I. INTRODUCTION
Admittance Space Stability
Analysis of Power Electronic
Systems
S. D. SUDHOFF, Member, IEEE
Purdue University
S. F. GLOVER, Student Member, IEEE
Purdue University
P. T. LAMM, Member, IEEE
Air Force Research Laboratory
D. H. SCHMUCKER, Member, IEEE
Naval Sea Systems Command
D. E. DELISLE, Member, IEEE
Power electronics based power distribution systems (PEDSs)
are becoming increasingly common, particularly in marine
and aerospace applications. Stability analysis of this class of
systems is crucial due to the potential for negative impedance
instability. Existing techniques of stability analysis introduce
artificial conservativeness, are sensitive to component grouping,
and at the same time do not explicitly address uncertainties
and variations in operating point. A new stability criterion,
which reduces artificial conservativeness and is also insensitive to
component grouping is described. In addition, a means of readily
establishing design specifications from an arbitrary stability
criterion which specifically includes a provision to incorporate
uncertainty, parameter variation, and nonlinearities is set forth.
The method is presented in the context of a hardware test system
and is experimentally validated.
Manuscript received October 4, 1999; revised February 1, 2000;
released for publication March 14, 2000.
IEEE Log No. T-AES/36/3/07814.
Refereeing of this contribution was handled by W. M. Polivka.
Authors’ addresses: S. D. Sudhoff and S. F. Glover, School of
Electrical and Computer Engineering, Purdue University, Electrical
Engineering Bldg., W. Lafayette, IN 47907-1285; P. T. Lamm,
Air Force Research Laboratory, Wright-Patterson Air Force
Base, Dayton, OH 45433; D. H. Schmucker, Naval Sea Systems
Command, 2531 Jefferson Davis Hwy., Arlington, VA 22242-5160.
c 2000 IEEE
0018-9251/00/$10.00 °
Power electronics based power distribution
systems (PEDSs) can offer excellent load regulation,
transient performance, and, with proper architecture,
a high degree of fault tolerance. They are excellent
candidates in power distribution systems that need to
be fully- or semi-autonomous. Typical applications
include ships [1], submarines [1], aircraft [2, 3], and
spacecraft [4].
Unfortunately, PEDSs also have disadvantages,
one of which is the potential for negative impedance
instability [5]. This type of instability is brought
about by the nearly ideal regulation capability of
modern power converters. As an example, consider a
dc/dc converter whose input is connected to a power
distribution bus and whose output supplies a load.
It is relatively easy to design a power converter in
which the output is controlled so well that the load
is essentially unaffected by perturbations at the input
of the power converter. As a result, the power going
into this converter is constant with respect to the input
voltage (at least for perturbations within a certain
range). In a small signal sense, it can be readily
shown that this constant power load appears as a
negative resistance–an intuitively destabilizing effect.
As a result, the stability analysis of PEDSs is a more
critical task than in conventional power distribution
systems.
Most stability analysis of PEDSs is based on
impedance/admittance methods. These are based
on the fact that small signal stability at a given
operating point can be determined by examining
the Nyquist contour of the product of the source
impedance (Zs ) and the load admittance (Yl ) in a
single bus dc system. The most straightforward of the
impedance/admittance methods is the Middlebrook
criterion [5] which states that such a system will
be stable provided that the Nyquist contour of Zs Yl
remains within the unit circle. The key feature of this
method is that it is very design oriented. For example,
given the source impedance it is easy to set forth a
bound on the load admittance that will ensure that
the system is stable in a small signal sense. This is
very important in view of the fact that most PEDSs
are developed not by one company or corporation,
but instead by several corporations (typically a
system integrator with multiple subcontractors or
vendors).
The primary disadvantage of the Middlebrook
criterion is that it leads to artificially conservative
designs. In particular, much of the region in the
Nyquist plane which is forbidden when using this
criterion actually has little influence on stability. This
leads to control loops which are slower than they need
to be, bus capacitance which is higher than it needs
to be, or the use of dampers (active or passive) for
the sake of fitting the mathematical formulation rather
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965
than meeting actual dynamic stability requirements.
