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Electric Circuits

James W. Nilsson, Susan A. Riedel

Chapter 16

Fourier Series - all with Video Answers

Educators

Chapter Questions

05:39

Problem 1

For each of the periodic functions in Fig. Pl6.1 specify
a) $\omega_{0}$ in radians per second
b) $f_{0}$ in hert
c) the value of $a_{v}$
d) the equations for $a_{k}$ and $b_{k}$
e) $v(t)$ as a Fourier series

Kajal Gautam
Kajal Gautam
Numerade Educator
01:06

Problem 2

Find the Fourier series expressions for the periodic voltage functions shown in Fig $\mathbf{P} 16.2 .$ Note that Fig $P 16.2(a)$ illustrates the square wave: $F i g . P 16.2(b)$ illustrates the full-wave rectified sine wave, where $v(r)=V_{m} \sin (\pi / T) t, 0 \leq t \leq T ;$ and $\mathrm{Fig}, \mathrm{P} 16.2(\mathrm{c})$
Illustrates the half-wave rectified sine wave, where $v(r)=V_{m} \sin (2 \pi / 7) l, 0 \leq t \leq T / 2$.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:06

Problem 3

Derive the Fourier series for the periodic voltage shown in Fig. $P 16.3,$ given that $$v(t)=200 \cos \frac{2 \pi}{T} t V, \quad \frac{-T}{4} \leq t \leq \frac{T}{4}.$$ $$v(t)=-100 \cos \frac{2 \pi}{T} t V, \quad \frac{T}{4} \leq t \leq \frac{3 T}{4}.$$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:27

Problem 4

Derive Eq. 16.5

Ajay Singhal
Ajay Singhal
Numerade Educator
02:39

Problem 5

a) Verify Eqs. 16.6 and 16.7
b) Verify Eq. 16.8 . Hint: Use the trigonometric identity $\cos \alpha \sin \beta=\frac{1}{2} \sin (\alpha+\beta)-\frac{1}{2} \sin (\alpha-\beta)$
c) Verify Eq. 16.9 . Hint: Use the trigonometric identity $\sin \alpha \sin \beta=\frac{1}{2} \cos (\alpha-\beta)-\frac{1}{2} \cos (\alpha+\beta)$
d) Verify Eq. 16.10. Hint: Use the trigonometric identity $\cos \alpha \cos \beta=\frac{1}{2} \cos (\alpha-\beta)+\frac{1}{2} \cos (\alpha+\beta)$.

Barsha Rana
Barsha Rana
Numerade Educator
02:16

Problem 6

Derive the expressions for the Fourier coefficients of an odd periodic function. Hint: Use the same technique as used in the text in deriving Eqs. $16.14-16.16$.

Kajal Gautam
Kajal Gautam
Numerade Educator
03:23

Problem 7

Show that if $f(t)=-f(t-T / 2),$ the Fourier coef ficients $b_{k}$ are given by the expressions
$b_{k}=0 \quad$ for $k$ even
$b_{k}=\frac{4}{T} \int_{0}^{T / 2} f(t) \sin k \omega_{o} t d t, \quad$ for $k$ odd Hint: Use the same technique as used in the text to derive Eqs. 16.28 and 16.29

James Kiss
James Kiss
Numerade Educator
01:50

Problem 8

Derive Eqs. $16.36 .$ Hint: Start with Eq. 16.29 and divide the interval of integration into 0 to $T / 4$ and $T / 4$ to $T / 2 .$ Note that because of evenness and quarter-wave symmetry, $f(t)=-f(T / 2-t)$ in the interval $T / 4 \leq t \leq T / 2 .$ Let $x=T / 2-t$ in the second interval and combine the resulting integral with the integration between 0 and $T / 4$.

Stephanie W
Stephanie W
Numerade Educator
06:14

Problem 9

Derive Eqs. 16.37 . Follow the hint given in Problem 16.8 except that because of oddness and quarter-wave symmetry, $f(t)=f(T / 2-t)$ in the interval $T / 4 \leq t \leq T / 2$.

