New Tools for Studying Voltage-Source Inverters E.Semail, C. Rombaut Author Affiliation: Electrotechnic and Power Electronics Laboratory (L2EP), University of Lille I, France Abstract: This letter describes a vectorial formalism useful for studying polyphase systems. For that purpose, mathematical tools as the concepts of barycenter, linear application and kernel are introduced. This formalism generalizes the Space Vector Theory (SVT). A two levels three legs voltage-source inverter (23VSI) is studying to prove this property. At the end, a graphic representation of the homopolar component is presented. Keywords: polyphase, formalism, space vector, inverter. Introduction: The graphic formalism of the space vector is commonly used for the study of classical three-phase systems whose voltage and current vectors belong to a plane. On the other hand, this formalism hardly allows us the study of more complex systems which are supplying by n legs inverters[3], [6], [9]. So, we have elaborated a method which generalizes the SVT and takes more advantage of the vectorial geometry tools. In the case of the three-phase classical loads, our method allows already to take into acccount graphically the freedom degree whose uses lead to optimal Pulse Width Modulation (PWM) controls,[4]. In [1], [2] and [5], this freedom degree is considered by modifying the conduction durations of the switches.
E v1
Ni -E
A1
A2
Ak
An other kind of control, more usual, consists in imposing the average value of vc by a PWM with a fixed period T of modulation. In this case, it is possible to precise the characterisation. The mean value at the kT instant can be expressed by formula:
v c (kT) =
r = P -1 r = P -1 1 kT tr tr v v OM r , with c (t) dt = å cr = å ò (k -1)T r =0 T r=0 T T
r = P -1
T = å t r and tr the activation duration of the vector vcr. r =0
The point M, define as OM = v c (kT ) , can be considered as the barycenter of the P points Mr, with positive barycentric coordinates tr/T. So, it is the entire volume of the polyedron B that characterizes the inverter. When a point M belongs to one face of the polyhedron, this means that one leg of the inverter is no more switched.
xc3
M5 M6
M7 M0
M1
xc1
o M2
ì O M0 = - E x c1 - E xc2 - E xc3 ï M4 ï O M1 = + E xc1 - E xc2 - E xc3 ïO M2 = + E x c1 + E x c2 - E x c3 xc2 ïïO M3 = - E xc1 + E xc2 - E xc 3 í O M4 = - E x c1 + E xc2 + E x c3 ï M3 ïO M5 = - E xc1 - E xc2 + E xc3 ï ïO M6 = + E x c1 - E xc2 + E x c3 ïO M7 = + E x c1 + E x c2 + E x c3 î
An
Figure 2. Characterization of a two levels three legs VSI (23VSI)
v k= v Ak-v Ni
Generalization of Space Vector Theory (SVT): We are showing the relation between the SVT and our formalism by studying a 23VSI which supplies a delta connected load. In this case, the relations between the voltages vk, imposed by the inverter, and these ones, u1, u2, u3 across terminals of the load lines, are:
Figure 1. Representation of a two levels n legs inverter (2nVSI)
Vectorial characterization of inverter: The n-legs inverter represented in Figure 1 imposes n voltages vk. So, we associate to this converter a vectorial space Fn with an orthonormal base of vectors (xc1, xc2,…, xcn). We can define then a voltage vector: vc = vc1 xc1 + vc2 xc2 +…+ vcn xcn. Each coordinate of vc can accept two values, +E and –E. Consequently, a family of P=2n vectors vcr characterizes the inverter. Let us consider, for geometric representation, the points, O, Mr such as OMr = vcr. The P points Mr are the vertex of a polyedron. For n = 3, we have represented them in Figure 2. The polyedron is then simply a cube. The instantaneous controls, like Direct Torque Control, consists in choosing between the P points Mr the which one is more convenient. The concept of Euclidean distance is then interesting to realize this choice.
u1 = v1 – v2 ; u2 = v2 – v3 ; u3 = v3 – v1. They define a linear application Ac, such as Ac(vc) = uc, with uc = uc1 xc1 + uc2 xc2 + uc3 xc3. Its kernel, noted Ker, whose definition is Ker = {v Î Fn /Ac(vc) = 0}, is a vectorial straight line directed by the vector xh = xc1+xc2+xc3. Consequently, for every number h, Ac( h xh) = 0. So, we find out that, vh = vc . xh = vc1+vc2+vc3, the value of the commonly so called "homopolar" component of vc, has no influence on the three voltages u1, u2 and u3. Consequently, it defines effectively a freedom degree for the control law. As vh is no matter to obtain desired voltages for the load u1, u2 and u3, it is possible to forget this component of vc. To do that, we consider Oker a vectorial space orthogonal to Ker, and vcp the projection on Oker of every vector vc. The inverter
can then be characterized by the points Mrp, projections of the vertex Mr of the polyhedron B. For the 23VSI we recognize, in Figure 3, the usual hexagon introduced by the SVT. So, our approach is effectively a generalization of this theory but is independent of the load. This is not the case for the SVT. For example, the study of a star connected threephase load with no isolated neutral [6], [10] is not possible with SVT because the vector uc doesn't belong to a plane.
M 3p (-E,E,-E)
M2p (E,E,-E)
x c2 p xc1p
M 7p
M4p M5p
M1p
M 0p
2E
(-E,E,E)
xc3p
(-E,-E,E)
(a)
(E,-E,-E)
M 6p (E,-E,E)
Figure 3. Space vector representation for a 23VSI. Projection of polyhedron on vectorial space Oker.
Graphic representation of homopolar component: If we project on the vectorial space Ker the vertex Mr of the polyhedron, we obtain the points Mrh. So, we have a geometric representation of the homopolar component values. In Figure 4, we can see the results for the 23VSI.
M7h
(E,E,E)
3E
M 2h M 4h M6 h 3E 3 3E 3
3E
O
M1h M 3h M 5h M 2h (-E,-E,-E)
Figure 4. Representation of the homopolar component for a 23VSI supplying three-phase delta connected load
Conclusion: We have presented a new approach of converters. The concepts of barycenter, linear application and kernel have been used to characterize inverters. We have particuliarly treated the classical 23VSI but the concepts are general. We can study the n-leg inverters whose supply either polyphase electrical machines or several three-phase electrical machines simultaneously. The multilevel inverters can be also treated. Besides, it is possible with others tools of vectorial geometry, to elaborate control laws and to calculate the duration of conduction of switches.
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