As a result, other stability criterion such as the Gain
and Phase Margin criterion (which is difficult to
reference because it was developed in parallel by a
number of groups), and the Opposing Component
criterion [6] were developed. However, although these
methods are less artificially conservative than the
Middlebrook criterion, they suffer from the fact that
in order to reduce artificial conservatism, it can be
necessary to move circuit elements across subsystem
boundaries. This is again problematic when multiple
engineering firms are involved in system synthesis.
Another difficulty which can be associated with
these methods is that they do not explicitly deal with
uncertainty, parameter variation, and nonlinearities.
This is indirectly dealt with by assignment of a gain
and phase margin; but if the margin is not adequate a
potential instability can be overlooked.
This paper sets forth an admittance/impedance
based design approach which explicitly incorporates
the uncertainty, the wide range of operating points,
and the high degree of reconfigurability found
in many dc distribution systems. There are two
aspects to this method, which can be employed
separately or together. The first is an alternate stability
criterion, which is less artificially conservative than
the heretofore available methods and at the same
time is less sensitive to component grouping. The
only disadvantage of this method is that in its raw
form it is not conducive to the formulation of design
specifications. This disadvantage is eliminated by the
second contribution of this paper–an admittance
space based design process, which allows design
specifications to be readily derived for an arbitrary
stability criterion. The advantages of this process
are that it is graphically interpretable and that it can
be used to explicitly address operating point and
parameter variations. The theoretical foundation for
this method is explained, and an example showing
the use of this method for determining the stability
of a sample system is provided and experimentally
validated.
II.
BACKGROUND. ADMITTANCE/IMPEDANCE
BASED STABILITY USING NYQUIST THEORY
Impedance based stability of a single bus dc power
system can be explained in terms of Fig. 1. Therein, vs
and Zs represent the Thevenin equivalent voltage and
impedance of a linearized source converter model, and
vl and Zl represent the Thevenin equivalent voltage
and impedance of the linearized load converter model,
and is and il represent the currents flowing into the
two converters, respectively. From Fig. 1, it is clear
that
Zl
Zs
v=
v +
v:
(1)
Zs + Zl s Zs + Zl l
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Fig. 1. Thevenin equivalent source and load converter model.
It is convenient to define
Zs =
Ns
Ds
(2)
Zl =
Nl
:
Dl
(3)
and
Substitution of (2) and (3) into (1) and manipulating
yields
N D v + Ns Dl vl
v= l s s
:
(4)
Nl Ds + Ns Dl
Assuming that the load operates in a stable fashion
if supplied from an ideal source, Nl will not have
any zeros in the right half plane. (Note that this
requirement does not say that the source must be
ideal, but rather that the load must be stable if it were
supplied by an ideal source). Likewise, assuming
that the source can operate in a stable fashion if
supplying a constant current load, at least in small
signal sense, then Ds will not have any zeros in
the right half plane. It should be observed that in
a mathematical sense these criteria define which
components are “sources” and which are “loads”,
not whether a component is supplying or consuming
power, though these two definitions often coincide.
Factoring these terms from the denominator of (4)
yields
N D v + Ns Dl vl
v= l s s
(5)
Nl Ds (1 + Zs Yl )
where Yl is the load admittance (1=Zl ). Since neither
Nl or Ds has any zeros in the right half plane, it
follows that the interconnected system of Fig. 1
is stable provided that 1 + Zs Yl does not have any
zeros in the right half plane. Then from Nyquist
theory, coupled with the fact that neither Zs or Yl
have any poles in the right half plane (because of
the conditions placed on Ds and Nl , respectively),
the number of unstable poles of the closed loop
system is equal to the number of clockwise
encirclements of ¡1 made by the Nyquist contour of
Zs Yl .
III. STABILITY CRITERIA
For the purposes of control design, there are
several methods of ensuring that the Nyquist contour
of Zs Yl does not encircle the ¡1 point, thereby
providing a guarantee of system stability. Several
of these methods are illustrated in Fig. 2. The first
of the methods set forth, the Middlebrook criterion
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Fig. 2. Stability criteria.
