A GENERAL UNIFIED APPROACH TO MODELLING SWITCHING-CONVERTER POWER STAGES
R. D. Middlebrook and Slobodan Cuk
California Institute of Technology
Pasadena, California
ABSTRACT
A method for modelling switching-converter
power stages is developed, whose starting point
is the unified state-space representation of the
switched networks and whose end result is either a
complete state-space description or its equivalent
small-signal low<-f requency linear circuit model.
A new canonical circuit model is proposed,
whose fixed topology contains all the essential
inputr-output and control properties of any dc-to-
dc switching converter, regardless of its detailed
configuration, and by which different converters
can be characterized in the form of a table con-
veniently stored in a computer data bank to
pro-
vide a useful tool for computer aided design and
optimization. The new canonical circuit model
predicts that, in general;switching action intro-
duces both zeros and poles into the duty ratio to
output transfer function in addition to those from
the effective filter network.
1. INTRODUCTION
1.1 Brief Review of Existing Modelling Techniques
In modelling of switching converters in
general,
and power stages in particular, two
main approaches - one based on state-space
modelling and the other using an averaging
technique - have been developed extensively,
but there has been little correlation between
them. The first approach remains strictly in
the domain of equation manipulations, and
hence relies heavily on numerical methods and
computerized implementations. Its primary
advantage is in the unified description of all
power stages regardless of the type (buck, boost,
buck-boost or any other variation) through
utilization of the exact state-space equations
of the two switched models. On the other hand,
the approach using an averaging technique is
This work was supported by Subcontract No. A72042-
RHBE from TRW Systems Group, under NASA Prime
Contract NAS3-19690 "Modeling and Analysis of Power
Processing Systems."
based on equivalent circuit manipulations,
resulting in a single equivalent linear circuit
model of the power stage. This has the distinct
advantage of providing the circuit designer with
physical insight into the behaviour of the
original switched circuit, and of allowing the
powerful tools of linear circuit analysis and
synthesis to be used to the fullest extent in
design of regulators incorporating switching
converters.
1.2 Proposed New State-Space Averaging Approach
The method proposed in this paper bridges the
gap earlier considered to exist between the state-
space technique and the averaging technique of
modelling power stages by introduction of state-
space averaged modelling. At the same time it
offers the advantages of botli existing methods -
the general unified treatment of the state-space
approach, as well as an equivalent linear circuit
model as its final result. Furthermore, it makes
certain generalizations possible, which otherwise
could not be achieved.
The proposed state-space averaging method,
outlined in the Flowchart of Fig. 1, allows a
unified treatment of a large variety of power
stages currently used, since the averaging step
in the state-space domain is very simple and clearly
defined (compare blocks la and 2a). It merely
consists of averaging the two exact state-space
descriptions of the switched models over a single
cycle T, where f
g
= 1/T is the switching frequency
(block
2a).
Hence there is no need for special
"knowr-how
M
in massaging the two switched circuit
models into topologically equivalent forms in order
to apply circuit-oriented procedure directly, as
required in [1] (block
lc).
Nevertheless, through
a hybrid modelling technique (block
2c),
the cir-
cuit structure of the averaged circuit model
(block 2b) can be readily recognized from the
averaged state-space model (block
2a).
Hence
all the benefits of the previous averaging
technique are retained. Even though this
out-
lined process might be preferred, one can proceed
from blocks 2a and 2b in two parallel but com<-
pletely equivalent directions: one following path
a strictly in terms of state-space equations, and
the other along path b in terms of circuit models.
In either case, a perturbation and linearization
18-PESC
76
RECORD
ί
α
\
statt
space
deScript
io
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:
ι
rtter
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7V
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âtate-
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:
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steady
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:
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O
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(qc
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j-
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r
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stole -
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averoyiny
éosic
state-space averaged model
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urhere:
6 & d
6<t
+ d'éz
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r
ê
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ttody
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and
line
transfer
functions
7~ττ
duty
ratio
d to I
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output
and/or
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state- voriqUel
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transfer
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function
state
-
space
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in
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-
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interval Td
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J
Iiî l
^-^«»-[{D-tyfal/W+vtf
Fig .
