IEEE Circuits and Systems Magazine - Q1 2020 - 35

i (t)

Example 2

Y (ω) = jB (ω)

The spurious excitation of natural oscillations can

+

be illustrated by means of the network considered in

v (t)
-

Example 1. Abruptly applying a sinusoidal voltage v (t )
Lossless
LC Network

starting with t = 0 yields additional natural oscillations
corresponding to the two poles of the susceptance4,
where the strength of these oscillations depends on the
initial phase of the sinusoid. This is demonstrated in the

v (t)

figures below for a cosine and a sine voltage. Integratt

ing the instantaneous power p (t ) = v (t ) i (t ) over time
yields the energy accumulated (stored) in the network
at every instant of time. It can be clearly seen that (i)
the characteristics depend on the excited natural oscillations and (ii) the energy is always larger or equal to

3

In the aforementioned simple example of a single inductor, these natural "oscillations" occur at zero frequency, i.e. DC.

FIRST QUARTER 2020

Voltage

1

v1(t) (cosine)
v2(t) (sine)

0.5
0
-0.5
-1

0

2

4

6

8

10 12
Time t

14

16

1
Current

of energy that is transferred to the network during the
transient phase. This can be easily demonstrated by
considering the difference in transferred energy when
abruptly switching on a cosine voltage or a sine voltage
to an inductance at time t = 0. In the latter case (sine voltage), a DC offset current is superposed to the stationary
current, which is associated to the well-known phenomenon of peak currents that may occur when transformers are connected to the line voltage. In lossy inductors,
this offset current decays with time; in case of an ideal
lossless inductor, however, it remains leading to a different amount of stored energy also in the stationary case
(see also Example 2). These differences are associated
with the natural oscillations of the network3 and can be
avoided, if these are not excited at all, which is why this
approach is considered in the following. Otherwise, also
valid conditions could be derived, however, these would
be network specific and would not yield the desired general Foster conditions introduced above.
To avoid such circumstances or ambiguities, we consider a sine (or cosine) signal, whose amplitude is only
gradually increasing to a stationary value after switching
on. In this case the corresponding spectrum occupies a
very narrow band around the frequency of the underlying sinusoid. We define the applied voltage as

zero, as expected.

18

20

i1(t)
i2(t)

0.5
0
-0.5
-1

Accumulated Energy

Figure 2. Switching on a gradually increasing voltage to a
lossless network yields buildup of the stationary solution for
sinusoidal excitation. If the transition is slow, the spurious
excitation of natural oscillations can be minimized and asymptotically vanishes if the bandwidth of the excitation signal
approaches zero.

0

2

4

6

8

10 12
Time t

14

16

18

20

Case: V = V0 cos (ωt)
Case: V = V0 sin (ωt)

1
0.5
0

0

2

4

6

8

10 12
Time t

v(t ) = v 0 (t ) cos ~ 0 t,

14

16

18

20

(3)

where the time-dependent amplitude (or envelope) v 0 (t )
represents a slowly varying function in time featuring a
very narrow spectrum V0 (~) in frequency domain. The
Fourier transform (FT) of v(t ) can then be written as

4
When exciting the network by an ideal current source, the natural oscillations are determined by the zeros of the susceptance, i.e. the poles
of the impedance.

IEEE CIRCUITS AND SYSTEMS MAGAZINE

35
IEEE Circuits and Systems Magazine - Q1 2020

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