Chapter 1

Carrier-Based Pulse Width Modulation for Two-level Three-phase

Voltage Inverters

1.1. Introduction

Two-level three-phase voltage inverters are very widely used for feeding alternating current electrical machines serving as actuators with variable input voltages (controllable for amplitude and frequency). However, they are also increasingly being used as sinusoidal current absorption rectifiers. A chapter from an earlier book [LAB 04] has already introduced these topics from a modeling perspective. Figure 1.1 recalls the basic principles of a two-level three-phase voltage inverter feeding a balanced three-phase load connected in a star configuration with isolated neutral; the diagram introduces the notations we will use; the input reference voltage is taken to be the mid-point between the direct current bus rails.

We can present the problem of control via PWM in the following manner:

– starting with the reference voltages , ,a ref b ref c refv v v to be imposed on terminals of the different phases of load, the first step is to determine the voltages , ,a b cP P P produced by the legs of the inverter, suitable reference

Chapter written by Francis LABRIQUE and Jean-Paul LOUIS.

2 Power Electronic Converters

Figure 1.1. Schematic diagram showing notations used

values , ,a ref b ref c refP P P such that the actual output voltages , ,a b cP P P lead to the desired values of the voltages , ,a b cv v v ;

– the next step is to transform the reference signals , ,a ref b ref c refP P P into binary (or PWM) signals [0,1], [ , , ]jx j a b c∈ ∈ corresponding to switches Sj being closed (if xj = 1) or Sj

* being closed (if xj = 0), and to the production of voltages , [ , , ]jP j a b c∈ taking the value +U/2 or −U/2 depending on whether xj = 1 or 0. Then, by dividing the time into intervals 1[ , ],k kt t k N− ∈ during each interval the fraction of the interval for which Pj is +U/2 (and hence the fraction for which Pj is −U/2) is altered in such a way that over each interval the mean value < Pj > of Pj matches the value of Pj ref.

In case of carrier-based modulation, which is the subject of this chapter, the transformation of the reference signals Pj ref into binary signals xj is achieved by comparing these signals to a carrier wave vp (triangular or saw-toothed) whose frequency determines the intervals over which we want <Pj> to match Pj ref (Figure 1.2). We have xj equal to 1 and therefore:

Carrier-Based Pulse Width Modulation 3

Figure 1.2. Sawtooth carrier modulation

PJ = U/2

if:

Pjref > vp

Alternatively, we have xj equal to 0 and therefore:

PJ = −U/2

if:

Pjref < vp

4 Power Electronic Converters

Carrier-based modulation also refers to any sort of modulation where the intervals where xj is equal to 1 and those where xj is equal to 0 are produced by a microcontroller or FPGA [MON 08] using a computation that emulates the intersection process between the reference and carrier as described earlier.

We will show how the reference voltages , ,a ref b ref c refv v v to be applied to the load can be represented, and then describe the conversion of these values to the reference voltages , ,a ref b ref c refP P P for each leg, and finally show how these values are transformed into binary control signals (PWM signals) for the switches. We will see that this enables us to:

– derive the various intersective modulation strategies described in the literature (sine-triangle modulation, sub-optimal modulation, centered modulation, and flat-top and flat-bottom modulation);

– establish the similarities between certain types of intersective modulation and other modulation strategies such as space vector modulation.

1.2. Reference voltages varef, vbref, vcref

Since we are assuming that the load fed by the inverter is a balanced three-phase load connected in a star configuration with isolated neutral, the constraint on the currents resulting from this connection pattern, which is

0a b ci i i+ + = , leads to an equivalent constraint on the voltages 0a b cv v v+ + = . Since , ,a ref b ref c refv v v are the desired values for the voltages

, ,a b cv v v , it is convenient to impose the same constraint on these such that 0a ref b ref c refv v v+ + = ; this implies that there are only two degrees of

freedom that must be set to determine the necessary reference values.

Traditionally, when discussing intersective PWM [KAS 91, LAB 95, MOH 89, and SEG 04] it is assumed that the reference values have the form:

( )( )

sin

sin 2 / 3 ,

sin 4 / 3

ref refa ref

b ref ref ref

c ref ref ref

Vv

v V

v V

θ

θ π

θ π

= − −

[1.1]

Carrier-Based Pulse Width Modulation 5

where refV is the desired amplitude for the voltages and refθ is an angular coordinate obtained by integrating the desired reference pulsation for the voltages:

0.

tref ref dtθ ω= ∫ [1.2]

We will introduce the rotation matrices P(θ) and the Clarke submatrix C32 [SEM 04]:

cos sin( )

sin cosP

θ θθ

θ θ−

=

32

1 0

1/ 2 3 / 2

1/ 2 3 / 2

C

= − + − −

Equation [1.1] can then be written as:

32 3200

( ) ( )1

a ref

b ref ref ref refref

c ref

v

v V C P C PV

v

θ θ

= = −−

[1.3]

or be represented by the diagram shown in Figure 1.3.

