Solved: a) Design an inverting amplifier using an ideal op amp that has a gain of 2.5

The op amp in the circuit in Fig. P5.1 is ideal. a) Label the five op amp terminals with their names. b) What ideal op amp constraint determines the value of ? What is this value? c) What ideal op amp constraint determines the value of ? What is this value? d) Calculate vo.

a) Replace the 2 V source in the circuit in Fig. P5.1 and calculate for each of the following source values: 6 V, 3.5 V, 1.25 V, 2.4 V, 4.5 V, 5.4 V. b) Specify the range of voltage source values that will not cause the op amp to saturate

Find (in milliamperes) in the circuit in Fig. P5.3.

The op amp in the circuit in Fig. P5.4 is ideal. a) Calculate if and b) Calculate if and c) Calculate if and d) Calculate if and v vb = 1 V. e) Calculate if and f) If , specify the range of such that the amplifier does not saturate.

Find in the circuit in Fig. P5.5 if the op amp is ideal.

The op amp in the circuit in Fig. P5.6 is ideal. Calculate the following: a) b) c) d) Figure P5.6 20 k 8 k 240 mV 15 V 15 V 60 k 30 k vo va io ia 40 k io vo va ia

A voltmeter with a full-scale reading of 10 V is used to measure the output voltage in the circuit in Fig. P5.7. What is the reading of the voltmeter? Assume the op amp is ideal.

a) Design an inverting amplifier with a gain of 4. Use an ideal op amp, a \(30 \ \mathrm{k} \Omega\) resistor in the feedback path, and \(\pm 12 \mathrm{~V}\) power supplies. b) Using your design from part (a), determine the range of input voltages that will keep the op amp in its linear operating region. c) Suppose you wish to amplify a 2 V signal, using your design from part (a) with a variable feedback resistor. What is the largest value of feedback resistance that keeps the op amp in its linear operation region? Using this resistor value, what is the new gain of the inverting amplifier?

a) Design an inverting amplifier using an ideal op amp that has a gain of 2.5. Use a set of identical resistors from Appendix H. b) If you wish to amplify signals between -2 V and +3 V using the circuit you designed in part (a), what are the smallest power supply voltages you can use?

a) The op amp in the circuit shown in Fig. P5.10 is ideal. The adjustable resistor has a maximum value of and is restricted to the range of Calculate the range of if b) If is not restricted, at what value of will the op amp saturate?

The op amp in the circuit in Fig. P5.11 is ideal. a) Find the range of values for \(\sigma\) in which the op amp does not saturate. b) Find \(i_{o}\) (in microamperes) when \(\sigma=0.272\).

The op amp in Fig. P5.12 is ideal. a) What circuit configuration is shown in this figure? b) Find \(v_{o}\) if \(v_{\mathrm{a}}=1 \mathrm{~V}, v_{\mathrm{b}}=1.5 \mathrm{~V}\), and \(v_{\mathrm{c}}=-4 \mathrm{~V}\). c) The voltages \(v_{\mathrm{a}}\) and \(v_{\mathrm{c}}\) remain at 1 V and V, respectively. What are the limits on \(v_{\mathrm{b}}\) if the op amp operates within its linear region?

Refer to the circuit in Fig. 5.12, where the op amp is assumed to be ideal. Given that \(R_{\mathrm{a}}=4 \ \mathrm{k} \Omega\), \(R_{\mathrm{b}}=5 \ \mathrm{k} \Omega, R_{\mathrm{c}}=20 \ \mathrm{k} \Omega, \quad v_{\mathrm{a}}=200 \ \mathrm{mV}\), \(v_{\mathrm{b}}=150 \ \mathrm{mV}, v_{\mathrm{c}}=400 \ \mathrm{mV}\), and \(V_{C C}=\pm 6 \mathrm{~V}\), specify the range of \(R_{f}\) for which the op amp operates within its linear region.

a) The op amp in Fig. P5.14 is ideal. Find \(v_{o}\) if \(v_{\mathrm{a}}=3 \mathrm{~V}, v_{\mathrm{b}}=9 \mathrm{~V}, v_{\mathrm{c}}=5 \mathrm{~V}\), and \(v_{\mathrm{d}}=6 \mathrm{~V}\). b) Assume \(v_{\mathrm{a}} v_{\mathrm{b}}\), and \(v_{\mathrm{d}}\) retain their values as given in (a). Specify the range of \(v_{\mathrm{c}}\) such that the op amp operates within its linear region.

