IEEE Circuits and Systems Magazine - Q1 2020 - 37

derived by applying a slowly increasing sinusoidal current rather than a voltage.

Appendix B: Evaluation of Integrals
The key lies in the asymptotic evaluation of the integrals

III. A Narrow-Band Equivalent Circuit and
Another Derivation of Foster's Condition
Inspired by the previous approach, which considered the
application of a narrow-band signal to build up a stationary
state, we can ask what equivalent circuit would be suitable
to represent a lossless network in a narrow band around
some frequency ~ 0 . It is well known that the admittance
(or impedance) of a lossless circuit at a particular frequency ~ 0 can be represented by a single equivalent inductance L eq or capacitance C eq by setting B (~ 0) = ~ 0 C eq or
B (~ 0) = -1/(~ 0 L eq). However, when tuning the frequency
~ away from ~ 0, the slope of the actual susceptance (reactance) will always be larger or equal than that of the equivalent component, which is a consequence of (2) and has
also been noted by Bode in [5]. By composing the equivalent circuit out of two components, this shortcoming can
be dealt with. In particular, in can be readily verified a parallel LC circuit with component values
C p = 1 (~ 0 Bl(~ 0) + B (~ 0))
2~ 0
1
Lp = 2
~ 0 ~ 0 Bl(~ 0) - B (~ 0)

which accounts for the fact that we consider that the
signals virtually assume the stationary state. Under
these conditions the following asymptotic relations (and
one identity), which will be used below, can be readily
verified (a stands for n / x with n = 1, 2 thus, in view of
the assumptions above, we have ~ 0 & a):
t0

#

0
t0

#
0

e - at sin(2~ 0 t ) dt - 1 ,
2~ 0

e - at cos(2~ 0 t ) dt - a 2 - 0,
4~ 0
t0

#
#
0

(11)

(12)

where we considered that the current in L eq is given by
i L = Vt sin(~ 0 t )/(~ 0 L p). Inserting the above values for the
equivalent components, Eqs. 10 and 11, we obtain

FIRST QUARTER 2020

period 2r/~ 0 of the sinusoidal function, and t 0 & x,

t0

w(t ) =

t2
B (~ 0)
w(t ) = V c Bl(~ 0) +
cos(2~ 0 t ) m $ 0,
4
~0

envelope is very slowly increasing compared to the

(10)

features the susceptance B (~ 0) and the slope Bl(~ 0) at
~ 0 and thus represents a narrow band approximation
for the original lossless circuit. When a narrowband
voltage is thus applied to this equivalent LC circuit, for
decreasing bandwidth of the excitation signal, it will
asymptotically take up the same power and thus also
energy as the original circuit. If we assume a sinusoidal voltage that slowly builds up such as considered in
the previous section (see Eqs. 3 and 8), we could thus
again calculate the energy that is transferred to the circuit. However, we can spare the efforts of integration by
simply summing up the well-known expressions for the
energies stored in the components L p and C p in the stationary state, which, again, have to be greater or equal
to zero at all times. In particular, for a stationary state
voltage of v(t ) = Vt cos(~ 0 t ), using the well-known expressions for energies in L and C, we have
C p v(t ) 2 L p i L (t ) 2
+
2
2
2
t
= V c C p cos 2 (~ 0 t ) + 21 sin 2 (~ 0 t ) m $ 0,
2
~0 L p

under the conditions ~ 0 x & 1, which means that the

(13)

0

e - at dt - 1 ,
a

sin(2~ 0 t ) dt = 1 (1 - cos(2~ 0 t 0)).
2~ 0

(17)

Inserting (6) and the envelope (8) in (7), we obtain for
the energy accumulated till time t 0
w (t 0) - Vt 20

t0

#
0

$ -B (~ 0) (1 - e - x ) 2 sin (~ 0t ) cos (~ 0t )
t

t
t
+ B l(~ 0) 1 e - x (1- e - x ) cos 2 (~ 0t ) 1 dt .
x
(18)

Expanding the brackets and using the trigonometric identities sin(2b) = 2 sin(b) cos(b) and cos 2(b) =
(cos(2b) + 1) / 2, by utilizing (17), we obtain the asymptotic solution given in (9).

where we used the identities cos 2 b + sin 2 b = 1 and
cos 2 b - sin 2 b = cos (2b ). The obtained inequality is the
same as the one obtained in (9) before and thus also
directly leads to Foster's condition (2).
Thus far, we have not considered the question,
whether the component values of the equivalent circuit
are positive and if the equivalent circuit thus would be
realizable5. Inserting the obtained conditions in (10)
and (11), it turns out that the component values Lp and
5

Note that this does not affect the energy consideration as a fictitious
negative inductor or capacitor would simply be a component that can
provide energy to the external circuitry. This energy could be overcompensated by the other component such that the equivalent circuit in
total could, in principle, still be passive.
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