A CONTINUOUS MODEL FOR THE TAPPED-INDUCTOR BOOST CONVERTER
R. D. Middlebrook
C a l i f o r n i a I n s t i t u t e of Technology
Pasadena, C a l i f o r n i a 91125
ABSTRACT
A continuous, 1ow-frequency, s m a l l - s i g n a l
averaged model for the tapped-inductor boost
converter with input f i l t e r i s developed and
experimentally v e r i f i e d , from which the dc
transfer function and the s m a l l - s i g n a l l i n e
input and duty r a t i o input describing functions
can e a s i l y be derived. A new effect due to
storage-time modulation in the t r a n s i s t o r switch
i s shown to explain observed excess f i l t e r damp­
ing resistance without associated l o s s in
conversion e f f i c i e n c y .
The presence of an
input f i l t e r can cause a severe disturbance,
even a n u l l , in the control duty r a t i o describing
function, with consequent potential performance
d i f f i c u l t i e s in a converter regulator.
1.
INTRODUCTION
A method of modeling switching converter
transfer functions has been described by Wester
and Middlebrook [ 1 ] , and applied to the basic
buck, boost, and buck-boost converters in the
continuous conduction mode. The p r i n c i p l e i s to
replace the several d i f f e r e n t , lumped, l i n e a r
models that apply in successive phases of the
switching cycle by a s i n g l e lumped, l i n e a r model
whose element values are appropriate averages
over a complete cycle of t h e i r successive values
within the c y c l e . The r e s u l t i n g "averaged" model
permits both the input-to-output ("line") and
duty r a t i o - t o - o u t p u t ("control") transfer func­
tions to be e a s i l y obtained f o r both dc (steady
state) and superimposed ac (describing function)
inputs. The nature of the model derivation
inherently r e s t r i c t s the v a l i d i t y to f r e ­
quencies below the switching frequency, and
model l i n e a r i t y i s ensured by independent
r e s t r i c t i o n of the superimposed ac signal to
small amplitudes. The r e s u l t i s therefore a
s m a l l - s i g n a l , low-frequency, averaged model.
The models obtained in [1] show that the
line and control describing functions contain
the anticipated low-pass LC f i l t e r response
characterized by a pair of l e f t half-plane poles
and, i f there i s nonzero resistance in s e r i e s
with the c a p a c i t o r , by a l e f t half-plane real
zero. The p o l e - p a i r p o s i t i o n s are conveniently
i d e n t i f i e d in terms of the f i l t e r corner f r e ­
quency and peaking f a c t o r , or Q-factor. Two
r e s u l t s of p a r t i c u l a r s i g n i f i c a n c e for the boost
and buck-boost converters are that the f i l t e r
corner frequency and Q-factor both vary with
steady-state duty r a t i o D and, even more impor­
tant, that the control describing function
acquires a r i g h t half-plane real zero.
Several extensions and developments of the
r e s u l t s of [1] are presented in t h i s paper.
1. A s m a l l - s i g n a l , low-frequency,
averaged model i s derived for the tappedinductor converter, of which the o r i g i n a l
("simple") boost converter i s a special case.
At the same time, the c i r c u i t being modeled
i s extended to include the l i n e input f i l t e r that
i s almost i n v a r i a b l y present in a p r a c t i c a l
system, and the modeling process i s refined to
include a more accurate representation of the
c i r c u i t losses that a f f e c t the Q-factor. The
motivation for the modeling refinement was to
explain measured Q-factors that were s i g n i f i ­
cantly lower than predicted by the o r i g i n a l model
even when the most generous values for the
physical l o s s - r e s i s t a n c e s were allowed. However,
although the refined model did indeed predict
lower Q - f a c t o r s , the quantitative effect was
s t i l l i n s u f f i c i e n t to explain the observed d i s ­
crepancies.
I t turned out that a quite different effect
was responsible for the lower Q - f a c t o r , namely,
modulation of the storage time of the t r a n s i s t o r
switch. The physical cause-and-effect sequence
i s as f o l l o w s : an applied s m a l l - s i g n a l ac
modulation of the switch duty r a t i o causes a
corresponding modulation of the current carried
by the switch at the instant of t u r n - o f f , and a
consequent modulation of the storage time.
The r e s u l t i s that the actual switch output
duty r a t i o modulation amplitude i s d i f f e r e n t
from the switch drive modulation amplitude.
It
may seem s u r p r i s i n g at f i r s t s i g h t that t h i s
effect would cause only a lowering of the Qf a c t o r , without a f f e c t i n g any of the other
q u a l i t a t i v e or quantitative features of the model;
PESC 75 RECORD—63
nevertheless, t h i s i s indeed confirmed by the
other principal extension presented in t h i s paper:
2. A storage-time modulation effect in the
switch i s shown to r e s u l t in an e f f e c t i v e series
resistance in the s m a l l - s i g n a l , low-frequency,
averaged model of the tapped-inductor boost
converter, whose presence lowers the apparent
Q-factor of the low-pass f i l t e r c h a r a c t e r i s t i c
contained in both the l i n e and control describing
functions.
Experimental r e s u l t s are presented for
various conditions chosen s p e c i f i c a l l y to expose
the functional dependence of the model element
values upon the several parameters, in order to
maximize the degree of model v e r i f i c a t i o n there­
by obtained.
In addition to such quantitative
v e r i f i c a t i o n , the following general conclusions
are of p a r t i c u l a r i n t e r e s t :
Some waveforms in the c i r c u i t of F i g . 1
under steady state operation are shown in F i g . 2.
The t r a n s i s t o r switch i s closed for a fraction
Ν turns
F i g , 1 . C i r c u i t of the tapped-inductor boost
converter with input f i l t e r .
1. The e f f e c t i v e s e r i e s resistance RM in
the model due to t r a n s i s t o r switch storage-time
modulation lowers the f i l t e r Q - f a c t o r , but i t
does not lower the conversion e f f i c i e n c y ; in
other words i t i s only an apparent resistance and
not an actual loss r e s i s t a n c e .
2. The presence of an LC input f i l t e r can
cause a serious modification of the control
describing function. A dip in the magnitude
of the control describing function can occur in
the neighborhood of the resonant frequency of
the input LC f i l t e r . This dip i s characterized
by a complex pair of zeros in the control d e s c r i b ­
ing f u n c t i o n . As the steady-state duty r a t i o i s
increased, as happens in normal regulation adjust­
ment of a closed-loop converter system, the complex
pair of zeros cannot only reach the imaginary
axis of the complex frequency plane, causing
the dip in the magnitude response to become a n u l l ,
but can move into the r i g h t half-plane causing a
large amount of additional phase l a g . A possible
null in the control describing function magnitude
response of course could severely degrade the
performance, and the excessive phase lag could
s e r i o u s l y a f f e c t the s t a b i l i t y , of a closed-loop
regulator system. The model presented here i s
useful in both the q u a l i t a t i v e and quantitative
design of such a system to guard a g a i n s t such
disastrous e v e n t u a l i t i e s .
2.
I _ amp-turns
* '
DT
Ν
(l-D)T = D T
,
s
S
s
DEVELOPMENT OF THE CONTINUOUS MODEL
The elements of a tapped-inductor boost
converter are shown in F i g . 1. An input f i l t e r ,
which would i n v a r i a b l y be used in a practical
system, i s included between the supply voltage V
and the input of the converter i t s e l f . As far
as the operation of the converter i s concerned,
only the f i l t e r elements L , R , and C need be
e x p l i c i t l y represented; the box around~the L i s
to imply that additional input f i l t e r elements may
be present without affecting the nature of the
converter operation. The resistances R and R
in F i g . 1 are always present in a practical
c i r c u i t even i f they represent only capacitance
esr.
q
s
S
s
$
§
64—PESC 75 RECORD
C
F i g . 2. Some waveforms in the c i r c u i t of
F i g . 1: v , i , and ν are continuous; ν
i s quasi-continuous; i i s discontinuous.
s
£
s
T D of i t s period T = l / f , where f i s the
switching frequency, and i s open for the
remaining fraction T D ' = T ( l - D ) .
While the
switch i s c l o s e d , the current i in the fraction
N / n of the total inductor turns Ν ramps up as
energy i s stored in the inductor; part of i
is
supplied from the input by i g , and the balance
comes from discharge of C causing a f a l l in
the capacitance voltage v . At the same time,
the diode i s open and the capacitance C d i s ­
charges into the load R causing a f a l l in the
voltage v . While the switch i s open, the
diode i s closed and the current i in the total
inductor turns Ν ramps down as the stored
inductor energy discharges into the load and
recharges C, causing a r i s e in v . Since i^
drops below i , C i s recharged causing a r i s e
in the voltage v .
S
s
s
S
s
s
s
x
s
s
s
c
s
c
q
s
s
Boundary conditions l i n k i n g the wave­
forms in the two i n t e r v a l s depend upon the
requirements that capacitance voltages and
inductor ampere-turns cannot change i n s t a n t a ­
neously at the switching i n s t a n t s . Consequently,
the voltages v and v are continuous at the
switching i n s t a n t s , whereas the output voltage
ν has steps at the switching i n s t a n t s because
of the drop in R . S i m i l a r l y , the inductor
ampere-turns i s continuous at the switching
i n s t a n t s , and i t i s convenient to i l l u s t r a t e
this in terms of a quantity i defined as
"ampere-turns per t u r n , " a l s o shown in F i g . 2.
Thus, i i s continuous but the actual inductor
current i has steps of r a t i o n at the switching i n s t a n t s . I d e n t i f i c a t i o n of i n i s an
important step in the derivation of the
continuous model.
