Page 70 - Modern Control of DC-Based Power Systems

Small-Signal Analysis of Cascaded Systems 35

2.4.5 Generalized Load Impedance
When using generalized impedance loads which follow the polynomial
structure given in (2.40) it is possible to derive with Eq. (2.19) a general-
ized load input impedance Z L of a closed-loop buck converter
m
P i
b i s
Z L s ðÞ 5 i50 ; a j ; b i AR (2.40)
n
P j
a j s
j50
m m m m n
P i13 P i12 P i11 P i P j12
b i s b i s b i s a j s
n 3 1 n 2 1 n 1 1 n 0 b i s 1 n d2
i50 i50 i50 i50 j50
s ðÞ 5
m n n n
Z IN CL
P i12 P j12 P j11 P j
b i s a j s a j s a j s
d n2 1 d 2 1 d 1 1 d 0
i50 j50 j50 j50
(2.41)
where in (2.41):
2
n 3 5 C 1 L; n 2 5 K D v in ; n 1 5 ð11K P v in1 Þ; n 0 5 K I v in ; n d2 5 L; d n2 5 C 1 D ;
2 2 2
d 2 52K D DV out1 ; d 1 5 D 2 K P D v in1 ; d 0 52K I D v in1
is
It is observed how the distribution of the poles and zeros of Z IN CL
influenced by the location of the poles and zeros of the generalized
load, as right-handed poles and zeros of the load can lead to power sys-
tem instability that are not obvious while analyzing a stable load
converter.

2.5 LINEAR CONTROL DESIGN AND VALIDATION
In this section, the stand-alone converter model is taken into
account one more time. It is shown how to design classical controls so
that a certain dynamic closed-loop performance is obtained. Moreover,
frequency domain validation techniques are presented to assess in the fre-
quency domain whether the closed-loop performance meets the
requirements.