CHAPTER 4 FOURIER SERIES 1 S A B A R I N A I S M A I L
Outline Introduction of the Fourier series. The properties of the Fourier series. Symmetry consideration Application of the Fourier series to circuit analysis. 2
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: f t f t nt Where n is an integer and T is the period of the function. 3
Trigonometric Fourier Series According to the Fourier theorem, any practical periodic function of frequency ω can be expressed as an infinite sum of sine or cosine functions. f t a a cos n t b sin n t n n dc n1 ac Where ω =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 a n and b n are called the Fourier coefficients. 4
Cont d To find a : a 1 T T o f t dt To find a n : To find b n : T 2 an f tcos ntdt T o T 2 b n f t sin ntdt T o 5
Harmonics The sinusoid sin(nω t) or cos(nω t) 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: 1. f(t) must be single valued everywhere. 2. It must have a finite number of finite discontinuities per period. 3. It must have a finite number of maximum and minima per period. 6
Cont d The last requirement is that t T t f t dt for anyt 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. 7
Example Find the Fourier series of the square wave given in figure below. 8
Amplitude-Phase Form An alternative is called the amplitude phase form: Where: f t a A cos n t n n n1 A a b tan 2 2 1 n n n n The frequency spectrum of a signal consists of the plots of amplitude and phases of the harmonics versus frequency b a n n 9
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 halfwave. 1
Even Symmetry The function is symmetrical about the vertical axis: f t f t 11 A main property of an even function is that: T/2 T/2 f t dt 2 f t dt e T /2 e
Cont d The Fourier coefficients for an even function become: T /2 2 a f tdt T T /2 Its become a Fourier cosine series. 4 an f tcos ntdt T b n 12
Odd Symmetry A function is said to be odd if its plot is antisymmetrical about the vertical axis. f t f t Examples; t, t 3, and sint An add function has this major characteristic: T /2 T /2 f t dt 13
Cont d This comes about because the integration from T/2 to is the negative of the integration from to T/2 The coefficients are: This gives the Fourier sine series. a a T /2 4 b n f t sin ntdt T n 14
Properties of Odd and Even 1. The product of two even functions is also an even function. 2. The product of two odd functions is an even function. 3. The product of an even function and an odd function is an odd function. 4. The sum (or difference) of two even functions is also an even function. 5. The sum (or difference) of two odd functions is an odd function. 6. The sum (or difference) of an even function and an odd function is neither even nor odd. 15
Half Wave Symmetry Half wave symmetry compares one half of a period to the other half. T f t f t 2 16 This means that each half-cycle is the mirror image of the next halfcycle.
Cont d The Fourier coefficients for the half wave symmetric function are: a a n 4 T T /2 f t cos n tdt for n odd for n even b n 4 T T /2 f t sin n tdt for n odd for n even Note that the half wave symmetric functions only contain the odd harmonics. 17
Example 18 Find the Fourier series expansion of f(t) in Figure below.
Exercise 19 Find the Fourier series expansion of f(t) in Figure below.
2 Common Functions
21 Cont d
Circuit Applications Fourier analysis can be helpful in analyzing circuits driven by non-sinusoidal waves. The procedure involves four steps: 1. Express the excitation as a Fourier series. 2. Transform the circuit from the time domain to the frequency domain. 3. Find the response of the dc and ac components in the Fourier series. 4. Add the individual dc and ac responses using the superposition principle. 22
Example A Fourier series expanded periodic voltage source. v t V V cos n t n n n1 23
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. 24
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Example 26 Find the response v o (t) of the given circuit if v s (t) is apply to the circuit.
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: v t V V cos n t dc n n n1 i t I I cos m t dc m m m1 27
For periodic voltages and currents, the total average power is the sum of the average powers in each harmonically related voltage and current: 1 P V I V I cos dc dc n n n n 2 n1 A RMS value is: 1 F a a b 2 2 2 rms n n 2 n1 28 Parseval s theorem defines the power dissipated in a hypothetical 1Ω resistor 1 p F a a b 2 2 2 2 1 rms n n 2 n1
Example Find the RMS value of the periodic current 29 i( t) 8 3cos2t 2sin 2t 15cos4t 1sin 4tA
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. 1 cos n t e e 2 sin 1 jn t jn t n t e jnt e jnt 2 j 3
The complex or exponential Fourier series representation and can be written as: c e jn f t n The values of c n are: n 1 T jn t cn f t e dt T t 31
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: A a jb 2c n n n n n 32
Example Obtain the complex Fourier series of the function f(t). 33
Exercise Obtain the complex Fourier series of the function f(t) and plot the amplitude and phase spectra. 34
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