1 .
Flowchar t
o f
average d modellin g approache s
proces s require d
t o
includ e
th e
dut y rati o
modulatio n effec t proceed s
i n a
ver y straightfor -
war d
an d
forma l manner , thu s emphasizin g
th e
corner-ston e characte r
o f
block s
2 a an d
2b .
A t
thi s stag e (bloc k
2 a
o r
2b )
th e
steady-stat e
(dc )
an d lin e
t o
outpu t transfe r function s
ar e
alread y
available ,
a s
indicate d
b y
block s
6 a an d
6 b
respectively , whil e
th e
dut y rati o
t o
outpu t
transfe r functio n
i s
availabl e
a t
th e
final-stag e
mode l
(4 a
o r
4b )
a s
indicate d
b y
block s
7 a an d 7b .
Th e
tw o
fina l stag e model s
(4 a an d
4b )
the n giv e
th e complet e descriptio n
o f th e
switchin g
converte r
b y
inclusio n
o f
bot h independen t
cons -
truis ,
th e
lin e voltag e variatio n
an d th e
dut y
rati o modulation .
Eve n thoug h
th e
circui t transformatio n pat h
b migh t
b e
preferre d fro m
th e
practica l desig n
standpoint ,
th e
state-spac e averagin g pat h
a i s
invaluabl e
i n
reachin g som e genera l conclusion s
abou t
th e
small-signa l low-frequenc y model s
o f
an y dc-to-d c switchin g converte r (eve n thos e
ye t
t o b e
invented) .
Whereas ,
fo r
pat h
b ,
on e
ha s
t o b e
presente d wit h
th e
particula r circui t
i n orde r
t o
procee d wit h modelling ,
fo r
pat h
a
th e fina l state-spac e average d equation s (bloc k
4a ) giv e
th e
complet e mode l descriptio n throug h
genera l matrice s
A- p
A2
an d
vector s b-^ ,
ï>2>
c,^- ,
an d
o f
th e tw o
startin g switche d model s
(Bloc k
la) .
Thi s
i s
als o
wh y
alon g pat h
b i n
th e Flowchar t
a
particula r exampl e
o f a
boos t
powe r stag e wit h parasiti c effect s
wa s
chosen ,
whil e alon g pat h
a
genera l equation s hav e bee n
retained . Specifically ,
fo r
th e
boos t powe r
stag e
b ^
=
\>2
s
b .
Thi s exampl e wil l
b e
late r
pursue d
i n
detai l alon g bot h
paths .
I n additio n
th e
state-spac e averagin g
approac h offer s
a
clea r insigh t int o
th e
quantitativ e natur e
o f
th e
basi c averagin g
approximation , whic h become s bette r
th e
furthe r
th e effectiv e low-pas s filte r corne r frequenc y
f
i s
belo w
th e
switchin g frequenc y
f
g
,
tha t
is ,
f
/ f «
1 .
Thi s
is ,
however , show n
t o
b e
c
s
equivalen t
t o
th e
requiremen t
fo r
smal l outpu t
voltag e ripple ,
an d
henc e doe s
no t
pos e
an y
seriou s restrictio n
o r
limitatio n
o n
modellin g
o f practica l dc-to-d c converters .
Finally ,
th e
state-spac e averagin g approac h
serve s
a s a
basi s
fo r
derivatio n
o f a
usefu l
genera l circui t mode l tha t describe s
th e
input -
outpu t
an d
contro l propertie s
o f
an y
dc-to-d c
converter .
PESC
76RECORD-19
1.3 New Canonical Circuit Model
The culmination of any of these deriva-
tions along either path a or path b in the
Flowchart of Fig. 1 is an equivalent circuit
(block
5),
valid for small-signal low-frequency
variations superimposed upon a dc operating
point,
that represents the two transfer functions
of interest for a switching converter. These
are the line voltage to output and duty ratio
to output transfer functions.