[5], consists of a circle of radius 1/GM in the s-plane
where GM denotes the gain margin. Clearly, if the
Nyquist plot of Zs Yl is always within the circle, then
encirclements of the ¡1 point cannot occur provided
the GM is greater than 1. This constraint has merit
in terms of a design criterion in that for a given Zs
the range of allowable Yl is readily established; in
particular
1
jYl j <
:
(6)
jZl jGM
Although convenient for design and readily
visualized, this approach tends to force artificially
conservative designs due to the demand for infinite
phase margin. For this reason, several other stability
constraints have been proposed. One alternate
approach is the Opposing Component criterion [6].
In this method the Nyquist diagram is required to fall
to the right of a line at s = 1=GM. This criterion has
an advantage over the Middlebrook criterion in that it
can be less artificially conservative because it allows
the Nyquist diagram to occupy a larger region of the
s-plane. The Gain Margin and Phase Margin (GMPM)
criterion depicted in Fig. 2 is another approach
which reduces the conservatism of the design. In this
method, the boundary consists of two line segments at
an angle of §PM from the negative real axis which
extend from infinity to the circle corresponding to
the Middlebrook criterion; the segment of this circle
connecting these two line segments is also apart of
the GMPM boundary. The region to the left of the
indicated boundary is considered forbidden. This
method is advantageous in that it allows both the
gain and phase margins to be arbitrarily specified,
thereby permitting the designer to specify the amount
of conservatism in a given design. In addition, this
method is also amenable to formulation of design
specifications. Again, assuming that the source
impedance is known this criterion will be satisfied
if either of the following conditions is satisfied for
every value of s in the path of the Nyquist contour
Fig. 3. Simple system. (a) Component grouping 1.
(b) Component grouping 2.
evaluation
jYl j <
1
jZl jGM
(7)
6
Yl (s) + 6 Zs (s) · (180± ¡ PM)
6
Yl (s) + 6 Zs (s) ¸ (¡180± + PM)
6
and
(8)
6
where 6 x ´ angle(Re(x) + j Im(x)) mapped to
(¡¼ ¼] and the operators +(¡) denotes the addition
6
6
(subtraction) of angles with the result mapped to
(¡¼ ¼]. Variations of this method are used extensively
by marine and aerospace vendors and system
integrators, and a form of the method is discussed in
the open literature in [7].
This paper proposes an alternate method, the
ESAC criterion, which is similar to the GMPM
criterion in that it allows both the gain and phase
margins to be specified, but it occupies a smaller
region of the s-plane. In this case the boundary
defining the forbidden region is formed by two line
segments which start at infinity, parallel the negative
real axis, and terminate at the boundary of the unit
circle. These segments are connected by two more line
segments which start at the unit circle and converge at
the point s = ¡1=GM. The advantage of this criterion
is that it opens up even more of the s-plane and hence
further reduces artificial conservativeness. A natural
extension of this method is to replace the horizontal
line segments with curves, which move toward the
real axis as s ! ¡1, but this refinement has been
noted to have little effect in practice.
The selection of the stability criterion has
considerable impact on not only the performance of
the design, but also on the design process itself. In
order to illustrate this, consider the simple system
shown in Fig. 3(a) and 3(b). In each case the system
itself is identical, but the grouping of the circuit
elements into a load and source differs. The circuit
itself consists of an ideal voltage source with a series
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967
Fig. 4. Nyquist contour for component grouping in Fig. 3(a).
Fig. 5. Nyquist contour for component grouping in Fig. 3(b).
RL impedance, a negative resistance representing the
dynamic linearization of a constant power load, and a
shunt capacitance. Upon writing the state equations of
this circuit it can be shown that stable operation will
occur provided that
R>r
(9)
example illustrates that a highly desirable feature for
any stability criterion which is defined as ¡Mr ! 1 is
that its slope djMi j=d(¡Mr ) goes to zero as ¡Mr ! 1.
This is the case with the ESAC criterion, but is not the
case with the GMPM criterion.
One way to allow the stable case to satisfy the
GMPM criterion is to represent the system as in
Fig. 3(b), which yields the Nyquist contour shown in
Fig. 5. Again, the unstable case violates both criteria.
However, with this component grouping the Nyquist
contour of the stable case (which occupies so small a
region of the s-plane that it is difficult to clearly see)
satisfies both criteria.
