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Principles and Applications of Electrical Engineering

Giorgio Rizzoni, James Kearns

Chapter 5

Frequency Response and System Concepts - all with Video Answers

Educators

Chapter Questions

03:15

Problem 1

Use trigonometric identities to show that the equalities in equations 5.27 and 5.28 hold.

Anas Venkitta
Anas Venkitta
Numerade Educator
31:03

Problem 2

Derive a general expression for the Fourier series coefficients of the square wave of Figure 5.15(a) in the text.

Mark Mathison
Mark Mathison
Numerade Educator

Problem 3

Compute the Fourier series coefficient of the periodic function shown in Eigure P5.3 and defined as:
$$x(t)= \begin{cases}0 & 0 \leq t \leq \xi \\ A & \xi \leq t \leq T\end{cases}$$
(FIGURE CAN'T COPY)
Figure P5.3

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Problem 4

Compute the Fourier series coefficient of the periodic function shown in Eigure P5.4 and defined as
(FIGURE CAN'T COPY)
Figure P5.4

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Problem 5

Compute the Fourier series expansion of the function shown in Figure P5.5, and express it in sine-cosine ( $a_n, b_n$ coefficients) form.
(FIGURE CAN'T COPY)
Figure P5.5

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Problem 6

Compute the Fourier series expansion of the function shown in Eigure P5.6, and express it in sine-cosine ( $a_n, b_n$ coefficients) form.
$$
x(t)=\left\{\begin{aligned}
\sin \left(\frac{2 \varepsilon}{T} t\right) & 0 \leq t<\frac{T}{2} \\
0 & \frac{T}{2} \leq t<T
\end{aligned}\right.
$$
(FIGURE CAN'T COPY)
Figure P5.6

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Problem 7

Write an expression for the signal shown in Figure P5.7, and derive a complete expression for its Fourier series.
(FIGURE CAN'T COPY)
Figure P5.7

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Problem 8

Write an expression for the signal shown in Figure P5.8 and derive its Fourier series.
(FIGURE CAN'T COPY)
Figure P5.8

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Problem 9

Find the Fourier series for the periodic function shown in Eigure P5.9. Determine integral expressions for the Fourier coefficients.
(FIGURE CAN'T COPY)
Figure P5.9

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Problem 10

Find the Fourier series for the periodic function shown in Figure P5.10. Determine integral expressions for the Fourier coefficients.
(FIGURE CAN'T COPY)
Figure P5.10

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Problem 11

a. Determine the frequency response $V_{\text {out }}(j \omega) / V_{\text {in }}(j \omega)$ for the circuit of Eigure P5.11. Assume $L=0.5 \mathrm{H}$ and $R=200 \mathrm{k} \Omega$.
b. Plot the magnitude and phase of the circuit for frequencies between 10 and $10^7 \mathrm{rad} / \mathrm{s}$ on graph paper, with a linear scale for frequency.
c. Repeat part b, using semilog paper. (Place the frequency on the logarithmic axis.)
d. Plot the magnitude response on semilog paper with magnitude in decibels.
(FIGURE CAN'T COPY)
Figure P5.11

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Problem 12

Repeat the instructions of Problem 5.11 for the circuit of Figure P5.12.
(FIGURE CAN'T COPY)
Figure P5.12

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Problem 13

Repeat the instructions of Problem 5. 11 for the circuit of Eigure 5.13.
(FIGURE CAN'T COPY)
Figure P5.13

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Problem 14

In the circuit shown in Figure P5.14, where $C=0.5 \mu \mathrm{~F}$ and $R=2 \mathrm{k} \Omega$,
a. Determine how the input impedance $\mathbf{Z}(j \omega)=\mathbf{V}_i(j \omega) / \mathbf{I}_i(j \omega)$ behaves at extremely high and low frequencies.
b. Find an expression for that impedance.
c. Show that this expression can be manipulated into the form $\mathbf{Z}(j \omega)=R[1- j(1 / \omega R C)]$.
d. Determine the frequency $\omega=\omega_C$ for which the imaginary part of the expression in part c is equal to 1 .
e. Estimate (without computing it) the magnitude and phase angle of $\mathbf{Z}(j \omega)$ at $\omega=10 \mathrm{rad} / \mathrm{s}$ and $\omega=10^5 \mathrm{rad} / \mathrm{s}$.
(FIGURE CAN'T COPY)
Figure P5.14