Figure 1.3. Generation of reference waves a refv , b re fv , c re fv

from the desired amplitude refV and pulsation refω

We can consider Vref and θref to represent a vector refVr

rotating with speed refω and whose projection onto three axes mutually separated by 2 / 3π gives the reference voltages a refv , b re fv , c re fv (Figure 1.4).

6 Power Electronic Converters

Figure 1.4. “Vector” representation of the generation of reference waves a refv ,

b re fv , c re fv

In the steady state case refV and refω have fixed values. In the transient case they may vary as a function of time.

This classical approach takes the two degrees of freedom required to fix the reference voltages , ,a b cv v v to be their amplitude refV and their pulsation refω (equal to 2π times their reference frequency reff ).

There are however many applications where the load is active and includes sources of pulsation 0ω . In this case, refω must be equal to 0ω in the steady state case, and the difference between that value and 0ω in the transient case [ 0( )ref ref∆ω ω ω= − ] can be treated, after the integration relating refθ to refω , as a phase shift refϕ added to an angle 0θ equal to:

0 00

tdtθ ω= ∫ [1.4]

This is equivalent to taking (Figure 1.5a):

0 0 00 0( )

t tref ref refdt dtθ ω ω ω θ ϕ= + − = +∫ ∫ [1.5]

Vref

b

Carrier-Based Pulse Width Modulation 7

It is then more helpful to consider refV and refϕ as the control parameters (Figure 1.5b) since, in contrast to ref∆ ω , refϕ is not required to be zero in the steady state regime but only to have a constant value.

Figure 1.5. Generation of reference waves a refv , b re fv , c re fv when their steady state pulsation is determined by the load; a) expression [1.4];

b) expression [1.5]

On the basis of Figure 1.5b we can write:

32 32 00 0

( ) ( ) ( ) ,1 1

a ref

b ref ref ref ref ref

c ref

v

v V C P V C P P

v

θ θ ϕ

= = − −

[1.6]

since a rotation by angle 0ref refθ θ ϕ= + is equivalent to a rotation by angle 0θ followed by a rotation by angle refϕ .

a)

b)

8 Power Electronic Converters

Equation [1.6] can then be written as:

32 0 32 0sin

( ) ( ) ,cos

a refd refref ref

b refref ref qref

c ref

vvV

v C P C PV v

v

ϕθ θ

ϕ

+

= = −

[1.7]

so that finally:

0 0

0 0

0 0

cos sin sin cos

cos sin( 2 / 3) sin cos( 2 / 3)

cos sin( 4 / 3) sin cos( 4 / 3)

a ref ref ref ref ref

b ref ref ref ref ref

ref ref ref refc ref

v V V

v V V

V Vv

ϕ θ ϕ θ

ϕ θ π ϕ θ π

ϕ θ π ϕ θ π

+

= − + − − + −

[1.8]

Figure 1.6. Generation of reference waves ,a refv ,b refv c re fv from their components dq

Equation [1.7] can be observed to be equivalent to the diagram in Figure 1.6, where the reference quantities are taken to be:

sind ref ref refv V ϕ=

and:

cosq ref ref refv V ϕ= −

instead of:

refV

Carrier-Based Pulse Width Modulation 9

and:

.refϕ

since:

32 323 ,2

tC C I=

where I is the 2 × 2 identity matrix, the voltages d refv and q refv can be determined from the desired voltages a refv , b re fv , c re fv if we left-multiply both sides of [1.7] by:

10 32

2 ( )3

tP Cθ−

then:

10 32

2 ( ) .3

a refd ref t

b refqref

c ref

vv

P C vv

v

θ−

=

[1.9]

Substituting sind ref ref refv v ϕ= and cosq ref ref refv v ϕ= − in [1.7] by their values as given by [1.9], we obtain the following equation:

32 322 ,3

a ref a reft

b ref b ref

c ref c ref

v v

v C C v

v v

=

[1.10]

which shows that the matrix:

32 32

2 / 3 1 / 3 1 / 32 1 / 3 2 / 3 1 / 33

1 / 3 1 / 3 2 / 3

tC C− −

= − − − −

10 Power Electronic Converters

acts as an identity matrix on the vector va ref, vb ref, vc ref as long as the constraint 0a ref b ref c refv v v+ + = is met (the homopolar component is zero).