Although it could be argued that the necessity to
rearrange circuit elements to satisfy a criterion is not
an issue, because the criterion could be satisfied by
merely dividing the source and load up as in Fig. 3(b),
it is in fact an issue from a design perspective. In
particular, for design purposes it is desirable to treat
circuit elements to be within those components in
which they physically reside (however that may
be) and not have to reorganize them in order to
facilitate a given analysis. This is particularly true
when different companies are designing the sources
and loads. Furthermore, aside from modularity in the
design process, artificial reconfiguration of circuit
elements also changes the physical interpretation of
the gain and phase margin. A GM of 6 dB implies
that the load could be doubled and the system would
still be stable (at least in a small signal sense). If
circuit elements are reorganized from their physical
association in order to satisfy a particular criterion, the
physical interpretation of the gain and phase margin
changes meaning. The conclusion of this exercise is
that the proposed ESAC criterion holds a significant
advantage over the GMPM criterion; not because the
grouping of Fig. 3(a) is in some way superior to Fig.
3(b), but rather because it is more amenable to use
without rearranging circuit elements across physical
component boundaries.
and
C>
L
:
Rr
(10)
Fig. 4 depicts the Nyquist contour of Zs Yl with the
circuit elements as shown in Fig. 3(a). Parameters of
the circuit are r = 300 m−, L = 10 mH, R = 24:3 −
(which corresponds to a 3.7 kW constant power load
at 300 V). Two cases are shown: a stable case (C =
40 mF) and an unstable case (C = 0:5 mF). The arcs
at jsj = 400¼ are approximations to arcs at jsj = 1.
From Fig. 4, the Nyquist contour for the unstable
case violates both criteria, as expected. However, it
is interesting to observe that although the stable case
satisfies the ESAC criterion it does not satisfy the
GMPM criterion.
In this example, note that the product of the source
impedance and the load admittance for the situation of
Fig. 3(a), may be expressed
N(s) = Zs (s)Yl (s) = (r + Ls)(Cs ¡ 1=R):
(11)
For a frequency s = j!, it is convenient to denote
the real and imaginary part of N(j!) as Nr and Ni ,
respectively, whereupon it can be shown that the
upper half of the Nyquist contour of (11) may be
expressed as
µ
¶1=2
¡Nr ¡ r=R
jNi j = jrC ¡ L=Rj
:
(12)
LC
It is also convenient to designate an arbitrary stability
criterion by a parameterized curve in the s-plane given
by Mr (x) + jMi (x) where x is an independent variable.
From (12) it is apparent that the slope djNi j=d(¡Nr )
! 0 as ¡Nr ! 1 for all values of parameter r, C, L,
and R. As a result, the Nyquist contour will violate
any stability criterion which has a non-zero slope
djMi j=d(¡Mr ) as ¡Mr ! 1. For the GMPM criterion,
djMi j=d(¡Mr ) is equal to tan(PM) as ¡Mr ! 1; the
conclusion is that for this system configuration, the
GMPM criterion cannot be satisfied for any set of
parameters values unless the PM is set to zero. This
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IV. FORMULATION OF A LOAD ADMITTANCE
CONSTRAINT FOR A KNOWN SOURCE
IMPEDANCE
Although the Middlebrook and the gain and phase
margin criteria lead to artificially conservative designs
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Fig. 7. Admittance constraint at a particular frequency.
Fig. 6. Construction of admittance constraint at a particular
frequency.
and are problematic in terms of component grouping,
they are extensively used because of the relatively
straightforward means of developing design criteria
by (6) or (7)—(8), respectively. This paper sets forth
a straightforward method whereby such a design
constraint can be derived for the ESAC or any other
criterion. In this method, a stability constraint is
developed in terms of three-dimensional admittance
space in which the three axis are frequency (x), phase
(y) and magnitude (z). The construction of the load
admittance constraint is illustrated in Fig. 6. Therein
the ESAC criterion stability constraint is shown along
with the Nyquist contour of the source impedance. In
this example, the source impedance is that shown in
Fig. 3(a). Point Zs,a in Fig. 6 corresponds to the value
of source impedance at a certain frequency fa (in this
case 24.8 Hz). Point sb in Fig. 6 represents a point on
the stability criterion. The load admittance which will
cause the Nyquist contour of Zs Yl to touch the stability
criterion curve at point sb at a frequency fa is given by
Yl,ab =
sb
:
Zs,a
(13)
By sweeping point sb over the entire stability
constraint curve (which is truncated at points sc to
sd for boundedness) a constraint is placed on the
load admittance at frequency fa as depicted in Fig. 7.