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Problem 15

In the circuit shown in Eigure P5.15, where $L=2 \mathrm{mH}$ and $R=2 \mathrm{k} \Omega$,
a. Determine how the input impedance $\mathbf{Z}(j \omega)=\mathbf{V}_i(j \omega) / \mathbf{I}_i(j \omega)$ behaves at extremely high and low frequencies.
b. Find an expression for the impedance.
c. Show that this expression can be manipulated into the form $\mathbf{Z}(j \omega)=R[1+ j(\omega L / R)]$.
d. Determine the frequency $\omega=\omega_C$ for which the imaginary part of the expression in part c is equal to 1 .
e. Estimate (without computing it) the magnitude and phase angle of $\mathbf{Z}(j \omega)$ at $\omega=10^5 \mathrm{rad} / \mathrm{s}, 10^6 \mathrm{rad} / \mathrm{s}$, and $10^7 \mathrm{rad} / \mathrm{s}$.
(FIGURE CAN'T COPY)
Figure P5.15

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05:39

Problem 16

In the circuit of Eigure P5 .16:
$$
\begin{array}{ll}
R_1=1.3 \mathrm{k} \Omega & R_2=1.9 \mathrm{k} \Omega \\
C=0.5182 \mu \mathrm{~F} &
\end{array}
$$
Determine:
a. How the voltage frequency response function
$$\mathbf{H}_V(j \omega)=\frac{\mathbf{V}_o(j \omega)}{\mathbf{V}_i(j \omega)}$$
behaves at extremes of high and low frequencies.
b. An expression for the voltage frequency response function and show that it can be manipulated into the form
$$\mathbf{H}_0(j a t)=\frac{H_0}{1+j f(a \theta)}$$
where
$$H_o=\frac{R_2}{R_1+R_2} \quad f(\omega)=\omega R_r C \quad R_r=\frac{R_1 R_2}{R_1+R_2}$$
c. The frequency at which $f(\omega)=1$ and the value of $H_o$ in decibels.
(FIGURE CAN'T COPY)
Figure P5.16

Narayan Hari
Narayan Hari
Numerade Educator

Problem 17

In the circuit shown in Eigure P5.17, determine the frequency response function in the form:
(FIGURE CAN'T COPY)
Figure P5.17

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Problem 18

The circuit shown in Figure P5.18 has
$$
\begin{array}{ll}
R_1=100 \Omega & R_2=100 \Omega \\
R_2=50 \Omega & C=80 \mathrm{nF}
\end{array}
$$
Determine the frequency response $\mathbf{V}_o(j \omega) / \mathbf{V}_{\text {in }}(j \omega)$.
(FIGURE CAN'T COPY)
Figure P5.18

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Problem 19

a. Determine the frequency response $\mathbf{V}_{\text {out }}(j \omega) / \mathbf{V}_{\text {in }}(j \omega)$ for the circuit of Figure P5.19.
b. Plot the magnitude and phase of the circuit for frequencies between 1 and $100 \mathrm{rad} / \mathrm{s}$ on graph paper, with a linear scale for frequency.
c. Repeat part b, using semilog paper. (Place the frequency on the logarithmic axis.)
d. Plot the magnitude response on semilog paper with magnitude in dB.
(FIGURE CAN'T COPY)
Figure P5.19

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Problem 20

Consider the circuit shown in Figure P5.20. Use the values for $R$ and $L$ given in Problem 5.15.
a. Sketch the amplitude response of $\boldsymbol{Y}=\mathbf{I} / \boldsymbol{V}_S$.
b. Sketch the amplitude response of $\boldsymbol{V}_1 / \boldsymbol{V}_S$.
c. Sketch the amplitude response of $\boldsymbol{V}_2 / \boldsymbol{V}_S$.
(FIGURE CAN'T COPY)
Figure P5.20

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01:01

Problem 21

Using a $15-\mathrm{k} \Omega$ resistance, design an $R C$ high-pass filter with a breakpoint at 200 kHz .

