FOURIER SERIES

1
CHAPTER 2

• FOURIER SERIES

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Chapter 2FOURIER SERIES - DEFINATION

• WHAT IS FOURIER SERIES ?
• It is a presentation that resolves the
  periodic function, f(t) into a dc component and
  an ac component comprising an infinite series of
  harmonic sinusoids.
• Periodic function-
• f(t) f(t nT)

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Chapter 2 FOURIER SERIES - COMPONENTS

• It can be defined mathematically as below
• ?0 is a Fundamental Frequency,
• ?0 2?f0 2?/T
• sin n?0t or cos n?0t is nth harmonic of f(t)
• a0 , an , bn are the Fourier Coefficient

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Chapter 2 FOURIER SERIES - COEFFICIENT

• Fourier Coefficients are defined as below
• a0 gt DC Component
• a0 1/T
• an gt Amplitude of Sinusoid in AC Component
• an 2/T

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Chapter 2 FOURIER SERIES FOURIER COEFFICIENT

• bn gt Amplitude of Sinusoid in AC Component
• bn 2/T
• The Fourier series equation can be written in the
  form of amplitude-phasor shown below
• f(t) a0 An cos (n?0t ?), where
• An an2 bn2 and ?n - tan-1 (bn /
  an)

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Chapter 2 FOURIER SERIES Function Values

• Cos 2n? ? 1
• Sin 2n? ? 0
• Cos n? ? (-1)n
• Sin n? ? 0
• Cos n?/2 ? (-1)n/2, n even
• 0, n odd
• 6. Sin n?/2 ? (-1)(n-1)/2, n odd
• 0, n
  even
• 7. ej2n? ? 1,
• 8. ejn? ? (-1)n
• 9. ejn?/2 ? (-1)n/2, n even,
• j(-1)(n-1)/2 , n odd

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Chapter 2 FOURIER SERIES

• EXAMPLE 1
• Refer to Example 17.1 (Page 762)
• Determine the Fourier Series for
• Square Wave
• Exercise 1
• Next, determine the Fourier Series
• For the Sawtooth signal.

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Chapter 2 FOURIER SERIES Symmetry Consideration

• The Fourier Series will be EVEN if it
• meets this equation f(t) f(-t)
• Thus, the Fourier Coefficient will be
• 1. a0 2/T
• 2. an 4/T
• 3. bn 0

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Chapter 2 FOURIER SERIES Symmetry Consideration

• The Fourier Series will be ODD if it
• meets the following equation f(t) -f(-t)
• Thus, The Fourier Coefficient will be
• 1. a0 0
• 2. an 0
• 3. bn 4/T

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Chapter 2 FOURIER SERIES Symmetry Consideration

• The properties of Even Odd function are-
• The product of 2 even function is also even
  function
• The product of 2 odd function is also odd
  function
• The product of even and odd function is odd
  function
• The sum or difference of 2 even functions is also
  even function
• The sum or difference of 2 odd functions is also
  odd function
• The sum or difference of even and odd functions
  is neither even or odd function

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Chapter 2 FOURIER SERIES Half-Wave Symmetry

• The function is half-wave symmetric if
• f(t T/2) - f(t)
• The Coefficient is defined as below
• 1. a0 0
• 2. an 4/T
  for n Odd
• 0 for n Even
• 3. bn 4/T
  for n Odd
• 0 for n Even

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Chapter 2 FOURIER SERIES Symmetry Consideration

• EXAMPLE 2
• Refer to Example 17.3 (Page 774)
• Exercise 2
• Next, determine the Fourier Series
• for the Triangular signal

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Chapter 2 FOURIER SERIES Application in RC, RL
RLC

• The input voltage can be expressed in term of
  Fourier Series as shown below
• v(t) V0 Vncos(n?0t ?n)
• The output voltage basically expressed in term of
  phasor such Impedance, Z or Admittance, Y (1/Z)
• For DC Component, ?n 0 or n 0,
• thus, Vs V0

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Chapter 2 FOURIER SERIES - Application in RC, RL
RLC

• EXAMPLE 3
• Determine the output voltage at inductor, vo(t)
• for the RL Circuit where the input voltage is in
• term of square wave.(R5? L 2H)
• Solution
• Determine the Fourier Series for Square Wave
• vs(t) ½ 2/? 1/n sin n?t, n 2k 1
• 2. Since the circuit is a series circuit, use
  voltage divider to obtain output voltage, Vo,
• Vo j?nL / (R j?nL) Vs j2n? /(5 j2n?)
  Vs

