EBK ELECTRIC CIRCUITS
EBK ELECTRIC CIRCUITS
10th Edition
ISBN: 8220100801792
Author: Riedel
Publisher: YUZU
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Chapter 16, Problem 1P

a.

To determine

Calculate the fundamental frequency ω0 for the both periodic functions.

a.

Expert Solution
Check Mark

Answer to Problem 1P

The fundamental frequency ω0 for the periodic functions in (a) and (b) are 785.4rad/s_and78.54krad/s_ respectively.

Explanation of Solution

Calculation:

Consider that the expression for the fundamental frequency ω0.

ω0=2πT        (1)

Substitute 8ms for T in equation (1).

ω0=2π8ms=785.4rad/s

Substitute 80ms for T in equation (1).

ω0=2π80ms=78.54krad/s

Conclusion:

Thus, the fundamental frequency ω0 for the periodic functions in (a) and (b) are 785.4rad/s_and78.54krad/s_ respectively.

b.

To determine

Calculate the frequency f0 for the both periodic functions.

b.

Expert Solution
Check Mark

Answer to Problem 1P

The fundamental frequency f0 for the periodic functions in (a) and (b) are 125Hz_and12.5Hz_ respectively.

Explanation of Solution

Calculation:

Consider that the expression for the frequency f0.

f0=1T        (2)

Substitute 8ms for T in equation (2).

f0=18ms=125Hz

Substitute 80ms for T in equation (2).

f0=180ms=12.5Hz

Conclusion:

Thus, the fundamental frequency f0 for the periodic functions in (a) and (b) are 125Hz_and12.5Hz_ respectively.

c.

To determine

Calculate the Fourier co-efficient av for the both periodic functions.

c.

Expert Solution
Check Mark

Answer to Problem 1P

The Fourier co-efficient av for the periodic functions in (a) and (b) are 25Vand0_ respectively.

Explanation of Solution

Calculation:

Calculate the Fourier co-efficient av for the periodic voltage in voltage in part (a).

av=VmT(4×103)=50(4×103)8×103{Vm=50VandT=8×103s}=25V

For the periodic voltage in part (b), the Fourier co-efficient av is 0 since it is the odd function with half-wave symmetry.

Conclusion:

Thus, the Fourier co-efficient av for the periodic functions in (a) and (b) are 25Vand0_ respectively.

d.

To determine

Calculate the Fourier co-efficients akandbk for the both periodic functions.

d.

Expert Solution
Check Mark

Answer to Problem 1P

The Fourier co-efficients akandbk for the periodic function in (a) are 100πksinπk2and0_ respectively. The Fourier co-efficients akandbk for the periodic function in (b) are 120πksinπk4and120πk[1cos(πk)]_ respectively.

Explanation of Solution

Calculation:

Consider that the periodic function in Figure P16.1(a). The Fourier co-efficient av is 25 V.

Calculate the Fourier co-efficient ak using Figure P16.1(a).

ak=2TT/4T/450cos2πktTdt=100TT2πksin2πktT|T/4T/4=100πksinπk2

Calculate the Fourier co-efficient bk using Figure P16.1(a).

bk=2TT/4T/450sin2πktTdt=100TT2πkcos2πktT|T/4T/4=0

Consider that the periodic function in Figure P16.1(b). The Fourier co-efficient av is 0.

Calculate the Fourier co-efficient ak using Figure P16.1(b).

ak={2T[0T/490cos2πktTdt+T/4T/230cos2πktTdt]2T[T/23T/490cos2πktTdt+3T/4T30cos2πktTdt]}={60TT2πk[3sin2πktT|0T/4+sin2πktT|T/4T/2]60TT2πk[3sin2πktT|T/23T/4+sin2πktT|3T/4T]}=30πk[2sinπk22sin3πk2]=120πksinπk2

Calculate the Fourier co-efficient bk using Figure P16.1(b).

bk={2T[0T/490sin2πktTdt+T/4T/230sin2πktTdt]2T[T/23T/490sin2πktTdt+3T/4T30sin2πktTdt]}={60TT2πk[3cos2πktT|0T/4+cos2πktT|T/4T/2]+60TT2πk[3cos2πktT|T/23T/4+cos2πktT|3T/4T]}=120πk[1cos(kπ)]

The value of bk is 0 for even values of k and the value of bk is 120(2)πk for off values of k.

Conclusion:

Thus, the Fourier co-efficients akandbk for the periodic function in (a) are 100πksinπk2and0_ respectively. The Fourier co-efficients akandbk for the periodic function in (b) are 120πksinπk4and120πk[1cos(πk)]_ respectively.

e.

To determine

Derive the Fourier series expression for the voltage v(t).

e.

Expert Solution
Check Mark

Answer to Problem 1P

The Fourier series expression of voltage v(t) for the periodic functions in (a) and (b) are 25+100πn=1(1nsinnπ2cosnω0t)Vand120πn=1,3,51n(sinnπ2cosnω0t+2sinnω0t)V_ respectively.

Explanation of Solution

Calculation:

Write the Fourier series expression of voltage v(t) for the periodic function in (a).

v(t)=av+n=1(ancosnω0t)V=25+n=1(100nπsinnπ2cosnω0t)V{av=25,an=100nπsinnπ2}=25+100πn=1(1nsinnπ2cosnω0t)V

Write the Fourier series expression of voltage v(t) for the periodic function in (b).

v(t)=av+n=1,2,3,(ancosnω0t+bnsinnω0t)V=0+n=1,2,3,(120nπsinnπ2cosnω0t+120(2)πnsinnω0t)V{av=0,an=120nπsinnπ2,bn=120(2)πn}=0+120πn=1,3,51n(sinnπ2cosnω0t+2sinnω0t)V=120πn=1,3,51n(sinnπ2cosnω0t+2sinnω0t)V

Conclusion:

Thus, the Fourier series expression of voltage v(t) for the periodic functions in (a) and (b) are 25+100πn=1(1nsinnπ2cosnω0t)Vand120πn=1,3,51n(sinnπ2cosnω0t+2sinnω0t)V_ respectively.

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EBK ELECTRIC CIRCUITS

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