The full-wave rectified cosine wave shown in Figure P6.6

Briefly discuss the fundamental concept of Fourier theory

Define the transfer function for a filter. Describe how the transfer function of a filter can be determined using laboratory methods.

Describe how a filter processes an input signal to produce the output signal in terms of the transfer function and the sinusoidal components.

ourier analysis shows that the sawtooth waveform of Figure P6.4 can be written as vst (t) = 1 2 sin (2000t) 2 2 sin (4000t) 2 3 sin (6000t) 2 n sin (2000nt) in which n assumes integer values. Use MATLAB to compute and plot the sum through n = 3 for 0 t 2 ms. Repeat for the sum through n = 50. 2 123 t (ms) vst(t) Figure P6.4

The triangular waveform shown in Figure P6.5 can be written as the infinite sum vt (t) = 1 + 8 2 cos(2000t) + 8 (3) 2 cos(6000t) + + 8 (n) 2 cos(2000nt) + in which n takes odd integer values only. Use MATLAB to compute and plot the sum through n = 19 for 0 t 2 ms. Compare your plot with the waveform shown in Figure P6.5. vt (t) 2 1 2 t (ms) Figure P

The full-wave rectified cosine wave shown in Figure P6.6 can be written as vfw = 2 + 4 (1) (3) cos(4000t) 4 (3) (5) cos(8000t) + + 4 (1) (n/2+1) (n 1) (n + 1) cos(2000nt) + in which n assumes even integer values. Use MATLAB to compute and plot the sum through n = 60 for 0 t 2 ms. Compare your plot with the waveform shown in Figure P6.6. vfw(t) = cos (2000pt) 1 0.5 1.0 t (ms) Figure P6.6

The Fourier series for the periodic waveform shown in Figure P6.7 is v(t) = $ n=1  sin(n/2) (n/2)2 cos(n/2) (n/2)  sin(500nt) Use MATLAB to compute and plot the sum through n = 3 for 4 t 4 ms. Then, plot the sum through n = 50. Compare your plots with the waveform in Figure P6.7. 1 1 1 2 3 2 1 4 3 t (ms) v (t) V Figure P6.7

The transfer function H(f) = Vout/Vin of a filter is shown in Figure P6.8. The input signal is given by vin (t) = 5+2 cos % 5000t + 30& +2 cos(15000t) Find an expression (as a function of time) for the steady-state output of the filter.

Repeat Problem P6.8 for the input voltage given by vin(t) = 4+5 cos(104t30)+2 sin(24000t)

Repeat Problem P6.8 for the input voltage given by vin(t) = 6 + 2 cos(6000t) 4 cos(12000t)

The input to a certain filter is given by vin (t) = 2 cos  104t 25 and the steady-state output is given by vout (t) = cos  104t + 20 Determine the (complex) value of the transfer function of the filter for f = 5000 Hz.

The input and output voltages of a certain filter operating in steady state with a sinusoidal input are shown in Figure P6.12. Determine the frequency and the corresponding value of the transfer function.

The input and output voltages of a filter operating under sinusoidal steady-state conditions are observed on an oscilloscope. The peak amplitude of the input is 5 V and the output is 15 V. The period of both signals is 4 ms. The input reaches a positive peak at t = 1 ms, and the output reaches its positive peak at t = 1.5 ms. Determine the frequency and the corresponding value of the transfer function.

The triangular waveform of Problem P6.5 is the input for a filter with the transfer function shown in Figure P6.14.Assume that the phase of the transfer function is zero for all frequencies. Determine the steady-state output of the filter. H( f) 2 500 f (Hz) Figure P6.14

Consider a circuit for which the output voltage is the running-time integral of the input voltage, as illustrated in Figure P6.15. If the input voltage is given by vin (t) = Vmax cos(2ft), find an expression for the output voltage as a function of time. Then, find an expression for the transfer function of the integrator. Plot the magnitude and phase of the transfer function versus frequency. vout vin(t) (t) = + + Integrator circuit vin (t) dt t 0 Figure P6.15