Guilherme Barros
Guilherme Barros
Numerade Educator
02:48

Problem 10

a) What is the fundamental frequency in hertz?
b) Is the function even?
c) Is the function odd?
d) Does the function have half-wave symmetry?
e) Does the function have quarter-wave symmetry?
f) Give the numerical expressions for $a_{v}, a_{k}$, and $b_{k}$

Anand Jangid
Anand Jangid
Numerade Educator
01:00

Problem 11

It is given that $v(t)=20 t \cos 0.25 \pi t \mathrm{V}$ over the interval $-6 \leq t \leq 6$ s. The function then repeats itself.
a) What is the fundamental frequency in radians per second?
b) Is the function even?
c) Is the function odd?
d) Does the function have half-wave symmetry?

Mayukh Banik
Mayukh Banik
Numerade Educator
04:57

Problem 11

It is given that $v(t)=20 t \cos 0.25 \pi t \mathrm{V}$ over the interval $-6 \leq t \leq 6$ s. The function then repeats itself.
a) What is the fundamental frequency in rad per second?
b) Is the function even?
c) Is the function odd?
d) Does the function have half-wave symmetry

Joanna Josey
Joanna Josey
Numerade Educator
04:17

Problem 12

Find the Fourier series of each periodic function shown in Fig. P16.12.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:35

Problem 13

a) Derive the Fourier series for the periodic voltage shown in Fig. P16.13.
b) Repeat (a) if the vertical reference axis is shifted $T / 2$ units to the left.

Kajal Gautam
Kajal Gautam
Numerade Educator
01:00

Problem 14

It is given that $f(t)=0.4 t^{2}$ over the interval $-5<t<5 \mathrm{s}$ a) Construct a periodic function that satisfies this $f(t)$ between -5 and $+5 s,$ has a period of $20 s$ and has half-wave symmetry.
b) Is the function even or odd?
c) Does the function have quarter-wave symmetry?
d) Derive the Fourier series for $f(t)$
e) Write the Fourier series for $f(t)$ if $f(t)$ is shifted 5 s to the right.

Raj Bala
Raj Bala
Numerade Educator
02:23

Problem 15

Repeat Probiem 16.14 given that $f(t)=0.4 t^{3}$ over the interval $-5<t<5$ s.

Uma Kumari
Uma Kumari
Numerade Educator
03:16

Problem 16

The periodic function shown in Fig. $P 16.16$ is even and has both half-wave and quarter-wave symmetry.
a) Sketch one full cycle of the function over the interval $-T / 4 \leq t \leq 3 T / 4$.
b) Derive the expression for the Fourier coefficients $a_{k}$.
c) Write the first three nonzero terms in the Fourier expansion of $f(t)$.
d) Use the first three nonzero terms to estimate $f(T / 8)$.

Kajal Gautam
Kajal Gautam
Numerade Educator
03:02

Problem 17

7 It is sometimes possible to use symmetry to find the Fourier coefficients, even though the original func. tion is not symmetrical! With this thought in mind consider the function in Assessment Problem 16.1 Observe that $v(t)$ can be divided into the two functions illustrated in Fig. P16.17(a) and (b). Furthermore, we can make $v_{2}(t)$ an even function by shifting it $T / 6$ units to the right. This is illustrated in Fig. $\mathrm{P} 16.17(\mathrm{c})$. At this point we note that $v(t)=v_{1}(t)+v_{2}(t)$ and that the Fourier series of $v_{1}(t)$ is a single-term series consisting of $V_{m}$. To find the Fourier series of $v_{2}(t),$ we first find the Fourier series of $v_{2}(t-T / 6)$ and then shift this series $T / 6$ units to the left. Use the technique just outlined to verity the Fourier series given as the answer to Assessment Problem $16.2(\mathrm{e})$.

