Lista de Exercícios Resolvidos - Circuitos Elétricos 2 - Análise de Fourier

UNIVERSIDADE FEDERAL DO PARÁ INSTITUTO DE TECNOLOGIA FACULDADE DE ENGENHARIA DA COMPUTAÇÃO E DE TELECOMUNICAÇÕES CIRCUITOS ELÈTRICOS 2 4ª LISTA PROFESSOR JOÃO WEYL Entrega Até dia do 2º teste Exercícios As questões abaixo são copiadas do livro texto Introduction to Electric Circuits James A Svoboda and Richard C Dorf I A 1ª são questões do cap 14 e segunda parte questão cap 15 151 Introduction This chapter introduces the Fourier series and the Fourier transform The Fourier series represents a nonsinusoidal periodic waveform as a sum of sinusoidal waveforms The Fourier series is useful to us in two ways The Fourier series shows that a periodic waveform consists of sinusoidal components at different frequencies That allows us to think about the way in which the waveform is distributed in frequency For example we can give meaning to such expressions as the highfrequency part of a square wave We can use superposition to find the steadystate response of a circuit to an input represented by a Fourier series and thus determine the steadystate response of the circuit to the periodic waveform We obtain the Fourier transform as a generalization of the Fourier series taking the limit as the period of a periodic wave becomes infinite The Fourier transform is useful to us in two ways The Fourier transform represents an aperiodic waveform in the frequency domain That allows us to think about the way in which the waveform is distributed in frequency For example we can give meaning to such expressions as the highfrequency part of a pulse We can represent both the input to a circuit and the circuit itself in the frequency domain the input represented by its Fourier transform and the circuit represented by its network function The frequencydomain representation of circuit output is obtained as the product of the Fourier transform of the input and the network function of the circuit Table 1531 Fourier Series and Symmetry SYMMETRY FOURIER COEFFICIENTS 1 Odd function ft ft an 0 for all n bn 4T int0T2 ft sin noot dt 2 Even function ft ft bn 0 for all n an 4T int0T2 ft cos noot dt 3 Halfwave symmetry ft ft T2 a0 0 an 0 for even n bn 0 for even n an 4T int0T2 ft cos noot dt for odd n bn 4T int0T2 ft sin noot dt for odd n 4 Quarterwave symmetry Halfwave symmetry and symmetry about the midpoints of the positive and negative half cycles A Odd function a0 0 an 0 for all n bn 0 for even n bn 8T int0T4 ft sin noot dt for odd n B Even function a0 0 bn 0 for all n an 0 for even n an 8T int0T4 ft cos noot dt for odd n Table 1541 The Fourier Series of Selected Waveforms FUNCTION TRIGONOMETRIC FOURIER SERIES Square wave w0 2piT ft A2 2Api sum from n1 to infinity sin2n1 w0t2n1 Pulse wave w0 2piT ft AdT 2Api sum from n1 to infinity sinnpi dTn cosnw0t FUNCTION TRIGONOMETRIC FOURIER SERIES Halfwave rectified sine wave w0 2piT ft Api A2 sin w0t 2Api sum from n1 to infinity cos2n w0t4n21 Fullwave rectified sine wave w0 2piT ft 2Api 4Api sum from n1 to infinity cosn w0t4n21 Sawtooth wave w0 2piT ft A2 4pi sum from n1 to infinity sinn w0tn Triangle wave w0 2piT ft A2 4Api2 sum from n1 to infinity cos2n1 w0t2n12 Table 1551 Complex Fourier Coefficients for Selected Waveform WAVEFORM