Minimisation of power losses in source-R-C
circuit with any (possibly nonlinear)
capacitor
E. Gluskin
In power electronics circuits, the capacitor, whose charging and discharging provides the energy transfer from a source to the load is,
usually, a linear one. It is interesting to consider also a nonlinear
capacitor having a smooth voltage-current characteristic, as, e.g. a ferroelectric capacitor, or a piecewise linear characteristic, as a switched
unit in which linear capacitors are switched. This can give a designer
an additional degree of freedom for optimisation of such a circuit,
first of all for minimisation of the power losses in it. The proof of
the reasonability to use a current (and not a voltage) source is provided
in terms of the variation principle that is widely used in classical mechanics. The use of the variational principle not just for optimisation of
some parameters (which is usual), but also for the qualitative
conclusion regarding the circuit structure – the use of the current
source – is another methodological point not to be missed.
Introduction: In power electronics, charging and discharging a capacitor
is the classical way of transferring energy from a source to a load ([1–5]
and references therein), and we have two relevant circuits – the charging
and the discharging ones. Since we consider here the minimisation of the
power losses, the charging circuit is our focus. Consider the S-R-C
circuit shown in (Fig. 1), including nonlinear capacitor [6,7] C, characterised, say, by a smooth charge-voltage characteristic v = ψ(q) which is
schematically shown in (Fig. 2).
R
S1
+
E(t)
–
+
S2
v
C
to the
load
–
Fig. 1 Power transfer circuit in which we are interested in the charging stage
The generally nonlinear capacitor C is characterised by a function
v = ψ(q), e.g. like that shown in Fig. 2. In our analysis, switch S2 is open
(non-conductive) all the time, and S1 is closed, becoming conductive
when the voltage v(t) on C equals v1. We assume that discharging the
capacitor into the load is totally associated with useful power consumption, i.e. the losses occur only in R, during the charging. We start with
the voltage source, but, considering the power losses in R, we shall come
to the necessity (reasonability) of passing from the voltage source to a
series source of constant current.
v
v =y (q)
(t2(mod T)) with some t1 and t2 prescribed, and, as usual, we can
speak about average powers of the energy transfer and of the energy
losses. The important requirement of partial charging is associated
with the requirement of sufficiently high frequency of the operation of
the device.
The average power transferred to the load through the capacitor is
q(t2 )
c(q)dq
(1)
PC = fWC = f
q(t1 )
where f is the frequency of the whole process, and WC is the portion of
energy received by the capacitor during single charging. This integral
expression directly follows from the basic thermodynamic (energy) definition of voltage as dW/dq.
Some notations: t1 and t2 – the given bounds (time instants) for the
interval of (partial) charging of capacitor C, in a T-periodic process;
t1 = t1(mod T), t2 = t2(mod T).
f = 1/T – frequency of the steady state operation;
v – voltage on C; q – charge accumulated in C;
ψ(.) – capacitor’s characteristic (v = ψ(q));
E(t) – voltage source
S – source of voltage or current (for instance S-R-C means a series
source-resistor-capacitor circuit);
S1, S2 – switches;
i(t) = dq/dt – the current charging the capacitor; also that of the series
current-source finally replacing E(t);
W – energy;
P – power.
Minimisation of power losses: We wish to minimise the average power
losses
t2 2
dq
dt
(2)
R
dt
t1
not only under the condition of v1 and v2 prescribed, but also under the
requirement that PC (the appropriately rewritten (1)),
t2
dq
(3)
c(q) dt
f
dt
t1
is prescribed.
According to (2,3) and the Lagrange multipliers method [8, 9] for the
constrained variation problem we can write for the function
L(q,
dq
dq
dq
) ; fR( )2 + lf c(q)
dt
dt
dt
(4)
the usual Euler-Lagrange variation equation. Let us recall that for
t2
dx
L(t, x, )dt
dt
t1
the Euler-Lagrange equation is [8]:
∂L d ∂L
=0
−
∂x dt dx
∂( )
dt
(5)
(here ‘x’ is our q). Using (4), we obtain (5) as
lf
q
Negative values are also permitted; ψ( − q) = ψq). Switched characteristics are also relevant because they can be approximated by
smooth characteristics.
We consider the partial-charging regime when the capacitor is
charged from voltage v1 > 0 up to a voltage v2, v1 < v2 < Emax. As
usual in power systems, the device is meant to be operated in some
steady-state (T-periodic) manner. Thus, v1 = v(t1(mod T)) and v2 = v
Techset CompositionLtd, Salisbury
(6)
However since
0
Fig. 2 A smooth nonlinear v = ψ(q) e.g. of ferroelectric capacitor ([6,7] and
references therein)
d c(q) dq
d2 q
d
− 2f R 2 − lf c(q) = 0
dq dt
dt
dt
d
d c(q) dq
c(q) ;
dt
dq dt
we finally obtain from (6) that
d 2 qopt
=0
dt 2
(7)
qopt(t) = A + Bt, t1 , t2
(8)
Thus
where constants A and B can be easily found, using the inverse
Doc: {EL}ISSUE/49-11/Pagination/EL20130006.3d
Circuits and systems
function ψ −1(.), from the boundary conditions:
Acknowledgment: The author is grateful to A. Ioinovici for a discussion.
−1
q1 = q(t = t1 ) = c (v1 )
and
q2 = q(t = t2 ) = c−1 (v2 )
(9)
Using (8) and (9), we have
iopt (t) =
dqopt
c−1 (v2 ) − c−1 (v1 )
=B=
t2 − t1
dt
E. Gluskin (Kinneret College on the Sea of Galilee (Jordan Valley),
Israel)
E-mail: gluskin@ee.bgu.ac.il
(10)
i.e. iopt (t) is constant. At the same time, one sees from the equation of
the charging-process circuit,
R
© The Institution of Engineering and Technology 2013
1 January 2013
doi: 10.1049/el.2013.0006
dq
+ c(q(t)) = E(t)
dt
that for q(t) ≡ qopt(t), Eopt(t) is not constant. If so, it is obviously reasonable not to try to adjust E(t) to qopt(t), but to replace the voltage source
by series source of the constant current given by (10).
Conclusions: We have shown that the use of a current source in a power
electronics circuit may be connected with the reduction of power losses,
and this conclusion is correct for different forms of the voltage-charge
characteristic of the capacitor that thus can be nonlinear. The variation
method is seen here to be a tool for a designer to consider the very structure of a circuit, not just to optimise some its parameters. The degree of
freedom in the creation of the capacitor’s characteristics in power transfer circuits should be interesting, because one can think not only about
analytical nonlinearities, but also about the use of switched capacitor
units. There is the hope that the present observation can be useful for
power electronics specialists and teachers.
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