1.3. Reference voltages Pa ref, Pb ref, Pc ref

In contrast to the voltages va ref, vb ref, vc ref, the voltages Pa ref, Pb ref, Pc ref are not required to have a sum of zero. We therefore have three degrees of freedom in defining these voltages. If we introduce the homopolar component of , ,a ref b ref c refP P P :

0 ( ) / 3ref a ref b ref c refP P P P= + +

then the quantities:

0a ref ref a h refP P P −− =

0b ref ref b h refP P P −− =

0c ref ref c h refP P P −− =

sum up to zero, just like the voltages , ,a ref b ref c refv v v do. Making use of the definition of 0 re fP we can write:

32 32

2 / 3 1/ 3 1/ 321/ 3 2 / 3 1/ 33

1/ 3 1/ 3 2 / 3

a h ref a ref a reft

b h ref b ref b ref

c h ref c ref c ref

P P P

P P C C P

P P P

−

−

−

− − = − − = − −

[1.11]

The matrix 32 322 / 3 tC C , which acts as an identity matrix with respect to

three quantities whose sum is zero (Equation [1.10]), acts to eliminate the homopolar component when it is applied to three quantities that do not sum up to zero. Using a process similar to that used for the quantities

, ,a ref b ref c refv v v , we can represent the quantities , ,a h ref b h ref c h refP P P− − − in terms of two reference quantities d refP , q re fP (Figure 1.7):

Carrier-Based Pulse Width Modulation 11

32 0( )a h ref

d refb h ref

q refc h ref

PP

P C PP

P

θ−

−

−

=

[1.12]

Figure 1.7. Generation of reference waves , ,a h ref b h ref c h refP P P− − −

from their dq components

By defining the matrix:

31

11 ,1

C =

[1.13]

we can express the reference quantities , ,a ref b ref c refP P P as a function of the reference quantities 0, ,d ref q ref refP P P (Figure 1.8):

Figure 1.8. Generation of reference waves , ,a ref b ref c refP P P from their d-q-0 components

a h refP −

b h refP − c h refP −

12 Power Electronic Converters

32 0 31 0( )a ref

d refb ref ref

q refc ref

P PP C P C P

PP

θ

= +

[1.14]

1.4. Link between the quantities a b cv , v , v and a b cP , P , P

Referring to the notations in Figure 1.1 we can write:

a b a bP P v v− = −

a c a cP P v v− = −

If we combine these two equations we obtain 2 2 3a b c a b c aP P P v v v v− − = − − = , with the final equality reflecting the fact that 0a b cv v v+ + = . Similarly, we can proceed to obtain bv and cv as a function of , ,a b cP P P , which leads to:

32 32

2 / 3 1 / 3 1 / 321 / 3 2 / 3 1 / 33

1 / 3 1 / 3 2 / 3

a aat

b b b

c c c

P Pvv P C C Pv P P

− − = − − =

− − −

[1.15]

Equation [1.15] states that the voltages , ,a b cv v v correspond to the voltages , ,a b cP P P after their homopolar component 0 ( ) / 3a b cP P P P= + + has been subtracted. We can therefore write:

0a av P P= −

0b bv P P= − [1.16]

0c cv P P= −

Therefore, the difference between Pi and vi stems from the homopolar component: the Pi values include a homopolar component while the vi values do not.

Carrier-Based Pulse Width Modulation 13

1.5. Generation of PWM signals

In order to determine the states , [ , , ]jx j a b c∈ of the switches of each leg from the reference waves j refP , we will consider sequentially:

– the case where these waves are compared to a reverse sawtooth carrier; – the case where these waves are compared to a conventional sawtooth

carrier; – the case where these waves are compared to a triangular carrier.

We will assume that the carrier is normalized and varies between −1 and +1, and that the reference waves also vary in this manner since they are divided by U/2:

, / ( / 2)j ref n j refP P U= [1.17]

1.5.1. Reverse sawtooth wave

Over each modulation period the carrier wave varies linearly from +1 to −1. It returns from −1 to +1 at the moment where one modulation period ends and the next begins. If Tp is the period of the carrier, over the (k+1) modulation period (from k pt kT= to ( )1 1k pt k T+ = + ) all the ' jS switches will be closed at kt since at this moment the carrier takes the value of +1 and therefore has a value greater than the value of all the reference waves, implying that , ,a b cx x x will be zero.

Each leg then undergoes a transition from ' jS closed to jS closed at time

jkt when the reference wave ,j ref nP intersects the carrier and takes a value greater than that of the carrier (Figure 1.9).

The order in which the switches commutate depends on the order in which the carrier intersects the reference waves.

There are six possible sequences: a, then b, then c; a, then c, then b; b, then c, then a; b, then a, then c; c, then a, then b; and c, then b, then a.