$
c
c
Ä
£
s
x
Waveforms shown in F i g . 2 are for the
"continuous conduction" mode of operation in
which the instantaneous inductor current does
not f a l l to zero at any point in the c y c l e , and
the entire discussion and r e s u l t s of t h i s paper
apply only to t h i s mode. Average, or d c ,
values of the waveforms are a l s o shown in F i g .
2. The dc output voltage V i s of course equal
to the dc voltage V on capacitance C.
c
A s u i t a b l e equivalent of the c i r c u i t of
F i g . 1 from which to begin derivation of the
averaged model i s shown in Fi g. 3, and includes
several p a r a s i t i c resistances of obvious
physical o r i g i n which are important in determining the e f f e c t i v e Q of the i m p l i c i t f i l t e r
response c h a r a c t e r i s t i c .
There are two
independent " d r i v i n g " s i g n a l s :
the line voltage
V g and the duty r a t i o d. Each i s taken to have
a dc part and s m a l l - s i g n a l ac p a r t , so that
Vg = V + v and d = D + d. Since the
"complementary" duty r a t i o 1-d frequently
occurs in the equations, i t i s given the symbol
d' = 1-d, so that d' = 1-D-d = D ' - d . As a
consequence of the dc and ac components of the
two d r i v i n g s i g n a l s , a l l other voltages and
currents in the c i r c u i t also have dc and ac
components, in p a r t i c u l a r the output voltage
ν = V + v. The a n a l y s i s objective i s to f i n d a
continuous model (unlike the switched model of
F i g . 3) from which the d c a n d ac components of
the output voltage V and ν can be found from
the dc and ac components of the two d r i v i n g
s i g n a l s V , v and D, ά.
g
g
g
g
The approach taken i s an extension of that
in [ 1 ] , in which the s m a l l - s i g n a l ac components
V g , a , e t c . are taken to be slow compared with
the switching frequency f .
This r e s t r i c t i o n
permits the nonlinear model of F i g . 3, which
c o n s i s t s of two switched l i n e a r models, to be
approximated by a l i n e a r model whose element
values are appropriate averages of t h e i r values
in the two i n t e r v a l s T d and T - d ' .
Furthermore,
t h i s approach permits most of the a n a l y s i s to
be performed through successive transformations
and reductions of c i r c u i t models; physical
i n s i g h t i s thus better retained, and under­
standing made e a s i e r , than i f a n a l y s i s i s
performed e n t i r e l y by a l g e b r a i c manipulation.
The r e s u l t of t h i s approach i s an "averaged"
model from which the two transmission charac­
t e r i s t i c s can be obtained f o r dc and for ac at
frequencies below the switching frequency.
s
s
The objective i s to combine the two
separate l i n e a r models of the piecewise-linear
model of F i g . 3 into a s i n g l e l i n e a r model.
The procedure c o n s i s t s of a number of steps in
manipulation of the model, which w i l l be
presented here for a s l i g h t l y s i m p l i f i e d version
of F i g . 3 in which R , R , R , Rj,and R are
set equal to zero. This i s done so that the
method may be i l l u s t r a t e d without the burden
of extra elements and terms which merely
complicate the diagrams and equations. The
effects due to these temporarily discarded
elements w i l l , however, be restored into the
final results.
A l s o , a number of comments
concerned with the s i g n i f i c a n c e and interpretation of certain steps w i l l be deferred u n t i l the
end of the d e r i v a t i o n .
$
v
ç
Step 1 . Draw separately the l i n e a r models
of F i g . 3 that apply during each of the
i n t e r v a l s T d , T d ' as shown in F i g s . 4(a) and
4(b) r e s p e c t i v e l y .
I d e n t i f y the voltages and
currents that are continuous across the switching i n s t a n t s , namely the capacitance voltages
v , v , and the ampere-turns per turn of both
the switched inductor, i , and the input
f i l t e r inductor, i .
s
F i g . 3. Switched model equivalent to the
c i r c u i t of F i g . 1 ; s t a r t i n g point for derivation of the continuous model of F i g . 10.
w
s
$
c
£
Q
PESC 75 RECORD—65
R
n i,
u
x
same topology i f the r a t i o i s 0:1 in F i g . 5(a)
and 1:1 in F i g . 5 ( b ) .
L/ηχ
Step 3. Replace the ideal transformers
in F i g . 5 by ideal dependent generators whose
c o n t r o l l i n g s i g n a l s are voltages or currents
that are continuous across the switching
i n s t a n t s . This i s done in F i g . 6.
(Notation:
squares are used to represent dependent
generators, c i r c l e s represent independent
generators.)
φ
Cs
(α) interval
Td
s
Φ
n .,mmn v
x
x
s
Cs
(b) interval T d ' = T ( l - d )
s
F i g . 4. Step
(parasitic
omitted):
during the
3
8
(a) interval T,d
1 in the model derivation
resistances in model of F i g . 3
separate l i n e a r models that apply
two i n t e r v a l s Τ d and Τ d ' .
s
s
Step 2. Manipulate the models of F i g . 4
to have both the same topology and the same
values of the continuous voltages and currents
at corresponding p o i n t s , as shown in F i g . 5.
In
the present case, t h i s requires s c a l i n g of the
current n i n in F i g . 4(a) to match the current
i
in F i g . 4 ( b ) , which i s done by introduction
of an ideal transformer of r a t i o l : n in F i g .
5 ( a ) ; since the voltage v i s already the same
in both models, the transformer i s introduced
to the r i g h t of C . To produce the same
topology, a 1:1 transformer i s introduced in
the same place in F i g . 5 ( b ) .
A l s o , since the
current i flows into C during interval T d '
but does not during interval T d , introduction
of another ideal transformer to the l e f t of C
allows t h i s condition to be realized with the
(b) interval T d'
x
Ä
x
s
s
F i g . 6. Step 3: replacement of ideal t r a n s ­
former by dependent generators controlled by
continuous v a r i a b l e s .
s
Ä
s
s
Step 4. Coalesce the two t o p o l o g i c a l l y
identical models of F i g s . 6(a) and 6(b) into a
s i n g l e model in which the various s i g n a l s have
the average of their values throughout the
entire period T , as shown in F i g . 7. For
s
(dn?+d')R
u
r:
1
%
^
Vc
c
1
Ο: I
F i g . 7. Step 4: coalesced models for the
two i n t e r v a l s into a s i n g l e averaged model;
implies imposition of the low-frequency
restriction.
example, the dependent current generator which
has a value n i ^ for an interval T d and a value
i£ for an interval T d ' has an average value
( d n + d ' ) i over the entire period T , and
s i m i l a r l y for the other dependent generators.
The p a i r on the r i g h t , of course, constitutes a
special case in which the s i g n a l i s zero for one
of the i n t e r v a l s . The coalescing process takes
a s l i g h t l y different form for an element whose
value i s not the same in the two i n t e r v a l s :
the
resistance ( d n + d ' ) R in F i g . 7 i s the value
x
s
s
x
(b) interval T d '
s
F i g . 5. Step 2 : introduction of ideal
transformers to e s t a b l i s h the same topology
and to expose the same continuous variables
for the two i n t e r v a l s .
66—PESC 75 RECORD
£
s
2
x
u
having a voltage drop across i t equal to the
average of the voltage drops in the two
intervals T d and T d ' .
s
R
DD'(n -l) R
x
u
•--
(η,-DR^d
u
F+H+
L
$
u
Vd
(n -l)V d
x
Step 5. Substitute dc and ac components
for the variables in F i g . 7. For example, the
left-hand dependent current generator i s expressed
as:
(dn +d')i
x
£
Φ
c
s
.
{n -\)l d
x
x
= [(D+u)n +(D'-d)](I^i )
x
Ä
|:D
=
(Dn +D')(I +i )+(n -l)I d+(n -l)i d
x
Ä
Ä
x
£
x
Fig.
Ä
D': I
X
9. Step 6: Restoration of ideal t r a n s ­
formers in place of the dependent generators
and adjustment of certain element p o s i t i o n s .
% (Dn +D')i +(n -l)I d
x
£
x
£
The approximation of the f i n a l line i s neglect
of the ac product term, which i s v a l i d f o r small
ac s i g n a l s superimposed on the dc. This
generator, therefore, may be decomposed into a
dependent generator proportional to i , and an
independent generator proportional to the d r i v i n g
signal d, as shown in F i g . 8. This figure shows
As mentioned at the beginning of the
d e r i v a t i o n , the resistances R , R , R , R , a n d
R were omitted.
I f these elements are
included, the model of F i g . 9 becomes extended
to that shown in F i g . 10, in which
s
w
y
d
c
£
( 2-i)R i,d
N
|_
u
V c
(D
;
(2)
è .[(D-D-)(n -l) R -(n
2
3
x
s
2
x
-l)R
- ("xVV - ' H
D (R
R ) ] I
C
Fig.
u
*
(3)
d
(4)
8. Step 5: s u b s t i t u t i o n of dc and ac
components; implies imposition of the s m a l l signal restriction.
corresponding manipulations of the other three
dependent generators of F i g . 7. The s u b s t i t u t i o n
process f o r the averaged resistance i s as f o l l o w s .
The voltage across ( d n + d ' ) R i s
(5)
RT
= DD'(n -l)[(n -l)(R +R )]+n R -R ]
(6)
R
E DD'(R +R ||R)
(7)
x
x
S
u
X
W
V
2
x
< V '> uV
d
+d
R
u
2
[(D+dJnZ+iD'-cD^d^^)
D
C
Dn R + D R
Dn + D' Λ
/
x
w
v
x
where again the product term in i^d i s omitted.
This voltage can be represented as that across a
resistance plus an independent generator proportional to 3 , as shown in F i g . 8. For conciseness,
the factor ΰη +ϋ' i s replaced by ϋ .
