Harmonic Analysis
The process of finding the Fourier series for a function given by numerical values is known as harmonic analysis.
ie, f(x) = (a0/2) + (a1 cosx + b1 sinx) + (a2 cos2x + b2 sin2x) + (a3cos3x + b3sin3x)+-------------…(1)
In (1), the term (a1cosx + b1 sinx) is called the fundamental or first harmonic, the term (a2cos2x + b2sin2x) is called the second harmonic and so on.
Example 16
Compute the first three harmonics of the Fourier series of f(x) given by the following table.
We exclude the last point x = 2Ï€.
Let f(x) = (a0/2) + (a1 cosx + b1 sinx) + (a2 cos2x + b2 sin2x) + …………
To evaluate the coefficients, we form the following table.
\ f(x) = 1.45 – 0.37cosx + 0.17 sinx – 0.1cos2x – 0.06 sin2x + 0.033 cos3x+…
Example 17
Obtain the first three coefficients in the Fourier cosine series for y, where y is given in the following table:
\ Fourier cosine series in the interval (0, 2π) is y = (a0 /2) + a1cosq+ a2cos2q+ a3cos3q+ …..
To evaluate the coefficients, we form the following table.
Now, a0 = 2 (42/6) = 14
a1 = 2 ( -8.5/6) = - 2.8
a2 = 2 (-4.5/6) =
a3 = 2 (8/6) = 2.7
y = 7 –2.8 cosq- 1.5 cos2q+ 2.7 cos3q+ …..
Example 18
The values of x and the corresponding values of f(x) over a period T are given below. Show that f(x) = 0.75 + 0.37 cosq+ 1.004 sinq,where q= (2Ï€x )/T
We omit the last value since f(x) at x = 0 is known.
Here q= 2Ï€x / T
When x varies from 0 to T, qvaries from 0 to 2Ï€ with2Ï€/6. an incre
Let f(x) = F(q) = (a0/2) + a1 cosq+ b1 sinq.
To evaluate the coefficients, we form the following table.
Now, a0 = 2 ( ∑ f(x)/6)=1.5
a1 = 2 (1.12 /6) = 0.37
a2 = 2 (3.013/6) = 1.004
Therefore, f(x) = 0.75 + 0.37 cosq+ 1.004 sinq
Exercises
1.The following table gives the variations of periodic current over a period.
t (seconds) : 0 T/6 T/3 T/2 2T/3 5T/6 T
A (amplitude): 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
Show that there is a direct current part of 0.75 amp in the variable current and obtain the
amplitude of the first harmonic.
2.The turning moment T is given for a series of values of the crank angle q°=75°
q° : 0 30 60 90 120 150 180
T° : 0 5224 8097 7850 5499 2626 0
Obtain the first four terms in a series of sines to represent T and calculate
T for q= 75°
3. Obtain the constant term and the co-efficient of the first sine and cosine terms in the
Fourier expansionfollowingof table„y‟. as given in th
X : 0 1 2 3 4 5
Y : 9 18 24 28 26 20
4. Find the first three harmonics of Fourier series of y = f(x) from the following data.
X : 0 ° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°
Y : 298 356373 337 254 15580 516093 147 221
2.8 Complex Form of Fourier Series
The series for f(x) defined in the interval (c, c+2Ï€)and satisfying
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