frequency, 4.2 kHz, from around 20 A to 10 A.
It can be seen that the effect of the trap filter on the DC
link current was found to be quite small under the conditions tested and would not influence significantly the
choice of filter capacitor ripple current rating. Under different design conditions, however, this conclusion would
not necessarily hold. For example, computed results show
that if the 0.25 mH series inductance were reduced to zero,
220

measured and calculated results is generally good and that,

as predicted by eqn. 16, only triplen harmonics occur in
the DC link.
The current waveforms for this example of a 3-phase
inverter were also computed in the time domain and are
compared with measurements in Figs. 6 and 7. Figs. 6a
and b illustrate the computed and measured shapes,
respectively, of the total DC current for one-half of the
fundamental cycle. It can be seen that there is close agreement between them. If the two waveforms were to be
superimposed it would be difficult to distinguish one from
the other. The quality of agreement in Fig. 6 is the same as
in Fig. 5 as both Figures contain the same information, the
former being in the time domain and the latter in the frequency domain.
The current waveform has both positive and negative
values as a consequence of the low power factor of the
load. For this reason also, the average DC current which is
responsible for real power transfer is small (around onefifth) of the peak DC link current. The current in the filter
capacitor consists of the alternating components only of
the DC link current and this is also contained in Fig. 6 by
displacing the zero level for the capacitor current to the
average DC current level. The current drawn by one phase
of the load, i.e. the leg current, is shown in Fig. 7, where
the computed and measured results in Fig. la and b,
respectively, are again in good agreement. The computed
results were obtained using eqn. 9 above. The leg PWM
IEE PROCEEDINGS,

Vol. 133, Pt. B, No. 4, JULY 1986

voltage was used as the reference phase for the calculations

so the positive going zero crossing of the current in Fig. 7

which occurs at just over 70 relates directly to the load

power factor of 0.4.

3.0 i -

2.5

2.0

Q.

E
o
. 1.5

1.0

0.5

12

18

24 30 36 42 48 54 60 66 72 78
harmonic order

84 90

Fig. 5
Comparison of theoretical and experimental results
for a 3-phase inverter
# experimental results
theoretical results

96 102 108

15r

10

< 5

A
f

- 0
o
Q

zero for
capacitor
current

-5

-10

20

40

60

80

100

120

160

140

180

degrees of fundamental period

15

10

/i ii

zero for
capacitor
current
MM*

-5
20

40

60

80

100

120

degrees of fundamental period

6

IEE PROCEEDINGS, Vol. 133, Pt. B, No. 4, JULY 1986

140

160

180

Fig. 6
Computed and measured DC link and
capacitor current for a 3-phase inverter
a Computed
b Measured
221

Generalised results for 3-phase inverters

Although Section 2 developed an equation to predict DC

link current under general conditions it is possible to
15r
10

-10
-15.

20

40
60
80
100
120 140
degrees of fundamental period

160 180

40
60
80
100
120 140
degrees of fundamental period
b

160 180

20

Fig. 7
Computed and measured half cycle of leg current for a 3-phase
inverter
a Computed
b Measured

derive some simplified results which have quite wide application under common conditions. This is done by applying
the substitution suggested in eqn. 17 into eqn. 16 and
results for a 3-phase inverter are given below. Under these
conditions and, assuming that the number of pulses per
cycle is a multiple of three, as is invariably the case with a
3-phase inverter with synchronous PWM, the harmonics
which appear in the DC link are approximately as follows.
Apart from the DC component, the most significant harmonics of current that appear are the even multiples of the
switching frequency, i.e. 2p, Ap etc., and the sidebands
around the odd multiples of three which are given by
P q, 3p q, 5p q etc. These conclusions apply except
for small values of frequency ratios where the separation
between modulating frequency and carrier frequency is so
small that sideband clusters overlap.
Computed results over a wide range of depth of modulation M and power factor, cos (p, are presented in Fig. 8,
for the normalised condition that / = 1.0. All the harmonic
amplitudes plotted are peak values.
Fig. 8a illustrates the variation of the zero order harmonic, or average current, in the DC link with M and cos
4>. There is a linear dependence with both these parameters
and it can be readily shown that the results are consistent
222