χ
χ
Step 6. Replace the dependent generators
by corresponding ideal transformers, and add a
resistance R M to the l e f t of the D transformer,
as shown in P i g . 9. To compensate f o r t h i s
a d d i t i o n , the reflected value D R must be
subtracted from the right-hand side o f the D
transformer so that the net resistance in t h i s
branch i s then [ ( D n + D ' ) - D ] R , which
reduces to D D ' ( n - l ; R as shown in F i g . 9.
One other change has been made in F i g . 9 , namely
the independent generator I d has been reflected
from the r i g h t to the l e f t side of the D' t r a n s ­
former.
x
2
X
U
x
2
x
D': I
10. Continuous, dc and low-frequency
s m a l l - s i g n a l ac,averaged model for the tappedinductor boost converter of F i g . 3
( p a r a s i t i c resistances r e s t o r e d ) ; D and d
refer to duty r a t i o at the t r a n s i s t o r c o l ­
lector.
2
x
x
2
x
I:D
Fig.
u
£
g
The
continuous
converter,
the dc and
can e a s i l y
c i r c u i t of F i g . 10 i s the complete
model of the tapped-inductor boost
including input f i l t e r , from which
ac l i n e and control transfer functions
be obtained.
PESC 75 RECORD—67
The ideal transformers operate down to
d c , and the dc output voltage V i s obtained as
a function of V and D by s o l u t i o n with the ac
generators set equal to zero.
A c t u a l l y , to
allow f o r an unspecified input f i l t e r , i t i s
more convenient to solve for the dc output voltage
V as a function of D and of the dc voltage at
the input f i l t e r output, which i s the same as
the input f i l t e r capacitance voltage V .
The
dc component I of the inductor ampere-turns per
turn current i s
DV
χ χ
(8)
D R + R
q
$
£
, 2
T
where
R
T
Ξ
D
i s a low-frequency averaged model, v a l i d for ac
signal frequencies much lower than the
switching frequency.
The steps leading up to step 4 have the
purpose not only of molding the separate l i n e a r
models f o r the two switching i n t e r v a l s into
s i m i l a r topological forms, but also of s e t t i n g
up the quantities that are to be averaged in
such a way that the current or voltage factor
in the product i s one that i s continuous across
the switching i n s t a n t s . Thus, in F i g s . 6 and
7 , the q u a n t i t i e s to be averaged a l l contain
i'o, ν , or v ( a c t u a l l y , the state variables of
the system) as one of the f a c t o r s , and the
inductor ampere-turns per turn i was s p e c i f i ­
c a l l y i d e n t i f i e d f o r t h i s purpose. In
c o n t r a s t , the actual inductor current i and the
output voltage v, for example, are not continuous
across the switching i n s t a n t s , as seen in F i g .
2, and t h e i r use i s therefore suppressed in the
manipulations leading up to step 4, although
the d e t a i l s were not e x p l i c i t because of
omission of the various p a r a s i t i c r e s i s t a n c e s .
c
£
x
2 [ R
u
+ ( D n
x w
R
+ D , R
v
) / D
x
] + R
l
+ R
2
+ D
'
2 R
d
(
9
)
i s the t o t a l e f f e c t i v e resistance referred to
the middle loop in F i g . 10. The dc output
voltage i s then
1
V = D'I R = - 4
D 1 + Rj/D
(10)
R
$
In the f i n a l r e s u l t of F i g . 10, a l l the
p a r a s i t i c resistances R , R , R , R , R , and
a l s o R which was retained throughout the deriva­
t i o n , appear in the same physical p o s i t i o n s as in
the o r i g i n a l c i r c u i t of F i g . 3.
(Although
since R and R each appears in the o r i g i n a l model
for only one of the two switching i n t e r v a l s , they
appear in F i g . 10 in an appropriately averaged
form with an obvious physical i n t e r p r e t a t i o n .
I t i s for t h i s reason that these averaged forms
are purposely inserted in the shown p o s i t i o n
in F i q . 10, j u s t as R(j in the adjacent
p o s i t i o n was purposely added in step 6.)
$
w
v
u
c
u
This equation represents the b a s i c boost
property of the converter; for a h i g h - e f f i c i e n c y
system, the e f f e c t i v e dc l o s s resistance Rj i s
small compared with the reflected load r e s i s ­
tance D R , so V ^ ( D / D ' ) V . For reduction to
the simple boost converter, n = 1 so D =
Dn +D' = 1 , and then V % ( 1 / D ' ) V .
, 2
X
S
x
x
x
S
The l i n e describing function i s
obtained from the model of F i g . 10 by s o l u t i o n
for v / V g ,
and the control describing function
i s obtained as ν/α through use of the generators
ê ] , e 2 > e g , j i , and
each of which i s proportional to d. Before consideration of such
a p p l i c a t i o n s of the model, however, some
comments w i l l be made concerning i t s form and
derivation.
The essence of the procedure of the model
derivation i s contained in step 4, in which the
two models of F i g . 6 for the i n t e r v a l s T d and
T d ' are coalesced into a s i n g l e model. The
exact average of the two current generators,
n i
from F i g . 6(a) and i
from F i g . 6 ( b ) , i s
<dn i + d'i ).
An e s s e n t i a l approximation i s
then made in the replacement of the average of
the product of two v a r i a b l e s by the product of
t h e i r averages, so that <dn i£ + d ' i n ) =
<dn i,> + < d ' i > %<d>n <i > + < d ' ) < i \ > =
(<d)n + < d ' ) ) < i > .
I t i s t h i s f i n a l form that
i s shown a g a i n s t the left-hand dependent current
generator in F i g . 7 , except that for s i m p l i c i t y
in notation the averaging s i g n s { ) have been
omitted.
As described in [ 1 ] , the above
approximation i s v a l i d i f a t l e a s t one of the
v a r i a b l e s i s continuous; in t h i s case, d i s
not continuous ( i t changes from 1 to 0 ) , but
i*£ i s continuous. As a l s o discussed in [ 1 ] ,
the averaging process imposes an upper f r e quency l i m i t on the v a l i d i t y of the r e s u l t , and
i t i s for t h i s reason that the model of F i g . 10
$
s
x
£
£
x
£
£
x
x
y
x
A
x
Ä
68—PESC 75 RECORD
Ä
w
y
In addition to the appearance of the
p a r a s i t i c resistances in the expected p l a c e s ,
the model of F i g . 10 shows the presence of
two additional resistances R] and R2 defined
by E q s . (6) and (7), which are related to
the p a r a s i t i c r e s i s t a n c e s . I t i s i n t e r e s t i n g
to note that R-j and Rj are zero at both zero
and unity dc duty r a t i o (D = 0 and D' = 0 ) ,
and have maximum values at D = D' = 0.5.
The additional resistances R-j and R2 appear
in the averaged model of F i g . 10 because of
s t r i c t adherence in the derivation to the
requirement that one of the two quantities in
an averaged product must be continuous across
the switching i n s t a n t .
However, as seen in
the waveforms of F i g . 2 , the output voltage
v, for example, while not continuous may be
considered quasi-continuous in that the steps
are small compared to the total value.
If
q u a s i - c o n t i n u i t y i s accepted instead of s t r i c t
continuity in step 4 of the d e r i v a t i o n , then
the resistances R-j and R2 do not appear in the
result.
This s i m p l i f i e d procedure was followed
in [1] in the derivation of an averaged model
for the simple boost converter.
To see the s i g n i f i c a n c e of the "extra"
resistances R-j and Rp, consider the model of
F i g . 10 reduced for determination of the l i n e
describing function v / V g . For t h i s case of
constant duty r a t i o , the ê ' s and j ' s are a i l
zero, and for further s i m p l i c i t y l e t the input
f i l t e r be omitted. Then, with the two t r a n s formers eliminated by appropriate r e f l e c t i o n
of the elements in the outer loops into the
center loop, the r e s u l t i n g reduced model i s as
shown in F i g . 1 1 . The line describing
function v / V g i s simply that of a l o s s y LC
Rr
D'v
D£R
U
D ( D n R + D ' R ) R,
x
x
w
v
R2
D' R
d
D R:
c
spite of the constant duty r a t i o base d r i v e .
Since the t r a n s i s t o r c o l l e c t o r t u r n - o f f i s
delayed a f t e r the base t u r n - o f f drive by the
storage time, a possible explanation of the
effect i s that the storage time i s being
modulated by the l i n e ac s i g n a l . This could
occur because the line ac signal modulates the
current carried by the t r a n s i s t o r during the ontime, and the storage time i s dependent upon the
c o l l e c t o r current to be turned o f f .
In an attempt to e s t a b l i s h a quantitative
model of t h i s e f f e c t , an obvious s t a r t i n g point
i s an expression for storage time t as a func­
tion of the c o l l e c t o r current I ç to be turned
o f f , and of the base drive c o n d i t i o n s . From
well-known charge-control c o n s i d e r a t i o n s , such
an expression i s [2]
s
φ
OS
_c_.
rv
2
'
Fig. 1 1 . Reduced version of averaged model
of F i g . 1 0 for determination of the line
describing function v / v , without input
f i l t e r ; R i s the total effective s e r i e s
resistance.
Β2
^1
i J — n ^ j
Σ
T M1
S
g
T
f i l t e r whose Q-factor i s determined (in part)
by the total effective series resistance R j ,
defined in Eq. ( 9 ) and in F i g . 11 as the sum
of the various p a r a s i t i c resistance elements
shown. C l e a r l y , R-j and Ro increase the
effective l o s s and lower the Q.
