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Small-Signal Analysis of Cascaded Systems 55

such an approach, based on the Opposing Argument Criterion, fails
when used with other less conservative criteria.)
More recent stability criteria include the Three-Step Impedance
Criterion (T-SIC) [40], the Unified Impedance Criterion (UIC) [41],and
the Maximum Peak Criteria (MPC) [42,43]. The T-SIC [40] relaxes the
conservativeness of previous criteria because it does not assume that G S in
(2.93) is necessarily a stable transfer function, which is typical of regulated
source subsystem. For this reason, the impedance criterion should not be
applied on the minor loop gain defined in (2.94), but rather on an
extended minor loop gain defined in [40]. All previous stability criteria
were developed for source subsystem interaction alone or load subsystem
interaction alone by using the Middlebrook’s Extra Element Theorem
(EET) [44]. Derived by using the 2EET [45], the UIC [41],particularly
suitable for cascade connected subsystems, constructs the minor loop gain
considering the simultaneous interaction of both source and load subsys-
tems. The last proposed stability criterion in order of time is the MPC
[42,43] which defines the minimum forbidden region for the minor loop
gain among all prior stability criteria (Fig. 2.29). Such a forbidden region
is determined by the maximum allowable peak of the sensitivity function,
providing a direct measure of the stability robustness. However, as also
demonstrated in [42,43], the state of the stability robustness strongly suffers
from the interface where the minor loop gain is measured.

2.6.1.1 Simulation Example
As an illustrative example, an averaged model simulation of a cascade of a
buck converter with a Voltage Source Inverter (VSI) in Fig. 2.30 is consid-
ered. The values of voltages and components for both buck converter and
VSI are also reported in Fig. 2.30. The controllers of both the buck converter
and VSI are designed according to the procedure presented in Section 2.5.
The load VSI, modeled in synchronous dq coordinates [46], is con-
trolled by an inner PI current mode (PICM subscript) loop with crossover
frequency f c_PICM 5 1 kHz and phase margin PM_ PICM 5 80 degrees, and
an outer PI voltage (PICM_FB subscript) loop with crossover frequency
f c_PICM_FB 5 0.1 kHz and phase margin PM_ PICM_FB 5 80 degrees. Due
to its importance in the small-signal stability analysis of the cascade system,
Fig. 2.31 depicts how the input impedance of the VSI is modified by effect
of the control action with respect to the open-loop (OL subscript) case.
Notice that by the addition of the PI current loop and then of the outer PI