IEEE Circuits and Systems Magazine - Q1 2020 - 36

V (~) = FT " v(t ) , = 1 (V0 (~ - ~ 0) + V0 (~ + ~ 0)),
2

(4)

where, due to the multiplication in time domain, the
spectral convolution of the FT of the cosine function
(represented by delta functions located at ! ~ 0) with
V0 (~) was considered. This spectrum represents a narrow-band signal around ! ~ 0 . When calculating the current that this voltage generates in a lossless admittance
Y (~) = jB(~), in view of the narrow band covered by the
excitation voltage, we can approximate the susceptance
function B(~) around ~ = ~ 0 as
B (~) . B (~ 0) + Bl(~ 0) (~ - ~ 0),

(5)

where Bl(~ 0) = [2B/2~]~ = ~0 . A corresponding approximation can be applied around ~ = -~ 0 . Performing the
inverse transform with these approximations then yields
i (t ) - - B (~ 0) v 0 (t ) sin(~ 0 t ) + Bl(~ 0) vo 0 (t ) cos(~ 0 t ), (6)
where vo 0 = 2v 0 /2t. Details of the underlying straightforward calculation are given in Appendix A.
Next, considering that the instantaneous p ow er delivered to the network can be obtained as
p(t ) = v(t ) i (t ), we observe that integrating p(t ) from
the moment the voltage starts building up, which we
choose to be t = 0, to a particular time t 0 provides the
energy that has been delivered to the network up to
this time and this energy always has to be greater or
equal than zero, i.e.

Appendix A: Inverse Transform of the Current

t0

w (t 0 ) =
The inverse Fourier transform of I (~) can be obtained
from the inverse Fourier transform of V (~) Y (~) =
jV (~) B (~), which, considering (4), reads
i (t ) =

+3

j
4r

#

-3

" V0 (~ - ~ 0) + V0(~ + ~ 0) , e j~t B (~) d~.

(14)

Using the earlier discussed narrow-band approximation
for the susceptance, we split the integral in two parts
yielding
i (t ) .

j
4r
+

+3

#

V0 (~ - ~ 0) {B (~ 0) + B l(~ 0) (~ - ~ 0)} e j~t d~

-3

j
4r

+3

#

V0 (~ + ~ 0) {B (-~ 0) + B l(-~ 0) (~ + ~ 0)}

-3

# e j~t d~.

(15)

Using6 B (-~) = -B (~) and B l(-~) = B l(~), we can rewrite the second integral by substituting ~ " -~. Considering further that 6 V0 (-~) = V 0) (~), we can write the
sum of both integrals in terms of the imaginary part of
one integral yielding
j
i (t ) .
2r
# Im )

+3

#

-3

(16)

Using the FT rules applying to frequency shifts and time
derivatives, the inverse transform is then readily obtained as given in (6).
6

The Fourier spectrum X (~) of a real valued function x (t ) (such as a volt)
age, a current or impulse response) fulfills X (- ~) = - X (~), where the
asterisk ()) denotes complex conjugation.

36

IEEE CIRCUITS AND SYSTEMS MAGAZINE

i (t) v 0 (t ) cos(~ 0 t ) dt $ 0.

(7)

0

This requirement is based on the conservation of energy and, in particular, the passive character of our network, which, according to our assumptions, can only
exchange energy with the outside world via the considered electrical port. Due to the assumed slowly varying envelope, the voltage is a narrow-band signal and
we can use the approximation (6) to evaluate the integral. The integration can be performed in a straightforward manner by choosing a specific envelope function.
A particularly simple result can be obtained by using
the envelope
v 0 (t ) = )

Vt 0 (1 - e -t/x ), for t $ 0,
0,
for t 1 0,

(8)

which, if ~ 0 x & 1, corresponds to a very slow increase
to the stationary state, which, in turn, is approached
when t 0 & x.
Inserting (8) and (6) in (7), after some straightforward calculations outlined in Appendix B, yields the following asymptotic stationary result valid for t 0 & x
w(t 0) -

V0(~ - ~ 0) " B (~ 0) + B l(~ 0) (~ - ~ 0) , e j~t d~ 3 .

#

Vt 02 ' B (~ 0)
cos(2~ 0 t 0) + Bl(~ 0) 1 $ 0.
4
~0

(9)

This inequality must be fulfilled for arbitrary t0. As the
cosine assumes values between -1 and +1 and the susceptance B(~ 0) may be positive or negative, in general,
we readily obtain the condition (2) for the susceptance.
Specifically, at times t 0 where the cosine equals +1, we
obtain the condition Bl(~ 0) $ - B(~ 0)/~ 0 . For times, where
the cosine equals -1, we obtain Bl(~ 0) $ B (~ 0) /~ 0 . Depending on the sign of the susceptance, one of these
conditions is more strict. As both must be fulfilled, we
can combine them into a single inequality, i.e. (2). Similarly the condition for the reactance function can be
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