The \(180 \ \mathrm{k} \Omega\) feedback resistor in the circuit in Fig. P5.14 is replaced by a variable resistor \(R_{f}\). The voltages \(v_{\mathrm{a}}-v_{\mathrm{d}}\) have the same values as given in Problem 5.14(a). a) What value of \(R_{f}\) will cause the op amp to saturate? Note that \(0 \leq R_{f} \leq \infty\). b) When \(R_{f}\) has the value found in (a), what is the current (in microamperes) into the output terminal of the op amp?

a) Design an inverting-summing amplifier using a \(120 \ \mathrm{k} \Omega\) resistor in the feedback path so that \(v_{o}=-\left(8 v_{\mathrm{a}}+5 v_{\mathrm{b}}+12 v_{\mathrm{c}}\right)\) Use \(\pm 15 \mathrm{~V}\) power supplies. b) Suppose \(v_{\mathrm{a}}=2 \mathrm{~V}\) and \(v_{\mathrm{c}}=-1 \mathrm{~V}\). What range of values for \(v_{\mathrm{b}}\) will keep the op amp in its linear operating region?

Design an inverting-summing amplifier so that \(v_{o}=-\left(8 v_{\mathrm{a}}+4 v_{\mathrm{b}}+10 v_{\mathrm{c}}+6 v_{\mathrm{d}}\right)\). Start by choosing a feedback resistor \(\left(R_{f}\right)\) from Appendix H. Then choose single resistors from Appendix H or construct resistor neworks from resistors in Appendix H to satisfy the design values for \(R_{\mathrm{a}}, R_{\mathrm{b}}, R_{\mathrm{c}}\), and \(R_{\mathrm{d}}\). Draw your final circuit diagram

The op amp in the circuit of Fig. P5.18 is ideal. a) What op amp circuit configuration is this? b) Calculate \(v_{o}\).

The op amp in the circuit of Fig. P5.19 is ideal. a) What op amp circuit configuration is this? b) Find \(v_{o}\) in terms of \(v_{s}\). c) Find the range of values for \(v_{s}\) such that \(v_{o}\) does not saturate and the op amp remains in its linear region of operation.

The op amp in the circuit shown in Fig. P5.20 is ideal, a) Calculate when equals 4 V. b) Specify the range of values of so that the op amp operates in a linear mode.c) Assume that equals 2 V and that the resistor is replaced with a variable resistor.What value of the variable resistor will cause the op amp to saturate?

a) Design a non-inverting amplifier (see Fig. 5.13) with a gain of 6, using a \(75 \ \mathrm{k} \Omega\) resistor in the feedback path. Draw your final circuit diagram. b) Suppose you wish to amplify input signals in the range \(-2.5 \mathrm{~V} \leq v_{g} \leq 1.5 \mathrm{~V}\). What are the minimum values of the power supplies that will keep the op amp in its linear operating region?

a) Design a non-inverting amplifier (see Fig. 5.13) with a gain of 2.5. Use resistors from Appendix H. You might need to combine resistors in series and in parallel to get the desired resistance. Draw your final circuit. b) If you use \(\pm 16 \mathrm{~V}\) power supplies for the op amp, what range of input values will allow the op amp to stay in its linear operating region?

The op amp in the circuit of Fig. P5.23 is ideal. a) What op amp circuit configuration is this? b) Find \(v_{o}\) in terms of \(v_{s}\). c) Find the range of values for \(v_{s}\) such that \(v_{o}\) does not saturate and the op amp remains in its linear region of operation.

The circuit in Fig. P5.24 is a noninverting summing amplifier. Assume the op amp is ideal. Design the circuit so that \(v_{o}=v_{\mathrm{a}}+2 v_{\mathrm{b}}+3 v_{\mathrm{c}}\). a) Specify the numerical values of \(R_{\mathrm{a}}\) and \(R_{\mathrm{c}}\). b) Calculate \(i_{\mathrm{a}}, i_{\mathrm{b}}\), and \(i_{\mathrm{c}}\) (in microamperes) when \(v_{\mathrm{a}}=0.7 \mathrm{~V}, v_{\mathrm{b}}=0.4 \mathrm{~V}\), and \(v_{\mathrm{c}}=1.1 \mathrm{~V}\).

a) Use the principle of superposition to derive Eq. 5.22. b) Derive Eqs. 5.23 and 5.24.

The op amp in the circuit of Fig. P5.26 is ideal. a) What op amp circuit configuration is this? b) Find an expression for the output voltage \(v_{o}\) in terms of the input voltage \(v_{a}\). c) Suppose \(v_{\mathrm{a}}=2 \mathrm{~V}\). What value of \(R_{\mathrm{f}}\) will cause the op amp to saturate?

The resistors in the difference amplifier shown in Fig. 5.15 are \(R_{\mathrm{a}}=24 \ \mathrm{k} \Omega, \quad R_{\mathrm{b}}=75 \ \mathrm{k} \Omega\), \(R_{\mathrm{c}}=130 \ \mathrm{k} \Omega\) and \(R_{\mathrm{d}}=120 \ \mathrm{k} \Omega\). The signal voltages and are 8 and 5 V, respectively, and \(V_{C C}=\pm 20 \mathrm{~V}\). a) Find \(v_{o}\). b) What is the resistance seen by the signal source \(v_{a}\)? c) What is the resistance seen by the signal source \(v_{b}\)?