The o r i g i n a l motivation for the model
derivation including s t r i c t continuity of one
of the variables in an averaged product, which
leads to the appearance of R] and Ro, was
to explain measured Q factors in both the line
and control describing functions that were
s i g n i f i c a n t l y smaller than those predicted
in the absence of R] and Ro. However, although
the improved model provided a q u a l i t a t i v e
change in the r i g h t d i r e c t i o n , the
quantitative lowering of the Q caused by the
addition R-j and Ro was i n s u f f i c i e n t to explain
the observed r e s u l t s . An entirely different
e f f e c t , discussed in the next s e c t i o n , was
found to be the cause of the observed low Q.
+
(Π)
in which τ i s the base c a r r i e r l i f e t i m e in the
saturated on" c o n d i t i o n , Iß] i s the forward
base current j u s t before t u r n - o f f , Iߣ i s the
turn-off (reverse) base d r i v e , and 3 i s the
active current gain for the c o l l e c t o r current
I q at the end of the storage time.
For typical
t u r n - o f f overdrive such that Iß2 >> I c / 3 the
log may be expanded to give
9
t
s
^ t
so
31
(12)
B2
where
x *n(l + I / I )
(13)
so
Hence, for constant base turn-on and t u r n - o f f
drive currents, the storage time decreases
l i n e a r l y with increasing l to be turned o f f .
s
B 1
B 2
Q
The next step i s to incorporate t h i s
r e s u l t into the duty r a t i o r e l a t i o n s h i p . I f the
base i s driven with duty r a t i o d so that the
on-drive i s present for an interval T d of
the switching period T , then the c o l l e c t o r w i l l
remain "on" for an interval T d given by
B
s
B
s
s
3.
MODEL EXTENSION FOR STORAGE TIME
MODULATION EFFECT
When a measurement of the ac l i n e
describing function i s made in the tappedinductor boosf converter shown in F i g . 1 , an
ac v a r i a t i o n v i s superimposed on the line
voltage V and the t r a n s i s t o r switch i s
driven from a modulator in such a way that the
base drive i s turned on and o f f with constant
(dc) duty r a t i o , without any control signal ac
modulation. As described above, such measurements showed a Q-factor s i g n i f i c a n t l y lower than
could be accounted for by reasonable values of
known p a r a s i t i c l o s s r e s i s t a n c e .
V • Vb
+
so that
B
d
T
+
(14)
*s
SO
(15)
where
3 l
g
M
g
B2 s
T
(16)
i s a "modulation parameter" that describes how
the c o l l e c t o r duty r a t i o d in Eq. ( 1 5 ) i s
affected by the c o l l e c t o r current I q .
In the context of the tapped-inductor
boost converter of F i g . 1 , the c o l l e c t o r
current to be turned o f f i s the inductor
current in the interval T d , namely
i = n „ i (Fig. 2).
In general, the base
drive duty r a t i o d has dc and s m a l l - s i g n a l ac
components dg = Dg + âo that contribute to the
corresponding components
= I + i , so t h a t ,
s
However, in the course of such measurements, i t was noted that the t r a n s i s t o r switch
duty r a t i o at the c o l l e c t o r was in f a c t being
modulated by the injected l i n e ac s i g n a l , in
s
£
B
£
Ä
PESC 75 RECORD—69
from Eq.
d =
(15),
so
D
d
M
B
" I
(17)
M
The dc and s m a l l - s i g n a l ac terms in the
above equation represent respectively the dc
and s m a l l - s i g n a l ac c o l l e c t o r duty r a t i o s D and
α that were employed in the development of the
continuous model of F i g . 10. A l l that i s
necessary, therefore, to account for the
c o l l e c t o r storage time modulation e f f e c t i s to
s u b s t i t u t e the above expressions wherever D and
α occur in the model of F i g . 10 and the
associated equations (1) through ( 7 ) .
The dc
s u b s t i t u t i o n represents merely a small o f f s e t
in the dc duty r a t i o , and i s of no q u a l i t a t i v e
concern.
In the ac s u b s t i t u t i o n , the term of
importance i s that in i £ , so that the f i v e ê and
3 generators of F i g . 10 and Eqs. (1) through
(5) each gains an additional generator as shown
e x p l i c i t l y in F i g . 12, in which
(18)
^Ml
R
M2
~
(19)
Vc^M
χ
χ
Μ
χ
1
x
£
Second, i t i s seen that each of the R^
dependent generators in F i g . 12 i s proportional
to the ac current flowing through i t , and can
therefore be d i r e c t l y replaced by the corres­
ponding r e s i s t a n c e .
The r e s u l t i n g model i s shown in F i g . 13,
in which^the K-j and K2 generators are dropped
and the e generators and R^ resistances are
condensed to
e = θ
ί
+ e
+ e
2
(23)
3
n [(D-D')(n -l) R -("x -l) u
2
^M3
represents the difference between the
c o l l e c t o r and base dc duty r a t i o s due to the
influence of the c o l l e c t o r dc current η Ι ^
upon the storage time. For normal designs t h i s
difference w i l l be s u f f i c i e n t l y small that
η Ι ^ / Ι << 1 . I t then follows from Eqs. (21)
and (22) that K-j << 1 and K2 << 1 , except
perhaps for extreme conditions where η >> 1
(very low inductor tap r a t i o ) , or D << 1
(approaching unity duty r a t i o D ) . In F i g . 12,
the ac current K-jî^ i s summed with the ac
current D i at point A ] , and the current
K2?
£ i s summed with
at point A 2 ; therefore,
to the extent that Κ ] , K2 << 1 both the K-j and
the K? generators in F i g . 12 have n e g l i g i b l e
effect and can be omitted from the model.
x
x
2
R
s
R
K
W
- d-(R || R ) 3 I / I „
c
(20)
A
(21
)
M
=
R
M1
+
R
M2
+
R
(24)
M3
Some reduction and s i m p l i f i c a t i o n in these
expressions can be achieved by comparison of
the contributing terms.
From Eq. (10), V =
V = D ' I R and i t can then be seen fron^Eqs. ( 2 ) ,
(3) and (19), (20) that the r a t i o s e / e 2 and
M3^ M2 are. on the order of the r a t i o between
c
Ä
— I /I
D' r M
(2
|K,i
3
R
êi
i.
)
e
3
ê
2
/
R
DrixRw+D'Rv RMI RM2+RM3
\Dn +D'
+
Λ
v
Iö
x
Φ.
A
Φ,
Φΐ
D = Drix+D'
F i g . 12. Extension of model of F i g . 10
to include generators that represent
the t r a n s i s t o r storage-time modulation
effect.
x
I:D
In F i g . 12, the e and j generators are s t i l l
given by E q s . (1) through ( 5 ) , with άβ s u b s t i ­
tuted for ά. However, now that the c o l l e c t o r
storage modulation e f f e c t has been e x p l i c i t l y
accounted for in the model, i t i s more con­
venient merely to drop the sub Β and to
redefine D and α to refer to the base drive duty
r a t i o rather than to the c o l l e c t o r duty r a t i o ,
so that the small difference between D and D
i s i m p l i c i t l y ignored and Eqs. (1) through (5)
remain applicable as they stand.
ß
Some further manipulation of F i g . 12
leads to a simpler and more useful model.
F i r s t , i t i s noted from Eq. (17) that η Ι / Ι
χ
70—PESC 75 RECORD
£
I,+î,
F i g . 13. Complete continuous, dc and lowfrequency s m a l l - s i g n a l a c , averaged model
for the tapped-inductor boost converter
of F i g . 3, including the "modulation
resistance" RM which i s present only
in the ac model; D and 3 refer to duty
r a t i o at the t r a n s i s t o r base.
some combination of the p a r a s i t i c resistances
and the load resistance R; consequently, the
contributions êo and R
can be dropped in
Eqs. (23) and (24).
Then, from Eq. (10) with
a s i m i l a r degree of approximation, V =
( D / D ' ) V and the remaining terms in Eqs. (23)
and (24) can be combined to give
M 3
X
Μ
D':l
x
S
3 l
:D * R
B2 s
T
The f i n a l model i s a continuous, averaged
model for the tapped-inductor boost converter
with input f i l t e r , including the t r a n s i s t o r
switch storage-time modulation e f f e c t accounted
for by the resistance R m .
I t i s v a l i d for
both dc and s m a l l - s i g n a l ac with the proviso
that the resistance R ^ i s to be included only
in the ac model and not in the dc model. This
i s necessary because of the step by which the
model of F i g . 13 was obtained from that of
F i g . 12: the total current through the R
generators of F i g . 12 i s l £ \ » but
generators^are functions only of the ac
component î . In the f i n a l model of F i g . 13,
the resistance R ^ i s enclosed in an oval as a
reminder that i t i s to be included only in the
ac model.
M
+ 1
t h e
R| = DD'(n -l)[(n -l)(R +R ) + n R - R ,
x
D V =D V
x
g
X
S
The storage-time modulation e f f e c t i s
manifested through two parameters: f i r s t , the
parameter Ijvj defined in Eq. (16) i s determined
by t r a n s i s t o r internal properties T and e , and
by the switch input c i r c u i t drive conditions
described by the period T and the t u r n - o f f
base current Iß2> second, the parameter RM
which, by Eqs. (18) through (20) and (23), i s
inversely proportional to I^j and i s a function
also of the switch output c i r c u i t , namely the
boost converter element values and operating
conditions.
4.
MODEL REDUCTION AND EXPERIMENTAL VERIFICATION
Although the q u a l i t a t i v e nature of the dc
c h a r a c t e r i s t i c V as a function of Vg and D and
of the ac line describing function v / V g are
obvious from the model of F i g . 13, the^nature
of the ac control describing function ν / α i s
not so obvious because the d r i v i n g signal^enters
through the three separate generators ê ,
and
J Understanding and interpretation of t h i s
c h a r a c t e r i s t i c i s therefore f a c i l i t a t e d by
some further manipulation and reduction of the
c i r c u i t of F i g . 13 for some special cases.