Chai Santi
Chai Santi
Numerade Educator
02:49

Problem 18

a) Derive the Fourier series for the periodic function shown in Fig. $P 16.18$ when $V_{m}=378 \pi \mathrm{V}$ Write the series in the form of Eq. 16.38
b) Use the first five nonzero terms to estimate $v(T / 8)$

Kajal Gautam
Kajal Gautam
Numerade Educator
04:17

Problem 19

For cach of the periodic functions in Fig. $P 16.1$ derive the Fourier series for $v(t)$ using the form of Eq. 16.38

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:17

Problem 20

Derive the Fourier series for the periodic function described in Problem $16.10,$ using the form of Eq. 16.38

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:17

Problem 21

Derive the Fourier series for the periodic function constructed in Problem 16.14 , using the form of Eq. 16.38

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:39

Problem 22

Derive Eqs. 16.69 and 16.70.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:56

Problem 23

$\begin{array}{lllll}\text { a) Derive } & \text { Eq. } & 16.71 . & \text { Hint: } & \text { Note that } & b_{k}=\end{array}$ $4 V_{m} / \pi k+k \omega_{o} R C a_{k} .$ Use this expression for $b_{k}$ to find $a_{k}^{2}+b_{k}^{2}$ in terms of $a_{k}$. Then use the expression for $a_{k}$ to derive Eq. 16.71
b) Derive Eq. 16.72

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
00:53

Problem 24

Show that when we combine Eqs. 16.71 and 16.72 with Eqs. 16.38 and $16.39,$ the result is Eq. 16.58 Hint: Note from the definition of $\beta_{k}$ that \[
\frac{a_{k}}{b_{k}}=-\tan \beta_{k}
\]
and from the definition of $\theta_{k}$ that
\[\tan \theta_{k}=-\cot \beta_{k}
\]Now use the trigonometric identity
\[\tan x=\cot (90-x)\]
to show that $\theta_{k}=\left(90+\beta_{k}\right)$.

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
03:08

Problem 25

a) Show that for large values of $C,$ Eq. 16.67 can be approximated by the expression $$v_{o}(t) \approx \frac{-V_{m} T}{4 R C}+\frac{V_{m}}{R C} t$$ Note that this expression is the equation of the triangular wave for $0 \leq t \leq T / 2$ Hints: (1)$\quad$ Let $\quad e^{-i / R C} \approx 1-(t / R C) \quad$ and $e^{-T / 2 R C} \approx 1-(T / 2 R C) ;(2)$ put the resulting expression over the common denominator $2-(T / 2 R C) ;(3)$ simplify the numerator; and
(4) for large $C,$ assume that $T / 2 R C$ is much less than 2
b) Substitute the peak value of the triangular wave into the solution for Problem 16.12 (see Fig. $\mathrm{P} 16.12(\mathrm{b})$ and show that the result is Eq. 16.59

Sachin Rao
Sachin Rao
Numerade Educator
02:56

Problem 26

The square-wave voltage shown in Fig. P16.26(a) is applied to the circuit shown in Fig. $P 16.26(b)$
a) Find the Fourier series representation of the steady-state current $i$.b) Find the steady-state expression for $i$ by straight forward circuit analysis.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:39

Problem 26

The square-wave voltage shown in Fig. $\mathrm{P} 16.26(\mathrm{a})$ is applied to the circuit shown in Fig. $P 16.26(b)$
a) Find the Fourier series representation of the steady-state current $i$
b) Find the steady-state expression for $i$ by straight. forward circuit analysis.

Ryan Hood
Ryan Hood
Numerade Educator
01:24

Problem 27

The periodic square-wave voltage seen in Fig $P 16.27(a)$ is applied to the circuit shown in He $\mathbf{Y} 16.27(\mathrm{b}) .$ Derive the first three nonzero terms in the I rourier series that represents the steady-state voltage $v_{0}$ if $V_{m}=60 \pi \mathrm{V}$ and the period of the input voltage is $\pi$ ms.