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 22

Using a $500-\Omega$ resistance, design an $R C$ low-pass filter that would attenuate a $120-\mathrm{Hz}$ sinusoidal voltage by 20 dB with respect to the DC gain.

Amit Srivastava
Amit Srivastava
Numerade Educator
08:15

Problem 23

At what frequency is the phase shift introduced by the circuit of Example 5.6 equal to $-10^{\circ}$ ?

Ramesh Singh
Ramesh Singh
Numerade Educator
05:25

Problem 24

At what frequency is the output of the circuit of Example 5.6 attenuated by 10 percent (that is, $V_o=0.9 V_i$ )?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 25

Assume the filter shown in Figure 5.11 is excited by the first two Fourier components of the sawtooth waveform in Example 5.3. Determine the output of the filter, and plot the input and output waveforms on the same graph. Assume the period $T=10 \mu \mathrm{~s}$ and the peak amplitude $A=1$ for the sawtooth waveform.

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Problem 26

Repeat Problem 5.25 with the square wave of Figure 5.15(a) as the input.

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Problem 27

Repeat Problem 5.25 for the pulse train of Example 5.4 as the input.

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Problem 28

Assume the circuit shown in Figure P5.12 is excited by the first three Fourier components of the sawtooth waveform in Example 5.3. Determine the output of the filter, and plot the input and output waveforms on the same graph. Assume $T =0.5 \mathrm{~s}$ and $A=2$ for the sawtooth waveform.

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Problem 29

Repeat Problem 5.28 with the square wave of Figure 5.15(a) as the input.

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Problem 30

Repeat Problem 5.28 with the pulse train of Example 5.4 as the input.

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Problem 31

Assume the filter shown in Figure P5.13 is excited by the first four Fourier components of the sawtooth waveform in Example 5.3. Determine the output of the filter, and plot the input and output waveforms on the same graph. Assume $T =0.1 \mathrm{~s}$ and $A=1$ for the sawtooth waveform.

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Problem 32

Repeat Problem 5.31 with the square wave of Figure 5.15(a) as the input.

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Problem 33

Repeat Problem 5.31 with the pulse train of Example 5.4 as the input.

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Problem 34

Repeat Problem 5. 11 for the circuit of Eigure P5.34. $R_1=300 \Omega, R_2=R_3=500 \Omega, L=4 \mathrm{H}, C_1=40 \mu \mathrm{~F}, C_2=160 \mu \mathrm{~F}$.
(FIGURE CAN'T COPY)
Figure P5.34

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Problem 35

Determine the frequency response of the circuit of Figure P5.35, and generate frequency response plots. $R_1=20 \mathrm{k} \Omega, R_2=100 \mathrm{k} \Omega, L=1 \mathrm{H}, C=100 \mu \mathrm{~F}$.
(FIGURE CAN'T COPY)
Figure P5.35

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Problem 36

In the circuit shown in Figure P5.36, if
$$
\begin{array}{ll}
L=190 \mathrm{mH} & R_1=2.3 \mathrm{k} \Omega \\
C=55 \mathrm{nF} & R_2=1.1 \mathrm{k} \Omega
\end{array}
$$
Determine how the input impedance behaves at extremely high or low
a. frequencies.
b. Find an expression for the input impedance in the form
$$
\begin{aligned}
Z(j \omega) & =Z_0\left[\frac{\left.1+j f_1(\omega)\right)}{1+j f_2(\omega)}\right] \\
Z_1 & =R_1+\frac{L}{R_2 C} \\
f_1(\omega) & =\frac{\omega^2 R_1 L C-R_1-R_2}{\omega\left(R_1 R_2 C+L\right)} \\
f_2(\omega) & =\frac{\omega^2 L C-1}{\omega C R}
\end{aligned}
$$
c. Determine the four frequencies at which $f_1(\omega)=+1$ or -1 and $f_2(\omega)=+1$ or -1 .
d. Plot the impedance (magnitude and phase) versus frequency.
(FIGURE CAN'T COPY)
Figure P5.36