2
I f the input f i l t e r i s omitted, the
generator becomes mmaterial, and one step in
reduction of the model of F i g . 13 for c a l c u l a tion of the control describing function may
be accomplished as in F i g . 14, in which the
u
x
D' R<
2
w
DV=D V
C
n / _y_
D I
D
M
TV
/3I T
~c
D'
s
=
x
x
B2
2
S
F i g . 14. P a r t i a l l y reduced version of
averaged model of F i g . 13 for determination
of the control describing function v / d ,
without input f i l t e r .
D and D' ideal transformers have been
eliminated by r e f l e c t i o n of the elements of
the l e f t and r i g h t loops into the center
loop. The c i r c u i t of Fig.14 contains both the
dc and ac models, and certain resistance
combinations of i n t e r e s t are i d e n t i f i e d in the
f i g u r e : R^ i s the ac resistance to the l e f t
of J2» T
'
total e f f e c t i v e dc s e r i e s l o s s
r e s i s t a n c e , and Rj. i s the total e f f e c t i v e ac
series damping r e s i s t a n c e . The d i s t i n c t i o n
between Rt and Rj occurs because the s t o r a g e time modulation resistance R|v| contributes to ac
damping, but does not contribute to l o s s
because i t i s absent in the dc model.
x
1 S
t
i e
A f i n a l reduction of the model i s made in
F i g . 15, in which the two remaining independent
D'(V+v)
I*H
Rj
Φv
>D"R
D )
E E - ( R , + sL)j
2
=^(1+^)
*
s
s
s
d
RM =
R
I t i s seen from F i g . 13 that i n c l u s i o n
of the storage-time modulation e f f e c t leads
to a modification in the model that i s at l e a s t
p o t e n t i a l l y capable of explaining the observed
properties: the appearance of the ac
resistance Rjvj lowers the Q of the i m p l i c i t
f i l t e r c h a r a c t e r i s t i c , and the f a c t that the
effect i s represented only by a resistance
confirms that no other property of the model
i s affected.
x
R = DD (R + R I!R)
2
C
_ n
C
Vd
D R
/ 2
x
D'V = D'V
C
D V =D V
x
g
x
s
D s D n + D'
x
x
F i g . 15. Fully reduced version of model of
F i g . 14, showing appearance of r i g h t h a l f plane zero ω ; Rj i s the total e f f e c t i v e
dc series loss r e s i s t a n c e , and R^ i s the
total e f f e c t i v e ac s e r i e s damping r e s i s t a n c e .
ά
generators are combined into a s i n g l e generator
given by
v
(R
dl
A
+ sL)J
(27)
2
With s u b s t i t u t i o n for e and j
from Eqs. (25)
and ( 5 ) , together with the dc r e l a t i o n
V = D ' I R from Eq. (10), Eq. (27) becomes
2
£
v
dl
— - (R
+ sL)
Vd
(28)
D R
, 2
Again with neglect of the r a t i o of p a r a s i t i c
resistances to the reflected load r e s i s t a n c e ,
the term in R may be dropped so that
£
(29)
Vd
D
PESC 75 RECORD—71
which i s a l s o shown in F i g . 15, where
D R
, 2
(30)
ing action and which a l s o changes with D. The
pole-zero pattern in the s-plane i s therefore
as shown in F i g . 16, in which the pole p a i r and
the zero ω are in the l e f t half-plane and the
zero ω i s in the r i g h t h a l f - p l a n e . The
arrows indicate motion with increasing D.
It
i s i n t e r e s t i n g to note that
i s in the r i g h t
h a l f plane because of the subtraction of tfve
contribution of the generator J2 from that of
the generator e in the s i n g l e generator Vçn in
F i g . 15.
ζ
The response of the ac output voltage ν
to the e f f e c t i v e d r i v i n g voltage v i in
F i g . 15 i s simply that of the l o s s y low-pass
filter.
In terms of s u i t a b l y normalized
q u a n t i t i e s , the r e s u l t for the overall control
describing function v / d i s
d
n
ν
x 0 s/u )0+sA> )
+
a
z
(31)
Vd
where
ζ _ Q
(32)
c
(33)
ω
a
ο
/
ά
Experimental v e r i f i c a t i o n of various
aspects of the continuous model of F i g . 13 have
been made. The f i r s t objective was to confirm
aspects of the model related to the s t o r a g e time modulation resistance R^. The c i r c u i t of
F i g . 17 was constructed, which corresponds to
that of F i g . 1 with the input f i l t e r omitted.
D'\(34)
^ R = 0.32i}
L=6.0mh
b
-Oh
remote,
sense
M ?
1_
Q'
(35)
D'
power
supply
and in which
J_
"o
= π L
(36)
= ~h
Q
/Ce
t
2
t
R
(37)
t
(39)
(38)
are normalized parameters related to the o r i g i n a l
elements in F i g . 3.
In the above r e s u l t s , terms
in the r a t i o R^/D'^R have been neglected com­
pared to unity.
Equation (31), referring to the model
of F i g . 15, shows t h a t , for the tapped-inductor
boost converter of F i g . 1 without input f i l t e r ,
the control transmission frequency response i s
q u a l i t a t i v e l y the same as previously obtained
for the simple boost converter in [ 1 ] ,
That
i s , the response i s characterized by a low-pass
f i l t e r whose corner frequency ω ' and peaking
f a c t o r Q' both change with dc duty r a t i o D, a
zero ω due to R which i s constant, and a
negative zero ω which r e s u l t s from the s w i t c h 0
ζ
1
c
9
s - plane
1 R =374^
B
VBI=TVB2F5.7V
F i g . 17. Experimental tapped-inductor boost
converter without input f i l t e r .
Base
drive duty r a t i o ac modulation i s imposed
by v - I , and the r e s u l t i n g c o l l e c t o r duty
r a t i o ac modulation i s monitored at V2In the c i r c u i t of F i g . 17, the inductor
consisted of 100 turns of #20 wire of
resistance R = 0.14Ω to the n = 2 t a p , plus 100
turns of #24 wire of resistance R^ = 0 . 3 2 Ω
wound on an Arnold A930157-2 MPP μ = 125 t o r o i d .
Independent measurements showed that the
inductance at typical dc current levels was
L = 6.0mh. The capacitor C was a combination
of s o l i d tantalums which independent measure­
ments showed to have a capacitance C = 45yf
and esr = 0.28Ω. The 2N2880 power t r a n s i s t o r
was switched through a base resistance Rß =
3 7 4 Ω between voltages +Vßi = +5.7v and - V £ =
-5.7v by a modulator that provided f i x e d frequency, t r a i l i n g - e d g e modulation controlled
by dc and s m a l l - s i g n a l ac input voltages V-j
and v-j.
a
x
B
χ
/
increasing D
\
A preliminary experiment was done to
measure the modulation parameter I^j determined
by the switch t r a n s i s t o r and i t s input c i r c u i t .
Combination of E q s . (12) and (16) shows that
the "normalized" storage time i s a l i n e a r
function of I :
r
so
(40)
M
F i g . 16. Pole-zero pattern in the s-plane
for the control describing function v / d
obtained from the model of F i g . 15.
72—PESC 75 RECORD
From the waveforms of F i g . 2, i t i s e a s i l y seen
that I Q , which i s the current in the t r a n s i s t o r
j u s t before t u r n - o f f , i s given by
V-V
D'T
c
(41)
From F i g . 15, V % ( D / D « ) V and I %
( D / D ' R ) V > s o the above equation reduces to
X
S
Ä
2
X
S
_ "xVs/ ,
1
D'
"X
\R
2
D D
'
2 T
S\
(42)
2D L /
Measurements of storage time t , taken d i r e c t l y
from an o s c i l l o s c o p e , were made f o r various
combinations of V , R and D in the c i r c u i t of
F i g . 17 (without the speedup capacitor Cß)
with switching frequency f = 1/T = 10kHz
and n = 1 (simple boost converter c o n f i g u r a t i o n )
Resulting data points of t / T v s . I Q from
Eq. (42) are shown in F i g . 18, from which the
measured slope gives Iyi = 40 amps.
s
s
s
S
x
s
-
^
1
s
= 0.0215
s
0.020
-without
C
speedup capacitor Cß = 0.0012yf. As expected,
t h i s reduces the storage time but does not
a l t e r i t s rate of change with Ι ς , so that the
modulation parameter I|vj remains the same.
B
0.015
I
4 0
M
x
x
As w i l l be seen, the l a r g e r part of the
e f f e c t i v e damping resistance R. i s the modula­
t i o n resistance R^; the e f f e c t i v e l o s s
resistance Rj i s therefore inaccurately deter­
mined from R t , and i s a l s o inaccurately
determined by c a l c u l a t i o n from the various
p a r a s i t i c resistances since they themselves
are not accurately known.
amps
I t happens that the e f f e c t i v e l o s s
resistance can be determined d i r e c t l y by a d i f ­
ferent measurement in the c i r c u i t of F i g . 17.
As i n d i c a t e d , the t r a n s i s t o r c o l l e c t o r wave­
form, in the presence of v-j, i s modulated both
in amplitude and duty r a t i o .
I f t h i s wave­
form i s put through a l i m i t e r , the output has
constant amplitude and the same duty r a t i o
modulation.
I f the l i m i t e r output i s applied
to the analyzer i n p u t , t h e ^ n a l y z e r narrow-band
voltmeter gives a reading V2 proportional to
the duty r a t i o modulation. Hence, V2 i s
proportional to the actual c o l l e c t o r duty r a t i o
modulation, which i s d i f f e r e n t from the base
drive duty r a t i o modulation becuase o f the
s t o r a g e - t i m e ^ o d u l a t i o n e f f e c t , and so a mea­
surement of v / V 2 in the c i r c u i t o f F i g . 17 gives
the control describing function that would
e x i s t i f there were no storage-time modulation.
