Explanation Problem design: Consider the periodic function f ( t ) as, f ( t ) = t 2 , 0 < t < T (1) Find the Fourier series for the special case T = 2 . Formula used: Write the expression to calculate the fundamental angular frequency. ω 0 = 2 π T (2) Here, T is the period of the function. The general expression for Fourier series representation of f ( t ) is, f ( t ) = ∑ n = − ∞ ∞ c n e j n ω 0 t Calculation: Substitute 2 for T in equation (2) to find the angular frequency ω 0 . ω 0 = 2 π 2 = π From equation (17.59) in the textbook, the coefficient of c n is as follows, c n = 1 T ∫ 0 T f ( t ) e − j n ω 0 t d t (3) Applying equation (1) in equation (3) as follows, c n = 1 2 [ ∫ 0 T t 2 e − j n π t d t ] { ∵ T = 2 , ω 0 = π } (4) Consider the following integration formula. ∫ a b u d v = [ u v ] a b − ∫ a b v d u (5) Compare the equations (4) and (5) to simplify the equation (4). u = t 2 d v = e − j n π t d t d u d t = 2 t v = e − j n π t − j n π Using the equation (5), the equation (4) can be reduced as, c n = 1 2 [ ( t 2 ) e − j n π t − j n π ] 0 2 − ∫ 0 2 e − j n π t − j n π ( 2 t ) d t = 1 2 [ ( 2 ) 2 e − j n π ( 2 ) − j n π − ( 0 ) 2 e − j n π ( 0 ) − j n π ] − ∫ 0 2 e − j n π t − j n π ( 2 t ) d t c n = 1 2 [ ( ( 4 ) e − j 2 n π − j n π − 0 ) − ∫ 0 2 e − j n π t − j n π ( 2 t ) d t ] (6) Assume the following to reduce the equation (6). x = 2 ∫ 0 2 ( t ) e − j n π t − j n π d t (7) Equation (6) becomes, c n = 1 2 [ ( ( 4 ) e − j 2 n π − j n π − 0 ) − x ] (8) Compare the equations (7) and (5) to simplify the equation (7). u = t d v = e − j n π t − j n π d u = d t v = e − j n π t ( − j n π ) 2 Using the equation (5), the equation (7) can be reduced as, x = 2 [ [ ( t ) e − j n π t ( − j n π ) 2 ] 0 2 − ∫ 0 2 e − j n π t ( − j n π ) 2 d t ] = 2 [ [ ( 2 ) e − j n π ( 2 ) ( − j n π ) 2 − ( 0 ) 2 e − j n π ( 0 )

6th Edition

ISBN: 8220102801448

Author: Alexander

Publisher: YUZU

See similar textbooks

Concept explainers

Fourier Transform

In mathematics, the Fourier transformation is a mathematical transformation that rotates responsibilities by using region or time into tasks depending on the local or temporal frequency, such as the rendering of a musical track in step with existing volu…

Question

Chapter 17, Problem 13P

To determine

Design a problem to make the better understand the Fourier series of a given periodic function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17, Problem 12P

Chapter 17, Problem 14P

Students have asked these similar questions

Don't use ai to answer I will report you answer

A transformer bank is composed of three single-phase transformers of10kVA, 20kV/200V; copper losses are 100 W and core lossesthey are 50 W. The bank is connected in Delta on the high voltage side and in star on the sidethe low-voltage side. A. What are the values of voltages B.What are the values of currents C. losses in the core and losses in copper. D. nominal power of the transformer bank.Solve by one of the experts, not using artificial intelligence

A 50 kVA, 13800/208 V transformer connected in Dy has an Rcc of 1% and aXcc of 7% per unit. A. What is the voltage regulation at full load and fp 0.8 in delay using the values of theimpedance (Ω).b. What is the voltage regulation under the same conditions using the system per unit Solve by one of the experts, not using artificial inteliggence

Chapter 17 Solutions

EBK FUNDAMENTALS OF ELECTRIC CIRCUITS

Ch. 17.2 - Find the Fourier series of the square wave in Fig....Ch. 17.2 - Determine the Fourier series of the sawtooth...Ch. 17.3 - Prob. 3PP Ch. 17.3 - Find the Fourier series expansion of the function...Ch. 17.3 - Prob. 5PP Ch. 17.4 - Prob. 6PP Ch. 17.4 - If the input voltage in the circuit of Fig. 17.24...Ch. 17.5 - The voltage and current at the terminals of a...Ch. 17.5 - Find the rms value of the periodic current i(t) =...Ch. 17.6 - Obtain the complex Fourier series of the function...