Suppose we have a circuit for which the output voltage is the time derivative of the input voltage, as illustrated in Figure P6.16. For an input voltage given by vin (t) = Vmax cos(2ft), find an expression for the output voltage as a function of time. Then, find an expression for the transfer function of the differentiator. Plot the magnitude and phase of the transfer function versus frequency. vout vin(t) (t) = + + Differentiator circuit dvin(t) dt Figure P6.16

The sawtooth waveform of Problem P6.4 is applied as the input to a filter with the transfer function shown in Figure P6.17. Assume that the phase of the transfer function is zero for all frequencies. Determine the steady-state output of the filter. 2500 3500 5 f (Hz) H(f) Figure P6.17

Suppose we have a system for which the output voltage is vo (t) = 1000 t t103 vin (t) dt Given the input voltage vin (t) = Vmax cos (2ft), find an expression for the output voltage as a function of time. Then, find an expression for the transfer function of the system. Use MATLAB to plot the magnitude of the transfer function versus frequency for the range from 0 to 2000 Hz. Explain why the magnitude of the transfer function is zero for multiples of 1000 Hz.

List the frequencies in hertz for which the transfer function of a filter can be determined given that the input to the filter is vin (t) = 2 + 3 cos(1000t) + 3 sin (2000t) + cos(3000t) V 336 Chapter 6 Frequency Response, Bode Plots, and Resonance and the output is vout (t) = 3 + 2 cos % 1000t + 30& + 4 sin (3000t) V Compute the transfer function for each of these frequencies.

Consider a system for which the output voltage is vo (t) = vin (t) + vin % t 2 103 & . (In other words, the output equals the input plus the input delayed by 2 ms.) Given that the input voltage is vin (t) = Vmax cos(2ft), find an expression for the output voltage as a function of time. Then, find an expression for the transfer function of the system. Use MATLAB to plot the magnitude of the transfer function versus frequency for the range from 0 to 2000 Hz. Explain in terms of the phasors for vin (t) and for vin % t 2 103 & why the transfer function has the magnitude it does for f = 250 Hz. Repeat for f = 500 Hz.

Draw the circuit diagram of a first-order RC lowpass filter and give the expression for the half-power frequency in terms of the circuit components. Sketch the magnitude and phase of the transfer function versus frequency.

Repeat Problem P6.21 for a first-order RL lowpass filter.

In Chapter 4, we used the time constant to characterize first-order RC circuits. Find the relationship between the half-power frequency and the time constant.

Suppose that we need a first-order RC lowpass filter with a half-power frequency of 2 kHz. Determine the value of the capacitance, given that the resistance is 15 k.

Consider a first-order RC lowpass filter. At what frequency (in terms of fB) is the phase shift equal to 1? 10? 89?

An input signal given by vin (t) = 5 cos(500t) + 5 cos(1000t) + 5 cos(2000t) is applied to the lowpass RC filter shown in Figure P6.26. Find an expression for the output signal. vout v (t) in(t)

The input signal of a first-order lowpass filter with the transfer function given by Equation 6.9 on page 288 and a half-power frequency of 400 Hz is vin (t) = 1 + 2 cos % 800t + 30& + 3 cos  20 103t Find an expression for the output voltage

Suppose we have a first-order lowpass filter that is operating in sinusoidal steady-state conditions at a frequency of 5 kHz. Using an oscilloscope, we observe that the positivegoing zero crossing of the output is delayed by 20s compared with that of the input. Determine the break frequency of the filter.

The input signal to a filter contains components that range in frequency from 10 Hz to 20 kHz. We wish to reduce the amplitude of the 20-kHz component by a factor of 100 by passing the signal through a first-order lowpass filter. What half-power frequency is required for the filter? By what factor is a component at 2 kHz changed in amplitude in passing through this filter?

Sketch the magnitude of the transfer function H(f) = Vout/Vin to scale versus frequency for the circuit shown in Figure P6.30. What is + + Vout Vin C = 10 mF 2 k 1 k 1 k 2 k Figure P6.30 Problems 337 the value of the half-power frequency? (Hint: Start by finding the Thvenin equivalent circuit seen by the capacitance.)