Therein the forbidden region at that frequency is
marked; it should be noted that in this case the region
extends upward to infinity at a certain phase–in this
case approximately 102± . Clearly, this procedure could
be conducted for any arbitrary criterion. Repeating
this process over all frequencies of interest results in
the three-dimensional load admittance constraint curve
depicted in Fig. 8. Therein, the x-axis is frequency,
the y-axis is phase, and z-axis is magnitude in dB.
In order to meet this constraint, the load admittance
must fall outside of the enclosed volume. To this end,
it should be observed that as the forbidden region
extends upward it transforms from a volume to a
surface and that this surface extends upward without
bound; its illustration is truncated outside the region
of interest.
Fig. 8. Admittance constraint in admittance space.
As an example of the use of this method,
superimposed in Fig. 8 is the load admittance of two
loads with a negative impedance load of R = 24:3 −
in parallel with C = 0:5 mF (unstable case of previous
discussion), and C = 40 mF (stable case of previous
discussion). As can be seen, in the unstable case the
load admittance profile penetrates the forbidden region
whereas in the stable case the forbidden region is
avoided. As a side observation, the inflection in the
load admittance profiles corresponds to the change in
the evaluation of the Nyquist contour from s = 0 to
s = +j1 to the arc at jsj = 1 connecting the s = +j1
“point” to the s = 1 “point” which together constitute
the upper half of the Nyquist contour. In Fig. 8, 1 is
taken to be a large but finite number for illustration
purposes.
V. GENERALIZATION TO NONLINEAR AND
UNCERTAIN SYSTEMS
The technique described in Section IV is based on
linearized component models. Power electronics based
systems exhibit nonlinear dynamics and therefore
the linearized component model changes as loading
conditions vary. Furthermore, parameter variations
result in further variations of the linearized plant
model. As an example, consider the laboratory dc
power system depicted in Fig. 9. This system is
typical of a turbine electric propulsion system. In
this system, a turbine prime mover (implemented in
the laboratory as a dynamometer emulating a turbine)
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Fig. 9. Example system.
Fig. 11. Generalized source impedance of generator rectifier.
Fig. 10. Linearized source impedance of synchronous machine
rectifier system.
drives a synchronous machine rectifier system whose
output voltage is regulated at 400 V. The rectifier
output is connected via a short transmission line
to an induction motor drive turning a mechanical
load (emulated by a second dynamometer in the
laboratory). In this case, the induction motor is
controlled using an indirect field oriented control
so that torque control is nearly instantaneous. The
parameters of this system and a detailed description
are set forth in [8]. That work described two control
systems, a nominal control and a dc link stabilizing
control (later referred to as a nonlinear stabilizing
control (NSC) in [9]–patent pending). The nominal
control is used herein.
Fig. 10 depicts the linearized source impedance
of the synchronous machine rectifier system as
power varies from 0 to 1 p.u. in 0.2 p.u. steps.
Not unexpectedly, the impedance is a function of
output power due to dynamic nonlinearities and will
also vary as other aspects of the operating point
(speed, voltage reference) and parameters vary. The
techniques of the previous section may be extended
to this practical system by representing the source
impedance and load admittance as generalized sets
(denoted ZsG (s) and YlG (s), respectively), which
encompass the full range of possible plants. This
is done by characterizing the source/load over the
full range of operating conditions and parameter
values and then bounding the result. The generalized
impedance/admittance is defined as this bounded
region. As an example, Fig. 11 depicts the generalized
source impedance of the generator rectifier. The
generalized impedance for this example was
constructed by defining a set of nominal speeds
970
Fig. 12. Generalized load admittance of tie line and induction
motor drive.
from 228—279 rad/s in 5 increments, defining a set
of output voltage references from 380—420 V in
5 increments, and defining a set of output powers
from 0 to 4.07 kW in 5 increments. The plant was
linearized about every combination of speed, voltage
reference, and output power obtainable using these
sets (therefore yielding 125 plants). The individual
source impedances of the resulting 125 plants were
then bounded and the generalized source impedance
was taken to be the interior region of this bound,
which provides a good representation of all possible
values of source impedance.
