Power Electronics_Chapter1

It can be shown that although the output voltage is again composed of a series of rectangular pulses, fundamental component of the output voltage is of the same frequency and magnitude a

( )

k

n

=

.

25 50 0 5

5 23

7 16 5

π

σ σ

σ σ

σ σ

A A

Stator current higher harmonics therefore have the same value regardless of the operating frequency This is so due to the fact that DC voltage and hence rms values of all the voltage harmonics change proportionally to operating frequency As leakage reactances are proportional to fundamental frequency as well, ratio of phase voltage harmonic rms value to sum of leakage reactances for the given harmonic remains constant regardless of the fundamental frequency.

In all the analysis so far it was assumed that the inverter operates in 180 degrees conduction mode and that variation of the inverter output voltage magnitude is achieved by rectifier control Such a solution is nowadays rarely applied Instead, the inverter is operated in pulse width modulated (PWM) mode and is supplied from a diode bridge rectifier (single-phase

or three-phase), so that input DC voltage of the inverter is constant Variation of both inverter output voltage and output frequency is now achieved by the inverter, due to operation in the PWM mode The idea of PWM is explained using single-phase VSI circuit of Fig 3.9 The most frequently utilised method of PWM is the so-called sinusoidal PWM technique, in which

a reference signal - a sine wave of desired amplitude and frequency - is compared with a triangular carrier wave of constant amplitude and frequency The instants for on and turn-off of semiconductors in Fig 3.9 are then determined with intersections of the reference signal and the carrier wave Switches are turned on and off in pairs: S1 and S2 are always together either on or off and similarly, S3 and S4 are always together either on or off The advantage of this approach is two-fold First of all, inverter now becomes capable of controlling both the frequency and the first harmonic magnitude, so that there is no need for application of a controllable rectifier Instead, a diode rectifier is used Secondly, switching now occurs at high frequency determined with the carrier wave frequency This enables faster control and simultaneously greatly improves harmonic spectrum of the load current Inverter output voltage now does not contain low order harmonics (the fifth, seventh, etc.) Instead, harmonics are situated around multiples of the switching frequency (i.e., triangular carrier wave frequency) Thus, if switching frequency is 5 kHz (typical value nowadays), then the inverter output voltage will contain, apart from fundamental, higher harmonics whose frequencies are around 5 kHz, 10 kHz, 15 kHz etc If the motor operates with 50 Hz fundamental frequency, then 5 kHz means that the order of the harmonic is around 100 (rather than 5, as it is in the simple VSI discussed previously) As reactance of the motor (Fig 6.14) is 100 times greater at

5 kHz than at 50 Hz (rather than only five times greater, as the case is for harmonic frequency

of 250 Hz), harmonic currents in the motor will be very small

The principle of sinusoidal PWM, as applied in the single-phase bridge inverter of Fig 3.9, is illustrated in Fig 6.15 The impact of variation of reference signal amplitude on output voltage waveform is evident from Fig 6.15 Widths of the pulses in the two cases shown differ although output frequency and triangular carrier wave frequency are the same The variation in pulse widths leads to subsequent difference in the values of the first harmonic of the output

voltage It can be shown that the fundamental harmonic of the output voltage waveforms shown in Fig 6.15 equals in frequency and in amplitude the reference signal

Reference and carrier signals

High amplitude of reference

V

Reference and carrier signals

Medium amplitude of reference signal

v

V

Fig 6.15: Sinusoidal PWM in single-phase bridge inverter for high and medium amplitudes of

the reference signal at same output frequencies

Extension of the principle of PWM from single-phase to three-phase voltage source inverter is rather straightforward The reference sinusoidal voltage (which equals desired fundamental voltage at the inverter output) is formed on the basis of the control law for given operating point (say, V/f = Vn/fn law) Thus the amplitude and the frequency of the reference sinusoidal signal are those that want to be obtained at machine terminals The reference sinusoidal signal is further compared with the carrier signal (again, high frequency triangular waveform) The instants of semiconductor switching in inverter legs are determined with intersections of sinusoidal reference and triangular carrier wave The three-phase inverter requires three reference sinusoidal signals in order to achieve operation with three-phase system of output voltages The three sinusoidal reference signals have mutual displacement of

120 degrees If triangular carrier frequency is sufficiently high, one carrier may be utilised for all the three phases It can be shown that although the output voltage is again composed of a series of rectangular pulses, fundamental component of the output voltage is of the same frequency and magnitude as the reference sinusoidal signal is

