Page 58 - Modern Control of DC-Based Power Systems

Small-Signal Analysis of Cascaded Systems 23

v
Analyzing Fig. 2.4 and setting ^ ref and ^ i Lo1 to zero the Eqs. (2.14)
(2.16) are obtained. The Eq. (2.17), which represents the relationship
^
between input current perturbation i in and inductor current perturbation
^ i L ðsÞ, is obtained by examining the circuit presented in Fig. 2.2.

^ 1
v
dsðÞ 52 ^ out1 HðsÞG c sðÞ (2.14)
V m
^
^ v out 5 dsðÞG vd sðÞ 1 ^v in ðsÞG vg sðÞ (2.15)
^ ^
i L sðÞ 5 dsðÞG id sðÞ 1 ^v in ðsÞG ig sðÞ (2.16)
^ (2.17)
^ i in 5 jsðÞdsðÞ 1 MðDÞ ^ i L sðÞ
After solving this system of equations the relationship for the closed-
loop input impedance (2.18) is obtained. This relationship will be further
used in the cascaded converter model. It can be generalized to in respect
to a closed-loop converter loaded with a generic impedance load Z
which is characterized by (2.19).
0 1

1 1 T
^ v in ðsÞ
Z IN CL sðÞ 5 5 Z IN sðÞ @ A (2.18)
1 2 T jsðÞ
^ v ref ; ^ i Lo1 50
^ i in ðsÞ
eðsÞMðDÞG ig sðÞ
2
s ðÞ 5 ðZC 1 L e s 1 L e s 1 ZÞV m 1 ZHðsÞG c sðÞMðDÞeðsÞÞ (2.19)
2
Z IN CL
V m MðDÞð Þ ð1 1 ZC 1 sÞ 2 jsðÞZHðsÞG c sðÞMðDÞ
2.2.3 Single Converter Closed Loop (PCMC)
Another popular control mode for DC DC converters is the PCMC
[11,15]. The procedure in which the PCMC controls the converter is
shown in Fig. 2.5. The inductor current I L is compared every cycle to a
control current I co . As soon as I L . I co , the switches of the converter are
actioned and the inductor current enters the downslope. This does not
necessarily lead to the depicted steady state. The control current I co is
again determined via feedback of the output voltage.
From this technique the first advantage of the PCMC can be con-
cluded: the inductor current is limited each cycle. A drawback of this
procedure is that it introduces a stability problem in the circuit. If steady-
state operation is considered and duty ratios higher than 0.5 are used then
disturbances are amplified until the converter enters a subharmonic opera-
tion mode [15]. This is called the mode limit as a stable operation cannot