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Problem 37

The circuit shown in Figure P5.37 is a second-order circuit because it has two reactive components ( L and $C$ ). A complete solution will not be attempted. However, determine:
a. The behavior of the voltage frequency response at extremely high and low frequencies.
b. The output voltage $\mathbf{V}_o$ if the input voltage has a frequency where:
$$
\begin{array}{ll}
\mathbf{V}_i=7.07 \angle \frac{\pi}{4} \mathbf{v} \quad R_1=2.2 \mathrm{k} \Omega \\
R_2=3.8 \mathrm{k} \Omega & X_c=5 \mathrm{k} \Omega \quad X_L=1.25 \mathrm{k} \Omega
\end{array}
$$
c. The output voltage if the frequency of the input voltage doubles so that
$$x_C=2.5 \mathrm{k} \Omega \quad x_L=2.5 \mathrm{k} \Omega$$
d. The output voltage if the frequency of the input voltage again doubles so that
$$X_c=1.25 \mathrm{k} \Omega \quad X_L=5 \mathrm{k} \Omega$$
(FIGURE CAN'T COPY)
Figure P5.37

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:39

Problem 38

In an RLC circuit, assume $\omega_1$ and $\omega_2$ such that $1\left(j \omega_1\right)-1\left(j \omega_2\right)-I_{\text {ma }} / \sqrt{2}$ and $\Delta \omega$ such that $\Delta \omega=\omega_2-\omega_1$. In other words, $\Delta \omega$ is the bandwidth of the current curve where the current has fallen to $1 / \sqrt{2}-0.707$ of its maximum value at the resonance frequency. At these frequencies, the power dissipated in a resistance becomes one-half of the dissipated power at the resonance frequency. In an RLC circuit with a high-quality factor, show that $Q=\omega_0 / \Delta \omega$.

MG
Miguel Angel Garcia Chavez
Numerade Educator
06:27

Problem 39

In an RLC circuit with a high quality factor:
a. Show that the impedance at the resonance frequency becomes a value of $Q$ times the inductive resistance at the resonance frequency.
b. Determine the impedance at the resonance frequency, assuming $L=280 \mathrm{mH}, C=0.1 \mu \mathrm{~F}, R=25 \Omega$.

Vishal Gupta
Vishal Gupta
Numerade Educator
05:25

Problem 40

At what frequencies is the output of the circuit of Example 5.10 attenuated by 10 percent (that is, $V_o=0.9 V_i$ )?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 41

At what frequencies is the phase shift introduced by the circuit of Example 5.10 equal to $20^{\circ}$ ?

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Problem 42

Assume the filter shown in Eigure P5.34 is excited by the first two Fourier components of the sawtooth waveform in Example 5.3. Determine the output of the filter, and plot the input and output waveforms on the same graph. Assume $T =50 \mathrm{~ms}$ and $A=2$ for the sawtooth waveform.

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Problem 43

Repeat Problem 5.42 for $T=0.5 \mathrm{~s}$ and 5 ms , and compare the results with $T=$ 50 ms .

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Problem 44

Repeat Problem 5.42 for the square wave of Figure 5.15(a).

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Problem 45

Repeat Problem 5.42 with the pulse train of Example 5.4 as the input.

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Problem 46

Assume the filter shown in Eigure P5.35 is excited by the first three Fourier components of the sawtooth waveform in Example 5.3 Determine the output of the filter, and plot the input and output waveforms on the same graph. Assume $T =5 \mathrm{~s}$ and $A=1$ for the sawtooth waveform.

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Problem 47

Repeat Problem 5.46 for $T=50 \mathrm{~s}$, and compare the results.

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Problem 48

Repeat Problem 5.46 with the square wave of Eigure 5.15(a) as the input.

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Problem 49

Repeat Problem 5.46 with the pulse train of Example 5.4 as the input.

