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CHAPTER 2: EXPERIMENTAL IMPLEMENTATION OF MATRIX CONVERTER DRIVE
the modulation matrix M(t) must produce the generic output:
⎡ ⎤
cos(ωot)
⎢ ⎥
[vo(t)] = qVi
⎢
⎣ cos(ωot−2π/3) ⎥
⎦
cos(ωot+2π/3)
[ii(t)] = q cos(ωo)
cos(ωi) Io
⎡
cos(ωit+φi)
⎢
⎣ cos(ωit+φi− 2π/3)
cos(ωit+φi+ 2π/3)
⎤
⎥
⎦
(2.2.4)
In these formulae q is the voltage ratio, ωi and ωo are the input and output angular
frequencies and φi and φo are the input and output displacement angles. Venturini
derived two solutions, the first one is under the assumption that φi = φo and the second
one that φi = −φo:
M1(t) = 1
⎡
⎤
1+2q cos(ωmt) 1+2q cos(ωmt−2π/3)1+2q cos(ωmt+2π/3)
⎢
⎥
⎢
3⎣1+2q
cos(ωmt+2π/3) 1+2q cos(ωmt) 1+2q cos(ωmt−2π/3) ⎥
⎦
1+2q cos(ωmt−2π/3)1+2q cos(ωmt+2π/3) 1+2q cos(ωmt)
with ωm = (ωo− ωi) (2.2.5)
M2(t) = 1
⎡
⎤
1+2q cos(ωmt) 1+2q cos(ωmt−2π/3)1+2q cos(ωmt+2π/3)
⎢
⎥
⎢
3⎣1+2q
cos(ωmt−2π/3)1+2q cos(ωmt+2π/3) 1+2q cos(ωmt) ⎥
⎦
1+2q cos(ωmt+2π/3) 1+2q cos(ωmt) 1+2q cos(ωmt−2π/3)
with ωm = −(ωo− ωi) (2.2.6)
The combination of these two solution will provide the control over the input displace-
ment factor in a way that
M(t) = α1 · M1(t)+α2·M2(t)
where α1+ α2 = 1
(2.2.7)
Setting α1 = α2 will provide unitary input displacement, no matter which is the load
displacement factor. Otherwise, changing this ratio will provide a means to control the
input displacement to have a lagging or leading factor, however for this case the load
displacement factor must be known. Opting for the unitary displacement condition,
the modulation functions can be simplified in a compact formula:
mKj = tKj
= 1
3
Tseq
1+ 2vKvj
V 2
i
for K = A,B,C and j = a,b,c
(2.2.8)
This solution allows a maximum modulation index q of 50%, visible in the top of Fig.
2.3a, representing the case of a standard three phase 50Hz input supply while the out-
put is set to be at 100Hz. Further work, carried out by Venturini and Alesina, improved
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