The resistor \(R_{\mathrm{f}}\) in the circuit in Fig. P5.28 is adjusted until the ideal op amp saturates. Specify \(R_{\mathrm{f}}\) in kilohms.

Design a difference amplifier (Fig. 5.15) to meet the following criteria: \(v_{o}=3 v_{\mathrm{b}}-4 v_{\mathrm{a}}\). The resistance seen by the signal source \(v_{\mathrm{b}}\) is \(470 \ \mathrm{k} \Omega\), and the resistance seen by the signal source \(v_{\mathrm{a}}\) is \(22 \ \mathrm{k} \Omega\) when the output voltage \(v_{o}\) is zero. Specify the values of \(R_{\mathrm{a}}, R_{\mathrm{b}}, R_{\mathrm{c}}\), and \(R_{\mathrm{d}}\) using single resistors or combinations of resistors from Appendix H.

The op amp in the adder-subtracter circuit shown in Fig. P5.30 is ideal. a) Find when and b) If and are held constant, what values of will not saturate the op amp? Figure P5.30 20 V 20 V vb vo 47 k

Select the values of \(R_{\mathrm{a}}\) and \(R_{\mathrm{f}}\) in the circuit in Fig. P5.31 so that \(v_{o}=8000\left(i_{\mathrm{b}}-i_{\mathrm{a}}\right)\). Use single resistors or combinations of resistors from Appendix H. The op amp is ideal.

The op amp in the circuit of Fig. P5.32 is ideal. a) Plot \(v_{o}\) versus \(\alpha\) when \(R_{f}=4 R_{1}\) and \(v_{g}=2 \mathrm{~V}\). Use increments of 0.1 and note by hypothesis that \(0 \leq \alpha \leq 1.0\). b) Write an equation for the straight line you plotted in (a). How are the slope and intercept of the line related to \(v_{g}\) and the ratio \(R_{f} / R_{1}\)? c) Using the results from (b), choose values for \(v_{g}\) and the ratio \(R_{f} / R_{1}\) such that \(v_{o}=-6 \alpha+4\).

In the difference amplifier shown in Fig. P5.33, compute (a) the differential mode gain, (b) the common mode gain, and (c) the CMRR.

In the difference amplifier shown in Fig. P5.34, what range of values of \(R_{x}\) yields a \(\mathrm{CMRR} \geq 1500\)?

The voltage \(v_{g}\) shown in Fig. P5.35(a) is applied to the inverting amplifier shown in Fig. P5.35(b). Sketch \(v_{o}\) versus t, assuming the op amp is ideal.

The signal voltage \(v_{g}\) in the circuit shown in Fig. P5.36 is described by the following equations: \(v_{g}=0, \quad t \leq 0\), \(v_{g}=4 \cos (\pi / 4) t \mathrm{~V}, \quad 0 \leq t \leq \infty\). Sketch \(v_{o}\) versus t, assuming the op amp is ideal.

a) Show that when the ideal op amp in Fig. P5.37 is operating in its linear region, \(i_{\mathrm{a}}=\frac{3 v_{g}}{R}\). b) Show that the ideal op amp will saturate when \(R_{\mathrm{a}}=\frac{R\left(\pm V_{C C}-2 v_{g}\right)}{3 v_{g}}\).

Assume that the ideal op amp in the circuit seen in Fig. P5.38 is operating in its linear region. a) Show that \(v_{o}=\left[\left(R_{1}+R_{2}\right) / R_{1}\right] v_{s}\). b) What happens if \(R_{1} \rightarrow \infty\) and \(R_{2} \rightarrow 0\)? c) Explain why this circuit is referred to as a voltage follower when \(R_{1}=\infty\) and \(R_{2}=0\).

The two op amps in the circuit in Fig. P5.39 are ideal. Calculate \(v_{o 1}\) and \(v_{o 2}\).

Assume that the ideal op amp in the circuit in Fig. P5.40 is operating in its linear region. a) Calculate the power delivered to the resistor. b) Repeat (a) with the op amp removed from the circuit, that is, with the resistor connected in the series with the voltage source and the resistor. c) Find the ratio of the power found in (a) to that found in (b). d) Does the insertion of the op amp between the source and the load serve a useful purpose? Explain.

The op amps in the circuit in Fig. P5.41 are ideal. a) Find \(i_{\mathrm{a}}\). b) Find the value of the left source voltage for which \(i_{\mathrm{a}}=0\).