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Problem 50

Consider the circuit shown in Figure P5.50. Determine the resonant frequency and the bandwidth for the circuit.
(FIGURE CAN'T COPY)
Figure P5.50

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Problem 51

Are the filters shown in Figure P5.51 low-pass, high-pass, bandpass, or bandstop (notch) filters?
(FIGURE CAN'T COPY)
Figure P5.51

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Problem 52

Determine if each of the circuits shown in Figure P5.52 is a low-pass, high-pass, bandpass, or bandstop (notch) filter.
(FIGURE CAN'T COPY)
Figure P5.52

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Problem 53

For the filter circuit shown in Figure P5.53:
a. Determine if this is a low-pass, high-pass, bandpass, or bandstop filter.
b. Determine the frequency response $\mathbf{V}_o(j \omega) / \mathbf{V}_{\mathrm{i}}(j \omega)$ assuming $L=10 \mathrm{mH}, C= 1 \mathrm{nF}, R_1=50 \Omega, R_2=2.5 \mathrm{k} \Omega$
(FIGURE CAN'T COPY)
Figure P5.53

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Problem 54

In the filter circuit shown in Eigure P5.54: $L=10 \mathrm{H}, C=1 \mathrm{nF}, R_S=20 \Omega, R_{\mathrm{C}}= 100 \Omega, R_o=5 \mathrm{k} \Omega$ Determine the frequency response $\mathbf{V}_o(j \omega) / \mathbf{V}_i(j \omega)$. What type of filter does this frequency response represent?
(FIGURE CAN'T COPY)
Figure P5.54

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Problem 55

In the filter circuit shown in Eigure P5.54: $L=0.1 \mathrm{mH}, C=8 \mathrm{nF}, R_S=300 \Omega$, $R_C=10 \Omega, R_o=500 \Omega$. Determine the frequency response $\mathbf{V}_o(j \omega) / \mathbf{V}_i(j \omega)$. What type of filter does this frequency response represent?

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Problem 56

In the filter circuit shown in Eigure P5.56:
$$
\begin{array}{ll}
R_s=5 \mathrm{k} \Omega & C=56 \mathrm{nF} \\
R_e=100 \mathrm{k} \Omega & L=9 \mu \mathrm{H}
\end{array}
$$
Determine:
a. The voltage frequency response
$$G_V(j \omega)=\frac{V_i(j \omega)}{V_i(j \omega)}$$
b. The resonant frequency.
c. The half-power frequencies.
d. The bandwidth and $Q$.
(FIGURE CAN'T COPY)
Figure P5.56

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Problem 57

In the filter circuit shown in Figure P5.56:
$$
\begin{array}{ll}
R_s=5 \mathrm{k} \Omega & C=0.5 \mathrm{nF} \\
R_e=100 \mathrm{k} \Omega & L=1 \mathrm{mH}
\end{array}
$$
Determine:
a. The voltage frequency response
$$
G_V(j \omega)=\frac{V_o(j \omega)}{V_i(j \omega t)}
$$
b. The resonant frequency.
c. The half-power frequencies.
d. The bandwidth and $Q$.

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Problem 58

In the filter circuit shown in Eigure P5.58:
$$
\begin{array}{ll}
R_S=500 \Omega & R_0=51 \Omega \\
R_C=4 \mathrm{k} \Omega & L=1 \mathrm{mH} \\
C=5 \mathrm{pF}^F &
\end{array}
$$
Determine the frequency response $\mathbf{G}_1(j \omega)$, where:
$$G_v(j o t)=\frac{V_o(j \omega)}{V_i(j \omega)}$$
What type of filter does this frequency response represent?
(FIGURE CAN'T COPY)
Figure P5.58

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Problem 59

In the notch filter circuit shown in Eigure P5.59, derive the voltage frequency response $\mathbf{G}_V(j \omega)$ in standard form, where:
$$G_v(j \omega)=\frac{V_o(j \omega)}{V_i(j \omega)}$$
Assume:
$$
\begin{array}{ll}
R_5=500 \Omega & R_0=5 \mathrm{k} \Omega \\
C=5 \mathrm{pF} & L=1 \mathrm{mH}
\end{array}
$$
(FIGURE CAN'T COPY)
Figure P5.59