Carson Merrill
Carson Merrill
Numerade Educator
03:35

Problem 27

The periodic square-wave voltage seen in Fig. $\mathrm{P} 16.27(\mathrm{a})$ is applied to the circuit shown in
in the fourier series that represents the steady-state voltage $v_{0}$ if $V_{m}=60 \mathrm{m} \mathrm{V}$ and the period of the inpul voltape is $\pi \mathrm{ms}$

Kajal Gautam
Kajal Gautam
Numerade Educator
03:35

Problem 28

The periodic square-wave voltage described in Assessment Problem 16.6 is applied to the circuit shown in Fig. $P 16.28$
a) Derive the first four nonzero terms in the Fourier series that represents the steady-state voltage $v_{o}$
b) Which frequency component in the input volt. age is eliminated from the output voltage? Explain why.

Kajal Gautam
Kajal Gautam
Numerade Educator
03:35

Problem 28

The periodic square-wave voltage described in Assessment Problem 16.6 is applied to the circuit shown in Fig. $P 16.28$
a) Derive the first four nonzero terins in the Fourier series that represents the steady-state voltage $v_{o}$
b) Which frequency component in the input volt. age is eliminated from the output voltage? Explain why.

Kajal Gautam
Kajal Gautam
Numerade Educator
01:27

Problem 29

The full-wave rectified sine-wave voltage shown in Fig. $P 16.29(a)$ is applied to the circuit shown in Fig. $P 16.29(b)$
a) Find the first four nonzero terms in the Founiet series representation of $v_{o^{\prime}}$
b) Does your solution for $v_{0}$ make sense? Explain.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:31

Problem 29

The full-wave rectified sine-wave voltage shown in
Fig. $P 16.29(a)$ is applied to the circuit shown
Tig. P16.29(b).
Find the first four nonzero terms in the Founict series representation of $v_{o}$
boes your solution for $v_{o}$ make sense? Explain.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:12

Problem 30

The periodic current shown in Fig $\mathrm{P} 16.30(\mathrm{a})$ is used to energize the circuit shown in Fig. $\mathrm{P} 16.30(\mathrm{b})$ Write the time-domain expression for the fifthharmonic current in the expression for $i_{o}$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:12

Problem 30

The periodic current shown in Fig $\mathrm{P} 16.30$ (a) is used to energize the circuit shown in Fig. $\mathrm{P} 16.30(\mathrm{b})$ Write the time-domain expression for the fifthharmonic current in the expression for $i_{o}$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:35

Problem 31

A periodic voltage having a period of $0.1 \pi \mathrm{ms}$ is given by the following Fourier series:$$v_{g}=45 \sum_{n=1,3,5 \ldots . .}^{\infty} \frac{\pi^{2} n^{2}-8}{n^{3}} \sin \frac{n \pi}{2} \cos n \omega_{o} t \mathrm{V}$$ This periodic voltage is applied to the circuit shown in Fig. P16.31. Find the amplitude and phase angle of the component of $v_{o}$ that has a frequency of $300 \mathrm{krad} / \mathrm{s}$

Kajal Gautam
Kajal Gautam
Numerade Educator
03:35

Problem 31

A periodic voltage having a period of $0.1 \pi \mathrm{ms}$ is $F$
given by the following Fourier series:$$v_{g}=45 \sum_{n=1,3,5 \ldots}^{\infty} \frac{\pi^{2} n^{2}-8}{n^{3}} \sin \frac{n \pi}{2} \cos n \omega_{o} t \mathrm{V}$$ This periodic voltage is applied to the circuit shown in Fig. $P 16.31 .$ Find the amplitude and phase angle of the component of $v_{o}$ that has a frequency of $300 \mathrm{krad} / \mathrm{s}$.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:56

Problem 32

The periodic voltage across a $81 \pi^{2} \mathrm{k} \Omega$ resistor is shown in Fig. P16.32.
a) Use the first four nonzero terms in the Fourier series representation of $v(t)$ to estimate the average power dissipated in the resistor.
b) Calculate the exact value of the average power dissipated in the $81 \pi^{2} \mathrm{k} \Omega$ resistor.
c) What is the percentage error in the estimated value of the average power dissipated?