In steady-state operation, a first-order RC lowpass filter has the input signal vin(t) = 5 cos(20 103t) and the output signal vout(t) = 0.5 cos(20 103t ). Determine the break frequency of the filter and the value of .

We apply a 5-V-rms 10-kHz sinusoid to the input of a first-order RC lowpass filter, and the output voltage in steady state is 0.2 V rms. Predict the steady-state rms output voltage after the frequency of the input signal is raised to 50 kHz and the input amplitude remains constant.

Consider the circuit shown in Figure P6.33(a). This circuit consists of a source having an internal resistance of Rs, an RC lowpass filter, and a load resistance RL. a. Show that the transfer function of this circuit is given by H (f) = Vout Vs = RL Rs + R + RL 1 1 + j % f # fB & (b) Vs Vout + Rs RL R C + Find Thvenin equivalent (a) Vs Vout + Rs C RL R + Figure P6.33 in which the half-power frequency fB is given by fB = 1 2RtC where Rt = RL (Rs + R) RL + Rs + R Notice that Rt is the parallel combination of RL and (Rs + R). [Hint: One way to make this problem easier is to rearrange the circuit as shown in Figure P6.33(b) and then to find the Thvenin equivalent for the source and resistances.] b. Given that C = 0.2 F, Rs = 2 k, R = 47 k, and RL = 1 k, sketch (or use MATLAB to plot) the magnitude of H(f) to scale versus f /fB from 0 to 3. P

a. Derive an expression for the transfer function H(f) = Vout/Vin for the circuit shown in Figure P6.34. Find an expression for the half-power frequency. b. Given R1 = 50 , R2 = 50 , and L = 15 H, sketch (or use MATLAB to plot) the magnitude of the transfer function versus frequency. Vout Vin + + R1 R2 L Figure P6.34

Perhaps surprisingly, we can apply the transfer-function concept to mechanical systems. Suppose we have a mass m moving through a liquid with an applied force f and velocity v. The motion of the mass is described by the first-order differential equation f = m dv dt + kv in which k is the coefficient of viscous friction. Find an expression for the transfer function H(f) = V F Also, find the half-power frequency (defined as the frequency at which the transfer function magnitude is 1/ 2 times its dc value) in terms of k and m. [Hint: To determine the 338 Chapter 6 Frequency Response, Bode Plots, and Resonance transfer function, assume a steady-state sinusoidal velocity v = Vm cos(2ft), solve for the force, and take the ratio of their phasors.

What is the main advantage of converting transfer function magnitudes to decibels before plotting?

What is the passband of a simple RC lowpass filter?

What is a logarithmic frequency scale? A linear frequency scale?

a.What frequency is halfway between 100 and 3000 Hz on a logarithmic frequency scale? b. On a linear frequency scale?

a. Find the frequency that is one octave lower than 500 Hz. b. Two octaves higher. c. Two decades lower. d. One decade higher.

What is a notch filter? What is one application?

a. Given |H(f)|dB = 10 dB, find |H(f)|. b. Repeat for |H(f)|dB = 10 dB.

Find the decibel equivalent for |H(f)| = 0.5. Repeat for |H(f)| = 2, |H(f)| = 1/ 2 = 0.7071, and |H(f)| = 2.

Explain what we mean when we say that two filters are cascaded.

We have a list of six successive frequencies, 5 Hz, f 1, f 2, f 3, f 4, and 50 Hz. Determine the values of f 1, f 2, f 3, and f 4 so that the frequencies are evenly spaced on: a. a linear frequency scale, and b. a logarithmic frequency scale

a. How many decades are between f 1 = 25 Hz and f 2 = 10 kHz? b. How many octaves?

We have two filters with transfer functions H1(f) and H2(f) cascaded in the order 12. Give the expression for the overall transfer function of the cascade. Repeat if the transfer function magnitudes are expressed in decibels denoted as |H1(f)|dB and |H2(f)|dB. What caution concerning H1(f) must be considered?