Fig. 12 depicts the generalized load admittance of
the tie line and the induction motor drive. In this case,
the generalized load admittance was based on 216
plants obtained by every combination of linearized
plants which could be arrived from the set of speed
values (ranging from 0—200 rad/s in 6 steps), torque
commands (ranging from 0—20.9 Nm in 6 steps), and
input voltage (ranging from 380—420 V in 6 steps).
The use of generalized impedance sets and
generalized admittance sets can be used to account
for nonlinearities. This is based on the fact that the
small signal stability of a nonlinear system can be
determined by the stability of the linearized plant
model [10]. By taking into account every possible
plant model, small signal stability of the plant can
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be guaranteed for all operating points considered.
Likewise, the use of generalized impedances and
admittances can be used to take into account
uncertainties, and variations in parameters and
operating conditions. For example, in the case the
synchronous machine / rectifier, the source impedance
set was formulated by linearizing the plant over the
entire range of operating conditions and therefore
represents a good approximation to the full range
of impedance values that could be encountered. In
the example shown, this involved varying the set
points and loading conditions, but could have just
as easily included parameter variation as well. There
are many methods to create such a boundary from the
set of linearized plants; herein the boundary is taken
to be an irregular N-sided polygon. This polygon
as a function of frequency forms the generalized
source impedance or generalized load admittance, as
appropriate.
From a computation point of view, the automatic
calculation of the impedances/admittances is readily
achievable by linearizing time-domain nonlinear
average value models (NLAMs) [11—13] of the
components. In particular, NLAMs are automatically
linearizable in a variety of state variable based
simulation languages such as ACSL [14] and Simulink
[15]. This facilitates rapid and easy calculation of the
generalized source impedance and the generalized
load admittance.
Although NLAM models are probably the easiest
method to use, the impedance/admittance sets required
can also be obtained experimentally (by measuring
impedance/admittance over a large range of operating
points) or by using a detailed computer simulation to
“measure” the input impedance.
The generalized source impedance can then
be used to generate a generalized load admittance
constraint. This is similar to the process described
in the previous section; however at each frequency
all source impedances within the generalized source
impedance set must be considered when determining
the boundary of allowable source admittances rather
than considering the source impedance to be a single
point.
Fig. 13 depicts the resulting load admittance
constraint and generalized load admittance. Therein,
the ESAC criterion was used with a 3 dB gain margin
and a 20± phase margin. In this case it can be seen
that the generalized load admittance intersects the load
admittance constraint, which could result in instability.
Fig. 14 depicts the measured system performance
during a ramp increase in torque command. Initially
the system is operating in the steady state with a
commanded bus voltage of 400 V, the generator speed
is at 253.4 rad/s, and the induction motor speed is
at 188.5 rad/s (the mechanical speed was fixed by a
dynamometer). Approximately 100 ms into the study,
a ramped torque command is issued. The traces are
Fig. 13. Load admittance constraint and generalized load
admittance.
Fig. 14. Measured system performance during ramp increase in
torque command.
the commanded current, the actual current, and the dc
bus voltage at the input to the converter. As can be
seen, an instability results, which is consistent with
Fig. 13.
It should be kept in mind that satisfying the
stability criterion is a sufficient but not necessary
condition for stability, and so it would have been
possible for the system to be stable even though a
violation of the stability criterion was observed. If a
necessary and sufficient condition criterion is desired,
this can be approximated by using the ESAC criterion
(because it introduces less artificial conservatism than
the other methods) with very low gain and phase
margins.
It is interesting to observe the changes that can
occur in the stability of the example system as the
control is modified. In particular, incorporating the
dc link stabilizing control first introduced in [8] and
later referred to as NSC in [9] yields a dramatic
change in the generalized load admittance, as seen
in Fig. 15. In this case, there is no intersection of
the generalized load admittance and so stability is
guaranteed (although there is an apparent intersection
in Fig. 15, rotation of the figure will reveal that the
intersection is merely a line-of-sight effect). The
experiment shown in Fig. 14 is repeated in Fig. 16
(with NSC) demonstrating that for this scenario
system operation is stable.
VI.