Mohit Khurana
Mohit Khurana
Texas A&M University
02:56

Problem 32

The periodic voltage across a $81 \pi^{2} \mathrm{k} \Omega$ resistor is shown in Fig. P16.32.
a) Use the first four nonzero terms in the Fourier series representation of $v(t)$ to estimate the average power dissipated in the resistor.
b) Calculate the exact value of the average power dissipated in the $81 \pi^{2} \mathrm{k} \Omega$ resistor.
c) What is the percentage error in the estimated value of the average power dissipated?

Mohit Khurana
Mohit Khurana
Texas A&M University
03:32

Problem 33

The triangular-wave voltage source is applied to the circuit in Fig. $P 16.33(a) .$ The triangular-wave voltage is shown in Fig. $P 16.33(b) .$ Estimate the average power delivered to the $20 \mathrm{kP}$ resistor when the circuit is in steady-state operation.

Kajal Gautam
Kajal Gautam
Numerade Educator
01:24

Problem 33

The triangular-wave voltage source is applied to the circuit in Fig. $\mathrm{P} 16.33(\mathrm{a})$. The triangular-wave voltage is shown in Fig, $P 16.33$ (b). Estimate the average power delivered to the $20 \mathrm{k} \Omega$ resistor when the circuit is in steady-state operation.

Anurag Kumar
Anurag Kumar
Numerade Educator
02:37

Problem 34

The periodic current shown in Fig. P16.34 is applied
to a $1 \mathrm{k} \Omega$ resistor.
a) Use the first three nonzero terms in the Fourier series representation of $i(t)$ to estimate the average power dissipated in the $1 \mathrm{k} \Omega$ resistor.
b) Calculate the exact value of the average power dissipated in the $1 \mathrm{k} \Omega$ resistor.
c) What is the percentage of error in the estimated 16.37 value of the average power?

Narayan Hari
Narayan Hari
Numerade Educator
02:23

Problem 34

The periodic current shown in Fig. P16.34 is applied
to a $1 \mathrm{k} \Omega$ resistor.
a) Use the first three nonzero terms in the Fourier series representation of $i(t)$ to estimate the average power dissipated in the 1 k $\Omega$ resistor.
b) Calculate the exact value of the average power dissipated in the $1 \mathrm{k} \Omega$ resistor.
c) What is the percentage of error in the estimated value of the a

Km Neeraj
Km Neeraj
Numerade Educator
02:23

Problem 35

38 a) Estimate the rms value of the full-wave rectified sinusoidal voltage shown in Fig. $P 16.38(a)$ by using the first three nonzero terms in the Fourier series representation of $v(t)$
b) Calculate the percentage of error in the estimation (see Problem 16.37).
c) Repeat (a) and (b) if the full-wave rectified sinusoidal voltage is replaced by the half-wave rectified sinusoidal voltage shown in Fig. $\mathrm{P} 16.38(\mathrm{b})$
e P1G 38

Narayan Hari
Narayan Hari
Numerade Educator
03:02

Problem 36

16 a) Find the rms value of the voltage shown in Fig. $P 16.36$ for $V_{m}=100 \mathrm{V} .$ Note that the Fourier series for this periodic volkge was found in Assessment Problem 16.3
b) Estimate the rms value of the voltage, using the first three nonzero terms in the Fourier series representation of $v_{g}(t)$

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:47

Problem 37

a) Estimate the rms value of the periodic square wave voltage shown in Fig. $\mathrm{P} 16.37(\mathrm{a})$ by using the first five nonzero terms in the Fourier series representation of $v(t)$
b) Calculate the percentage of error in the estimation if
$\%$ error $=\left[\frac{\text { cstimated value }}{\text { exact value }}-1\right] \times 100$
c) Repeat parts $(a)$ and $(b)$ if the periodic squart Wave voltage is replaced by the periodic triang lar voltage shown in $\mathrm{Fig}, \mathrm{P} 16.37(\mathrm{b})$