Two first-order lowpass filters are in cascade as shown in Figure P6.48. The transfer functions are H1 (f) = H2 (f) = 1 1 + j % f # fB & a. Write an expression for the overall transfer function. b. Find an expression for the half-power frequency for the overall transfer function in terms of fB. (Comment: This filter cannot be implemented by cascading two simple RC lowpass filters like the one shown in Figure 6.7 on page 287 because the transfer function of the first circuit is changed when the second is connected. Instead, a buffer amplifier, such as the voltage follower discussed in Section 14.3, must be inserted between the RC filters.) Vout Vin + + H1(f) H2(f) RL Figure P6.48

Two filters are in cascade. At a given frequency f 1, the transfer function values are |H1(f 1)|dB = 25 and |H2(f 1)|dB = +15. Find the magnitude of the overall transfer function in decibels at f = f 1.

What is a Bode magnitude plot?

What is the slope of the high-frequency asymptote for the Bode magnitude plot for a first-order lowpass filter? The low-frequency asymptote? At what frequency do the asymptotes meet?

Suppose that four filters, having identical first-order lowpass transfer functions, are cascaded, what will be the rate at which the overall transfer function magnitude asymptote declines above the break frequency? Explain.

A transfer function is given by H (f) = 100 1 + j(f /1000) Sketch the asymptotic magnitude and phase Bode plots to scale. What is the value of the half-power frequency?

A transfer function is given by H(f) = 0.1[1 + j(f /200)] Sketch the asymptotic magnitude and phase Bode plots to scale. What is the value of the break frequency?

Sketch the asymptotic magnitude and phase Bode plots to scale for the transfer function H(f) = 10 1 j(f /100) 1 + j(f /100)

Solve for the transfer function H(f) = Vout/Vin and sketch the asymptotic Bode magnitude and phase plots to scale for the circuit shown in Figure P6.56. Vout Vin + + R = 100 C = 270 pF Figure P6.56

The circuit shown in Figure 6.57 has R1 = R2 = 2 k? and C = (1/?)?F. Solve for the transfer function H(f) = Vout/Vin and draw the asymptotic Bode magnitude and phase plots.

Consider a circuit for which a. Assume that Vout(t) = A cos(2?ft), and find an expression for Vin(t). b. Use the results of part (a) to find an expression for the transfer function H(f) = Vout/Vin for the system. c. Draw the asymptotic Bode plot for the transfer function magnitude.

The circuit shown in Figure P6.59 has R = 20 and L = 20 mH. Solve for the transfer function H(f) = Vout/Vin and draw the Bode magnitude and phase plots for the circuit. Vout Vin + + L R Figure P6.59

In solving Problem P6.15, we find that the transfer function of an integrator circuit is given byH (f) = 1/(j2f). Sketch the Bode magnitude and phase plots to scale. What is the slope of the magnitude plot?

In solving Problem P6.16, we find that the transfer function of a differentiator circuit is given by H(f) = j2f. Sketch the Bode magnitude and phase plots to scale. What is the slope of the magnitude plot?

What is the slope of the high-frequency asymptote for the Bode magnitude plot for a first-order highpass filter? The low-frequency asymptote? At what frequency do the asymptotes meet?

Draw the circuit diagram of a first-order RC highpass filter and give the expression for the half-power frequency in terms of the circuit components.

Consider the circuit shown in Figure P6.64. Sketch the asymptotic Bode magnitude and Vout Vin + + R1 = 9 k C = 1 mF R2 = 1 k Figure P6.64 phase plots to scale for the transfer function H(f) = Vout/Vin.