NAME OF WAVEFORM AND EQUATION SYMMETRY Cn 1 Square wave ft A T4 t T4 A T4 t 3T4 Even A sin npi2npi2 n odd 0 n 0 and n even 2 Rectangular pulse ft A d2 t d2 Even A deltaT sinnpidTnpidT 3 Triangular wave Even A sin2npi2npi22 n not equal 0 0 n0 4 Sawtooth wave ft 2AtT T2 t T2 Odd Aj1nnpi n not equal 0 0 n0 5 Halfwave rectified sinusoid ft sin w0t 0 t T2 0 T2 t 0 None 1pi1n2 n even j4 n 1 0 otherwise 6 Fullwave rectified sinusoid ft sin w0t Even 2pi1n2 n even 0 otherwise Table 1591 The Fourier Transform Pair EQUATION NAME PROCESS Fjw integral from infinity to infinity ft ejwt dt Transform Time domain to frequency domain Conversion of ft into Fjw ft 12pi integral from infinity to infinity Fjw ejwt dw Inverse transform Frequency domain to time domain Conversion of Fjw into ft Table 15101 Selected Properties of the Fourier Transform NAME OF PROPERTY FUNCTION OF TIME FOURIER TRANSFORM 1 Definition ft Fω 2 Multiplication by constant Aft AFω 3 Linearity af₁ bf₂ aF₁ω bF₂ω 4 Time shift ft t₀ ejωt₀Fω 5 Time scaling fat a 0 1a Fωa 6 Modulation ejω₀tft Fω ω₀ 7 Differentiation dftdt jωⁿFω 8 Convolution to f₁x f₂t x dx F₁ω F₂ω 9 Time multiplication tⁿ ft jⁿ dⁿFωdωⁿ 10 Time reversal ft Fω 11 Integration to t ft dt Fωjω πF0δω Table 15102 Fourier Transform Pairs ft WAVEFORM fω 1 Pulse f₁t Aut Δ2 Aut Δ2 2 Impulse δt t₀ 3 Decaying exponential Aeat ut 4 Symmetric decaying exponential Aeat 5 Tone burst gated cosine Af₁t cos ω₀t 6 Triangular pulse 7 A Sabt A sin bt bt 8 Constant dc ft A 9 Cosine wave A cos ω₀t ΔΔSaωΔ2 ejωt₀ Aajω 2aAa² ω² ΔΔ2 Saω ω₀ Saω ω₀ Δ AΔSa²ωΔ2 πb ω b 0 ω b 2πA δω πAδω ω₀ δω ω₀ ft WAVEFORM fω 10 Signum ft 1 t 0 1 t 0 11 Step input Aut 2jω Aπδω 1jω Note Sax sin xx Table 15131 Obtaining the Fourier Transform Using the Laplace Transform CASE METHOD A ft nonzero for positive time only and ft 0 t 0 Step 1 Fs ℒft 2 Fω Fsₛⱼω B ft nonzero for negative time only and ft 0 t 0 Step 1 Fs ℒft 2 Fω Fsₛⱼω C ft nonzero over all time Step 1 ft ft ft 2 Fs ℒft Fs ℒft 3 Fω Fsₛⱼω Fsₛⱼω Note The poles of Fs must lie in the lefthand splane FIGURE 1572 a An RC circuit excited by a periodic voltage vst FIGURE E 1532 The period T pi s Answer ft 24pi2 sum from n1 to N of 1n2 sin n pi 3 sin n omega 0 t and n odd omega 0 2 rads EXERCISE 15101 Find the Fourier transform of fat for a 0 when Fomega Fft Answer Ffat 1a Fomegaa EXERCISE 15102 Show that the Fourier transform of a constant dc waveform ft A for infinity t infinity is Fomega 2 pi A delta omega by obtaining the inverse transform of Fomega EXERCISE 15131 Derive the Fourier transform for ft t ea t t 0 t ea t t 0 Answer j 4 a omega a2 omega22 P 1571 Determine the steadystate response vot for the circuit shown in Figure P 1571 The input to this circuit is the voltage vs t shown in Figure P 1532 Answer vot 6 sum from n1 to infinity 240 n pi sqrt400 n2 pi2 sinpi2 n t pi2 n2 tan1n pi 20 Figure P 1571 P 1595 Determine the Fourier transform of the pulse shown in Figure P 1595 Answer Fj omega 23 sin omega sin 2 omega 23 cos omega cos 2 omega Figure P 1595 P 15124 Find the output voltage vo t using the Fourier transform for the circuit of Figure P 15124 when vt e ut ut V Figure P 15124 1516 SUMMARY Periodic waveforms arise in many circuits For example the form of the load current waveforms for selected