Narayan Hari
Narayan Hari
Numerade Educator
02:23

Problem 38

38 a) Estimate the rms value of the full-wave rectified sinusoidal voltage shown in Fig. P16.38(a) by using the first three nonzero terms in the Fourier series representation of $v(t)$
b) Calculate the percentage of error in the estimation (see Problem 16.37 ).
c) Repeat (a) and (b) if the full-wave rectified sinusoidal voltage is replaced by the half-wave rectified sinusoidal voltage shown in Fig. $\mathrm{P} 16.38(\mathrm{b})$

Narayan Hari
Narayan Hari
Numerade Educator
02:49

Problem 39

a) Derive the expressions for the Fourier coefficients for the periodic current shown in Fig. $P 16.39$
b) Write the first four nonzero terms of the series using the alternative trigonometric form given by $\mathrm{kg}, 16.39$ c) Use the first four nonzero terms of the expression derived in (b) to estimate the rms value of $i_{x}$
d) Find the exact rms value of $i_{r}$
e) Calculate the percentage of error in the estimated rms value.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:49

Problem 40

a) Use the first four nonzero terms in the Fourier series approximation of the periodic voltage shown in Fig. $\mathrm{P} 16.40$ to estimate its rms value
b) Calculate the true rms value of the voltage.
c) Calculate the percentage of error in the estimated value.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:11

Problem 41

Assame the periodic function described in Problem 16.16 is a current $i_{g}$ with a peak ampli tude of 2 A.
a) Find the rms value of the current.
b) If this current exists in a $54 \Omega$ resistor, what is the average power dissipated in the resistor?
c) If $i_{\mathrm{g}}$ is approximated by using just the fundamen tal frequency term of its Fourier series, what is the average power delivered to the $54 \Omega$ resistor?
What is the percentage of error in the estimation of the power dissipated?

Kajal Gautam
Kajal Gautam
Numerade Educator
01:06

Problem 42

The rms value of any periodic triangular wave having the form depicted in Fig. $P 16.42(a)$ is independent of $t_{a}$ and $t_{b} .$ Note that for the function to be single valued, $t_{a} \leq t_{b}$. The rms value is equal to $V_{p} / \sqrt{3} .$ Verify this observation by finding the rms value of the three waveforms depicted in Fig. $P 16.42(b)-(d)$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:56

Problem 43

Use the exponential form of the Fourier series to write an expression for the voltage shown in Fig. $P 16.43$

Kajal Gautam
Kajal Gautam
Numerade Educator
00:55

Problem 44

Derive the expression for the complex Fourier coefficients for the periodic current shown in Fig. $P 16.44$

Kajal Gautam
Kajal Gautam
Numerade Educator
07:57

Problem 45

a) The periodic current in Problem 16.44 is applied to a $60 \Omega$ resistor. If $I_{m}=20$ A what is the average power delivered to the resistor?
b) Assume $i(t)$ is approximated by a truncated exponential form of the Fourier series consisting of the first seven nonzero terms, that is $n=0,1,2,3,4,5,6$ and $7 .$ What is the rms value of the current, using this approximation?
c) If the approximation in part (b) is used to represent $i$ what is the percentage of error in the cal culated power?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:04

Problem 46

The periodic voltage source in the circuit shown in Fig. $P 16.46(a)$ has the waveform shown in Fig. $P 16.46(b)$
a) Derive the expression for $C_{n}$
b) Find the values of the complex coefficients $C_{0}, C_{-1}, C_{1}, C_{-2}, C_{2}, C_{-3}, C_{3}, C_{-4},$ and $C_{4}$ for the
input voltage $v_{R}$ if $V_{m}=72 \pi \mathrm{V}$ and $T=50 \pi \mu \mathrm{s}$
c) Repeat (b) for $v_{p^{*}}$
d) Usc the complex coefficients found in (c) to estimate the average power delivered to the $200 \mathrm{kg}$ resintor.