Consider the first-order highpass filter shown in Figure P6.65. The input signal is given by vin(t) = 5 + 5 cos(2000t) Find an expression for the output vout(t) in steady-state conditions. vout v (t) in(t) + + C = 1 mF R = 1000 2p Figure P6.65

Repeat Problem P6.65 for the input signal given by vin(t) = 10 cos(400t) + 20 cos(4000t) + 5 sin(4 104t)

The circuit shown in Figure P6.67 has R = 200/ and C = 10 F. Sketch the Bode magnitude and phase plots to scale for the transfer function H(f) = Vout/Vin. Vin Vout + + C R Figure P6.67

Consider the circuit shown in Figure P6.68. Sketch the Bode magnitude and phase plots to scale for the transfer function H(f) = Vout/Vin. Vout Vin +

Suppose we need a first-order highpass filter (such as Figure 6.19 on page 300) to attenuate a 60-Hz input component by 80 dB. What value is required for the break frequency of the filter? By how many dB is the 600-Hz component attenuated by this filter? If R = 100 , what is the value of C?

What can you say about the impedance of a series RLC circuit at the resonant frequency? How are the resonant frequency and the quality factor defined?

Consider the series resonant circuit shown in Figure P6.71, with L = 20 H, R = 14.14 , and C = 1000 pF. Compute the resonant frequency, the bandwidth, and the half-power frequencies. Assuming that the frequency of the source is the same as the resonant frequency, find the phasor voltages across the elements and sketch a phasor diagram. Vs = VR 1 0 VL VC R C L + + + + Figure P6.71

Work Problem P6.71 for L = 80 H, R = 14.14 , and C = 1000 pF.

Suppose we have a series resonant circuit for which B = 15 kHz, f 0 = 300 kHz, and R = 40 . Determine the values of L and C.

Derive an expression for the resonant frequency of the circuit shown in Figure P6.74. R C L Figure P6.74 (Recall that we have defined the resonant frequency to be the frequency for which the impedance is purely resistive.)

At the resonant frequency f 0 = 1 MHz, a series resonant circuit with R = 50 has |VR| = 2 V and |VL| = 20 V. Determine the values of L and C. What is the value of |VC|?

Suppose we have a series resonant circuit for which f 0 =12MHz andB=400 kHz. Furthermore, the minimum value of the impedance magnitude is 5 . Determine the values of R, L, and C.

What is a bandpass filter? How is its bandwidth defined?

What can you say about the impedance of a parallel RLC circuit at the resonant frequency? How is the resonant frequency defined? Compare the definition of quality factor for the parallel resonant circuit with that for the series resonant circuit.

A parallel resonant circuit has R = 5 k, L = 50 H, and C = 200 pF. Determine the resonant frequency, quality factor, and bandwidth.

A parallel resonant circuit has f 0 = 20 MHz and B = 200 kHz. The maximum value of |Zp| is 5 k. Determine the values of R, L, and C.

A parallel resonant circuit has f 0 = 100 MHz, B = 4 MHz, and R = 1 k. Determine the values of L and C.

Consider the parallel resonant circuit shown in Figure 6.29 on page 309. Determine the L and C values, given R = 2 k, f0 = 8 MHz, and B = 500 kHz. If I = 103 0 , draw a phasor diagram showing the currents through each of the elements in the circuit at resonance.

Name four types of ideal filters.

An ideal bandpass filter has cutoff frequencies of 9 and 11 kHz and a gain magnitude of two in the passband. Sketch the transferfunction magnitude to scale versus frequency. Repeat for an ideal band-reject filter.

An ideal lowpass filter has a cutoff frequency of 10 kHz and a gain magnitude of two in the passband. Sketch the transfer-function magnitude to scale versus frequency. Repeat for an ideal highpass filter.

Suppose that sinewave interference has been inadvertently added to an audio signal that has frequency components ranging from 20 Hz to 15 kHz. The frequency of the interference slowly varies in the range 950 to 1050 Hz. A filter that attenuates the interference by at least 20 dB and passes most of the audio components is desired. What type of filter is needed? Sketch the magnitude Bode plot of a suitable filter, labeling its specifications.

In an electrocardiograph, the heart signals contain components with frequencies ranging from dc to 100 Hz. During exercise on a treadmill, the signal obtained from the electrodes also contains noise generated by muscle contractions. Most of the noise components have frequencies exceeding 100 Hz. What type of filter should be used to reduce the noise? What cutoff frequency is appropriate?