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Problem 60

In the notch filter circuit shown in Eigure P5.59, derive the voltage frequency response $\mathbf{G}_V(j \omega)$ in standard form, where:
$$G_v(j \omega)=\frac{V_o(j \omega)}{V_t(j \omega)}$$
Assume:
$$
\begin{array}{ll}
R_s=500 \Omega & R_n=5 \mathrm{k} \Omega \\
\omega_n=12.13 \mathrm{Mrad} / \mathrm{s} & C=68 \mathrm{nF} \\
L=0.1 \mu \mathrm{H} &
\end{array}
$$
Also, determine the half-power frequencies, bandwidth, and $Q$.

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Problem 61

In the notch filter circuit shown in Figure P5.59, derive the voltage frequency response $\mathbf{G}_V(j \omega)$ in standard form, where:
$$G_V(j \omega)=\frac{V_a(j \omega)}{V_i(j \omega)}$$
Assume:
$$
\begin{array}{lll}
R_s=4.4 \mathrm{k} \Omega & R_a=60 \mathrm{k} \Omega & \omega_n=25 \mathrm{Mrad} / \mathrm{s} \\
C=0.8 \mathrm{nF} & L=2 \mu \mathrm{H} &
\end{array}
$$
Also, determine the half-power frequencies, bandwidth, and $Q$.

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Problem 62

In the bandstop (notch) filter shown in Figure P5.62:
$$
\begin{array}{ll}
L=0.4 \mathrm{mH} & R_c=100 \Omega \\
C=1 \mathrm{pF} & R_s=R_s=3.8 \mathrm{k} \Omega
\end{array}
$$
Determine:
a. An expression for the voltage frequency response:
$$G_v(j \omega)=\frac{\mathbf{V}_0(j \omega)}{\mathbf{V}_i(j \omega)}=G_0 \frac{1+j f_1(\omega)}{1+j f_2(\omega)}$$
b. The magnitude of the frequency response at very high and very low frequencies and at the resonant frequency.
c. The magnitude of the frequency response at the resonant frequency.
d. The resonant and half-power frequencies.
(FIGURE CAN'T COPY)
Figure P5.62

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Problem 63

In the filter circuit shown in Figure P5.56, assume:
$$
\begin{array}{ll}
R_S=5 \mathrm{k} \Omega & C=5 \mathrm{nF} \\
R_e=50 \mathrm{k} \Omega & L=2 \mathrm{mH}
\end{array}
$$
Determine:
a. An expression for the voltage frequency response function
$$\mathbf{G}_V(j \omega)=\frac{\mathbf{V}_a(j \omega)}{\mathbf{V}_i(j \omega \theta)}$$
b. The resonant frequency.
c. The half-power frequencies.
d. The bandwidth and $Q$.

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Problem 64

Many stereo speakers are two-way speaker systems; that is, they have a woofer for low-frequency sounds and a tweeter for high-frequency sounds. To get the proper separation of frequencies going to the woofer and to the tweeter, crossover circuitry is used. A crossover circuit is effectively a bandpass, highpass, or low-pass filter. The system model is shown in Eigure P5 64. The function of the crossover circuitry is to channel frequencies below a given crossover frequency, $f_c$, into the woofer and frequencies higher than $f_c$ into the tweeter. Assume an ideal amplifier such that $R_S=0$ and that the desired crossover frequency is $1,200 \mathrm{~Hz}$. Find $C$ and $L$ when $R_1=R_2=8 \Omega$.
(FIGURE CAN'T COPY)
Figure P5.64

Victor Salazar
Victor Salazar
Numerade Educator

Problem 65

Determine the frequency response $\mathbf{V}_{\text {out }}(\omega) / \mathbf{V}_S(\omega)$ for the network in Eigure P5.65. Generate the Bode magnitude and phase plots when $R_S=R_o=5 \mathrm{k} \Omega, L= 10 \mu \mathrm{H}$, and $C=0.1 \mu \mathrm{~F}$.
(FIGURE CAN'T COPY)
Figure P5.65

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Problem 66

Refer to Problem 5.64 but assume that $L=2 \mathrm{mH}, C=125 \mu \mathrm{~F}$, and $R_S=R_1=R_2 =4 \Omega$ in Eigure P5.64.
a. Determine the impedance seen by the amplifier as a function of frequency. At what frequency is maximum power transferred by the amplifier?
b. Generate the Bode magnitude and phase plots of the currents through the woofer and tweeter.