loads is shown in Figure15161 Whereas the load current for motors and incandescent lamps is of the same form as that of the source voltage it is significantly altered for the power supplies dimmers and variablespeed drives as shown in Figures 15161b c Electrical engineers have long been interested in developing the tools required to analyze circuits incorporating periodic waveforms FIGURE 15161 Load current waveforms for a motors and incandescent lights b switchmode power supplies and c dimmers and variablespeed drives The vertical axis is current and the horizontal axis is time Source Lamare 1991 The brilliant mathematicianengineer JeanBaptisteJoseph Fourier proposed in 1807 that a periodic waveform could be represented by a series consisting of cosine and sine terms with the appropriate coefficients The integer multiple frequencies of the fundamental are called the harmonic frequencies or harmonics The trigonometric form of the Fourier series is ft a0 sum from n1 to N an cos n omega 0 t sum from n1 to N bn sin n omega 0 t The coefficients of the trigonometric Fourier series can be obtained from a0 1T integral from 0 to T ft dt an 2T integral from 0 to T ft cos n omega 0 t dt n 0 bn 2T integral from 0 to T ft sin n omega 0 t dt n 0 An alternate form of the trigonometric form of the Fourier series is ft c0 sum from n1 to N cn cos n omega 0 t theta n where c0 a0 average value of ft and cn sqrtan2 bn2 and theta n tan1 bn an if an 0 180circ tan1 bn an if an 0 The Fourier coefficients of some common periodic signals are tabulated in 1541 Symmetry can simplify the task of calculating the Fourier coefficients The exponential form of the Fourier series is ft sum from infinity to infinity Cn ejn omega 0 t where Cn is the complex coefficients defined by Cn 1T integral from 0 to T ft ejn omega 0 t dt The line spectra consisting of the amplitude and phase of the complex coefficients of the Fourier series when plotted against frequency are useful for portraying the frequencies that represent a waveform The practical representation of a periodic waveform consists of a finite number of sinusoidal terms of the Fourier series The finite Fourier series exhibits the Gibbs phenomenon that is although convergence occurs as n grows large there always remains an error at the points of discontinuity of the waveform To determine the response of a circuit excited by a periodic input signal vst we represent vst by a Fourier series and then find the response of the circuit to the fundamental and each harmonic Assuming the circuit is linear and the principle of superposition holds we can consider that the total response is the sum of the response to the dc term the fundamental and each harmonic The Fourier transform provides a frequencydomain description of an aperiodic timedomain function A circuit with an impulse response ht and an input ft has a response yt that may be determined from the convolution integral The table of Laplace transforms Table 1421 developed in Chapter 14 can be used to obtain the Fourier transform of a function ft EXERCISE 1551 Find the exponential Fourier coefficients for the function shown in Figure E 1551 Answer Cn 0 for even n and Cn 2 j pi n for odd n EXERCISE 1552 Determine the complex Fourier coefficients for the