Each AM radio signal has components ranging from 10 kHz below its carrier frequency to 10 kHz above its carrier frequency. Various radio stations in a given geographical region are assigned different carrier frequencies so that the frequency ranges of the signals do not overlap. Suppose that a certain AM radio transmitter has a carrier frequency of 980 kHz. What type of filter should be used if we want the filter to pass the components from this transmitter and reject the components of all other transmitters? What are the best values for the cutoff frequencies?

Draw the circuit diagram of a second-order highpass filter. Suppose that R = 1 k, Qs = 1, and f 0 = 100 kHz. Determine the values of L and C.

Draw the circuit diagram of a second-order lowpass filter. Given that R = 50 , Qs = 0.5, and f 0 = 30 MHz, determine the values of L and C.

Consider the filter shown in Figure P6.91. a. Derive an expression for the transfer function H(f) = Vout/Vin. b. Use MATLAB to obtain a Bode plot of the transfer-function magnitude for R1 = 9 k, R2 = 1 k, and C = 0.01F. Allow frequency to range from 10 Hz to 1 MHz. c. At very low frequencies, the capacitance becomes an open circuit. In this case, determine an expression for the transfer function and evaluate for the circuit parameters of part (b). Does the result agree with the value plotted in part (b)? d. At very high frequencies, the capacitance becomes a short circuit. In this case, determine an expression for the transfer function and evaluate for the circuit parameters of part (b). Does the result agree with the value plotted in part (b)? + Vin R2

Repeat Problem P6.91 for the circuit of Figure P6.92. + Vin Vout R1 + R2 C Figure P6.92

Suppose that we need a filter with the Bode plot shown in Figure P6.93(a). We decide to cascade a highpass circuit and a lowpass circuit as shown in Figure P6.93(b). We choose R2 = 100R1 so that the second (i.e., righthand) circuit looks like an approximate open circuit across the output of the first (i.e., left-hand) circuit. a. Which of the components form the lowpass filter? Which form the highpass filter? b. Compute the capacitances needed to achieve the desired break frequencies, making the approximation that the left-hand circuit has an open-circuit load. c. Write expressions that can be used to compute the exact transfer function H(f) = Vout/Vin and use MATLAB to produce a Bode magnitude plot for f ranging from 1 Hz to 1MHz.The result should be a close approximation to the desired plot shown in Figure P6.93(a).

Suppose that we need a filter with the Bode plot shown in Figure P6.93(a). We decide to cascade a highpass circuit and a lowpass circuit, as shown in Figure P6.94. We choose C2 = C1/100 so that the second (i.e., righthand) circuit looks like an approximate open circuit across the output of the first (i.e., left-hand) circuit. a. Which of the components form the lowpass filter? Which form the highpass filter? b. Compute the resistances needed to achieve the desired break frequencies, making the approximation that the left-hand circuit has an open-circuit load. c. Write expressions that can be used to compute the exact transfer function H(f) = Vout/Vin and use MATLAB to produce a Bode magnitude plot for f ranging from 1 Hz to 1 MHz. The result should be a close approximation to the desired plot shown in Figure P6.93(a). Vin Vout + + C1 C2 R2 R1 0.1 mF 1000 pF Figure P6.94

Other combinations of R, L, and C have behaviors similar to that of the series resonant circuit. For example, consider the circuit shown in Figure P6.95. a. Derive an expression for the resonant frequency of this circuit. (We have defined the resonant frequency to be the frequency for which the impedance is purely resistive.) b. Compute the resonant frequency, given L = 1 mH, R = 1000 , and C = 0.25 F. c. Use MATLAB to obtain a plot of the impedance magnitude of this circuit for f ranging from 95 to 105 percent of the resonant frequency. Compare the result with that of a series RLC circuit. R L C Figure P6.95

Consider the circuit of Figure P6.74 with R = 1 k, L = 1 mH, and C = 0.25 F. a. Using MATLAB, obtain a plot of the impedance magnitude of this circuit for f ranging from 9 to 11 kHz. b. From the plot, determine the minimum impedance, the frequency at which the impedance is minimum, and the bandwidth (i.e., the band of frequencies for which the impedance is less than 2 times the minimum value). c. Determine the component values for a series RLC circuit having the same parameters as those found in part (b). d. Plot the impedance magnitude of the series circuit on the same axes as the plot for part (a).