14 Power Electronic Converters

Figure 1.9. Modulation with a reverse sawtooth carrier

,a ref nP

,b ref nP

,a ref nP

Carrier-Based Pulse Width Modulation 15

Each commutation causes one of the components of the vector ( , , )a b cx x x to transition from 0 to 1; the vector starts with the value (0, 0, 0)

at the start of the period, with ' ' ', ,a b cS S S all closed; it ends the period with the value (1, 1, 1).

The voltage jP , ( , , ),j a b c∈ is −U/2 over the interval [ , ]k jkt t ,

k = 1, 2,…., where jx is zero and where 'jS is closed. It is U/2 over the

interval 1[ , ]jk kt t + where jx is 1 and where jS is closed. The voltages jv , ( , , )j a b c∈ can be determined from the voltages jP through equation [1.15].

The time jkt , where ,j ref nP intersects with the carrier is the root of the following equation:

, ( ) 1 2 ; ( , , )jk pj ref n jk

p

t kTP t j a b c

T−

= − ∈ [1.18]

The mean value j kP< > of jP over the modulation period (note: 1k k pt t T+ = + ) is therefore:

( )

( )

1

,

1 ( )2

1 2 ( ) ( )2 2

j k k jk jk kp

k jk p j ref n jk j ref jkp

UP t t t tT

U Ut t T P t P tT

+ < > = − − −

= − + = =

[1.19]

(NOTE: Remember that ,j ref nP is normalized using equation [1.17]). Equation [1.19] shows that the PWM process, over which jP takes the value −U/2 and then +U/2, results in the mean value of jP over the modulation period being equal to the value of j re fP at one particular point within an interval of this period: the point where the carrier intersects

,j ref nP . If the reference waves j refP vary only slightly over a modulation period, the sequence of samples , ( )j ref n jkP t will provide a good representation of the reference waves. The same goes for the mean values

j kP< > of the voltages jP .

16 Power Electronic Converters

Instead of the natural sampling of the jP waves that we have just described, an alternative synchronous sampling is possible, where the values of ,j ref nP for each period are based on their values , , ( )j ref n k j ref n pP P kT= at the start of the period (Figure 1.10). We note that this synchronous sampling process for the reference waves will occur naturally if the modulation is performed numerically within a microprocessor using a calculation emulating the intersection process, or if the reference waves ,j ref nP are obtained by digital to analog conversion at the output of a computation unit whose sampling period is synchronized with the period of the carrier.

Figure 1.10. Modulation using a reverse sawtooth wave with synchronous sampling of the reference waves

For the rest of this section we will assume that the reference waves are sampled at the start of each modulation period, so that we have:

,

,

,

; 1, 2,...2

a ref nk aref ka k

b k b ref nk bref k

c k c ref nk c ref k

P PPUP P P k

P P P

< > < > = = = < >

[1.20]

Using equation [1.14] we can write:

32 0 31 0( ) ,a ref k

d ref kb ref k k ref k

q ref kc ref k

PP

P C P C PP

P

θ

= +

[1.21]

Carrier-Based Pulse Width Modulation 17

where 0( )kP θ , d ref kP , q ref kP are the values of their corresponding quantities at k pt kT= . Substituting [1.21] into [1.20] we obtain:

32 0 31 0( )a k

d ref kb k k ref k

qref kc k

P PP C P C P

PP

θ< > < > = + < >

[1.22]

Equation [1.15], which connects the voltages , ,a b cv v v to the instantaneous values of the voltages , ,a b cP P P can also be applied to the mean values (over each modulation period) of these quantities. This gives us:

32 3223

a ka kt

b k b k

c k c k

Pvv C C Pv P

< >< > < > = < >

< > < >

[1.23]

Substituting [1.22] into [1.23] we obtain:

32 0( ) ,a k

d ref kb k

q ref kc k

v Pv C P k

Pv

θ< > < > = < >

[1.24]

given that:

– 32 323 ,2

tC C I= where I is (as mentioned earlier) the 2 × 2 identity

matrix:

– ( )32 32 31 0 0 0 ttC C C = .

1.5.2. Conventional sawtooth carrier

Over each modulation period the carrier now varies linearly from −1 to +1 and returns from +1 to −1 at the boundary between one period and the next. Over the (k + 1)th modulation period from:

k pt kT=

18 Power Electronic Converters

to:

( )1 1k pt k T+ = +

every switch jS will close at kt since at this moment the carrier takes the value −1 and therefore has a value smaller than that of each of the reference waves, implying that ,ax bx and cx are equal to 1.

Each leg then undergoes a transition from jS closed to 'jS closed at the

time jkt when the reference wave ,j ref nP intersects the carrier pv (Figure 1.11). Each transition causes one component of the vector ( ax , bx , cx ) to move from 1 to 0, starting from a value [1,1,1] at kt and finishing with a value [0,0,0] at the end of the period.