Analysis of the PWM inverter output voltages is always conducted using the notion of

the ‘modulation index’ Modulation index m is defined as the ratio of the sinusoidal signal

reference amplitude to the amplitude of the triangular carrier wave Another frequently utilised

term is the frequency ratio F, which is defined as the ratio of the triangular carrier frequency to

the frequency of the reference signal Hence

If the drive is controlled using V/f = const law, than the amplitude and the frequency of the sinusoidal reference signal are varied to satisfy this law Full PWM operation is possible only for modulation index values between zero and one If modulation index exceeds the value of one, so called pulse-dropping region is entered In this region reference signal is of higher amplitude than the carrier signal and some of the intersections between the reference and the carrier do not take place any more (i.e some of the pulses are dropped - hence the name pulse dropping) Eventually, if the modulation index is sufficiently high, the inverter essentially reverts to six-step operation In general, depending on the application, it is possible to realise the PWM inverter fed drive in two ways: full PWM operation is maintained in the whole base speed region (i.e from zero to rated frequency) or full PWM operation takes place in part of the base speed region only, while at rated frequency the inverter operates either with six-step output voltage or with partial PWM (i.e some pulses are dropped) This region of operation is frequently called over-modulation Note that the beneficial feature of PWM inverter, non-existence of low order harmonics, holds true only so long as the full PWM operation is preserved The described two ways of operating the PWM inverter will require different input

DC voltage at rated output frequency The amplitude of the fundamental output line-to-neutral voltage for operation in the full PWM mode is given with

( )

( )

DC

1

1

2

2 2

=

while the corresponding output line-to-line fundamental rms voltage value for the six-step operation is given in equation (6.19)

Example:

A three -phase 415 V, 50 Hz star-connected induction motor is to be supplied from a three-phase voltage source inverter that is controlled using sinusoidal PWM Calculate the required inverter input DC voltage if: a) the inverter operates in the whole base speed region with full PWM; b) inverter reverts to six-step operation at rated output frequency

Solution:

Fundamental component of the inverter line-to-line voltage has to be at 50 Hz equal to 415 V, rms, regardless of the applied method of control, so that required phase to neutral fundamental component has to be of 240 V rms If full PWM operation takes place at 50 Hz, then from (6.25) one has

( )

m

=

1

If the operation at 50 Hz is in six-step mode, then from (6.19)

Considering that the normal three-phase rectifier input voltage is 415 V line-to-line, then, assuming almost infinite capacitance in the DC link, the DC voltage could at most be equal to the peak of the input line-to-line voltage, 587 V The standard practice is therefore to operate the inverter at rated frequency in the pulse dropping mode, close to six-step waveform If full PWM operation is needed at rated frequency, then specially designed motors have to be used, whose rated voltage is below 415 V but whose insulation is of higher rating.

It is usually said that DC voltage utilisation in the PWM mode is poorer than in the six-step mode Given the DC voltage, maximum fundamental line-to-line voltage rms values are 61% and 78% of the

applied DC voltage for sinusoidal PWM (with m = 1) and six-step voltage, respectively.

In the past, when available switching frequencies of semiconductors were rather low, it

was customary to keep the frequency ratio F constant, so that good feature of the sinusoidal

PWM with regard to harmonic content was preserved at all operating frequencies Nowadays, semiconductor switching frequencies are for major part of the power region in the kHz region Typically, for small to medium powers, inverter switching frequency is of the order of 2 to 20 kHz Consequently, carrier frequency is nowadays kept normally constant and only one carrier

is used for all the three reference signals

In summary, it can be stated that operation of a VSI in PWM mode yields two substantial benefits, when compared to operation in 180 degrees conduction mode A diode rectifier can be used instead of a controllable rectifier, since the inverter is now capable of controlling both the frequency and the rms value of the fundamental component of the output voltage Additionally, higher harmonics of the voltage are now of substantially higher frequencies, meaning that current is much closer to a true sine waveform

One special type of PWM, that is nowadays extremely frequently applied, is the so-called ‘voltage space vector modulation’ For reasons that are beyond the scope here, this PWM method is the prevailing one in closed-loop control of induction motors fed from PWM inverters Explanation of this method however requires at first the introduction of the notion of the space vector

Let us at first suppose that a three-phase supply is purely sinusoidal and balanced, so that the system of phase voltages can be given with

a

b

c

=

2

cos

ω

(6.26)

Space vector of phase voltages is defined as

,

and is obviously a complex quantity that simultaneously represents all the three voltages of the three-phase supply applied to the machine