The circuit inside the shaded area in Fig.P5.42 is a constant current source for a limited range of values of \(R_{L}\). a) Find the value of \(i_{L}\) for \(R_{L}=4 \ \mathrm{k} \Omega\). b) Find the maximum value for \(R_{L}\) for which \(i_{L}\) will have the value in (a). c) Assume that \(R_{L}=16 \ \mathrm{k} \Omega\). Explain the operation of the circuit. You can assume that \(i_{n}=i_{p} \approx 0\) under all operating conditions. d) Sketch \(i_{L}\) versus \(R_{L}\) for \(0 \leq R_{L} \leq 16 \ \mathrm{k} \Omega\).

Derive Eq. 5.60.

Repeat Assessment Problem 5.6, given that the inverting amplifier is loaded with a \(500 \ \Omega\) resistor.

a) Find the Thévenin equivalent circuit with respect to the output terminals a, b for the inverting amplifier of Fig. P5.45. The dc signal source has a value of 880 mV. The op amp has an input resistance of \(500 \ \mathrm{k} \Omega\), an output resistance of \(2 \ \mathrm{k} \Omega\) and an open-loop gain of 100,000. b) What is the output resistance of the inverting amplifier? c) What is the resistance (in ohms) seen by the signal source \(v_{s}\) when the load at the terminals a, b is \(330 \ \Omega\)?

Repeat Problem 5.45 assuming an ideal op amp.

Assume the input resistance of the op amp in Fig. P5.47 is infinite and its output resistance is zero. a) Find \(v_{o}\) as a function of \(v_{g}\) and the open-loop gain A. b) What is the value of \(v_{o}\) if \(v_{g}=1 \mathrm{~V}\) and A = 150? c) What is the value of \(v_{o}\) if \(v_{g}=1 \mathrm{~V}\) and \(A=\infty\)? d) How large does A have to be so that \(v_{o}\) is 99% of its value in (c)?

The op amp in the noninverting amplifier circuit of Fig. P5.48 has an input resistance of \(560 \ \mathrm{k} \Omega\), an output resistance of \(8 \ \mathrm{k} \Omega\), and an open-loop gain of 50,000. Assume that the op amp is operating in its linear region. a) Calculate the voltage gain \(\left(v_{o} / v_{g}\right)\). b) Find the inverting and noninverting input voltages \(v_{n}\) and \(v_{p}\) (in millivolts) if \(v_{g}=1 \mathrm{~V}\). c) Calculate the difference \(\left(v_{p}-v_{n}\right)\) in microvolts when \(v_{g}=1 \mathrm{~V}\). d) Find the current drain in picoamperes on the signal source \(v_{g}\) when \(v_{g}=1 \mathrm{~V}\). e) Repeat (a)–(d) assuming an ideal op amp.

Suppose the strain gages in the bridge in Fig. 5.21 have the value \(120 \ \Omega \pm 1 \%\). The power supplies to the op amp are \(\pm 15 \mathrm{~V}\), and the reference voltage, \(v_{\text {ref }}\), is taken from the positive power supply. a) Calculate the value of \(R_{f}\) so that when the strain gage that is lengthening reaches its maximum length, the output voltage is 5 V. b) Suppose that we can accurately measure 50 mV changes in the output voltage. What change in strain gage resistance can be detected in milliohms?

a) For the circuit shown in Fig. P5.50, show that if the output voltage of the op amp is approximately vo L Rf R2 (R + Rf) (R + 2Rf) (- R)vin.b) Find if and c) Find the actual value of in (b).

a) If percent error is defined as \(\% \text { error }=\left[\frac{\text { approximate value }}{\text { true value }}-1\right] \times 100\), show that the percent error in the approximation of \(v_{o}\) in Problem 5.50 is \(\% \text { error }=\frac{\Delta R}{R} \frac{\left(R+R_{\mathrm{f}}\right)}{\left(R+2 R_{\mathrm{f}}\right)} \times 100\). b) Calculate the percent error in \(v_{o}\) for Problem 5.50.

Assume the percent error in the approximation of \(v_{o}\) in the circuit in Fig. P5.50 is not to exceed 1%. What is the largest percent change in R that can be tolerated?

Assume the resistor in the variable branch of the bridge circuit in Fig. P5.50 is \(R-\Delta R\) instead of \(R+\Delta R\). a) What is the expression for \(v_{o}\) if \(\Delta R \ll R\)? b) What is the expression for the percent error in as a function of \(R, R_{\mathrm{f}}\), and \(\Delta R\)? c) Assume the resistance in the variable arm of the bridge circuit in Fig. P5.50 is \(9810 \ \Omega\) and the values of \(R, R_{\mathrm{f}}\), and \(v_{\mathrm{in}}\) are the same as in Problem 5.50(b). What is the approximate value of \(v_{o}\)? d) What is the percent error in the approximation of \(v_{o}\) when the variable arm resistance is \(9810 \ \Omega\)?

a) Using the circuit in Fig. 15.1, design a low-pass filter with a passband gain of 10 dB and a cutoff frequency of 1 kHz. Assume a 750 nF capacitor is available. b) Draw the circuit diagram and label all components.

a) Using only three components from Appendix H, design a low-pass filter with a cutoff frequency and passband gain as close as possible to the specifications in Problem 15.1(a). Draw the circuit diagram and label all component values. b) Calculate the percent error in this new filters cutoff frequency and passband gain when compared to the values specified in Problem 15.1(a).