with the continuity of power flow through a lossless

3-phase inverter.
Fig. 8b shows that both | B2p \ and | B4p | also vary linearly with power factor and that | B2p | is always greater
than | B 4p |. The more complex relationship between these
currents and M, which is not readily discernible in this
Figure, is illustrated more clearly in Fig. 8c, which presents
| B2p | and | B 4p | plotted against M for the power factor
condition cos (f> = 0.8. It is apparent from Fig. Sb,
however, that these general shapes apply for all power
factors. Most notable in Fig. 8c is that | B2p | is a
maximum at M = 0.5-0.6. This is, presumably, similar to
the condition in a simple chopper when the ripple element
in the filter capacitor is a maximum when the mark/space
ratio is 0.5.
Fig. $d displays \Bpq\ against M because these harmonics are very closely independent of power factor. They
increase with M and at higher values of M they are of the
same order of magnitude as I B?
'2p\ n | and
'4pl
Fig. Se shows that the variation |B 3 p J with M and
cos <f> is relatively complex. These harmonics have a generally decreasing trend with improving power factor. Their
dependence on M shows a maximum with M = 0.5-0.6,
decreasing with higher and lower depths of modulation.
There are two points of detail which should be noted
about these results. First, it is recognised that there is, in
principle, an inconsistency in assuming that the load time
constant is long compared to the PWM switching period
so that the fundamental component of current only is considered and, at the same time, quoting results for a perfectly resistive load, i.e. cos 0 = 1.0. However, this is
thought not to detract from the illustrative trends of the
general curves in Fig. 4. Secondly, with the approximate
form of sinusoidal PWM used, as described in Appendix 8,
results in sidebands either side of p and 3p being slightly
different in amplitude and so the average of each pair has
been plotted in Fig. 4 above. If the pure form of sinusoidal
PWM has been used that obtained by intersecting a
sinusoidal reference wave with a triangular carrier each
pair of sidebands would have the same amplitude and this
amplitude is very close to the average value used in the
present results.
5

Conclusions

A general analytical method of calculating the shape of the

DC link current in a polyphase inverter has been described
and justified by favourable comparisons between measured
and calculated conditions for both single- and 3-phase
inverters. Although it has been discussed primarily in
terms of sinusoidally pulse-width modulated inverters it is
valid for any balanced form of control providing that the
appropriate harmonic coefficients are used in the switching
function and voltage waveform. It has been shown that in
an inverter with q legs, the only harmonics appearing in
the DC current waveform are integer multiples of q, i.e. in
a 3-phase inverter all harmonics in the DC current are
multiples of three.
An attempt has been made to present some generalised
results for a 3-phase inverter which allows the DC link
current harmonics to be scaled from graphs which show
the variation of the harmonic currents with both depth of
modulation of the waveform and the power factor of the
load.
The results presented, either the general analytical
expressions or the generalised curves, enable a ripple
current specification for the main filter capacitor of a
PWM inverter to be defined.
1EE PROCEEDINGS,

Vol. 133, Pt. B, No. 4, JULY 1986

M=1.0
0-5

03
0-2
0-1
0-1
0.1

0.2

0.3 0.4 0.5 0.6 0.7

power factor

0-2

0-3
0-4 0-5 0-6 07
depth of modulation.M

0.8 0.9

08

09

10

0.24
0.22
0.20
018
0.16
0.14
012
0.10
008
0.06
0.04
0.02

0.2

0.8

0.6

0.4

1.0

depth of modulation. M
d

"0

0.1 0.2

03

04 0.5 0.6 0.7

power factor
o

0.8 0.9 1.0

Fig. 8

Three-phase inverter: variation of parameters

a Variation
b Variation
c Variation
d Variation
e Variation

of average DC link current with M and cos <j>

of | B2p | and | B 4 p | with M and cos <t>
of | Blp\ and | B4p\ with depth of modulation for cos 4> = 0.8
of \Bpq\ with depth of modulation
of | B3pq\ with M and cos <t>

