Vectorial PWM for Basic Three-Phase Inverters
5.1 REVIEW OF SPACE VECTOR THEORY .1 H istory and E volution of tHE C onCEpt.1 Historyand EvolutionoftHE ConCEpt
5.1.2 t HEory : v ECtorial t ransforms and a dvantagEs
5.1.2.2 Park Transform
This transform is not directly necessary for the presentation of space vector modula-tion algorithm, but is very used by electrical engineers in vector control of power converters. The two coordinates (α, β) are next transformed through a vector rota-tion with the rotarota-tional frequency identical to the electrical frequency.
U
Frequently, these two transforms are used in a single transform stage that coin-cides with the vector space theory introduced in the beginning (5.7):
U
Each of these transforms has an inverse that allows transformation to the phase measures from the orthogonal coordinates. The general inverse transform is given by
u t u u t
Particular forms are also available for Park and Clarke inverse transforms.
–120
FIGURE 5.1 Derivation of a vector equivalent to a three-phase system.
5.1.3 appliCationto tHrEE-pHasE Control systEms
This mathematical representation of three-phase measures treats the analysis of the three-phase system as a whole, instead of considering equations for each phase indi-vidually. Many control methods for three-phase systems have been derived from this mathematical approach.
Electrical drives (induction machine or synchronous machine drives) are con-trolled by the so-called field-orientation principle (Figure 5.2a). The three-phase grid interfaces or AC/DC converters (Figure 5.2b) are nowadays seen as active filtering systems controlled by the instantaneous power components (p–q) theory. All these systems use PWM algorithms in the final control stage.
controlFlux
Torque control
PI
PI idref
iqref
εd
εq
Vds
vqs
Vxs
PWM
Transf3/2 e+jθ
e–jθ 3-PH
INV
idm
iqm
IM vys
S1…S6
Power
stage Load
Ref anglePLL
Current transform
ia,ib ea,eb
PI PI
PI udc-ref
udc
id
iq-ref = 0
iq Compensation
volt limit transform
PWM (a)
(b)
FIGURE 5.2 Examples of application of vectorial representation in control.
5.2 VECTORIAL ANALYSIS OF THE THREE-PHASE INVERTER 5.2.1 matHEmatiCal dErivationof CurrEnt spaCE vECtor trajECtoryin
ComplEx planEsfor six-stEp opEration (witH rEsistivEand rEsistivE -induCtivE loads)
The three-phase inverter presented in Figure 5.3 is built of six insulated gate bipo-lar transistors (IGBTs), but it can be made with any other power-switching device, depending on the voltage and current ratings.
Figure 5.4 presents the appropriate output voltages without PWM. This is the so-called six-step operation and it is also the simplest and oldest control method for this type of power converter. Different switching states are given in the figure with a digital code (for example: 1 0 1), showing whether the high-side IGBT is ON (for 1) or the low-side device is ON (for 0). Another possible notation uses a sign to show where the pole terminals (A, B, C) are connected.
Each state of the power converter leads to a switching vector in the complex plane. There are thus six active switching vectors V1,...,V6 equally sharing six sectors within the complex plane (Figure 5.5).
The vectorial analysis of the operation of this system is first developed for an inductive three-phase load. Each phase current waveform can be derived by the inte-gral of each phase voltage equation.
v L i
Applying transform (5.2) to these allows writing the same equations for the vector space variables.
FIGURE 5.3 Basic topology for the three-phase voltage-source inverter.
The variation slope of the phase current is doubled when the phase voltage gets doubled. The maximum value of the phase current is denoted here by IM. During the time interval [t1, t2], the output voltage vector is V1 and the phase voltages are 2/3 Vdc,
−1/3 Vdc, and −1/3 Vdc.
Choosing the time origin in t1(t1 = 0), the load current and voltage expressions can be mathematically expressed as linear variations during the interval (t1, t2). From Figure 5.6, it yields:
i t I V
L t
i t I V
L t
i t I V
a M dc
b M dc
c M dc
( ) . ( ) .
( )
= − ⋅ + ⋅⋅ ⋅
= − ⋅ − ⋅ ⋅
= −
0 5 2
3
0 5 3
33⋅ ⋅
L t
(5.14)
100 va
vb vc
101 110 010 011 001
+––
+–+ ++– –+– –++ ––+
State:
V6 V1 V2 V3 V4 V5
2/3 Vdc 1/3 Vdc
FIGURE 5.4 Output voltage waveforms and state coding for the six-step operation.
V3 (0,1,0)
0 Re
Im
V4 (0,1,1) V1 (1,0,0)
V6 (1,0,1) V5 (0,0,1)
V2 (1,1,0)
FIGURE 5.5 Switching vectors corresponding to the six-step operation of the inverter.
Applying definition (5.1) to the space vectors associated to the current and voltage
It can be seen from Equation 5.18 that the magnitude of each voltage switching vector is (2/3)Vdc. The tip of the current space vector has a linear variation in the complex plane, with the Real part varying linearly from –0.5IM to 0.5IM during the time interval [t1, t2] of duration T/6 (Figure 5.7).
FIGURE 5.6 Output phase current and voltage waveforms: (a) calculation; (b) full-cycle trajectory.
This trajectory is oriented so that the current space vector is quasi-perpendicular on the voltage space vector and this was expected from the integral form of the inductive load equation. The vector projection on the Real axis represents the value of the first phase current. Considering the variation of the phase current during the interval [t1, t2] allows determination of the maximum value of the current (IM). It
The voltage space vector is identical with the switching vector V2 at the next inter-val. The current space vector moves between the position along the vectors V6 (at t2) and V1 (at next interval, t3). Similar calculation proves the linearity of this trajectory.
Extending the same reasoning for all six possible voltage-switching vectors defines the trajectory of the tip of the current vector in the complex plane.
The resulting trajectory is a hexagon oriented along the voltage-switching vectors.
The projection on the real axis is at any time equal to the current on the first phase. Currents or voltages on the other two phases can be graphically derived by the projection on two fictitious axes at 120° and 240°, respectively. This vectorial analy-sis provides information about all the currents and voltages in the system. Moreover, because of the 60° symmetry of the operation, it is enough to limit the vectorial analysis to a 60° sector.
Extending this analysis to the general case of an R–L load (Figure 5.8), consider the vectorial equation for the load circuit
Vs = ⋅is Rs + Ls⋅ dtis
Applied vector V1
Linear current
FIGURE 5.7 Current vector trajectory in the complex plane for a pure L load.
with the generic solution load. In other words, for each switching vector applied to the load, the current trajec-tory is a portion of exponential. To better define such trajectrajec-tory, let us suppose the initial value as being equal to i(0) = I and the final value after T/6 is
i 2 I
These conditions help defining the constant C and initial value I:
t I tip of the current vector for large
R/L constant
Trajectory of the tip of the current vector for small
R/L constant
FIGURE 5.8 Current vector trajectory in the complex plane for an R−L load.
Replacing complex constant C in Equation 5.23 yields:
i t( )= R⋅V ⋅ − I
+ ⋅
− −
1 s 1 eτt eτt (5.25)
5.2.2 dEfinitionof fluxofa (voltagE) vECtorand idEal flux trajECtory