1. CHAPTER 4
FOURIER SERIES

2. analyze electric circuit
Course Outcome
CO2
Ability to
analyze electric circuit
using
Fourier Series
for the circuit comprising passive elements.

3. Outline Introduction of the Fourier series.
The properties of the Fourier series.
Symmetry consideration
Application of the Fourier series to circuit analysis.

4. Fourier Series
While studying heat flow, Fourier discovered that a nonsinusoidal periodic function can be expressed as an infinite sum of sinusoidal functions.
Recall that a periodic function satisfies:
Where n is an integer and T is the period of the function.

5. Trigonometric Fourier Series
According to the Fourier theorem, any practical periodic function of frequency ω0 can be expressed as an infinite sum of sine or cosine functions.
Where ω0=2 / T is called the fundamental frequency in radians per second.
Its resolves the function into a dc component and an ac component.
The constants an and bn are called the Fourier coefficients.

6. Cont’d
To find a0:
To find an:
To find bn:

7. Harmonics
The sinusoid sin(nω0t) or cos(nω0t) is called the n’th harmonic of f(t).
If n is odd, the function is called the odd harmonic.
If n is even, the function is called the even harmonic.
For a function to be expressed as a Fourier series it must meet certain requirements:
f(t) must be single valued everywhere.
It must have a finite number of finite discontinuities per period.
It must have a finite number of maximum and minima per period.

8. Cont’d The last requirement is that
These conditions are called the Dirichlet conditions.
A major task in Fourier series is the determination of the Fourier coefficients.
The process of finding these is called Fourier analysis.

9. Fourier Analysis
To evaluate the Fourier coefficients and we often need to apply the following integrals:

10. Con’t
Values of the cosine, sine, and exponential functions for integral multiples п, where n is an integer.

11. Fourier Analysis: Example
Find the Fourier series of the square wave given in figure below.

12. The f(t) is ODD function;
Therefore

13. Amplitude-Phase Form
An alternative form is called the amplitude phase form:
Where:
The frequency spectrum of a signal consists of the plots of amplitude and phases of the harmonics versus frequency

14. Symmetry Considerations
The series consists of only sine terms.
If the series contains only sine or cosine, it is considered to have a certain symmetry.
There is a technique for identifying the three symmetries that exist, even, odd, and half-wave.

16. Even Symmetry The function is symmetrical about the vertical axis:
A main property of an even function is that:

17. Cont’d The Fourier coefficients for an even function become:
Its become a Fourier cosine series.

18. Odd Symmetry
A function is said to be odd if its plot is antisymmetrical about the vertical axis.
Examples; t, t3, and sint
An odd function has this major characteristic:

19. Cont’d
This comes about because the integration from –T/2 to 0 is the negative of the integration from 0 to T/2
The coefficients are:
This gives the Fourier sine series.

20. Properties of Odd and Even
The product of two even functions is also an even function.
The product of two odd functions is an even function.
The product of an even function and an odd function is an odd function.
The sum (or difference) of two even functions is also an even function.
The sum (or difference) of two odd functions is an odd function.
The sum (or difference) of an even function and an odd function is neither even nor odd.

21. Half Wave Symmetry
Half wave symmetry compares one half of a period to the other half.
This means that each half-cycle is the mirror image of the next half-cycle.

22. Cont’d
The Fourier coefficients for the half wave symmetric function are:
Note that the half wave symmetric functions only contain the odd harmonics .

23. Example Find the Fourier series expansion of f(t) in Figure below.
Even or Odd symmetry?

24. The f(t) is ODD function;
Therefore

25. Exercise (PP17.4 pg.777)
Find the Fourier series expansion of f(t) in Figure below.
Even or Odd symmetry?

26. The f(t) is EVEN function;
Therefore

28. Common Functions

29. Cont’d

30. Circuit Applications
Fourier analysis can be helpful in analyzing circuits driven by non-sinusoidal waves.
The procedure involves four steps:
Express the excitation as a Fourier series.
Transform the circuit from the time domain to the frequency domain.
Find the response of the dc and ac components in the Fourier series.
Add the individual dc and ac responses using the superposition principle.

31. Example A Fourier series expanded periodic voltage source.
Amplitude-Phase Form

32. On inspection, this can be represented by a dc source and a set of sinusoidal sources connected in series.
Each source would have its own amplitude and frequency.
Each source can be analyzed on its own by turning off the others.
For each source, the circuit can be transformed to frequency domain and solved for the voltage and currents.
The results will have to be transformed back to the time domain before being added back together by way of the superposition principle.

34. Revision: Phasor
A phasor is a complex number that represents the amplitude and phase of a sinusoid.
where
Rectangular
Polar
Exponential

35. Revision: Phasor (time domain) (phasor domain)
Transform a sinusoid to and from the time domain to the phasor domain:
(time domain) (phasor domain)
Phasor will be defined from the cosine function.
If a voltage or current expression is in the form of a sine, it will be changed to a cosine by subtracting from the phase.

36. Revision: Cont. Mathematic operation of complex number: Addition
Subtraction
Multiplication
Division

37. Example
Find the response vo(t) of the given circuit if vs(t) is apply to the circuit.

39. 𝑓𝑜𝑟 𝑛=2𝑘−1

40. Using voltage divider rule
For dc component, ω = 0,
For the nth harmonic, first convert Vs from sine into cos form:
Rectangular (x+jy) function
Phasor form (see slide 36)

41. From:
And using

42. Average Power and RMS
Fourier analysis can be applied to find average power and RMS values.
To find the average power absorbed by a circuit due to a periodic excitation, we write the voltage and current in amplitude-phase form:

43. For periodic voltages and currents, the total average power is the sum of the average powers in each harmonically related voltage and current:
A RMS value is:
Parseval’s theorem defines the power dissipated in a hypothetical 1Ω resistor

44. Example
Find the RMS value of the periodic current
RMS value

45. Exponential Fourier Series
A compact way of expressing the Fourier series is to put it in exponential form.
This is done by representing the sine and cosine functions in exponential form using Euler’s law.

46. The complex or exponential Fourier series representation and can be written as:
The values of cn are:

47. The exponential Fourier series of a periodic function describes the spectrum in terms of the amplitude and phase angle of AC components at positive and negative harmonic frequencies.
The coefficients of the three forms of Fourier series (sine-cosine, amplitude-phase, and exponential form) are related by:

48. Example
Obtain the complex Fourier series of the function f(t).

49. Therefore

50. Exercise
Obtain the complex Fourier series of the function f(t) and plot the amplitude and phase spectra.

51. Therefore
𝑢𝑠𝑖𝑛𝑔 𝑡 𝑒 𝑎𝑡 𝑑𝑡= 𝑒 𝑎𝑡 𝑎 2 𝑎𝑥−1

52. When n = 0;

53. End of Chapter 4