Christoph Haederli - Les thèses en ligne de l'INP - Institut National ...

3-L DC Link ML Converter Properties 53
i = * i
(32)
S1 d1
* iout
= α
out
i = d2*
i = (1 − ) i
(33)
S 2 out
α
u = α * U
out
out_
DC− DC
(34)
In flying capacitor stages, multiple commutation cells are connected in series. Each of the cells
can have its own duty cycle α x.
U DC
α1 α2 α3
U out
Figure 46, 4-L MC converter
To keep the flying capacitor voltages balanced, all upper switch average current must be equal
and all lower switch average currents must be equal. According to Kirchhoff’s law, the resulting
flying capacitor average currents will be zero. As the switch currents are directly related to the duty
cycles (32), (33), all duty cycles must be equal on average. Instantaneous values may differ. For a
multi cell converter with n cells we can state that there is one common α:
x ∈{ 1,.., N}
→ α = α , N = number of cells (35)
x
To get optimum modulation, the multiple cells need to be operated interleaved. There needs to
be an according phase shift between the switching patterns of the individual cells. One simple way
to implement that is with a carrier based PWM with phase shifted carriers. All of the phase angles
in the set A are applied, none twice. The order does not need to be specific to assure balancing of
the capacitors.
A = { 0,2π
/ n,4π
/ n,..,2(
n −1)
π / n}
(36)
x ∈ { 1,.., N}
∧ y ∈{1,..,
N}
∧ x ≠ y → ϕ ∈ A ∧ ϕ ∈ A ∧ϕ
≠ ϕ
(37)
x
y
x
y
This type of modulation ensures the desired output voltage according to (34). The same
average output voltage can also be obtained with different phase displacement, but no optimum
modulation is achieved (for example: phase shift zero leads to synchronized switching and 2-L
behavior, the advantage of multilevel operation is lost). The same concepts for calculations with
average currents, voltages and duty cycles also apply to all combinations of commutation cells used
in the 3-L DC link topologies. Specifically, we can also define a duty cycle α L for a complete phase