The frequency response of t h i s c h a r a c t e r i s t i c
would therefore be predicted by the model of
Fig. 15 with R|vj absent, so that a d i r e c t
measurement of Rj can be obtained.
0.010
0.005
Fig. 18. Determination o f the "modulation
parameter" 1^ by o s c i l l o s c o p e observation
of the c o l l e c t o r storage time t in the
c i r c u i t of F i g . 17 ( n = 1 ) .
s
x
As a matter of i n t e r e s t , the t r a n s i s t o r
parameters T and 3 may also be deduced from
these r e s u l t s . From F i g . 18, the v e r t i c a l
axis intercept i s t / T c = 0.0215, so t
=
2.15ysec. From Eq. ( 1 3 ) , x = t / i n ( ] + l ^ / l
A group of measurements was then made
for the purpose of exposing the e f f e c t of
the storage-time modulation resistance R^ upon
the e f f e c t i v e damping of the f i l t e r character­
i s t i c in the control describing f u n c t i o n . The
c i r c u i t of F i g . 17 was used, reduced t o the
simple boost converter configuration with
n = 1 , so that D = D n „ + D' = 1 f o r a l l D.
A Hewlett-Packard 302A Wave Analyzer in the
BFO mode was used as an o s c i l l a t o r and
automatically tracking narrow-band voltmeter.
The o s c i l l a t o r output provides the ac control
s i g n a l v-j, and the voltmeter reads the ac
output voltage v. There i s no s i g n a l frequency
dependence in the modulator, so that the ac
duty r a t i o α at the modulator output i s
proportional to v-j, independent of frequency.
The measurements o f the control describing
function v/v-j should therefore have a
frequency response predicted by v / d from the
model of F i g . 15.
s
S 0
0
l
s o
r
S 0
s
in which
I i
= (V i
B
B
-
VDC)/R
B
S
2
)
=T3ma and"
Iß2
( B2 + B F ) / ^ B
' ' > where VßE % 0.7v
i s the forward base-emitter threshold voltage
of the t r a n s i s t o r , which leads to T = 3.8ysec.
=
V
=
V
m a
s
Then,
from Eq.
(16),
3 = T IM/IB? S
T
S
=
9
0
-
Also as a matter of i n t e r e s t , data are a l s o
shown in F i g . 18 f o r base drive including the
A measurement of the c h a r a c t e r i s t i c
| v / v o J made in t h i s way on the c i r c u i t of F i g .
17 with η = 1 i s shown as curve (a) in F i g .
19. The measurement conditions were f = 20kHz,
χ
s
D = 0.25,
V = 50v,
and R = 2 4 0 Ω .
From Eq.
(36),
the normalizing frequency i s f = ωρ/2π = 310Hz,
and from Eq. (32) the e f f e c t i v e f i l t e r corner
0.75 χ 310 = 230Hz.
frequency i s f
= D'fo As seen from curve (a) in F i g . 19, the measured
0
r
PESC 75 RECORD—73
ι
1
JF Q ' = I 3 d b
to the switching period T , I = 20 amps under
the condition f = 20kHz for curve (b) in F i g .
19, and then R = V / I = 50/20 = 2.5Ω. With
use of the value Rj = 1.2ω already found, the
total e f f e c t i v e s e r i e s damping resistance in
the model of F i g . 15 i s R = R^j + R T =
2.5 + 1.2 = 3.7Ω, which from Eq. (3/) gives
Q = i o L / R = 11.4/3.7 = 3.08.
Then, from
Eq. (35) with Q = 2 1 , Q = 4 1 , and D = 0.25,
the t o t a l e f f e c t i v e peaking factor i s
Q' = 1.94 -> 6 db. The d i r e c t measurement of
curve (b) shows a Q' of about 5db, so that
the presence of the modulation resistance
R|vj = 2.5Ω in the model of F i g . 15 accounts
both q u a l i t a t i v e l y and q u a n t i t a t i v e l y for the
considerable lowering of the peaking factor
by about 8db.
s
M
s
M
M
t
db
t
0
t
L
c
Another r e s u l t due to R^ i s a change in
e f f e c t i v e f i l t e r peaking factor Q with change
of voltage operating level or of switching
frequency, which would not occur in the absence
of the storage-time modulation e f f e c t .
In the
simple boost converter with η = 1 , Rjvj =
V/ÏM
s V / ^ B 2 s and i s therefore proportional
to both V and t = 1/T .
Curve (c) in F i g .
19 shows the control transmission c h a r a c t e r i s t i c
under the same conditions as for curve (b)
except that V i s halved, which s i m i l a r l y scales
a l l the dc v a l u e s , including the output voltage
V.
Hence, for I M = 20 amps, Rjvj i s halved to
1.25Ω and Rj stays the same at 1.2Ω, so that
Rt = 2.45Ω. With a l l other q u a n t i t i e s the
same, Q' = 2.90 + 9db which agrees well with
the measured value of 8db in curve ( c ) .
A l t e r n a t i v e l y , i f V i s restored to 50v but
the switching frequency i s halved to f = 10kHz,
I
doubles to 40 amps and Rjvj = 1.25Ω a g a i n ,
so that Q remains at 9db, which i s the value
measured in curve (d) in F i g . 19.
1
χ
=
T
T
s
I
ι
10Hz
20
Fig.
ι
ι
ι
I
ι
ι
4 0 6 0 80100
ι
ι
I
I kHz
s
19. Exposure of e f f e c t of modulation
resistance RM on the control describing
function peaking f a c t o r Q' in the c i r c u i t
of F i g . 17 with n = 1 and D = 0.25;
(a) determination of R t = 1.2Ω from
observed Q' = 13db; ( b ) , ( c ) , (d) predicted
lower Q' when various calculated values
of RM are i n c l u d e d , and observed data
points.
x
corner frequency i s e s s e n t i a l l y equal to t h i s
predicted value. Curve (a) a l s o i n d i c a t e s that
the peaking f a c t o r i s Q' = 13db -> 4.47.
From
Eq. (35), the total e f f e c t i v e damping i s due to
Qt, Qi » and Q ; for the present case, from Eqs.
(38) and (39), Q = u ) L / R = 11.4/0.28 = 41
in which R = 0.28Ω i s the capacitance e s r , and
Q = R/cu L = 240/11.4 = 2 1 . Then, with use of
the measured Q' = 4.47, Eq. (35) may be solved
to give Q = 9.39.
F i n a l l y , from Eq. (37) with
R replaced by R t because Rjvj i s absent under
the conditions of t h i s measurement, Rj =
g L / Q = 1.2Ω. This i s a reasonable v a l u e ,
since for n = 1 R i s the total inductor
resistance R = R
k
0.46Ω, which leaves
1.2 - 0.46 % 0.7Ω for the remaining p a r a s i t i c
resistances.
c
c
0
c
c
L
0
t
t
W
t
x
u
+ R
=
a
With R thus d i r e c t l y determined from the
response v/v2> attention can be returned to the
actual control describing function v/v-i in
which the modulation resistance R^ i s present
in the model of F i g . 15. Curve (b) in F i g . 19
shows I v/v-j I for the c i r c u i t of F i g . 17 under
the same conditions as for curve ( a ) , namely
f = 20kHz, D = 0.25, R = 240Ω. For n = 1 ,
the expression given by Eq. (26) i s R^ = V/1|vj =
t V/3IB?T .
A s previously determined by
independent measurement, Ijvj = 40 amps for
f = 10kHz; therefore, since 1^ i s proportional
T
s
S
x
S
s
74—PESC 75 RECORD
S
s
M
1
Attention i s now turned to measurements
under conditions that expose the r i g h t h a l f plane zero O K in the model of F i g . 15.
Still
for the simple boost converter configuration
η = 1 , measurements of the control describing
function ν/ν·| were taken on the c i r c u i t of
Fig.
17 under the condition f = 20kHz, D = 0 . 6 ,
V = 25v, and R = 162Ω. ^ a t a points of both
magnitude and phase of v/v-j are shown in F i g .
20. The phase measurements were a l s o taken
with the Hewlett-Packard 302A Wave Analyzer,
by techniques that have been described e l s e ­
where [ 3 ] .
The control describing function
predicted by the model of F i g . 15 i s obtained
as f o l l o w s .
χ
s
The normalizing frequency i s f = 310Hz,
and from E q . (32) the e f f e c t i v e f i l t e r corner
frequency i s f ' = D ' f = 0.4 χ 310 = 130Hz.
From Eqs. (38) and ( 3 9 ) , Q = 41 and Q, = 14.
At f = 20kHz, Ijyj = 20 amps and the correspond­
ing Modulation resistance i s Ry| = V / I ^ =
25/20 =1,25ω. With the p a r a s i t i c l o s s
resistance s t i l l Rj = 1.2ω, the total
e f f e c t i v e s e r i e s damping resistance i s R =
R M + Rj = 2.45Ω, so from Eq. (37) Q = 4 . 6 1 .
men,
from Eq. (35), the e f f e c t i v e peaking
factor i s Q' = 1.37 -> 3db. Next, from Eq.
ç
0
0
c
t
t
ι
1
I
'
.•eQ'=3db
1
l l |
1
'
ι
' I
ι
i i |
ι
_
magnitude^
„ ΝΛ'=Ι30Ηζ\
k o
— predicted
0
" Ι ν 1.
--
--90°
\
NR.
\°
--180°
lOdb
#
\
\·
Ο
°
phase'-
--270°
~°
y /
ο ο
u
ο
" "0*73 ΰ" υ
Ο
-
— predicted
° measured
fA
(
b
)
.