waveform shown in Figure E 1552 EXERCISE 1571 Find the response of the circuit of Figure 1572 when R 10 kΩ C 04 mF and vs is the triangular wave considered in Example 1531 Figure 1533 Include all terms that exceed 2 percent of the fundamental term Answer vot approx 020 sin4t 86 0008 sin 12t 89 V 14813 3s 2000Ω 02s 2000Ω Is I1s 5000Ω J0 107 2000 s 2000 107 s 107 s 5000 Is 1 0 Xs 107s 5000 Xs 107 s 5000 Vs Xs 5000 5000 02s 107 5000 s 5000 1 5000 02s 251011 s5000s 25000 Vs 325107 s 25000 325107 s 5000 Vf 325107 e25000t 325107 e5000t V Vf 325107 e5000t e25000t V P 14301 Vot 34 1 e100t ut V 1s 5Ω sL R Vot a Vos R s2 L s 5R R s sL 5 R Vos R s R5 LR R5sL R5 1 R5 Rs RL sL R5 Vot R R5 1 eR5L t 34 1 e100t RR5 34 R5 4R3 4R3 R 5 4R 3R 5 R 15Ω L 02H P 143019 Hs 20 s8 Vouts Vins a 20 e8t Az um Voutt b Is 1 sL 20 1 02s 20 Vos 15 02s 20 75 s 00 Vot 75 e100t V c Vos 05 15 s2 1002 02s 20 75s s2 1002 02s 20 375s s3 100s2 100s2 1000000 Vos 1875s 1875 s2 30000 1875 s 300 P 14824 Hs 32 s2 8s 16 32 s42 Vouts Vins Vouts 1 Vouts 32 s42 Voutt 32 t e4t 14819 Vit R1 R2 L C Vot VisR1 xR2 xsL 1sC xR1 x1R2 1sL 1sC 1R1 1sR1 Vos 1sC xsL 1sC xs R2 s2 CL R2 s s2 CL R1 s2 CL R2 s R2 R1 C R1 R2 Vos R2 s3 LC R1 R2 s2 R1 R2 C s R1 R2 P 14826 Is 1 x 50mH 005Ω 450Ω 1μF 1s106 30k s 1 x450 x005s 30ks x s2 2107 450 s2 9000s 2107 Ios x 005s 30ks 9000s s2 9000s 2107 Ios 45000 s 5000 36000 s 4000 9000 5s5000 4s4000 Ios Iot 9000 5 e5000t 4 e4000t 1551 T2 w0pi100 Cu1T 0T fte jwt dt Cu120² ftejpI t dt Fp 0² ftejpI t dt0¹ t² ejpi t dt Cu1 T 0T ftejwot dt1T 0T t² ejwot dt 12 ejwt t²jw ejwpt 2tjw dt 12 ejwpI t²jw 2jw t ejwt dt 12 ejwpt t²jw 2jw ejwt1jw²t 1jw² dt 12 ejwpt t²jw 2jw ejwt1jw²t 1jw² 12 ejwpt t²jw 2jw ejwt 1jw²t 1jw² ejwt 1jw² dt Cu 1jw t² ejwt 2jw 1jw² t ejwt 1jw² ejwt dt Cu 1jw t² ejwt 2jw 1jw² t ejwt 1jw² 1jw ejwt dt Cu 1jw t² ejwt 2jw 1jw² t ejwt 1jw²w ejwt dt Cu Cu2w³ Cu 1552 Cu1T 0T ftejwt dt Cu12 0² t² ejwpt dt 12 t² ejwt tjw 2jw ejwt t ejwt dt 12 t² ejwt tjw 2jw ejwt 1jw² t 1jw² dt 12 t² ejwt tjw 2jw ejwt 1jw² t 1jw² 12 t² ejwt tjw 2jw ejwt 1jw² t 1jw² 12 t² ejwt tjw 1595 Fw ft ejwtdt 2 0 t e t dt 15126 usw Fw maw mw 𝜀r 0 𝜀𝑟 0 0 usd F g ejwt 0 1w coswt g j w t dt Vgw 80 ejwt V0t 12 32 0 m mjwt 2 t 0 mt² 15331 ft t eγ t t 0 Fw ft ejwt dt t 0 γ 0 0 t eat ejwt dt 0 t eajw t dt ddw Fw 1 1 1 ajw² 1 ajw² Fw 1 ajw² 1 ajw² ddw Fw 1 ajw² 1 ajw² Fw 11ajw³ 1 2ajw ajw³ a² j 2aw 3a w² j w³ a² j 2aw 3a w² j w³ a² w²² j 4aw a² w²² j 4aw a² w²² 15301 ft aoo Fw ft ejwt dt ft at dfdt at ejwt t t dfdt at ejwt t t t dfdt at ejwt FLw ft ejwt dt t ft ejwt dt ddw ft ejwt dt 1j ddw Fw ft ejwt j t dt 1j ddw ft ejwt 15302 ft A t A ejwt dt Aejwt 1 j w ejwt A A ejwt Fw j A w 1j w A δw π 2π A δw 15302 1571 30 Ω 001F Vct V0t 24 π2 n1 1 n² sen nπ 3 sen nω0 t Wo2 𝜈𝑜t Vct 1 jω0 001 30 1 jω 001 24 π2 1 n² sen nπ 3 sen n2t 1 jw001 30 1 1ω0001 𝜈𝑜t 60 10 t n1 640 n π² n² 30² n² π² 2 sen n π 2 t n π 2 tg¹n π 30 𝜈𝑜t 6 t n1 240 n π 400 n² π 2 sen n π 2 t n π 2 tg¹ n π 30 pω eʲωt w₀₁ω w₀ 1 jω ω₀ pt 1 jω p₀t eʲωt dτ ₀ eʲωτ fτ dτ ₀ Fmjω e 0 e⁰⁰ 0 e 0 e⁰ dτ ₀ eʲωτ fτ dτ 1 jω e⁰ 0 0 e⁰ Θ f t 1 jω 1 jω w₀ 1 jω w₀ w₀1 jω w₀ 14 f m jω τ₀ jω 1 jω jω τ₀ 1 jω w₀ w₀ jω jω τ₀ jω τ₀ jω w₀ jω jω τ₀ jω 1 jω 1 jω Θ f t 1 jω ₀ Fjω dΩ mτ mτ mτ mτ 1 mτ ft Fjω 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