Design an op amp-based low-pass filter with a cutoff frequency of 2500 Hz and a passband gain of 5 using a 10 nF capacitor. a) Draw your circuit, labeling the component values and output voltage. b) If the value of the feedback resistor in the filter is changed but the value of the resistor in the forward path is unchanged, what characteristic of the filter is changed?

The input to the low-pass filter designed in Problem 15.3 is 3.5 cos t V. a) Suppose the power supplies are . What is the smallest value of that will still cause the op amp to operate in its linear region? b) Find the output voltage when . c) Find the output voltage when . d) Find the output voltage when .

Find the transfer function for the circuit shown in Fig. P15.5 if is the equivalent impedance of the feedback circuit, is the equivalent impedance of the input circuit, and the operational amplifier is ideal.

a) Use the results of Problem 15.5 to find the transfer function of the circuit shown in Fig. P15.6. b) What is the gain of the circuit as ? c) What is the gain of the circuit as ? d) Do your answers to (b) and (c) make sense in terms of known circuit behavior?

Repeat Problem 15.6, using the circuit shown in Fig. P15.7.

a) Use the circuit in Fig. 15.4 to design a high-pass filter with a cutoff frequency of 8 kHz and a passband gain of 14 dB. Use a 3.9 nF capacitor in the design. b) Draw the circuit diagram of the filter and label all the components.

Using only three components from Appendix H, design a high-pass filter with a cutoff frequency and passband gain as close as possible to the specifications in Problem 15.8. a) Draw the circuit diagram and label all component values. b) Calculate the percent error in this new filters cutoff frequency and passband gain when compared to the values specified in Problem 15.10(a).

Design an op amp-based high-pass filter with a cutoff frequency of 4 kHz and a passband gain of 8 using a 250 nF capacitor. a) Draw your circuit, labeling the component values and the output voltage. b) If the value of the feedback resistor in the filter is changed but the value of the resistor in the forward path is unchanged, what characteristic of the filter is changed?

The input to the high-pass filter designed in Problem 15.10 is 2.5 cos t V. a) Suppose the power supplies are . What is the smallest value of that will still cause the op amp to operate in its linear region? b) Find the output voltage when . c) Find the output voltage when . d) Find the output voltage when .

The voltage transfer function for either high-pass prototype filter shown in Fig. P15.12 is H(s) = s s + 1 . Show that if either circuit is scaled in both magnitude and frequency, the scaled transfer function is

The voltage transfer function of either low-pass prototype filter shown in Fig. P15.13 is Show that if either circuit is scaled in both magnitude and frequency, the scaled transfer function is

The voltage transfer function of the prototype bandpass filter shown in Fig. P15.14 is H(s) = a 1 Qbs s2 + a 1 Q bs + 1 Show that if the circuit is scaled in both magnitude and frequency, the scaled transfer function isH(s) = a 1 Qb a s kf b a s kf b 2 + a 1 Qb a s kf b + 1

a) Specify the component values for the prototype passive bandpass filter described in Problem 15.14 if the quality factor of the filter is 20. b) Specify the component values for the scaled bandpass filter described in Problem 15.14 if the quality factor is 20; the center, or resonant, frequency is and the impedance at resonance is c) Draw a circuit diagram of the scaled filter and label all the components.

An alternative to the prototype bandpass filter illustrated in Fig. P15.14 is to make , and henrys. a) What is the value of C in the prototype filter circuit? b) What is the transfer function of the prototype filter? c) Use the alternative prototype circuit just described to design a passive bandpass filter that has a quality factor of 16, a center frequency of and an impedance of at resonance. d) Draw a diagram of the scaled filter and label all the components. e) Use the results obtained in Problem 15.14 to write the transfer function of the scaled circuit.

The passive bandpass filter illustrated in Fig. 14.22 has two prototype circuits. In the first prototype circuit, and In the second prototype circuit, farads, and henrys. a) Use one of these prototype circuits (your choice) to design a passive bandpass filter that has a quality factor of 25 and a center frequency of The resistor 50 krad>s. R is 40 k.b) Draw a circuit diagram of the scaled filter and label all components.

The passive bandreject filter illustrated in Fig. 14.28(a) has the two prototype circuits shown in Fig. P15.18. a) Show that for both circuits, the transfer function is b) Write the transfer function for a bandreject filter that has a center frequency of and a quality factor of 10.

The transfer function for the bandreject filter shown in Fig. 14.28(a) is Show that if the circuit is scaled in both magnitude and frequency, the transfer function of the scaled circuit is equal to the transfer function of the unscaled circuit with s replaced by where is the frequency scale factor.