The voltage jP , [ , , ]j a b c∈ is U/2 from kt to jkt over the interval where

jx is 1 and jS is closed. It is −U/2 from jkt to 1kt + over the interval where

jx is zero and 'jS is closed. The voltages ju are linked to the voltages jP

by equation [1.15].

The time jkt when jP intersects the carrier is the solution of the following equation:

2( )( ) 1 ; ( , , )jk p

jref n jkp

t kTP t j a b c

T−

= − + ∈ [1.25]

The mean value of jP over the (k+1) th modulation period is therefore:

( )

( )

11 ( )

2

1 2 ( ) ( )2 2

j k jk k k jkp

p jk k jref n jk jref jkp

UP t t t tT

U UT t t P t P tT

+ < > = − − −

= − + − = =

[1.26]

If we adopt a synchronous sampling scheme for the reference waves, we obtain:

Carrier-Based Pulse Width Modulation 19

Figure 1.11. Modulation by a conventional sawtooth carrier

,a ref nP ,b ref nP

,c ref nP

20 Power Electronic Converters

( ); 1,2,...j k j ref k j ref pP P P kT k< > = = = [1.27]

When the reference waves are sampled at the beginning of the modulation period, equations [1.20] to [1.24] apply equally well to the case of a conventional sawtooth carrier as to the reverse sawtooth carrier.

1.5.3. Triangular carrier

Modulation by a triangular carrier can be considered as equivalent to repeated modulation, first by a reverse sawtooth wave and then by a conventional sawtooth wave.

The period of the carrier is twice the duration / 2pT of each of the ramps (first decreasing and then increasing) that constitute the carrier (Figure 1.12).

If the period starts with modulation by a decreasing ramp, at the start of the period all the switches '

jS are closed since at this point in time the

carrier has a value greater than that of every reference wave.

Each leg then undergoes a transition from 'jS closed to jS closed at the

time when the corresponding reference wave crosses the carrier wave; by the end of the decreasing ramp all the switches jS are closed.

During the increasing ramp each arm undergoes a transition from jS

closed to 'jS closed, such that at the end of the period the situation is once

again when all the 'jS switches are closed.

There is no longer, as was the case with sawtooth waves, a moment where all the legs commutate simultaneously at the point between one modulation period and the next.

If the reference waves are sampled, this may occur: – at the start of each modulation period, as was the case with sawtooth

carriers (Figure 1.13);

Carrier-Based Pulse Width Modulation 21

Figure 1.12. Modulation by a triangular carrier

,a ref nP

,b ref nP

,c ref nP

22 Power Electronic Converters

Figure 1.13. Modulation by a triangular carrier with synchronous reference sampling at the

start of each carrier period

– at the start of each sawtooth component of the carrier, which means that the waves jP match the mean values of their reference waves j re fP on the scale of every half-period of the modulation (Figure 1.14).

Figure 1.14. Modulation by a triangular carrier with synchronous reference sampling at the start of each half-period

Depending on the way the sampling is performed, we have either:

– from k pt kT= to ( )1 1k pt k T+ = + , k = 1,2, …

[ ]j k j ref k j ref pP P P kT< > = = [1.28]

– or from / 2k pt kT= to ( )1 1 / 2k pt k T+ = + , k = 1,2, …

Carrier-Based Pulse Width Modulation 23

[ / 2]j k j ref k j ref pP P P kT< > = = [1.29]

Over each half-period of the carrier we have, as with sawtooth carriers, six possible switching sequences, depending on the values of ,aref kP ,bref kP and c ref kP over this half-period.

For a half-period consisting of a decreasing ramp, the transitions from 'jS closed to jS closed occur first on the leg whose reference voltage is

largest, and then on the leg with the intermediate reference voltage, and finally for the leg whose reference voltage has the smallest value. Thus, if

a ref kP > b ref kP > c ref kP the commutations will occur first on leg a, then on leg b, and finally on leg c and the vector [ ,ax ,bx cx ] representing the states

of the switches on each leg ( 'jS closed for 0;jx = jS closed for 1jx = )

moves from (0,0,0) to (1,0,0), then to (1,1,0), and finally to (1,1,1). A similar process can be used to determine the sequence of values of the vector ( ,ax ,bx cx ) and hence the states of the switches for each of the five other cases.

For a half-period consisting of an upward ramp, the vector ( ,ax ,bx cx ) moves gradually from (1,1,1) to (0,0,0) with the transitions from jS closed

to 'jS closed, acting first on the leg whose reference voltage is smallest, then

on the one with the middle reference voltage, and finally on the leg with the largest reference voltage.

It can be seen that the twelve switching sequences we have just defined are identical to those that are obtained using space vector modulation (Chapter 2 and reference [LAB 98]).