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Chapter 2 FOURIER SERIES - Application in RC, RL
RLC

• EXAMPLE 3 (Continue)
• 3. For dc component, ?n 0, thus
• Vs ½,
• 4. For ac component, ?n ? 0
• Vs 2/n? ?-90o
• 5. Now, calculate output voltage, Vo
• Vo 2n? ?-90o / sqrt(25
• 4n2?2)?tan-1 2n?/5 (2/n? ?-90o)
• (4?-tan-1 2n?/5)/sqrt(25 4n2?2)
• 6. In time domain,
• vo (t) 4 / sqrt(25 4n2?2)cos(n?t-tan-
  12n?/5)

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Chapter 2 FOURIER SERIES -Application in RC, RL
RLC

• Exercise 3
• Next, determine the output voltage,
• vo(t) of RC circuit if the input voltage
• is in of term of sawtooth wave (Refer
• to page 779, practice problem 17.6).

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Chapter 2 FOURIER SERIES AVERAGE POWER RMS
VALUE

• The power in the electrical circuit due to the
  periodic input voltage is defined as
• P VdcIdc 1/2 VnIn cos (?n - ?n)
• The rms value of periodic function is defined as
  -
• Frms sqrt (1/T )

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Chapter 2 FOURIER SERIES AVERAGE POWER RMS
VALUE

• EXAMPLE 4
• Determine the average power
• supplied to the RC circuit if the input
• current is i(t) 2 10cos(t 10o)
• 6cos(3t 35o) A. (Refer page 782, Example
  17.8).
• NEXT EXERCISES
• Do practice problem 17. 8

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Chapter 2 FOURIER SERIES AVERAGE POWER RMS
VALUE

• Practice Problem 17.8
• Given,
• v(t) 80 120cos 120?t
• 60cos (360?t 30o)
• i(t) 5cos(120?t - 10o)
• 2cos(360?t 60o)
• Find the Average Power absorbed by
• the circuit.

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Chapter 2 FOURIER SERIES AVERAGE POWER RMS
VALUE

• SOLUTION
• Since Power is defined as PVI Power due to
  periodic input voltage is defined as
• P VdcIdc 1/2 VnIn cos (?n - ?n)
• 2. Now, Calculate Power
• P 80(0) ½ (120)(5)cos(10o) ½
• 60 cos(30o)(2)
• 295.44 51.962
• 347.4 W

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Chapter 2 FOURIER SERIES EXPONENTIAL FOURIER
SERIES

• Use Euler Identity to express FS in term of
  exponential
• 1. cos n?0t ½ ejn?0t e-jn?0t
• 2. sin n?0t ½j ejn?0t - e-jn?0t
• Thus, periodic function, f(t) can be expressed
  as
• f(t) cn ejn?0t
• cn 1/T

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Chapter 2 FOURIER SERIES EXPONENTIAL FOURIER
SERIES

• The exponential FS Coefficient, cn can be
  expressed in term of amplitude-phase form as
  shown below
• cn cn ? ?n sqrt (an2 bn2)/2
• ?-tan-1 bn / an

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Chapter 2 FOURIER SERIES EXPONENTIAL FOURIER
SERIES

• EXAMPLE 5
• Find the exponential FS of
• f(t) et , 0 lt t lt 2? with
• f(t2?) f(t)
• Solution
• Refer Example 17.10 (Page 787)

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Chapter 2 FOURIER SERIES EXPONENTIAL FOURIER
SERIES

• EXERCISE
• Do Practice Problem 17.10 (Page
• 788). Find the complex FS for the
• Square Wave signal.

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Chapter 2 FOURIER SERIES EXPONENTIAL FOURIER
SERIES

• SOLUTION
• 1. For the Square Wave signal,
• f(t) 1, 0 lt t lt 1
• 0, 1 lt t lt 2
• 2. Now, define the period, T ?
• T 2, ? gt 2? / 2 ?
• 3. Next, calculate exponential FS
  Coefficient, cn
• 4. cn 1/2
  j/(2n?)(e-jn? -1)

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Chapter 2 FOURIER SERIES EXPONENTIAL FOURIER
SERIES

• SOLUTION
• 5. Since e-jn? cos(n?) jsin(n?) gt (-1)n
• 6. cn j/2?n(-1)n 1
• gt 0 if n even
• gt -j/(n?) if n odd
• 7. for n 0, c0 0.5
• 8. Thus, the exponential FS for f(t) is -
• f(t) ½ - j/n? ejn?t