Show that the observation made in Problem 15.19 with respect to the transfer function for the circuit in Fig. 14.28(a) also applies to the bandreject filter circuit (lower one) in Fig. 14.31.

The two prototype versions of the passive bandreject filter shown in Fig. 14.31 (lower circuit) are shown in Fig. P15.21(a) and (b). Show that the transfer function for either version is H(s) = s2 + 1 s2 + a 1 Q bs + 1

The circuit in Fig. P9.24 is scaled so that the resistor is replaced by resistor and the 400 mH inductor is replaced by a 20 mH inductor. a) What is the scaled value of the capacitor? b) Find the frequency for which the impedance Zab is purely resistive for the scaled circuit. c) How is the frequency you found in part (b) related to the frequency for which the impedance Zab is purely resistive in the un-scaled circuit?

Scale the inductor and capacitor in Fig. P9.66 so that the magnitude and the phase angle of the output current do not change when the input frequency is changed from 250 rad/sec to 10,000 rad/sec. a) What are the scaled values of the inductor and capacitor? b) What is the steady-state value of the output current, , when the input current is 60 cos 10,000t mA?

Scale the bandpass filter in Problem 14.18 so that the center frequency is 25 kHz and the quality factor is still 8, using a inductor. Determine the values of the resistor and the capacitor, and the two cutoff frequencies of the scaled filter.

Scale the bandreject filter in Problem 14.38 to get a center frequency of 16 krad/s, using a 50 nF capacitor. Determine the values of the resistor, the inductor, and the bandwidth of the scaled filter.

a) Show that if the low-pass filter circuit illustrated in Fig. 15.1 is scaled in both magnitude and frequency, the transfer function of the scaled circuit is the same as Eq. 15.1 with s replaced by , where is the frequency scale factor. b) In the prototype version of the low-pass filter circuit in Fig. 15.1, and What is the transfer function of the prototype circuit? c) Using the result obtained in (a), derive the transfer function of the scaled filter.

a) Show that if the high-pass filter illustrated in Fig. 15.4 is scaled in both magnitude and frequency, the transfer function is the same as Eq. 15.4 with s replaced by where is the frequency scale factor. b) In the prototype version of the high-pass filter circuit in Fig. 15.4, and What is the transfer function of the prototype circuit? c) Using the result in (a), derive the transfer function of the scaled filter.

a) Using capacitors, design an active broadband first-order bandpass filter that has a lower cutoff frequency of 200 Hz, an upper cutoff frequency of 2000 Hz, and a passband gain of 20 dB. Use prototype versions of the low-pass and high-pass filters in the design process (see Problems 15.26 and 15.27). b) Write the transfer function for the scaled filter. c) Use the transfer function derived in part (b) to find where is the center frequency of the filter. d) What is the passband gain (in decibels) of the filter at ? e) Using a computer program of your choice, make a Bode magnitude plot of the filter.

a) Using capacitors, design an active broadband first-order bandreject filter with a lower cutoff frequency of 80 Hz, an upper cutoff frequency of 800 Hz, and a passband gain of 0 dB. Use the prototype filter circuits introduced in Problems 15.26 and 15.27 in the design process. b) Draw the circuit diagram of the filter and label all the components. c) What is the transfer function of the scaled filter? d) Evaluate the transfer function derived in (c) at the center frequency of the filter. e) What is the gain (in decibels) at the center frequency? f) Using a computer program of your choice, make a Bode magnitude plot of the filter transfer function.

Design a unity-gain bandpass filter, using a cascade connection, to give a center frequency of 200 Hz and a bandwidth of 1000 Hz. Use capacitors. Specify and RH.

Design a parallel bandreject filter with a center frequency of , a bandwidth of , and a passband gain of 6. Use capacitors, and specify all resistor values.

Show that the circuit in Fig. P15.32 behaves as a bandpass filter. (Hintfind the transfer function for this circuit and show that it has the same form asthe transfer function for a bandpass filter. Use the result from Problem 15.1.) a) Find the center frequency, bandwidth and gain for this bandpass filter. b) Find the cutoff frequencies and the quality for this bandpass filter.

For circuits consisting of resistors, capacitors, inductors, and op amps, involves only even powers of To illustrate this, compute for the three circuits in Fig. P15.33 when Figure P15.33 R 1 sC (c) Vi Vo (a) (b) Vo Vi R1 R1 R2 R3 R2 1 sC 1 sC 1 sC Vo Vi sL H(s) =

a) Determine the order of a low-pass Butterworth filter that has a cutoff frequency of 2000 Hz and a gain of no more than at 7000 Hz. b) What is the actual gain, in decibels, at 7000 Hz?

