Page 88 - Modern Control of DC-Based Power Systems

Small-Signal Analysis of Cascaded Systems 53

1
V out L Z in L
G SL 5 5 G S G L U 5 G S G L U (2.93)
V in S Z in L 1 Z out S 1 1 T MLG
where the minor loop gain T MLG is defined as

Z out S
T MLG 5 (2.94)
Z in L
It is called minor loop gain because it can be shown that the output
impedance of the source subsystem Z out S and input impedance of the
load subsystem Z in L form a type of negative feedback system as depicted
in Fig. 2.28.
Since G S and G L are stable transfer functions, the minor loop gain
term is the one responsible for stability. Therefore, a necessary and suffi-
cient condition for stability of the cascade system can be obtained by
applying the Nyquist Criterion to T MLG , i.e., the interconnected system
is stable if and only if the Nyquist contour of T MLG does not encircle the
ð2 1; j0Þ point. From the control theory of linear systems, two quantities
called Gain Margin (GM) and Phase Margin (PM) can be defined. These
quantities quantify “how far” the Nyquist contour of T MLG is from the
critical ð2 1; j0Þ point, and, therefore, “how far” the system is from being
unstable. The quantities GM and PM are also related to the dynamic
response of the system: the higher the damping, the larger GM and PM
are (and vice versa the lighter the damping, the smaller GM and PM are)
[22]. Specifying GM and PM provides the engineers a way to design for
system stability with certain stability margins linked to the desired
dynamic time domain performance. Based on this concept, what is practi-
cally needed is to guarantee that the Nyquist diagram of T MLG does not
encircle the ð2 1; j0Þ point with sufficient stability margins. For this rea-
son, many practical stability criteria for the cascade system of Fig. 2.28
were proposed. These stability criteria define various boundaries between
forbidden and allowable regions for the polar plot of T MLG . The bound-
aries are defined by a certain GM and PM and are shown in Fig. 2.29.
The forbidden regions are the ones that include the ð2 1; j0Þ point.
System stability can be ensured by keeping the contour of T MLG outside
the forbidden regions. Based on the definition of the forbidden regions,
design formulations can be specified which relate the desired GM and
PM to the system parameters. Note that these criteria give only sufficient,
but not necessary stability conditions.