/
Q
f = 20 kHz
-720 Hz
=
s
/
1
1—
*»
s
\
·
. 0'=2dbSj
•
\
\ ·
\
·
V = 25v:
R =4.on
\ ·
M
- I
f = 20 kHz
V = 25v
M
-
V.·
•
t
( R absent)
—ι
•
•
( a )
1^1
• measured
τ "
l l |
^ Q'=9db
•
\
lOdb
\ ·
(c)
N
X
" •«&yQ =-2db\»
/
—· ^ .
simple boost : n = 1
x
\f
f = 13 kHz
,
z
1
-
V=50v:
r = s.on
,
Λ
î \
m
V
v,
( R present)
\
f' =230Hz\*
\
0
M
ι I
100 Hz
ι
,
Ι
1
Ι
ι 1
10 kHz
Ι
I kHz
F i g . 20. Exposure of effect of r i g h t h a l f plane zero ω on the control describing
function in the c i r c u i t of F i g . 17 with
n = 1 and D = 0.6:
the maximum phase lag
exceeds 180°.
\·
\
tapped boost: n = 2
x
V
\
ά
f =-5.8 kHz"
a
V
1
I
,
x
(33), f = Q f = 41 χ 310 = 13kHz, and from
Eq. (347 (with n = 1 and D = 1) f =
- D Q , f = - 0 . 4 χ 14 χ 310 = -720Hz.
Thus,
a l l the parameters in the control frequency
response of Eq. (31) are known, and the
corresponding predicted magnitude and phase
asymptotes are shown in F i g . 20 for comparison
with the measured data p o i n t s . I t i s seen
that agreement i s quite good; in p a r t i c u l a r , i t
may be noted that the phase exceeds 180° lag
as i s expected from the r i g h t half-plane zero
f ; the phase f a i l s to reach 270° l a g because
of the l e f t half-plane zero ω due to the
capacitance e s r .
c
x
, 2
x
a
2
0
4
ζ
Further s e r i e s of measurements were made
on the c i r c u i t of F i g . 17 in the tapped-inductor
condition with n = 2. A f i r s t set of data
points for the control describing function v/v2
was taken under the conditions f = 20kHz,
D = 0.25, V = 25v, and R = 240Ω, and the r e s u l t s
are shown in curve (a) in F i g . 2 1 . A S in the
previous example with n = 1 , the v/V2
c h a r a c t e r i s t i c does not include the effect of
RM, so that a direct measurement of the p a r a s i ­
t i c loss resistance may be obtained. From curve
(a) in F i g . 2 1 , the measured effective peaking
factor i s Q' = 9db -> 2.82, and Q = 41 and Q =
21 as in the example with n = 1 .
Therefore,
from Eq. (35), Q. = 4.9 and hence R x = 2.3Ω.
Since the normalizing frequency i s f = 310Hz,
the effective f i l t e r corner frequency i s f ' =
D ' f = 230Hz.
x
Λ
s
λ
x
c
L
x
0
0
0
Λ
The actual control describing function
v / V ] was then measured under the same c o n d i t i o n s ,
so that the effect of RM was included, and the
r e s u l t s are shown in curve (b) in F i g . 2 1 . The
predicted value of R i s obtained from Eq. ( 2 6 ) ,
RM = n 2 v / D I :
for f = 20kHz, I = 20 amps as
before; for n = 2 and D = 0.25, D = D n + D' =
(0.25 ,x 2) + 0.75 = 1.25, so for V = 25v,
R = (22 25)/(1.25x20) = 4.0Ω. The predicted
value of the total effective s e r i e s damping
Λ
M
x
x
M
x
M
X
.
ι
s
M
x
x
,
,
1
- lι
1
1
100 Hz
0
i Ul l
,
L_J
I kHz
10
F i g . 2 1 . Exposure of effect of
and the
r i g h t half-plane zero ω on the control
describing function in the c i r c u i t of
F i g . 17 with n = 2 and D = 0.25:
(a) determination of Rj = 2.3Ω from observed
Q' = 9db; ( b ) , (c) predicted lower Q'
when various calculated values of R|v| are
included, and observed data p o i n t s .
x
resistance i s then R = R + R = 4.0 + 2.3 =
6.3Ω, which leads to Q = 1.81 and a t o t a l
effective peaking factor Q' = 1.22 ^ 2db,
in good agreement with the observed value in
curve (b) of F i g . 2 1 . From Eq. (34), the r i g h t
half-plane zero i s f = - ( n D ' 2 Q , / D ) f =
-(2 χ 0.752 χ 21/1.25)310 = -5.8kHz, a l s o in
good agreement with curve (b) of F i g . 2 1 .
t
M
T
t
a
x
x
As a further check on the model, the
previous set of measurements was repeated with
a l l dc levels doubled, so that V = 50v. The
r e s u l t s are shown in curve (c) in F i g . 2 1 ;
the only predicted change i s that Rjvj i s doubled
because V i s doubled, so Rjv| = 8.0Ω. Then,
Rt = 8.0 + 2.3 = 10.3Ω so Qt = 1.1 which leads
to Q' = 0.77 -> -2db, in good agreement with
curve (c) of F i g . 2 1 .
F i n a l l y , some sets of measurements of
the control describing function were made for
the generalized tapped-inductor boost converter
with an input f i l t e r .
The experimental c i r c u i t
with inductor tap at η = 2 i s shown in F i g . 22,
and the t r a n s i s t o r drive conditions and the
converter inductor and capacitor were the same
as in F i g . 17.
Independent measurements led
to the input f i l t e r element values L = 3.2mh,
C = 12yf, R = 3.5Ω including the capacitor
e s r . The switching frequency was 20kHz.
χ
s
s
5
Prediction of r e s u l t s from the c i r c u i t
of F i g . 22 can be made from the general con-
PESC 75 RECORD—75
I_s= 3.2mh
v
(R
dl
sL)J
+
A
(43)
2
(44)
i l •+ — Vd
x \
"a.
jt
D
and an additional generator v ^ r e s u l t i n g from
the presence of Z given by
$
\i2
F i g . 22. Experimental tapped-inductor boost
converter with input f i l t e r .
=
D
x
2 Z
s ( f c 3]
(45)
h)
+
From Eqs. (4) and ( 5 ) , t h i s generator can be
expressed as
tinuous model of F i g . 13. The dc and l i n e
transmission c h a r a c t e r i s t i c s are straightforward
and w i l l not be discussed further. The nature
of the control describing function i s not quite
so obvious, and understanding of i t s s a l i e n t
q u a l i t a t i v e form i s f a c i l i t a t e d by some further
steps in reduction of the model in order to f i n d
a simple equivalent d r i v i n g generator propor­
t i o n a l to the ac duty r a t i o d.
2
η
V
νd2
.ο = D / Z
— — 5 - d
χ s
,2
Λ
(46)
c
D
D
R
in which the dc relation I n = V/D'R has been
used. Hence, the total effective driving
generator in F i g . 24 i s
\2
Ί
7
Figures 23 and 24 show reduced forms of
the general ac model of F i g . 13 that are analogous
to those of F i g s . 14 and 15, but with retention
of the input f i l t e r whose effect i s represented
by i t s source impedance Z looking back into
U
ω
\ d ' /
ά
Vd
(47)
With s u b s t i t u t i o n for the frequency dependence
of the source impedance Z , Eq. (47) becomes
s
s
ν,
η
d _ χ
Rt
Rj
Vd
2
1 +
— - ( ^
D
κ
^Νβ)]
' * i(k) * s(k)
' -J
D'
(48)
where
(49)
F i g . 23. P a r t i a l l y reduced version of
averaged model of F i g . 13 for determination
of the control describing function ν / α , with
input f i l t e r .
The s a l i e n t features of the control
describing function in the presence of the input
f i l t e r can now be seen by inspection of the
reduced model of F i g . 24, and Eq. (48).
The source impedance Z goes through a maximum
value of about Q ^ /
t approximately i t s
resonant frequency ω , so that Vd goes through a
corresponding minimum value. In f a c t , the
minimum could a c t u a l l y be a null i f conditions
are such that
s
-AV^
RT
dl
Λ
d 2
= D / Z ( - j, + j
s
n Γ
x
s . /D \ Ζ Ί
c
2
x
5
2
R
s
a
s
5
V ,2
>D R
Rt
(±) v = e - ( R , + sL)7
(î)v
··
D (V + v)
Vd
D R>
2
D V = D'V
C
-±r D V = D X
X
0
D = Dn+D'
x
F i g . 24.
Fully reduced version of model of
F i g . 23, showing appearance of a minimum
or p o s s i b l y a null in the e f f e c t i v e d r i v i n g
s i g n a l v , in the neighborhood of the input
f i l t e r resonant frequency ω where Z
reaches a maximum.
d
δ
s
the power supply as indicated in F i g . 22. As
seen from F i g . 24, the total e f f e c t i v e d r i v i n g
generator Vç| c o n s i s t s of the generator v ^ ,
previously i d e n t i f i e d in F i g . 15 as
76—PESC 75 RECORD
This condition i s a function of dc duty r a t i o
D. For the numerical values in the experimental
c i r c u i t of F i g . 22, Q = 4.66 and Eq. (51)
predicts that a null should occur at D ^ 0.28.
Computer s o l u t i o n of the model of F i g . 24 with
Vd given by Eq. (48) showed that a null in
the control describing function v / d a c t u a l l y
occurred at D = 0.29.
Further i n s i g h t into
the r e s u l t s was obtained by computer s o l u t i o n
for the pole-zero locations of the control
describing function for t h i s value of D and
for another on either side of t h i s value,
namely D = 0.20, 0.29, and 0.5.
