A NOTE ON FOURIER SERIES OF HALF WAVE RECTIFIER, FULL WAVE
RECTIFIER AND UNRECTIFIED SINE WAVEJambunatha Sethuraman*Vinayaka
Missions Kirupananda Variyar Engineering CollegeSalem Tamil Nadu
India
ABSTRACT: There is always an inherent phase difference between a
sinusoidal input and output (response) for a linear passive causal
system. This is explained in detail and even in the Fourier series
of a periodic causal function, this principle can be elegantly used
with profit and better understanding. Several illustrations are
give in support of this novel idea. There need not be a special
section for Fourier cosine/sine transforms as this approach covers
them also.
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INTRODUCTION: is a periodic function with period It can be
expended as a Fourier series in the interval : (1) spanned by the
basic set of orthogonal functions the function receives the
co-ordinates . Each sinusoidal wave has angular frequency The
fundamental frequency The set of Fourier coefficients has Time
Space
Temporal frequencySpatial frequency
(2)Half-wave rectifier output wave form and its Fourier series
The coefficients are evaluated as
It is convenient to write the coefficients as Note that there is
only one coefficient which survives in sine series, namely all the
other coefficients vanish. It is interesting. Why it is so, will be
clear later. See Fig. 1. A scale factor of 2 is due to doubling the
function in the period.: The wave form can be represented as .. For
the full-wave rectifier the Fourier coefficients are given by
is due to doubling the existence of the function.
sine wave: This is a pure sine wave given by , namely . This is
expected since it is a pure sine wave and has only one Fourier
component, viz., and hence only contributes. Again a factor 2 is
present due to doubling the function in the period. See Fig. 3.
The assertion is the the half-wave rectifier contains the
coefficients of full-wave rectifier and (unrectified) pure sine
wave. This is interesting. A closer analysis shows that full-wave
rectifier and pure sine wave are respectively even and odd
extensions of half-wave rectifier! If half-wave rectifier is
extended as an even function (full-wave rectifier) only the cosine
coefficients survive and sine coefficients (odd) vanish. A factor 2
arises due to the period is doubled. If the half-wave rectifier is
extended as an odd function, i.e., pure sine wave only the odd sine
coefficients survive and all even coefficients vanish. This is an
important concept and can be applied to all so called Fourier
sine/cosine series. In both extensions, a factor 2 arises due to
the function is doubled in the period.REDUNDANT EXERCISES: It is
not necessary to teach Fourier sine /cosine series and they are
redundant in the sense that they are special cases of Fourier
series of a causal periodic function.Our approach is further
strengthened by the following exercise: See. Fig. 4.
Fig. 4. A rectangle periodic function with duty cycle 50% has
only When extended as an odd function the are simply doubled. When
extended as an even function, it becomes a continuous straight line
with constant value 1. Hence There is consistency in this approach.
It is well-known that a constant function has only dc term as there
is no undulation or change in the function. In Fourier analysis, it
is sometimes regarded as useless term having no information; but it
is not so. Its role is important and serves as a canvas for
painting. The next illustration with a causal triangle function is
also self evident and proves beyond doubt our assertion. See. Fig.
5.
Fig. 5. A function is said to be causal if it is zero for
negative range. For example the Heaviside function is causal :.
(3)Exponential function used to describe radioactivity is also
causal. (4)Half wave rectifier is causal because for negative
duration of the period, the wave is zero.CAUSALITY AND QUADRATURE
RESPONSE: One might have noticed that when a cosine periodic force
is acting on a damped harmonic oscillator , in the response
(displacement) there is a component proportional to cosine periodic
force and also a component of displacement proportional to sine of
the periodic force! This is surprising as no sine periodic force
was applied. Yet the system responds as if a sine periodic force
were also applied. This displacement is said to be in quadrature
response and that proportional to cosine force is said to be in
phase response. The differential equation of a damped harmonic
oscillator (DSHO) is [5, 6] (5)
where is the natural frequency and is the damping constant per
unit mass and C is the strength of the impulse. is the displacement
and is the velocity of the particle. It can be shown that [2]. The
velocity is given by [2] . (6) If a periodic force per unit
mass.The velocity is now given by, (7) there are two components for
the velocity: first in-phase component and in-quadrature component
. It is surprising that there is a response proportional to and
that oscillating force was not applied. Nevertheless the response
has that component also. A causal linear passive system always
produces an impulse response with both in-phase and in-quadrature
response. We can write eq. (.) as
and tan (8)The in-phase and in quadrature responses are not
independent of each other as the the principle of causality
ascertains that that for a physically realizable system it is not
possible to give an arbitrary characteristic for the in-phase
response without setting up a definite in-quadrature response and
hence a definite phase characteristic. For more information the
reader is referred to ref. 2 and 3 in Academia.edu.References: 1.
E. Butkov, Mathematical physics, Addison-Wesley Pub Co (Reading),
1968, Ch. 72. Jambunatha Sethuraman, A convolution approach to
damped harmonic oscillator,
https://www.academia.edu/5385350/CONVOLUTION_APPROACH_TO_DAMPED_HARMONIC_OSCILLATOR
3. Jambunatha Sethuraman, Hilbert transform, Causality,
analyticity, and Fraunhofer diffraction,
https://www.academia.edu/5453386/Causality_Analyticity_Hilbert_transform_and_Fraunhofer_diffraction