The circuit in Fig. 15.21 has the transfer function given by Eq. 15.34. Show that if the circuit in Fig. 15.21 is scaled in both magnitude and frequency, the transfer function of the scaled circuit is H(s) = 1 R2 C1C2 a s kf b 2 + 2 RC1 a s kf b + 1 R2 C1C2

a) Write the transfer function for the prototype low-pass Butterworth filter obtained in Problem 15.34(a). b) Write the transfer function for the scaled filter in (a) (see Problem 15.35). c) Check the expression derived in part (b) by using it to calculate the gain (in decibels) at 7000 Hz. Compare your result with that found in Problem 15.34(b).

a) Using resistors and ideal op amps, design a circuit that will implement the low-pass Butterworth filter specified in Problem 15.34. The gain in the passband is one. b) Construct the circuit diagram and label all component values.

The purpose of this problem is to illustrate the advantage of an nth-order low-pass Butterworth filter over the cascade of n identical low-pass sections by calculating the slope (in decibels per decade) of each magnitude plot at the corner frequency To facilitate the calculation, let y represent the magnitude of the plot (in decibels), and let Then calculate at for each plot. a) Show that at the corner frequency of an nth-order low-pass prototype Butterworth filter, b) Show that for a cascade of n identical low-pass prototype sections, the slope at is c) Compute for each type of filter for and . d) Discuss the significance of the results obtained in part (c).

Verify the entries in Table 15.1 for and n = 6

The circuit in Fig. 15.25 has the transfer function given by Eq. 15.47. Show that if the circuit is scaled in both magnitude and frequency, the transfer function of the scaled circuit is Hence the transfer function of a scaled circuit is obtained from the transfer function of an unscaled circuit by simply replacing s in the unscaled transfer function by where is the frequency scaling factor.

a) Using resistors and ideal op amps, design a high-pass unity-gain Butterworth filter that has a cutoff frequency of 2.5 kHz and has a gain of no more than at 500 Hz. b) Draw a circuit diagram of the filter and label all the components.

a) Using 250 nF capacitors and ideal op amps, design a low-pass unity-gain Butterworth filter with a cutoff frequency of 40 kHz and is down at least at 200 kHz. b) Draw a circuit diagram of the filter and label all component values.

The high-pass filter designed in Problem 15.41 is cascaded with the low-pass filter designed in Problem 15.42. a) Describe the type of filter formed by this interconnection. b) Specify the cutoff frequencies, the midfrequency, and the quality factor of the filter. c) Use the results of Problems 15.35 and 15.40 to derive the scaled transfer function of the filter. d) Check the derivation of (c) by using it to calculate where is the midfrequency of the filter.

a) Design a broadband Butterworth bandpass filter with a lower cutoff frequency of 200 Hz and an upper cutoff frequency of 2500 Hz. The passband gain of the filter is 40 dB. The gain should be down at least 40 dB at 40 Hz and 12.5 kHz. Use 1 F capacitors in the high-pass circuit and resistors in the low-pass circuit. b) Draw a circuit diagram of the filter and label all the components.

a) Derive the expression for the scaled transfer function for the filter designed in Problem 15.44.b) Using the expression derived in (a), find the gain n = 6. (in decibels) at 40 Hz and 1000 Hz. c) Do the values obtained in part (b) satisfy the filtering specifications given in Problem 15.44?

Derive the prototype transfer function for a fifthorder high-pass Butterworth filter by first writing the transfer function for a fifth-order prototype low-pass Butterworth filter and then replacing s by in the low-pass expression.

The fifth-order Butterworth filter in Problem 15.46 is used in a system where the cutoff frequency is a) What is the scaled transfer function for the filter? b) Test your expression by finding the gain (in decibels) at the cutoff frequency.

Show that if and in the circuit in Fig. 15.26, the prototype values of and are R3 = 2Q. R2 = Q 2Q2 - K , R

a) Use 20 nF capacitors in the circuit in Fig. 15.26 to design a bandpass filter with a quality factor of 16, a center frequency of 6.4 kHz, and a passband gain of 20 dB. b) Draw the circuit diagram of the filter and label all the components.

The purpose of this problem is to guide you through the analysis necessary to establish a design procedure for determining the circuit components in a filter circuit.The circuit to be analyzed is shown in Fig. P15.50. a) Analyze the circuit qualitatively and convince yourself that the circuit is a low-pass filter with a passband gain of b) Support your qualitative analysis by deriving the transfer function (Hint: In deriving the transfer function, represent the resistors with their equivalent conductances, that is, and so forth.) To make the transfer function useful in terms of the entries in Table 15.1, put it in the form c) Now observe that we have five circuit components and and three transfer function constraintsK, and At first glance, it appears we have two free choices among the five components. However, when we investigate the relationships between the circuit components and the transfer function constraints, we see that if is chosen, there is an upper limit on in order for to be realizable. With this in mind, show that if the three conductances are given by the expressions For to be realizable, d) Based on the results obtained in (c), outline the design procedure for selecting the circuit components once K, and are known. Figure P15.50

Assume the circuit analyzed in Problem 15.50 is part of a third-order low-pass Butterworth filter having a passband gain of 4. (Hint: implement the gain of 4 in the second-order section of the filter.) a) If in the prototype second-order section, what is the upper limit on ? b) If the limiting value of is chosen, what are the prototype values of and ? c) If the corner frequency of the filter is 2.5 kHz and is chosen to be 10 nF, calculate the scaled values of and d) Specify the scaled values of the resistors and the capacitor in the first-order section of the filter. e) Construct a circuit diagram of the filter and label all the component values on the diagram.

