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Fundamentals of Electric Circuits Chapter 17 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Overview • This chapter introduces the Fourier series. • The definition and properties of the series will be introduced. • Symmetry considerations for different waveforms will be covered. • The general concept of applying the Fourier series to circuit analysis is discussed. 2 Trigonometric 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 ω0 can be expressed as an infinite sum of sine or cosine functions. f t a0 an cos n0t bn sin n0t dc n 1 ac • Where ω0=2/T is called the fundamental frequency in radians per second. 4 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. • The equation on the last slide is called the trigonometric Fourier series of f(t). • The constants an and bn are called the Fourier coefficients. 5 Fourier Series • The Fourier series of a function is a representation that resolves the function into a dc component and an ac component. • 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 Trigonometric Fourier Series • The last requirement is that t 0 T f t dt t0 • For any t0 • 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 Trigonometric Fourier Series • To find a0: T 1 a0 f t dt To • To find an: T 2 an f t cos n0tdt To • To find bn: T 2 bn f t sin n0tdt To 8 Amplitude-Phase Form • An alternative is called the amplitude phase form: f t a0 An cos n0t n n 1 • Where: An a b 2 n 2 n bn n tan an 1 • The frequency spectrum of a signal consists of the plots of amplitude and phases of the harmonics versus frequency 9 Example 1 Determine the Fourier series of the waveform shown below. Obtain the amplitude and phase spectra 10 Solution: 1, 0 t 1 f (t ) and f (t ) f (t 2) 0, 1 t 2 2 T an f (t ) cos( n0t )dt 0 and T 0 2 / n , n odd 2 T bn f (t ) sin( n0t )dt n even T 0 0, 2 / n , n odd An n even 0, 90, n odd n n even 0, a) Amplitude and b) Phase spectrum 1 2 1 f (t ) sin( nt ), n 2k 1 2 k 1 n Truncating the series at N=11 11 Symmetry Considerations • If one looks at a typical Fourier series, for the square wave, for example. • 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. 12 Even Symmetry • In the case of even symmetry, the function is symmetrical about the vertical axis: f t f t • A main property of an even function is that: T /2 T /2 T /2 fe t dt 2 f e t dt 0 13 Even Symmetry • The Fourier coefficients for an even function become: 2 T /2 a0 4 an T T f t dt 0 T /2 f t cos n tdt 0 0 bn 0 • Note that this become a Fourier cosine series. • Since Cosine is an even function, one can see how this series is called even. 14 Odd Symmetry • A function is said to be odd if its plot is antisymmetrical about the vertical axis. f t f t • Examples of odd functions are t, t3, and sin t • An add function has this major characteristic: T /2 f0 t dt 0 T /2 15 Odd Symmetry • 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: a0 0 an 0 4 bn T T /2 f t sin n tdt 0 0 • This gives the Fourier sine series, 16 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. 17 Properties of Odd and Even II 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. 18 Half Wave Symmetry • Half wave symmetry compares one half of a period to the other half. • In this context, a half wave (odd) symmetric function has the following property: T f t f t 2 • This means that each half-cycle is the mirror image of the next half-cycle. 19 Half Wave Symmetry • The Fourier coefficients for the half wave symmetric function are: a0 0 4 T /2 f t cos n0tdt for n odd an T 0 for n even 0 4 T /2 f t sin n0tdt for n odd bn T 0 for n even 0 • Note that the half wave symmetric functions only contain the odd harmonics . 20 Common Functions 21 Common Functions 22 17.2 Symmetry Considerations (4) Example 2 Find the Fourier series expansion of f(t) given below. Ans: f (t ) 1 n n 1 cos t sin n 1 n 2 2 2 *Refer to in-class illustration, textbook 23 24 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. 25 Circuit Applications • The first step is to determine the Fourier series expansion of the excitation. • For example a periodic voltage can be expressed as a Fourier series as follows: v t V0 Vn cos n0t n n 1 • 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. 26 Circuit Applications II • Example of a Fourier series expanded periodic voltage source. 27 Circuit Applications III • 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. 28 Circuit Applications IV 29 Example 4 Find the response v0(t) of the circuit below when the voltage source vs(t) is given by 1 2 1 vs (t ) sin nt , n 2k 1 2 n 1 n 30 Solution Phasor of the circuit V0 j 2n Vs 5 j 2n For dc component, (n=0 or n=0), Vs = ½ => Vo = 0 For nth harmonic, 2 4 tan 1 2n / 5 VS 90, V0 Vs 2 2 n 25 4n In time domain, v0 (t ) k 1 4 25 4n 2 2 cos(nt tan 1 2n ) 5 Amplitude spectrum of the output voltage 31 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 Vdc Vn cos n0t n n 1 i t I dc I m cos m0t m m 1 32 Average Power and RMS • 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 Vdc I dc Vn I n cos n n 2 n 1 • A RMS value is: Frms 1 2 2 a an bn 2 n1 2 0 • Parseval’s theorem defines the power dissipated in a hypothetical 1Ω resistor 2 p1 Frms 1 a02 an2 bn2 2 n 1 33 Example 5: Determine the average power supplied to the circuit shown below if i(t)=2+10cos(t+10°)+6cos(3t+35°) A Ans: 41.5W * 34 35 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 jn0t cos n0t e e jn0t 2 1 jn0t jn0t e sin n0t e 2j 36 Exponential Fourier Series • This can be rewritten as: f t n cn e jn0t • This is the complex or exponential Fourier series representation. • The values of cn are: T 1 cn f t e jn0t dt T0 37 Exponential Fourier Series • 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 coeffcients of the three forms of Fourier series (sine-cosine, amplitude-phase, and exponential form) are related by: An n an jbn 2cn 38 39 40