The equivalent circuit is a canonical model
that contains the essential properties of any
dc-to-dc switching converter, regardless of the
detailed configuration. As seen in block 5 for
the general case, the model includes an ideal
transformer that describes the basic dc-to-dc
transformation ratio from line to output; a
low-pass filter whose element values depend upon
the dc duty ratio; and a voltage and a current
generator proportional to the duty ratio modula-
tion input.
The canonical model in block 5 of the Flow-
chart can be obtained following either path a or
path b, namely from block 4a or 4b, as will be
shown later. However, following the general
description of the final averaged model in block
4a,
certain generalizations about the canonical
model are made possible, which are otherwise not
achievable. Namely, even though for all currently
known switching dc-to-dc converters (such as the
buck, boost, buck-boost, Venable
[3],
Weinberg [4]
and a number of others) the frequency dependence
appears only in the duty-ratio dependent voltage
generator but not in the current generator, and then
only as a first-order (single-zero) polynomial in
complex frequency s; however, neither circumstance
will necessarily occur in some converter yet to be
conceived. In general, switching action introduces
both zeros and poles into the duty ratio to output
transfer function, in addition to the zeros and
poles of the effective filter network which
essentially constitute the line voltage to output
transfer function. Moreover, in general, both
duty-ratio dependent generators, voltage and cur-
rent,
are frequency dependent (additional zeros
and
poles).
That in the particular cases of the
boost or buck-boost converters this dependence
reduces to a first order polynomial results from
the fact that the order of the system which is
involved in the switching action is only two.
Hence from the general result, the order of the
polynomial is at most one, though it could reduce
to a pure constant, as in the buck or the Venable
converter [3].
The significance of the new circuit model is
that any switching dc-to-dc converter can be
reduced to this canonical fixed topology form,
at least as far as its input-output and control
properties are concerned, hence it is valuable for
comparison of various performance characteristics
of different dc-to-dc converters. For example, the
effective filter networks could be compared as to
their effectiveness throughout the range of dc
duty cycle D (in general, the effective filter
elements depend on duty ratio
D),
and the confi-
guration chosen which optimizes the size and
weight.
Also, comparison of the frequency depen-
dence of the two duty-ratio dependent generators
provides insight into the question of stability
once a regulator feedback loop is closed.
1.4 Extension to Complete Regulator Treatment
Finally, all the results obtained in modelling
the converter or, more accurately, the network
which effectively takes part in switching action,
can easily be incorporated into more complicated
systems containing dc-to-dc converters. For
example,
by modelling the modulator stage along the
same lines, one can obtain a linear circuit model
of a closed-loop switching regulator. Standard
linear feedback theory can then be used for both
analysis and synthesis, stability considerations,
and proper design of feedback compensating
net-
works for multiple loop as well as single-loop
regulator configurations.
2.
STATE-SPACE AVERAGING
In this section the state-space averaging
method is developed first in general for any dc-
to-dc switching converter, and then demonstrated
in detail for the particular case of the boost
power stage in which parasitic effects (esr of
the capacitor and series resistance of the in-
ductor) are included. General equations for
both steady-state (dc) and dynamic performance
(ac) are obtained, from which important transfer
functions are derived and also applied to the
special case of the boost power stage.
2.1 Basic State-Space Averaged Model
The basic dc-to-dc level conversion function
of switching converters is achieved by
repetitive
switching between two linear networks consisting
of ideally lossless storage elements, inductances
and
capacitances.
In practice, this function may
be obtained by use of transistors and diodes
which Operate as synchronous switches. On the
assumption
that the circuit operates in the so-
called "continuous conduction" mode in which the
instantaneous inductor current does not fall to
zero at any point in the cycle, there are only
two different "states" of the circuit. Each state,
however, can be represented by a linear circuit
model (as shown in block lb of Fig. 1) or by a
corresponding set of state-space equations (block
la).
Even though any set of linearly independent
variables can be chosen as the state variables,
it is customary and convenient in electrical
networks to adopt the inductor currents and capa-
citor voltages. The total number of storage
elements thus determines the order of the system.