Donald Albin
Donald Albin
Numerade Educator
02:47

Problem 47

a) Find the rms value of the periodic voltage in Fig. $P 16.46(b)$
b) Use the complex coefficients derived in Problem $16.46(\mathrm{b})$ to estimate the rms value of $v_{R}$
c) What is the percentage of error in the estimated rms value of $v_{g} ?$

Narayan Hari
Narayan Hari
Numerade Educator
01:44

Problem 48

a) Make an amplitude and phase plot, based on Eq. 16.38 , for the periodic voltage in Example 16.3 Assume $V_{m}$ is $40 \mathrm{V}$. Plot both amplitude and phase versus $n \omega_{0},$ where $n=0,1,2,3, \dots$
b) Repeat
(a), but base the plots on Eq. 16.82

Narayan Hari
Narayan Hari
Numerade Educator
06:34

Problem 49

a) Make an amplitude and phase plot, based Eq. $16.38,$ for the periodic voltage Problem $16.32 .$ Plot both amplitude and phase versus $n \omega_{0}$ where $n=0,1,2, \dots$
b) Repeat
(a), but base the plots on Eq. 16.82 .

Kajal Gautam
Kajal Gautam
Numerade Educator
07:52

Problem 50

A periodic voltage is represented by a truncated Fourier scries. The amplitude and phase spectra are shown in Fig. $P 16.50(a)$ and $(b),$ respectively.
a) Write an expression for the periodic voltage using the form given by Eq. 16.38
b) Is the voltage an even or odd function of $t ?$
c) Does the voltage have half-wave symmetry?
d) Does the voltage have quarter-wave symmetry?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:49

Problem 51

A periodic function is represented by a Fourier series that has a finite number of terms, The amplitude and phase spectra are shown in Fig. P16.51(a) and (b), respectively.
a) Write the expression for the periodic current using the form given by Eq. 16.38
b) Is the current an even or odd function of $n$
c) Does the current have half-wave symmetry?
d) Calculate the rms value of the current in milliamperes.
e) Write the exponential form of the Fourier series.
1) Make the amplitude and phase spectra plots on the basis of the exponential series

Kajal Gautam
Kajal Gautam
Numerade Educator
View

Problem 52

The input signal to a third-order low.pas Butterworth filter is the periodic triangular-wale voltage shown in Fig P16.52. The corner frequency of the filter is 1 rad/s. Write the first three termso the Fourier series that represents the steadystate output voltage of the filter.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:11

Problem 53

The input signal to a second-order low-pass Figure $P$ Butterworth filter is a full-wave rectified sine wave with an amplitude of $2.5 \pi \mathrm{V}$ and a fundamental frequency of 5000 rad/s. The corner frequency of the filter is $1 \mathrm{krad} / \mathrm{s}$. Write the first two terms in the Fourier series that represents the steady-state output voltage of the filter.

Narayan Hari
Narayan Hari
Numerade Educator
08:11

Problem 54

The transfer function $\left(V_{a} / V_{g}\right)$ for the narrowband bandpass filter circuit in Fig. $\mathrm{P} 16.54(\mathrm{a})$ is
\[
H(s)=\frac{-K_{o} \beta s}{s^{2}+\beta s+\omega_{o}^{2}}
\]
a) Find $K_{o}, \beta,$ and $\omega_{o}^{2}$ as functions of the circuit parameters $R_{1}, R_{2}, R_{3}, C_{1},$ and $C_{2}$
b) Write the first three terms in the Fourier series that represents $v_{o}$ if $v_{g}$ is the periodic voltage in Fig. $P 16.54(b)$

Amit Srivastava
Amit Srivastava
Numerade Educator