Page 266 - Rashid, Power Electronics Handbook

14 Inverters 255

FIGURE 14.44 The three-phase CSI. Feedback control based on dq0 transformation.

state error is achieved, the per phase equations of the converter A ®rst approximation is to assume that the decoupling
are written as block is not there; in other words, ‰i Š ¼‰mŠ . On the other
c dq dq
hand, the model of the controllers can be written as
d
C ‰v Š ¼‰i Š ÿ‰i Š ð14:76Þ 1 ð t
p abc
l abc
o abc
dt ‰mŠ ¼ kf‰v Š ÿ‰v Š gþ ð‰v Š ÿ‰v Š Þdt
dq p;ref dq p dq p;ref dq p dq
T ÿ1
d
L ‰i Š ¼‰v Š ÿ R‰i Š ð14:77Þ ð14:82Þ
l abc
l abc
p abc
dt
where k and T are the proportional and integrative gains of
the ac line currents are in fact imposed by the modulator and the PI controller that are chosen to achieve a desired dynamic
they satisfy response. Combining the model of the controllers and the
model of the converter in dq coordinates and using the
Laplace transform, the following relationship between the
‰i Š ¼ i ‰i Š ð14:78Þ
o abc
i c abc
reference and actual load-phase voltages is found:

Replacing Eq. (14.78) into the model of the converter i i 1 R R
equations (14.76) and (14.77), using the dq0 transformation ‰v Š ¼ sk þ sI þ W þ I sI þ W þ I
p dq
C T L L
and assuming null zero component, the model of the conver-
ÿ1
ter becomes 2 i i i i s
Š
s I þ sW þ kI þ I þ I ‰v p;ref dq
C CT LC
d i i 1 ð14:83Þ
‰v Š ¼ÿW‰v Š þ ‰i Š ÿ ‰i Š ð14:79Þ
l dq
c dq
p dq
p dq
dt C C
In order to ®nally prove that zero steady-state error is
d 1 R achieved for step inputs in either the d or q component of
‰i Š ¼ÿW‰i Š þ ‰v Š ÿ ‰i Š ð14:80Þ
p dq
l dq
l dq
l dq
dt L L the load-phase voltage reference, the previous expression is
evaluated in s ¼ 0. This results in the following:
where W is given by

ÿ1
i i 1 R R i i
‰v Š ¼ W þ I W þ I I ‰v Š
p dq C T L L CT p;ref dq
0 ÿo
W ¼ ð14:81Þ
o 0 ¼‰v p;ref dq ð14:84Þ
Š