Hence, for the case of the sinusoidal supply

s

s

s

=

2

2

2

The result is obtained after relatively simple trigonometric manipulations This is an equation of

a circle in the complex plane It describes a complex number of constant amplitude (for given

V) whose phase continuously changes in time Space vector is therefore a complex number that

is time-dependent Space vector of stator voltages, for constant V value and constant

frequency, travels uniformly along a circle in the complex plane One revolution of the space vector corresponds to one period of the supply frequency Complex plane and the space vector

of phase voltages are illustrated in Fig 6.16

In the previously discussed sinusoidal PWM, the three reference signals were three sine waves of appropriate amplitude and frequency Hence the calculated space vector corresponds

to what one wishes to impose to the machine terminals: three-phase system of sinusoidal

voltages of appropriate variable amplitude and frequency Indeed, in all the cases where voltage space vector modulation is applied, the reference voltage is the one given with (6.28)

β

v s( ωt=120°)

v s( ωt=0°) α

v s( ωt=270°)

Fig 6.16: Space vector of phase voltages for sinusoidal supply (superscript ‘s’ omitted)

It can be easily shown that the space vector of the line-to-line voltages travels uniformly along another circle whose radius is√3 larger than the radius of phase voltage circle, and that there is a 30 degrees phase shift between the two space vectors

Situation described above corresponds to the steady-state operation During a transient space vector of stator three-phase voltages will change its locus from one circle of certain diameter along which it has travelled with constant speed (initial steady-state), to another circle

of a different diameter along which it will travel ultimately again with new constant speed (final steady-state)

Let us consider now situation in a three-phase PWM voltage source inverter (VSI) From the point of view of the distinct non-zero voltage values that can be obtained, there is no difference between six-step VSI and a PWM VSI A six-step VSI is therefore analysed Power circuit of the VSI and associated voltage wave-forms, valid for six-step operation, are for convenience shown again in Figs 6.17 and 6.18

Leg voltages (voltages between points A, B and C and the negative rail of the dc supply ‘n’) are denoted in what follows with v A,v B,v C As can be seen From Fig 6.18, change

in any one of the three leg voltages takes place after every sixty degrees Leg voltages have constant values within sixty degrees intervals Thus it follows that the space vector of leg voltages will have six distinct and discrete values and that, instead of uniformly rotating in the complex plane, it will be jumping from one position to the other

Table 6.1 summarises values of leg voltages in the six sixty degrees intervals, lists switches that are on, and defines a corresponding space vector for each interval Apart from the six non-zero voltage space vectors, that can be obtained in the six-step mode of operation, two additional vectors (no 7 and 8) are added at the bottom of the Table These two vectors can be obtained only in PWM operation of the VSI and they describe the condition when the induction motor terminals are short circuited either through the positive rail of the dc supply (vector 7) or through the negative rail of the dc supply (vector 8)

Calculation of the leg voltage space vectors is rather simple From the definition of the voltage space vector in (6.27) one gets by substituting of individual leg voltages of Table 6.1 for each of the six sixty degrees intervals the following:

Table 6.1 Leg voltages

switching state switches on space vector Leg voltage vA Leg voltage vB Leg voltage vC

p

C

n

IM

Fig 6.17: Power circuit of the voltage source inverter

The two remaining space vectors are identically equal to zero as either all the leg voltages are zero or all the leg voltages have the same value (1 + +a a2 = 0) Hence

It follows from (6.29) that all the non-zero space vectors have identical amplitudes However, they are stationary, indicating that only discrete values of the leg voltage space vector are possible in a VSI In the six-step mode of operation transition from one space vector to the other takes place after each sixty degrees interval (for 50 Hz output, after 3.33 ms) In the PWM mode of operation the non-zero values remain to be given with (6.29) PWM mode adds two more possible vectors, called zero vectors, (6.30) Additionally, transition from one vector

to the other takes place at much higher frequency than the output frequency and each vector is utilised many times in creation of the output voltage of the given frequency

Distinct values of leg voltages (6.29) can be described with a single equation

v leg = 2 3V DC éëêj k− 1 ùûú

3

vA VDC vAB VDC

0 60 120 180 240 300 360 ωt [°]

Leg voltages

Line-to-line voltages

2/3 VDC

vb

Phase to neutral voltages

vc

Fig 6.18: Leg, line-to-line and phase to neutral voltages in VSI fed induction machine

For k = 7 and k = 8 leg voltage space vector equals zero As can be seen from (6.31),

time is not present in this equation, confirming that the vector does not travel continuously in time Frequency is not present either, so that the rate at which certain vector value is applied will be governed by switching frequency in the PWM VSI