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Problem 67

For the notch filter shown in Eigure P5,67 assume that $R_S=R_0=500 \Omega, L=10$ mH , and $C=0.1 \mu \mathrm{~F}$.
a. Determine the frequency response $\mathbf{V}_{\text {out }}(j \omega) / \mathbf{V}_s(j \omega)$.
b. Generate the associated Bode magnitude and phase plots.
(FIGURE CAN'T COPY)
Figure P5.67

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Problem 68

It is very common to see interference caused by power lines, at a frequency of 60 Hz . This problem outlines the design of the notch filter shown in Figure $\underline{\mathrm{P} 5.68}$ to reject a band of frequencies around 60 Hz .
a. Determine the impedance $\mathbf{Z}_{a b}(j \omega)$ between nodes $a$ and $b$ for the filter of Eigure P5.68, $r_L$ represents the resistance of a practical inductor.
b. For what value of $C$ will the center frequency of $\mathbf{Z}_{a b}(j \omega)$ equal 60 Hz when $L=100 \mathrm{mH}$ and $r_L=5 \Omega$ ?
c. Would the "sharpness," or selectivity, of the filter increase or decrease if $r_L$ were increased?
d. Assume that the filter is used to eliminate the $60-\mathrm{Hz}$ noise from a $1-\mathrm{kHz}$ sine wave. Evaluate the frequency response $\mathbf{V}_o / \mathbf{V}_{\text {in }}(j \omega)$ at both frequencies when:
$$
\begin{array}{ll}
v_R(t)=\sin (2 \pi 1,000 t) \mathrm{V} & r_R=50 \Omega \\
v_n(t)=3 \sin (2 \pi 60 t) & R_n=300 \Omega
\end{array}
$$
Assume $L=100 \mathrm{mH}$ and $r_L=5 \Omega$. Use the value of $C$ found in part b.
e. Generate the Bode magnitude and phase plots for $\mathbf{V}_o / \mathbf{V}_{\text {in }}$. Mark the plots at 60 Hz and $1,000 \mathrm{~Hz}$.
(FIGURE CAN'T COPY)
Figure P5.68

Lainey Roebuck
Lainey Roebuck
Numerade Educator

Problem 69

The circuit of Figure P5.69 is representative of an amplifier-speaker connection. The crossover filter allows low-frequency signals to pass to the woofer. The filter's topography is known as a $\pi$ network.
a. Find the frequency response $\mathbf{V}_o(j \omega) / \mathbf{V}_s(j \omega)$.
b. If $C_1=C_2=C, R_S=R_o=600 \Omega$, and $1 / \sqrt{L C}=R / L=1 / R C=2,000 \pi$, generate the Bode magnitude and phase plots in the range $100 \mathrm{~Hz} \leq f \leq 10 \mathrm{kHz}$.
(FIGURE CAN'T COPY)
Figure P5.69

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Problem 70

For the circuit shown in Figure P5.70:
a. Determine the frequency response:
$$G_v(j a+)=\frac{\left.V_{o u}(j a)\right)}{V_a(j a t)}$$
b. Sketch, by hand, the associated Bode magnitude and phase plots. List all the steps in constructing the plot. Clearly show the break frequencies on the frequency axis.
c. Use the MatLab command "Bode" to generate the same plots. Verify your sketch. Assume $R_1=R_2=2 \mathrm{k} \Omega, L=2 \mathrm{H}, C_1=C_2=2 \mathrm{mF}$.
(FIGURE CAN'T COPY)
Figure P5.70

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03:05

Problem 71

Repeat all parts of Problem 5.70 for the frequency response:
$$H(j \omega)=\frac{I_{\text {ou }}(j \omega)}{V_{\mathrm{a}}(j \omega)}$$
Use the same component values as in Problem 5.70.