Let us denote such a choice of a vector of state-
variables by x.
It then follows that any switching dc-to-
dc converter operating in the continuous conduc-
tion mode can be described by the state-space
equations for the two switched models:
20-PESC 76 RECORD
(i) interval Td:
χ = A-χ + b,v
1 lg
(ii) interval Td
f
:
χ = A
0
x + b
0
v
2 2 g
(D
y
l
where Td denotes the interval when the switch is
in the on state and T(l-d) Ξ Td
1
is the interval
for which it is in the off state, as shown in
Fig.
2. The static equations y-^ - c^x and
y
9
-
c
2^
x are
necessary in order to account for
tne case when the output quantity does not
switch
on
Td
off
ti>y>e
Fig.
2. Definition of the two switched intervals
Td and Td
1
.
coincide with any of the state variables, but
is rather a certain linear combination of the
state variables^
Our objective now is to replace the state-
space description of the two linear circuits
emanating from the two successive phases of the
switching cycle Τ by a single state-space
des-
cription which represents approximately the beha-
viour of the circuit across the whole period T.
We therefore propose the following simple avera-
ging step: take the average of both dynamic and
static equations for the two switched intervals
(1) ,
by summing the equations for interval Td
multiplied by d and the equations for interval
Td
f
multiplied by d'. The following linear
continuous system results
:
χ - dCA-x+b-v ) + d'(A
0
x+b
0
v )
1 1 g 2 L g
y = dy. + d'y
9
= (dc-
T
+d
f
c
9
T
)x
(2)
After rearranging (2) into the standard
linear continuous system state-space description,
we obtain the basic averaged state-space descrip-
tion (over a single period T):
χ -
(dA-+d
?
A
0
)x
+(db-+d
f
b
0
)v
12 1 l g
y =
(dcZ+d'cJ^x
(3)
This model is the basic averaged model which
is the starting model for all other derivations
(both state-space and circuit
oriented).
Note that in the above equations the duty
ratio d is considered constant; it is not a time
dependent variable
(yet),
and particularly not a
switched discontinuous variable which changes
between 0 and 1 as in [1] and [2], but is merely
a fixed number for each cycle. This is evident
from the model derivation in Appendix A. In
particular, when d - 1 (switch constantly on)
the averaged model (3) reduces to switched
model
(li) ,
and when d = 0 (switch off) it
reduces to switched model
(Iii),
In essence, comparison between (3) and (1)
shows that the system matrix of the averaged
model is obtained by taking the average of two
switched model matrices A- and its control is
the average of two control vectors b-, and b
0
, and
vectors b^ and ,
its output is the average of two outputs y^ and
y 2 over a period T.
The justification and the nature of the
approximation in substitution for the two switched
models of (1) by averaged model (3) is indicated
in Appendix A and given in more detail in [6].
The basic approximation made, however, is that
of approximation of the fundamental matrix
e
At = ι + At +
·* *
by its first-order linear
term. This is, in turn,shown in Appendix Β to
be the same approximation necessary to obtain the
dc condition independent of the storage element
values (L,C) and dependent on the dc duty ratio
only. It also coincides with the requirement for
low output voltage ripple, which is shown in
Appendix C to be equivalent to f /f « 1,
namely the effective filter corner frequency
much lower than the switching frequency.
The model represented by (3) is an averaged
model over a single period T. If we now assume
that the duty ratio d is constant from cycle to
cycle,
namely, d = D (steady state dc duty
ratio),
we
get :
Ax + bv
where
Τ
y = c χ
g
(4)
(5)
A - DA
X
+
D
f
A
2
b - Db
1
+ D*b
2
c
T
= Dc^ + D
f
c
2
T
Since (4) is a linear system, superposition
holds and it can be perturbed by introduction of
line voltage variations ν as ν • V + ν , where
V is the dc line input voltage? cauling §
corresponding perturbation in the state vector
χ • X + x, where again X is the dc value of the
state vector and χ the superimposed ac pertur-
bation.