CONCLUSIONS
A method of approaching system stability from a
three-dimensional admittance space point of view has
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[2]
Quigley, R. F., Jr. (1993)
More electric aircraft.
In Proceedings of the 1993 Applied Power Electronics
Conference, 906—911.
[3]
Elbuluk, M. E., and Kankam, M. D. (1997)
Potential starter/generator technologies for future
aerospace applications.
IEEE Aerospace and Electronic Systems Magazine, 12, 5
(May 1997), 24—31.
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switching regulators.
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[6]
Carrol, J. (1992)
An input impedance stability criterion allowing more
flexibility for multiple loads which are independently
designed.
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An experimentally validated nonlinear stabilizing control
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Fig. 15. Load admittance constraint and generalized load
admittance when utilizing NSC.
Fig. 16. Measured system performance during ramp increase in
torque command when utilizing NSC.
been set forth. One of the key features of the method
is the translation of the stability criterion specified by
the designer to a constraint on the load admittance as
viewed from admittance space. Another key feature
of the method is that nonlinearities and uncertainties
are treated by viewing the source impedance and load
admittance as sets rather than as known quantities.
Although the proposed methodology can be used with
any of the stability criterion discussed, the ESAC
criterion has considerable advantage from a design
point of view since circuit elements may be lumped
into the components in which they physically reside
and because less artificial conservatism is introduced.
The mechanics of using the proposed method
is a two-step process. The basis for being able to
use this proposed approach is the ability to obtain
sets of impedances of power system components
representing all possible operating conditions. Such
sets may be readily obtained from NLAM represented
in state-variable based simulation languages such as
ACSL and Simulink, obtained from detailed computer
simulations, or measured. In order to facilitate the
numerical processing of the linearized models, the
authors have created a Matlab toolbox to formulate
the stability criterion, generate generalized source
impedances and load admittances, parallel loads and
sources, include the effects of transmission lines, etc.
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Scott D. Sudhoff (M’88) received the B.S.E.E., M.S.E.E., and Ph.D. degrees from
Purdue University, Lafayette, IN, in 1988, 1989, and 1991, respectively.
From 1991—1993 he served as a half-time visiting faculty and half-time
consultant for P. C. Krause and Associates. From 1993—1997 he served as an
Assistant Professor at the University of Missouri—Rolla (UMR) and became an
Associate Professor at UMR in 1997. Later in 1997, he joined the faculty at
Purdue University as an Associate Professor. His interests include the analysis,
simulation, and design of electric machinery, drive systems, and finite inertia
power systems.
Dr. Sudhoff has authored or co-authored over twenty journal papers.
Steven F. Glover (S’94) graduated as an Honor Scholar, Summa Cum
Laude, with his B.S. in electrical engineering in 1995 from the University of
Missouri—Rolla (UMR) and completed his M.S.E.E. at UMR in 1997.
In 1997 he served as an associate research engineer for UMR and Paul C.
Krause and Associates (P.C.K.A.). Since 1998 he has worked as a research
engineer for Purdue University and P.C.K.A.. He is currently working towards his
Ph.D. in electrical engineering at Purdue University. His interests include analysis,
design, and control of electromechanical systems, drives, and power systems.
Peter T. Lamm (M’00) has a B.S. in engineering physics and an M.S. in
engineering physics from Wright State University, Dayton, OH.
He is employed as a research electrical engineer for the Air Force Research
Laboratory at Wright-Patterson AFB. His duties encompass research and
development of electrical power components and subsystems and their subsequent
integration into military aerospace systems.
Mr. Lamm is a member of the American Physical Society, AIAA, SAE, Sigma
Pi Sigma, and Tau Beta Pi.
Donald H. Schmucker (S’70–M’78) received his B.S.E.E. from the University
of Kentucky, Lexington, in 1970 and an M.S.E.E. from the University of
Maryland, College Park, in 1985.
He has worked for the U.S. Navy since 1971 specializing in naval electric
propulsion and power generation systems. He is currently a Senior Electrical
Engineer with the Naval Sea Systems Command in Crystal City, VA, working on
the development and application of high power density electrical equipment for
integrated propulsion and ship service power generation systems.
SUDHOFF ET AL.: ADMITTANCE SPACE STABILITY ANALYSIS OF POWER ELECTRONIC SYSTEMS
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