IEEE Solid-State Circuits Magazine - Fall 2017 - 10

C IRCU IT INTU ITIONS

Ali Sheikholeslami

Reinventing the Wheel

W

Welcome to the 15th article in this
column series. As the title suggests,
each article provides insights and
intuitions into circuit design and
analysis. These articles are aimed
at undergraduate students but may
serve the interests of other readers as well. I would appreciate your
comments and feedback as well as
your requests and suggestions for
future articles in this series. Please
send your comments to ali@ece
.utoronto.ca.
In the column in the Spring 2017
issue, we discussed why sinusoids
are so important to circuit designers. In this article, we explore methods of generating sinusoids, trying
to make intuitive sense of sinusoidal waveforms.

Generating Sinusoids

The real sinusoids, such as cos ^~t h
and sin ^~t h, appear far more complicated when compared to rectangular and triangular waveforms. In
fact, a rectangular waveform is just
a derivative with respect to time of
a triangular waveform. The derivative of a rectangular waveform is
mostly zero, except at discontinuities. In contrast, cos ^~t h is a very
smooth function; its derivative is
well defined, and its second-order
derivative resembles the original
function. If we were to create a sinusoidal function in a one-dimensional
space, we would soon find that it is
almost impossible. However, this
same function is easily understood
Digital Object Identifier 10.1109/MSSC.2017.2745921
Date of publication: 16 November 2017

10

FA L L 2 0 17

and constructed if one resorts to a
two-dimensional space.
Consider a ball moving at a constant speed on a circle, as shown in
Figure 1(a). The constant speed is
captured by a triangular waveform
representing the angle of rotation
as a function of time [see Figure 1(b)].
If we project the movement of this
ball along the x-axis (or the y-axis),
i.e., a one-dimensional space, we
will have a sinusoid, as shown in
Figure 1(c). This is how a sinusoid
is defined using Euler's formula:
cos ^~t h = Re " e j~t , . It is also how
a sinusoid is generated by an ac
generator, powering up buildings
and cities.
In an ac generator, as pictorially
shown in Figure 2, the windings of
a coil are rotated in a circle in the presence of a constant magnetic field.
Alternatively, a magnet can be rotated
in the middle of a coil. In either case,
there is a projection of a circular magnetic field along the winding of the
coil. This is indeed Euler's formula in
action, where the vector representing the winding area (A) is rotating in
a circle and being projected along the
magnetic field vector.
The presence of circular movement is essential in creating a sinusoidal waveform. It is not always
obvious, however, that there is a
circle behind the creation of every
sinusoid. We look at LC oscillators to
illustrate this point.
Figure 3 shows a simplified version of an LC oscillator consisting
of an ideal capacitor and an ideal
inductor, with no resistive losses
in either of the two. If the capaci-

IEEE SOLID-STATE CIRCUITS MAGAZINE

y

e jθ
sin(θ )

θ

x

cos(θ )

(a)

θ
2π

Time

(b)
y (t )

x (t )

Time

(c)
FIGURE 1: (a) A point moving at a constant
speed on a circle, (b) i as a function of time,
and (c) projection of the point movement
along the x-axis and y-axis as functions
of time.

tor is charged to an initial voltage,
V0, closing the switch will create a
current through the inductor. The
inductor current initially discharges
the capacitor, reducing its voltage to
zero. When this occurs, the current
in the inductor will be at its peak,
charging the capacitor in the op posite direction. The current in the
inductor then reverses direction,
and the same trend continues in the
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Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Fall 2017