Ch. 17.6 - Obtain the complex Fourier series expansion of...Ch. 17.7 - Prob. 12PP Ch. 17.8 - Rework Example 17.14 if the low-pass filter is...Ch. 17 - Which of the following cannot be a Fourier series?...Ch. 17 - If ft=t,0t,ft+n=ft, the value of 0 is (a) 1 (b) 2...Ch. 17 - Which of the following are even functions? (a) t +...Ch. 17 - Prob. 4RQ Ch. 17 - Prob. 5RQ Ch. 17 - If f(t) = 10 + 8 cos t + 4 cos 3t + 2 cos 5t + ,...Ch. 17 - Prob. 7RQ Ch. 17 - The plot of |cn| versus n0 is called: (a) complex...Ch. 17 - Prob. 9RQ Ch. 17 - Prob. 10RQ Ch. 17 - Evaluate each of the following functions and see...Ch. 17 - Using MATLAB, synthesize the periodic waveform for...Ch. 17 - Given that Fourier coefficients a0, an, and bn of...Ch. 17 - Find the Fourier series expansion of the backward...Ch. 17 - Prob. 5P Ch. 17 - Find the trigonometric Fourier series for f t =...Ch. 17 - Determine the Fourier series of the periodic...Ch. 17 - Using Fig. 17.51, design a problem to help other...Ch. 17 - Determine the Fourier coefficients an and bn of...Ch. 17 - Find the exponential Fourier series for the...Ch. 17 - Obtain the exponential Fourier series for the...Ch. 17 - Prob. 12P Ch. 17 - Prob. 13P Ch. 17 - Find the quadrature (cosine and sine) form of the...Ch. 17 - Express the Fourier series...Ch. 17 - The waveform in Fig. 17.55(a) has the following...Ch. 17 - Prob. 17P Ch. 17 - Prob. 18P Ch. 17 - Obtain the Fourier series for the periodic...Ch. 17 - Prob. 20P Ch. 17 - Prob. 21P Ch. 17 - Calculate the Fourier coefficients for the...Ch. 17 - Using Fig. 17.61, design a problem to help other...Ch. 17 - Prob. 24P Ch. 17 - Determine the Fourier series representation of the...Ch. 17 - Find the Fourier series representation of the...Ch. 17 - For the waveform shown in Fig. 17.65 below, (a)...Ch. 17 - Obtain the trigonometric Fourier series for the...Ch. 17 - Prob. 29P Ch. 17 - Prob. 30P Ch. 17 - Prob. 31P Ch. 17 - Prob. 32P Ch. 17 - Prob. 33P Ch. 17 - Prob. 34P Ch. 17 - Prob. 35P Ch. 17 - Prob. 36P Ch. 17 - If the periodic current waveform in Fig. 17.73(a)...Ch. 17 - Prob. 38P Ch. 17 - Prob. 39P Ch. 17 - The full-wave rectified sinusoidal voltage in Fig....Ch. 17 - Prob. 42P Ch. 17 - The voltage across the terminals of a circuit is...Ch. 17 - Prob. 44P Ch. 17 - A series RLC circuit has R = 10 , L = 2 mH, and C...Ch. 17 - Prob. 46P Ch. 17 - Prob. 47P Ch. 17 - Prob. 48P Ch. 17 - Prob. 49P Ch. 17 - Prob. 50P Ch. 17 - Prob. 51P Ch. 17 - Prob. 52P Ch. 17 - Prob. 53P Ch. 17 - Find the exponential Fourier series for the...Ch. 17 - Obtain the exponential Fourier series expansion of...Ch. 17 - The Fourier series trigonometric representation of...Ch. 17 - Prob. 57P Ch. 17 - Find the exponential Fourier series of a function...Ch. 17 - Prob. 59P Ch. 17 - Obtain the complex Fourier coefficients of the...Ch. 17 - The spectra of the Fourier series of a function...Ch. 17 - Prob. 62P Ch. 17 - Plot the amplitude spectrum for the signal f2(t)...Ch. 17 - Prob. 64P Ch. 17 - Prob. 65P Ch. 17 - Prob. 66P Ch. 17 - Prob. 67P Ch. 17 - Prob. 68P Ch. 17 - Prob. 69P Ch. 17 - Design a problem to help other students better...Ch. 17 - Prob. 71P Ch. 17 - Prob. 72P Ch. 17 - Prob. 73P Ch. 17 - Prob. 74P Ch. 17 - Prob. 75P Ch. 17 - Prob. 76P Ch. 17 - Prob. 77CP Ch. 17 - Prob. 78CP Ch. 17 - Consider the full-wave rectified sinusoidal...Ch. 17 - Prob. 82CP

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, electrical-engineering and related others by exploring similar questions and additional content below.