Modulation by a triangular carrier has the property that it is indiscernible in terms of the switching sequences from space vector modulation.

1.5.4. Note

A modulation based on a random carrier is sometimes used, selecting in a non-deterministic manner for each period, either a conventional or a reverse sawtooth.

24 Power Electronic Converters

1.6. Determination of the reference waves aref kP , bref kP , and cref kP from the reference waves aref k bref k cref kv , v , v

As we saw in section 1.5, with PWM, jP will only match j re fP when averaged over a given period of modulation. The same clearly applies to the voltages , ,a b cv v v with respect to the reference waves , ,a ref b ref c refv v v .

Here again, the problem is to determine over each modulation period the values of the waves ,aref kP ,bref kP c ref kP (or the ,d ref kP ,qref kP 0 ref kP com-ponents of these waves) such that we obtain:

a ref ka k

b k bref k

c k c ref k

vvv vv v

< > < > = < >

[1.30]

Substituting [1.30] into [1.24] we obtain the equation that must be used to link the reference values for the voltages of each phase and the reference values for the dq components of the voltages in each leg:

32 0( )a ref k

d ref kbref k

q ref kcref k

vP

v C P kP

v

θ

=

[1.31]

If we multiply both sides of [1.31] by:

10 32

2 ( )3

tP k Cθ−

we obtain:

10 32

2 ( ) .3

aref kd ref k t

bref kqref k

cref k

vP

P k C vP

v

θ−

=

[1.32]

Carrier-Based Pulse Width Modulation 25

We then substitute [1.32] into [1.21] to obtain:

32 32 31 02 .3

aref k aref kt

bref k bref k ref k

cref k cref k

P v

P C C v C P

P v

= +

[1.33]

Since the matrix 32 3223

tC C is equivalent to an identity matrix for the

quantities , ,aref k bref k cref kv v v that sum up to zero, equation [1.33] can be reduced to:

31 0

aref k a ref k

bref k bref k ref k

c ref k c ref k

P v

P v C P

P v

= +

[1.34]

Equation [1.34] shows that the reference values , ,a ref b ref c refv v v fix the values of ,aref kP ,bref kP c ref kP except for their homopolar component 0 ,ref kP which is a remaining degree of freedom, which can be manipulated to optimize the modulation to match some desired quality criterion. This result is consistent with the statement given at the end of section 1.4.

1.6.1. “Sine” modulation

“Sine” modulation is obtained if in equation [1.34] we take the homopolar component 0 re fP of the reference waves j re fP to have a value zero, which makes these waves equal to the reference waves j refv :

aref k aref k

bref k bref k

cref k cref k

P v

P v

P v

=

[1.35]

In the steady state case the j re fP waves then form a balanced three-phase system of sinusoidal voltages, just like the waves j refv , and therefore the term “sine” modulation is given (Figure 1.15).

26 Power Electronic Converters

Figure 1.15. Sinusoidal modulation with triangular carrier

The amplitude of the j re fP waves, and hence the amplitude of the sinusoidal waves that can be produced in the steady state case at the three-phase terminals of the load, cannot be greater than U/2 if we wish to avoid the emergence of an effect known as overmodulation1.

Compared to full-wave control, which gives voltages at the three-phase

terminals of the load whose fundamental component has amplitude 2Uπ

,

“sine” modulation incurs a reduction in amplitude of:

/ 2 / 42 /UU

ππ

= [1.36]

or a reduction of 21%. This reduction in amplitude is known as “voltage drop due to pulse width modulation” [LAB 95].

1. This effect represents the absence of any intersection between the carrier and a reference wave for one or more modulation periods, because the value of a reference wave is greater than the maximum value of the carrier (or less than its minimum value). The voltage Pj is thus equal to +U/2 (or −U/2) over the entire interval. When overmodulation occurs, the equality <Pj>k = Pjref k is not maintained. For a detailed analysis of overmodulation see Chapter 3.

Carrier-Based Pulse Width Modulation 27

1.6.2. “Centered” modulation

“Centered” modulation is when a value for the homopolar component in equation [1.34] is taken to be equal to + minus half the sum of the largest and smallest of the reference waves , ,a ref k b ref k c ref kv v v .