Similarly, y = Y + y, and
χ = ΑΧ + bV + Αχ + bv
g g
Τ Τ
Λ
Y + y= cX+cx
(6)
PESC
76RECORD-21
Separation of the steady-state (dc) part
from the dynamic (ac) part then results in the
steady state (dc) model
AX + bV = 0;
g
Τ
c Χ
Φ·
Õ
-c
T
A
"4>ν
(7)
and the dynamic (ac) model
χ = Ax + bv
y - c χ
g
(8)
It is interesting to note that in (7) the
steady state (dc) vector X will in general only
depend on the dc duty ratio D and resistances
in the original model, but not on the storage
element values (L
f
s and
C
!
s).
This is so
because X is the solution of the linear system
of equations
AX + bV
g
(9)
in which L
f
s and C!s are proportionality con-
stants.
This is in complete agreement with the
first-order approximation of the exact dc
conditions shown in Appendix B, which coincides
with expression (7).
From the dynamic (ac) model, the line
voltage to state-vector transfer functions can
be easily derived as
:
4^
= (el-ATH
v
g
(s)
y(s)
v
g
(s)
(10)
Τ
-1
c (si-Α) b
Hence at this stage both steady-state
(dc) and line transfer functions are available,
as shown by block 6a in the Flowchart of Fig. 1.
We now undertake to include the duty ratio
modulation effect into the basic averaged
model (3).
Õ + y — c Χ + c χ + (c
x
-c
2
A
)Xd + (c
±
-c
2
)xd
dc ac
term term
ac term
nonlinear term
The perturbed state-space description is
nonlinear owing to the presence of the product
of the two time dependent quantities χ and d.
2.3 Linearization and Final State-Space Averaged
Model
Let us now make the small-signal approxima-
tion,
namely that departures from the steady state
values are negligible compared to the steady state
values themselves:
ν
_£
V
g
« 1,
(12)
Then,
using approximations (12) we neglect all
nonlinear terms such as the second-order terms in
(11) and obtain once again a linear system, but
including duty-ratio modulation d. After sepa-
rating steady-state (dc) and dynamic (ac) parts
of this linearized system we arrive at the follow-
ing results for the final state-space averaged
model.
Steady-state (dc) model:
Y X = -A
1
bV
g
c
T
X
=
-cV^bV
(13)
g
Dynamic (ac small-signal) model:
χ = Ax + bv + [(A-AJX + (b-b
9
)V Jd
g 12 1 l g
Τ , Τ Τ"
y = c χ + (c
1
-c
2
)Xd
(14)
In these results, A, b and c are given as before
by (5).
Equations (13) and (14) represent the small-
signal low-frequency model of any two-state
switching dc-to-dc converter working in the con-
tinuous conduction mode.
2.2 Perturbation
Suppose now that the duty ratio changes from
cycle to cycle, that is, d(t) = D + â where D
is the steady-state (dc) duty ratio as before and
d is a superimposed (ac) variation. With the
corresponding perturbation definition χ - X + χ,
y = Y + y and v„ = ν
σ
+ í
ó
the basic model (3)
b
O
Ο 6
ecomes
:
k = AX+bV 4- Ax+bv + [(A -A
9
)X + (b..-b
9
)V ]d
o
g LZ. .I Ζ g
de term line duty ratio variation
variation
+ [(A^A^x + (b
1
-b
2
)v
g
]d (11)
nonlinear second-order term
It is important to note that by neglect of
the nonlinear term in (11) the source of harmonics
is effectively removed. Therefore, the linear
description (14) is actually a linearized
describing function result that is the limit of
the describing^function as the amplitude of the
input signals ν and/or d becomes vanishingly
small.
The significance of this is that the
theoretical frequency response obtained from (14)
for line to output and duty ratio to output
transfer functions can be compared with experi-
mental describing function measurements as
explained in [1], [2], or [8] in which small-
signal assumption (12) is preserved. Very good
agreement up to close to half the switching
frequency has been demonstrated repeatedly
([1],
[2], [3], [7]).
22-PESC
76
RECORD