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 72

Repeat all parts of Problem 5.70 for the circuit of Figure P5.72 and the frequency response:
$$H(j \omega)=\frac{\bar{V}_{\omega t}(j \omega)}{I_\omega(j \omega)}$$
Let $R_1=R_2=1 \mathrm{k} \Omega, C=1 \mu \mathrm{~F}, L=1 \mathrm{H}$.
(FIGURE CAN'T COPY)
Figure P5.72

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Problem 73

Repeat all parts of Problem 5.70 for the circuit of Eigure P5.72 and the frequency response:
$$G_i(j o r)=\frac{I_{\mathrm{axt}}(j \omega)}{I_{\mathrm{ax}}(j e l)}$$
Use the same values as in Problem 5.72 .

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Problem 74

For the circuit of Figure P5.74 determine the frequency response $\mathbf{H}(j \omega)= \mathbf{V}_{\text {out }} / \mathbf{I}_{\text {in }}$. Repeat all parts of Problem 5.70. Assume $R_1=R_2=2 \mathrm{k} \Omega, C_1=C_2=1$ mF.
(FIGURE CAN'T COPY)
Figure P5.74

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Problem 75

Repeat all parts of Problem 5.70 for the circuit of Figure P5.74 and the frequency response:
$$G_l(j \omega)=\frac{I_{\omega a}(j \omega)}{I_n(j \omega)}$$
Use the same component values as in Problem 5.74.

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Problem 76

Refer to Figure P5.34 and assume $R_1=300 \Omega, R_2=R_3=500 \Omega, L=4 \mathrm{H}, C_1= 40 \mu \mathrm{~F}, C_2=160 \mu \mathrm{~F}$.
a. Determine the frequency response:
$$G_v(j \omega)=\frac{V_{e s u}(j \omega)}{V_{i n}(j \omega v)}$$
b. Sketch, by hand, the associated Bode magnitude and phase plots. List all the steps in constructing the plot. Clearly show the break frequencies on the frequency axis.
c. Use the MatLab command "Bode" to generate the same plots. Verify your sketch.

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Problem 77

Refer to Figure P5.34 and the parameter values listed in Problem 5.76.
a. Determine for the frequency response:
$$\mathbf{G}_V(j \omega)-\frac{\mathbf{V}_C(j \omega)}{\mathbf{V}_{-}(j \omega)}$$
b. Repeat parts b and c of Problem 5.76 for this frequency response.

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Problem 78

Refer to Eigure P5,35 and repeat the instructions of parts b and c of Problem $\underline{5.76}$. Assume $R_1=20 \mathrm{k} \Omega, R_2=100 \mathrm{k} \Omega, L=1 \mathrm{H}, C=100 \mu \mathrm{~F}$.

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Problem 79

Assume in a certain frequency range that the ratio of output amplitude to input amplitude is proportional to $1 / \omega^3$. What is the slope of the Bode magnitude plot in this frequency range, expressed in $\mathrm{dB} /$ decade?

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Problem 80

Assume that the amplitude of an output voltage depends on frequency according to:
$$\mathbf{V}(j \omega)=\frac{A \omega+B}{\sqrt{C+D \omega^2}}$$
Find:
a. Each break frequency in terms of $A, B, C$ and $D$.
b. The slope (in dB/decade) of the Bode magnitude plot at the highfrequency end.
c. The slope (in dB/decade) of the Bode plot at the low-frequency end.

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Problem 81

Determine the equivalent impedance $\mathbf{Z}_{\text {eq }}$ in standard form as defined in Figure P5.81(a). Choose the Bode plot from Figure P5.81(b) that best describes the behavior of the impedance as a function of frequency. Describe how to find the resonant and cutoff frequencies, and the magnitude of the impedance for those ranges where it is constant. Label the Bode plot to indicate which feature you are discussing.
(FIGURE CAN'T COPY)
Figure P5.81

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