M= 0.6

"0

Acknowledgment

The authors are indebted to the UK Science and Engineering Research Council for their financial support.
7

References

1 PELLY, B.R.: Thyristor phase controlled convertors and cycloconvertors (Wiley 1971), Chap. 11
2 BOWES, S.R., and MOUNT, M.J.: 'Microprocessor control of PWM
inverters', IEE Proc. B, Electr. Power Appl., 1981, 128, (6), pp. 293-305

0.1 0.2 0.3

0.4 05 06
power factor

0.7

0.8 0.9

Su and 5Zi are modulated appropriately. <50 is a convenient

parameter, which represents the quarter pulse width. If
$li = S2i = So then the average voltage over the pulse, is
clearly zero.
Working with the left-hand half pulse, the leading edge
of the pulse, then the average voltage over this half pulse
Vu can be shown to be
(19)

?u = jMO.5 V)
where

Appendix

(20)

The basis of a common sinusoidal pulse-width modulation

strategy from which the harmonic coefficients of both the
voltage and the switching function were calculated, and
which was also employed in the experimental equipment,
is described below.
It may be assumed that the fundamental period is subdivided into p short pulses of equal width A, such that:
A = 2n/p

t\
+ 0.5V

0-

(18)

The voltage at the output terminal of a leg with respect to

the notional midpoint of the DC source can be controlled
by asymmetrical double edge modulation over the ith
pulse which is centred at a,- with respect to the origin for
the fundamental period, as shown in Fig. 9. The two angles
IEE PROCEEDINGS,

1.0

Vol. 133, Pt. B, No. 4, JULY 1986

-0.5V

. 6,,>,
Fig. 9

80

i0

Characteristic pulse in cycle of PWM waveform

223

i.e. the average voltage over a half pulse is proportional to

the normalised angle f}u which varies in the range
- l ^ ^ ^ l

(21)

In most power applications it is sufficient to make the

average voltage during a half pulse of the modulated waveform proportional to the amplitude of the sine wave to
which the waveform is approximating at the midpoint of
the half pulse, i.e.
/?lt(0.5 V) oc Vm sin (a,- <50)

(22)

If we define the depth of modulation index M as the constant of proportionality in the usual way so that
Vm = M(0.5 V)

(24a)

Similarly
p2i = M sin (a,- + 50)

(24b)

By substituting for ft using eqn. 20, eqns. 24 are often

expressed in an alternative form of the type [2]
u = <50[l + M sin (a, - <50)]

224

(25b)

Eqns. 25 define the modulating law for this PWM strategy.

The switching function is the set of p pulses with modulating angles calculated according to eqns. 25 above, but
level shifted and attenuated so that its amplitude is unity
and represents switches of state between zero and unity
instead of between 0.5 V as shown in Fig. 9.
The amplitude of the nth harmonic due to the ith pulse,
An. is given by
An. = (l/j)(l/2nn){e-Jnd2i

jndu
e

}e-Jnai

(26)

For the p pulses in the fundamental period

(23)

then
Pu = M sin (a, - <50)

+ M sin (a,- + <50)]

(27)

There is no simple analytical summation of eqn. 27

because of the complex dependence of <51( and <52l on a, as
expressed in eqns. 25. However, it is a simple matter to
evaluate An by computation.
The harmonic coefficients of the voltage waveform indicated in Fig. 9 are also derived from eqn. 27, except that
the DC term is absent, and the amplitudes are multiplied
by V.

(25a)

IEE PROCEEDINGS,

Vol. 133, Pt. B, No. 4, JULY 1986