Numerical
values used were V = 14v f o r D = 0.20, 0.29
and V = 9.9v for D = 0 . 5 ; R = V / I with
s
s
s
M
M
IM = 20 amps; and the value Rj = 2.3Ω p r e ­
viously determined for the n = 2 converter
was used for a l l three values of D, even
though i t s value a c t u a l l y varies s l i g h t l y with D.
The r e s u l t s for the pole-zero p o s i t i o n s
are shown in F i g . 25. The l e f t half-plane zero,
lower-frequency complex pole p a i r , and the r i g h t
half-plane zero represent the basic response of
the converter e f f e c t i v e low-pass f i l t e r , with
the expected position-dependence upon D. The
higher-frequency complex pole p a i r and complex
zero pair represent the response due to the
input f i l t e r , whose p o s i t i o n s also depend upon
D. In p a r t i c u l a r , i t i s noted that the complex
x
s-plane
f = 20kHz i
300
8?
a,b,c
-13
kHz
-Ar-
s
200
100 kHz
b
0
-300-200-100 A
I
X f-
N
a
-o-
0 ΙΟΟΗζ I kHz 2
100 Hz
10 d b
a: D = 0 . 2 0 ,
V
b: D = 0 . 2 8 ,
V
c: D = 0 . 5 0 ,
V
s
s
s
=Ι4ν
= I4v
=9.9v
100 Hz
I kHz
10 k H z
F i g . 27. Computer-calculated magnitude and
phase of the control describing function for
pole-zero pattern (b) in F i g . 25, and data
points obtained from the experimental c i r ­
c u i t of F i g . 22, showing the null in the
magnitude response.
X
X
F i g . 25. Computer-calculated pole-zero pattern
in the s-plane for the control describing
function ν/α obtained from the model of
F i g . 24 for the c i r c u i t of F i g . 22. A
null occurs at D = 0.29 when the complex
zero p a i r crosses the imaginary a x i s .
magnitude:
—predicted •
-lOdb
measured
•0'
-90°
f =20kHz^.
s
-180'
f
-270°
= 2 0 kHz.
phase:
- 3 6 0 ° /
D = 0 . 2 0 , V , = 14 ν
s
—
predicted
°
measured
D = 0.50, V = 9.9v
s
10 db
100 Hz
1 0 0 Hz
I kHz
10 k H z
F i g . 26. Computer-calculated magnitude and
phase of the control describing function
for pole-zero pattern (a) in F i g . 25,
showing the minimum in the magnitude response.
I kHz
10 k H z
F i g . 28. Computer-calculated magnitude and
phase of the control describing function
for pole-zero pattern (c) in F i g . 25, and
data points obtained from the experimental
c i r c u i t of F i g . 22, showing the large phase
lag at high frequencies.
PESC 75 RECORD—77
zero p a i r crosses from the l e f t half-plane to
the r i g h t half-plane as D i n c r e a s e s , with the
expected null at D = 0.29.
The magnitude and phase p l o t s correspond­
ing to the three sets of computed pole-zero p o s i ­
tions are shown in F i g s . 26 through 28. Data
points d i r e c t l y measured in the c i r c u i t of F i g .
22 are also shown for two of the sets of condi­
tions.
I t i s seen that good agreement i s obtained
between the measured r e s u l t s and the prediction
of the model of F i g . 24. From a practical point
of view, the chief s i g n i f i c a n c e i s that a
minimum in the control describing function can
e x i s t as a consequence of the presence of an
input f i l t e r , and for a certain dc duty r a t i o
the minimum could become a n u l l .
I t i s note­
worthy that a null can occur in spite of f i n i t e
Q (nonzero R ) in the input f i l t e r . Thus, i f
such a converter were part of a r e g u l a t o r ,
normal internal adjustment of the dc duty ratio
could cause a null in the loop g a i n . Even
more s e r i o u s , values of dc duty r a t i o D greater
than that for which the null occurs lead to a
very large phase lag at high frequencies, much
larger than for smaller values of D. This i s
a consequence of the movement of the complex
zero p a i r from the l e f t half-plane to the r i g h t
h a l f - p l a n e . Therefore, very severe regulator
s t a b i l i t y problems could be experienced unless
great care i s exercised in the design.
s
?
measured values of the effective f i l t e r Q-factor
that were s u b s t a n t i a l l y lower than those
predicted by the o r i g i n a l model. However, i t
was found that t h i s refinement provided i n s u f ­
f i c i e n t quantitative c o r r e c t i o n , and that
instead an e n t i r e l y different effect was
responsible for these lower Q - f a c t o r s .
This new effect i s shown to be due to
storage-time modulation in the t r a n s i s t o r power
switch, in which the effective duty r a t i o
modulation at the c o l l e c t o r i s different from
the d r i v i n g duty r a t i o modulation at the base.
The model of F i g . 13 incorporates t h i s effect in
a modulation resistance R|v|, which i s shown
enclosed in an oval as a reminder that i t i s
to be included in the ac model only. The
s i g n i f i c a n c e of t h i s r e s u l t i s that the
modulation resistance Rjv| contributes to the
effective f i l t e r ac damping resistance but does
not contribute to the effective dc l o s s r e s i s ­
tance; consequently, the f i l t e r Q-factor i s
lowered, but the converter e f f i c i e n c y i s un­
a f f e c t e d , by the presence of the modulation
resistance R ^ . This r e s u l t maybe considered
a rare exception to Murphy's law, since the
storage-time modulation effect produces a
desirable damping effect without an associated
undesirable l o s s of e f f i c i e n c y .
Experimental measurements of both magnitude
and phase of the control describing function are
presented to verify the averaged model both
q u a l i t a t i v e l y and q u a n t i t a t i v e l y .
Experimental
conditions were chosen s p e c i f i c a l l y to expose
certain features of the model for individual
verification:
for the simple boost converter
of F i g . 17 with n = 1 , F i g . 19 shows the
damping effect due to the modulation resistance
RJVJ and F i g . 20 exposes the r i g h t half-plane
zero and associated excess phase l a g ; F i g . 21
shows the effect of Rjv| in the tapped-inductor
converter with η = 2.
x
5.
CONCLUSIONS
A s m a l l - s i g n a l , low-frequency, averaged
model for the tapped-inductor boost converter
with input f i l t e r has been developed and
experimentally v e r i f i e d .
The general model i s
shown in F i g . 13, and from i t the dc transfer
function and the ac line and control describing
functions can be obtained.
χ
Experimental r e s u l t s are also presented
for the control describing function of the
system of F i g . 22, a tapped-inductor boost
converter with η = 2 and with an input f i l t e r .
The corresponding averaged model i s shown in
F i g . 24, and the important feature i s that the
control describing function acquires a complex
p a i r of zeros that can move from the l e f t h a l f plane to the r i g h t half-plane as the dc duty
r a t i o D i s increased, as would happen during
normal adjustment in a practical closed-loop
regulator. The s i g n i f i c a n c e of t h i s i s
i l l u s t r a t e d in the magnitude and phase r e s u l t s
of F i g s . 26 through 28, in which the magnitude
plot has a minimum, in the neighborhood of the
input f i l t e r resonant frequency, which a c t u a l l y
becomes a null when the complex pair of zeros
l i e s on the imaginary a x i s . Furthermore, as
the pair of zeros moves into the r i g h t h a l f plane, the magnitude null retreats to a minimum
but the phase lag becomes exceedingly l a r g e .
Unless recognized and properly accounted f o r ,
t h i s e f f e c t could have disastrous effects upon
the s t a b i l i t y of a closed-loop regulator.
χ
The model of F i g . 13 i s obtained
p r i n c i p a l l y by manipulations of the c i r c u i t
diagrams rather than by a l g e b r a i c a n a l y s i s , so
that physical i n s i g h t into the s i g n i f i c a n c e of
the steps i s retained throughout the d e r i v a t i o n .
In the absence of an input f i l t e r , the ac l i n e
and control describing functions are
characterized by an effective low-pass f i l t e r
described by a p a i r of l e f t h a l f - p l a n e poles and
a l e f t half-plane zero; the control describing
function has in addition a r i g h t half-plane zero.
The p o s i t i o n s o f the poles and of the r i g h t
half-plane zero change with dc duty r a t i o D.
The method of derivation of the averaged
model i s a refinement of that described for the
simple boost converter in [ 1 ] ; the refinement
leads to a more accurate representation of the
p a r a s i t i c l o s s resistances in the model, and
was an attempt to explain experimentally
78—PESC 75 RECORD
The physical i n s i g h t into the nature of
the properties of a tapped-inductor boost
converter makes the averaged model a useful and
e a s i l y applied design t o o l . The method of
model derivation can be applied in a s i m i l a r
manner to numerous other c i r c u i t c o n f i g u r a t i o n s .
Thanks are due to Howard Ho and Alan
C a s s e l , graduate students of the C a l i f o r n i a
I n s t i t u t e of Technology, for t h e i r assistance
in constructing the experimental converters
and for the computer plots shown in F i g s . 26
through 28.
REFERENCES
[1]
G. W. Wester and R. D. Middlebrook,
"Low-Frequency Characterization of
Switched dc-dc Converters," IEEE Power
Processing and Electronics S p e c i a l i s t s
Conference, 1972 Record pp. 9-20; a l s o ,
IEEE Trans. Aerospace and Electronic
Systems, v o l . A E S - 9 , no. 3, pp. 376-385,
May 1973.
[2]
C. L. Searle et a l , Elementary C i r c u i t
Properties of T r a n s i s t o r s , SEEC v o l . 3,
p. 284; John Wiley & Sons, 1964.
[3]
R. D. Middlebrook, "Measurement of Loop
Gain in Feedback Systems," International
Journal of E l e c t r o n i c s , v o l . 38, no. 4,
pp. 485-512, April 1975.
PESC 75 RECORD—79