Page 55 - Modern Control of DC-Based Power Systems

20 Modern Control of DC-Based Power Systems

L e s
^ v out ðsÞ
Z out2t ðsÞ 5 5 (2.3)
2
CL e s 1 L e s 1 1
2 ^ i Lo1 ðsÞ ^ d;^v in 50
Z
1

^ i L ðsÞ MDðÞeðsÞð 1 CsÞ
Z
G Ld2t ðsÞ 5 5 (2.4)
^ CL e s 1 L e
2
dðsÞ ^v in ; ^ i Lo1 50 s 1 1
Z
^ 1
Z
G Lg2t ðsÞ 5 i L ðsÞ 5 MDðÞð 1 CsÞ (2.5)
2
CL e s 1 L e s 1 1
^ v in ðsÞ ^ d; ^ i Lo1 50
Z
^ 1
G Lo2t ðsÞ 5 i L ðsÞ 5 (2.6)
2
CL e s 1 L e
^ i Lo1 ðsÞ ^ d;^v in 50 s 1 1
Z

1
^ i in ðsÞ
Z in2t sðÞ 5 5 (2.7)

^ v in ðsÞ ^ d; ^ i Lo1 50 MDðÞG Lg ðsÞ
With these equations a complete description of the buck and the
boost converter is found to which the control system can be added and
which then fully describes a single converter and its behavior with small-
signal disturbances. A more compact notation of the Eqs. (2.1) (2.7) is
given by the following mathematical representation of the converter in
(2.8):
0 10 1
2Z out2t G vg2t G vd2t ^
i Lo1
^ v out
5 @ Z out A@ ^ v in A (2.8)
^ i L G Lg2t G Ld2t ^
d
sL e
This matrix of (2.8) is depicted in a control block representation of
Fig. 2.2, where the open-loop small-signal model can be derived by set-
^
ting dðsÞ equal to zero.
2.2.2 Single Converter—Closed Loop (VMC)
In order to derive a cascaded closed-loop converter system, the small-
signal-closed loop transfer functions of the single converter are first
required. Those transfer functions are needed for the closed-loop input
and output impedance Z in CL , Z out CL and the transfer function repre-
senting the closed-loop input to output voltage perturbation G vg CL [13].
To illustrate the concept, the most common control system operation for
aDC DC converter was selected, i.e., VMC [11].