If max( )j ref kv denotes the operation of selecting the largest of the reference waves , ,a ref k b ref k c ref kv v v and min( )j ref kv the operation for selecting the smallest of the waves, we obtain:

311 max( ) min( )2

aref k a ref k

bref k bref k j ref k j ref k

c ref k c ref k

P v

P v C P P

P v

= − +

[1.37]

It can be seen that the value of the homopolar component has the effect of causing the largest and smallest of the reference waves j ref kP to lie symmetrically on each side of the horizontal axis, and hence the term “centered” is given. In the case where the reference waves j refv form a balanced three-phase system with sinusoidal values of amplitude V and pulsation ω:

sina ref ref refv V tω=

sin( 2 / 3)b ref ref refv V tω π= −

sin( 4 / 3),cref ref refv V tω π= −

equation [1.37] gives the following voltages j re fP (Figure 1.16):

– between / 6ref tω π= − and refω t = π/6, the voltage c refv is the most positive and brefv is the most negative; we therefore have:

31/ 2( ) sin2aref aref bref cref ref refP v v v V tω= − + =

31/ 2( ) cos2bref bref bref cref ref refP v v v V tω= − + = −

28 Power Electronic Converters

31/ 2( ) cos2cref cref bref cref ref refP v v v V tω= − + = +

– between refω t = π/6 and refω t = π/2 , the voltage a re fv is the most positive and brefv is the most negative, such that:

31/ 2( ) cos( / 3)2aref aref aref bref ref refP v v v V tω π= − + = −

31/ 2( ) cos( / 3)2bref bref aref bref ref refP v v v V tω π= − + = − −

31/ 2( ) sin( / 3)2cref cref aref bref ref refP v v v V tω π= − + = − −

and so on.

Figure 1.16. Centered modulation with a triangular carrier

The amplitude of the reference waves , ,aref bref crefP P P is never greater than U/2 and there are no saturation effects as long as:

32 2ref

UV <

Carrier-Based Pulse Width Modulation 29

or when:

3refUV <

The voltage drop is not more than 9% [LAB 95]. We observe that the increase in amplitude for refV relative to U/2 when using this technique is the same as with space vector modulation [LAB 98].

In addition, with synchronous sampling of the reference waves, centering gives (over each period of the carrier in the case of sawtooth carriers or over each half-period of the carrier in the case of triangular carriers) the same duration for the time over which the vector ( , , )a b cx x x is (0,0,0) and the time over which it is (1,1,1), in other words, the time interval over which all the switches '

jS are closed and the time interval over which all the switches

jS are closed.

As a result, centered modulation using a triangular carrier and synchronous sampling of the reference waves over each half-period is indiscernible from space vector modulation in the case of a two-level three-phase voltage inverter [LAB 98].

1.6.3. “Sub-optimal” modulation

This method can produce a result close to that of centered modulation in terms of maximum amplitude that can be achieved for the reference waves when they form a balanced three-phase system of sinusoidal voltages. It takes the homopolar component of the voltages j re fP to be a sinusoidal wave of amplitude 0.09 U whose pulsation is three times that of the reference waves [LAB 95] (Figure 1.17):

31

sin

sin( 2 / 3) 0, 09 sin 3

sin( 4 / 3)

aref ref ref

bref ref ref ref

c ref ref ref

P V t

P V t C U t

P V t

ω

ω π ω

ω π

= − + ⋅ ⋅ −

[1.38]

30 Power Electronic Converters

Figure 1.17. Sub-optimal modulation with a triangular carrier

Vref can then achieve amplitudes of up to 1.15 U/2 without introducing any overmodulation effects [LAB 95].

1.6.4. “Flat top” and “flat bottom” modulation

Flat top modulation involves setting the largest of the reference waves j re fP to be equal to 1, by requiring the homopolar component to have a

value equal to:

0 1 max( )ref j refP P= − [1.39]

This strategy (Figure 1.18) is intended to reduce switching losses by avoiding any switching from taking place in a given leg over the time period where its reference voltage j ref nP is largest.

Setting the voltage j ref nP equal to 1 over this interval is equivalent to keeping jS constantly closed, since we must have:

2 2j j ref nU UP P= =

Carrier-Based Pulse Width Modulation 31

Figure 1.18. Flat-top modulation with a triangular carrier

Similarly, flat-bottom modulation sets the most negative of the reference waves j re fP equal to −1 by setting the homopolar component equal to:

0 1 min( ),ref j refP P= − − [1.40]

which means that 'jS is kept constantly closed for each leg over the intervals

where j re fP is most negative (Figure 1.19).

Figure 1.19. Flat-bottom modulation with a triangular carrier

32 Power Electronic Converters

Flat-top modulation (or flat-bottom modulation) implies an unequal distribution of current between the two switches of each leg, since current flows in switches jS (or '

jS ) over an interval equivalent to one third of the period of reference waves j re fu in the case of a balanced sinusoidal three-phase system.

This drawback can be addressed by combining these two types of modulation: the most positive (largest) of the reference waves j re fP is set to 1 when this wave is greater than the absolute value of the smallest of those waves, and the most negative (smallest) of the reference waves is set to −1 when its absolute value is greater than that of the largest of the reference waves (Figure 1.20).