Use the method of Example 6.9 to obtain a magnitude bode plot of the transfer function H(f) = Vout# Vin for the lowpass filter of Figure P6.97 with frequency ranging from 100 kHz to 10 MHz. Manually verify the plotted values for high and low frequencies. Also, determine the half-power frequency for this filter. Rs L + C1 C2 RL + V1 V2 Vin = Vout 1 V Rs = RL = 50 C1 = C2 = 3.1831 109 F L = 15.915 mH Figure P6.97

Repeat Problem P6.97 for the highpass filter shown in Figure P6.98. Rs C + L1 L2 + V1 V2 Vin = Vout 1 V Rs = RL = 50 L1 = L2 = 7.9577 mH C = 1591.5 pF Figure P6.98

Other combinations of R, L, and C have behaviors similar to that of the parallel circuit. For example, consider the circuit shown in Figure P6.99. a. Derive an expression for the resonant frequency of this circuit. (We have defined the resonant frequency to be the frequency for which the impedance is purely resistive. However, in this case you may find the algebra easier if you work with admittances.) b. Compute the resonant frequency, given L = 1 mH, R = 1 , and C = 0.25 F. c. Use MATLAB to obtain a plot of the impedance magnitude of this circuit for f ranging from 95 to 105 percent of the resonant frequency. Compare the result with that of a parallel RLC circuit. R L C Figure P6.99

Consider the filter shown in Figure P6.100. a. Derive an expression for the transfer function H(f) = Vout/Vin. b. Use MATLAB to obtain a Bode plot of the transfer function magnitude for R = 10 , L = 10 mH, and C = 0.02533F. Allow frequency to range from 1 kHz to 100 kHz. c. At very low frequencies, the capacitance becomes an open circuit and the inductance becomes a short circuit. In this case, determine an expression for the transfer function and evaluate for the circuit parameters of part (b). Does the result agree with the value plotted in part (b)? d. At very high frequencies, the capacitance becomes a short circuit and the inductance becomes an open circuit. In this case, deter- + Vin Vout + R L C Figure P6.100 mine an expression for the transfer function and evaluate for the circuit parameters of part (b). Does the result agree with the value plotted in part (b)?

Repeat Problem P6.100 for the circuit of Figure P6.101. + Vin Vout R + C L Figure P6.101

a. Develop a digital filter that mimics the action of the RL filter shown in Figure P6.102. Determine expressions for the coefficients in terms of the time constant and sampling intervalT. (Hint: If your circuit equation contains an integral, differentiate with respect to time to obtain a pure differential equation.) b. Given R = 10 and L = 200 mH, sketch the step response of the circuit to scale. c. UseMATLAB to determine and plot the step response of the digital filter for several time constants. Use the time constant of part (b) and fs = 500 Hz. Compare the results of parts (b) and (c). x(t) y(t) + + L R Figure P6.102

Repeat Problem P6.102 for the filter shown in Figure P6.103. x(t) y(t) + + L R Figure P6.103

Consider the second-order bandpass filter shown in Figure P6.104. a. Derive expressions for L and C in terms of the resonant frequency 0 and quality factor Qs. b. Write the KVL equation for the circuit and use it to develop a digital filter that mimics the action of the RLC filter. Use the results of part (a) to write the coefficients in terms of the resonant frequency 0, circuit quality factor Qs, and sampling interval T. (Hint: The circuit equation contains an integral, so differentiate with respect to time to obtain a pure differential equation.) x(t) y(t) + + L C i(t) = y(t) 1 Figure P6.104