Space vector values for the leg voltage are shown in Fig 6.19

Consider next line-to-line voltages at the output of the inverter, shown in Fig 6.18 Table 6.2 summarises values of line-to-line voltages in different sixty degree intervals and again lists two more states, obtainable in PWM mode only, when all the line-to-line voltages are

zero There are again six non-zero values of the voltage space vector and two conditions that yield zero value of the voltage space vector

7,8

Fig 6.19: Discrete values of the leg voltage space vector

Table 6.2 Line-to-line voltages

switching state switches on space vector Line-to-line

voltage vAB

Line-to-line

voltage vBC

Line-to-line

voltage vCA

Space vector of line-to-line voltages is again calculated using the definition of the space vector, (6.27) Substitution of individual line-to-line voltages into (6.27) for each of the six sixty degrees intervals produces the following result:

(6.32)

Space vector of line-to-line voltages is again equal to zero in states 7 and 8,

Line-to-line voltages are therefore characterised with six discrete values, whose amplitude is

√3 larger than for the leg voltages, and they are shifted in phase by 30 degrees with respect to the corresponding values of the leg voltage space vector Values of the line-to-line voltage space vector are shown in Fig 6.20

Space vector of line-to-line voltages, whose discrete values are given in (6.32), can be described with an expression similar to (6.31)

v L = V DC éj k−

ëê

ù ûú

2

3 3 exp 2 1 6

π

For k = 7 and k = 8 line-to-line voltage space vector equals zero As expected, (6.34) is

independent of time Hence the time interval during which the space vector remains in one position is determined with the inverter switching frequency

2

Re

5

Fig 6.20: Discrete values of the line-to-line voltage space vector

Finally, let us consider phase to neutral voltages of the motor, whose wave-forms are given in Fig 6.18 Table 6.3 summarises values of phase to neutral voltages for the six sixty degrees intervals

Table 6.3 Phase to neutral voltages

switching state switches on space vector Phase voltage va Phase voltage vb Phase voltage vc

1 1,4,6 v1phase (100) (2/3)V DC -(1/3)V DC -(1/3)V DC

3 2,3,6 v3phase (010) -(1/3)V DC (2/3)V DC -(1/3)V DC

5 2,4,5 v5phase (001) -(1/3)V DC -(1/3)V DC (2/3)V DC

Applying once more the same procedure, one finds the non-zero values of the space vector of the phase voltages as equal to:

exp( / )

π

(6.35)

The two remaining space vectors are again identically equal to zero Hence

Values of the space vector of phase voltages are identically equal to the values of the leg voltage space vector Thus

v phase= 2 3V DC éëêj k− 1 ùûú

3

For k = 7 and k = 8 phase voltage space vector equals zero Space vector values for the phase

to neutral voltages are shown in Fig 6.21, which is identical to Fig 6.19

7,8

Fig 6.21: Discrete values of the phase voltage space vector

As expected for a balanced three-phase system of voltages, line-to-line voltages have

√3 larger amplitude than phase voltages and are leading corresponding phase voltages by 30 degrees

Once when the space vectors are introduced, it becomes now possible to explain the principle of the voltage space vector modulation As already pointed out, control system will generate a reference voltage space vector, that corresponds to the ideal sinusoidal three-phase supply of certain frequency and amplitude Hence the reference voltage space vector is

On the other hand, the PWM inverter can generate only six discrete non-zero voltage vectors and two zero voltage vectors It is therefore not possible to directly impose the required reference voltage vector (6.38) However, the reference vlaue of the voltage space vector can

be obtained on average, during one switching period, by imposing the two neighbouring

available space vectors and a zero space vector for appropriate time intervals during the switching period Consider the situation shown in Fig 6.22 The reference voltage space vector is shown in a particular instant of time as being positioned in the first sextant of the plane During the switching period this desired reference value can be achived on average by imposing the available space vectors 1 and 2 and the zero voltage space vector for appropriate time intervals

According to Fig 6.22,

Let the switching period be T s Similarly, let the two zero voltage vectors be denoted as v o In other to achieve during one switching period on average required reference voltage (6.38) by

means of (6.39), it is necessary to impose non-zero voltage vectors 1 and 2 for the times a T s

and b T s, respectively Zero voltage vector will be imposed for the remainder of the switching

period, i.e for time interval c T s Therefore

Proportions of the switching period during which appropriate switching vectors are imposed are governed by the amplitude and phase of the reference voltage space vector It can be shown that