Figure 1.20. Combined flat-top-flat-bottom modulation

1.7. Conclusion

In this chapter we have derived the equations connecting the desired reference values for the phase voltages with the reference values for the leg voltages in case of a two-level three-phase voltage inverter feeding a balanced three-phase load connected in a star configuration when the legs are controlled using carrier-based PWM.

Carrier-Based Pulse Width Modulation 33

In particular, we have shown that centered PWM with a triangular carrier is indistinguishable from space vector PWM and that the flat-top and flat-bottom strategies can be used to reduce switching losses at a given PWM frequency by avoiding the need to switch for each leg during certain intervals.

We have not considered issues such as harmonic content of the voltages produced using these techniques and the influence on this content of the type of modulation chosen (sine, centered, sub-optimal, or flat-top-flat-bottom) or of the type of carrier wave used2 (conventional or reverse sawtooth, triangular, or random). Discussion of these issues would require a dedicated chapter on the topic.

1.8. Bibliography

[BOO 88] BOOST M.A., ZIOGAS P.D., “State-of-the-art carrier PWM techniques: a critical evaluation”, IEEE Trans. Ind. Appl., 24(2), 271–280, 1988.

[HAU 99] HAUTIER J.P., CARON J.P., Convertisseurs statiques: méthodologie causale de modélisation et de commande, Edition Technip, Paris, 1999.

[HOL 93] HOLZ J., “On the Performance of optimal pulse width modulation technique”, EPE Journal, 3, (1), 17–6, 1993.

[HOU 84] HOULDSWORTH J.A., GRANT D.A., “The use of harmonic distorsion to increase the output of a three-phase PWM inverter”, IEEE Trans. Ind. Appl., 20(5), 1224-1228, 1984.

[KAS 91] KASSAKIAN J.G., SLECHT M.F., VERGHESE G.C., Principles of Power Electronics, Addison Wesley, Reading, MA, 1991.

[KAZ 94] KAZMIERKOWSKI M.P., DZIENAKOWSKI M.A., “Review of Current Regulation technique for three-phase PWM Inverter”, IEEE-IECON, Bologne, vol. 1, p. 567–575, 1994.

[LAB 95] LABRIQUE F., BAUSIÈRE R., SÉGUIER G., Les convertisseurs de l’électronique de puissance 4: la conversion continu-continu, Lavoisier, Paris, 1995.

[LAB 98] LABRIQUE F., SÉGUIER G., BUYSE H., BAUSIÈRE R., Les convertisseurs de l’électronique de puissance 5, Lavoisier, Paris, 1998.

2. In all cases we have taken the example of a triangular carrier wave.

34 Power Electronic Converters

[LAB 04] LABRIQUE F., LOUIS J.P., Modélisation des onduleurs de tension en vue de leur commande en MLI, Chapter 4. In: LOUIS J.P. (ed.), Modèles pour la commande des actionneurs électriques, p. 185–213, Hermès, Paris, 2004.

[LOU 04a] LOUIS J.P. (ed.), Modélisation des machines électriques en vue de leur commande: Concepts généraux, Hermes, Paris, 2004.

[LOU 04b] LOUIS J.P. (ed.), Modèles pour la commande des actionneurs électriques, Hermes, Paris, 2004.

[LOU 95] LOUIS J.P., BERGMANN C., “Commande numérique des ensembles convertisseurs-machines, (1) Convertisseur-moteur à courant continu”, Techniques de l’ingénieur, D 3641 and D 3644, 1995, “(2) Systèmes triphasés : régime permanent”, Techniques de l’ingénieur, D 3642, 1996, “(3) Régimes intermédiaires et transitoires”, Techniques de l’ingénieur, D 3643 and D 3648, 1997.

[MOH 89] MOHAN N., UNDELAND T., ROBBINS W., Power Electronics, John Wiley & Sons, Chichester, 1989.

[MON 93] MONMASSON E., HAPIOT J.C., GRANDPIERRE M., “A digitalc Control system based on field programmable gate array for AC drives”, EPE Journal, vol. 3, n° 4, p. 227–234, 1993.

[MON 08] MONMASSON E., CIRSTEA M.N., “FPGA Design Methodology for Industrial Control Systems-A Review”, IEEE Transactions on Industrial Electronics, vol. 54, n° 4, p. 1824–1842, 2007.

[SEG 04] SÉGUIER G., BAUSIÈRE R., LABRIQUE F., Electronique de puissance, 8th edition, Dunod, Paris, 2004.

[SEM 04] SEMAIL E., LOUIS J.P., Propriétés vectorielles des systèmes électriques triphasés, chapitre 4. In: LOUIS J.P. (ed.), Modélisation des machines électriques en vue de leur commande: Concepts généraux, p. 181–246, Hermes, Paris, 2004.