Interchange the Rs and Cs in the circuit in Fig. P15.50; that is, replace with with with with and with R2 C . a) Describe the type of filter implemented as a result of the interchange. b) Confirm the filter type described in (a) by deriving the transfer function Write the transfer function in a form that makes it compatible with Table 15.1. c) Set and derive the expressions for and in terms of K, and (See Problem 15.50 for the definition of and ) d) Assume the filter described in (a) is used in the same type of third-order Butterworth filter that has a passband gain of 8. With calculate the prototype values of and in the second-order section of the filter.

a) Use the circuits analyzed in Problems 15.50 and 15.52 to implement a broadband bandreject filter having a passband gain of 20 dB, a lower corner frequency of 1 kHz, an upper corner frequency of 8 kHz, and an attenuation of at least 24 dB at both 2 kHz and 4 kHz. Use 25 nF capacitors whenever possible. b) Draw a circuit diagram of the filter and label all the components.

a) Derive the transfer function for the bandreject filter described in Problem 15.53. b) Use the transfer function derived in part (a) to find the attenuation (in decibels) at the center frequency of the filter.

The purpose of this problem is to develop the design equations for the circuit in Fig. P15.55. (See Problem 15.50 for suggestions on the development of design equations.) a) Based on a qualitative analysis, describe the type of filter implemented by the circuit. b) Verify the conclusion reached in (a) by deriving the transfer function Write the transfer function in a form that makes it compatible with the entries in Table 15.1. c) How many free choices are there in the selection of the circuit components? d) Derive the expressions for the conductances and in terms of and the coefficients and (See Problem 15.50 for the definition of and ) e) Are there any restrictions on or ? f) Assume the circuit in Fig. P15.55 is used to design a fourth-order low-pass unity-gain Butterworth filter. Specify the prototype values of and in each second-order section if capacitors are used in the prototype circuit.

The fourth-order low-pass unity-gain Butterworth filter in Problem 15.55 is used in a system where the cutoff frequency is 3 kHz. The filter has 4.7 nF capacitors. a) Specify the numerical values of and in each section of the filter. b) Draw a circuit diagram of the filter and label all the components.

Interchange the Rs and Cs in the circuit in Fig. P15.55, that is, replace with with and vice versa. a) Analyze the circuit qualitatively and predict the type of filter implemented by the circuit. b) Verify the conclusion reached in (a) by deriving the transfer function Write the transfer function in a form that makes it compatible with the entries in Table 15.1. c) How many free choices are there in the selection of the circuit components? d) Find and as functions of and e) Are there any restrictions on and ? f) Assume the circuit is used in a third-order Butterworth filter of the type found in (a). Specify the prototype values of and in the secondorder section of the filter if C1 = C2 = 1 F.

a) The circuit in Problem 15.57 is used in a thirdorder high-pass unity-gain Butterworth filter that has a cutoff frequency of 800 Hz. Specify the values of and if 5 F capacitors are available to construct the filter. b) Specify the values of resistance and capacitance in the first-order section of the filter. c) Draw the circuit diagram and label all the components. d) Give the numerical expression for the scaled transfer function of the filter. e) Use the scaled transfer function derived in (d) to find the gain in dB at the cutoff frequency.

a) Show that the transfer function for a prototype narrow band bandreject filter is b) Use the result found in (a) to find the transfer function of the filter designed in Example 15.13.

a) Using the circuit shown in Fig. 15.29, design a narrow-band bandreject filter having a center frequency of 4 kHz and a quality factor of 10. Base the design on b) Draw the circuit diagram of the filter and label all component values on the diagram. c) What is the scaled transfer function of the filter?

Using the circuit in Fig. 15.32(a) design a volume control circuit to give a maximum gain of 14 dB and a gain of 11 dB at a frequency of 50 Hz. Use a 10 k resistor and a 50 k potentiometer. Test your design by calculating the maximum gain at and the gain at using the selected values of R1, R2, and C1.

Use the circuit in Fig. 15.32(a) to design a bass volume control circuit that has a maximum gain of 20 dB that drops off 3 dB at 75 Hz.

Plot the maximum gain in decibels versus when for the circuit designed in Problem 15.61. Let vary from 0 to 1 in increments of 0.1.

a) Show that the circuits in Fig. P15.64(a)and (b) are equivalent. b) Show that the points labeled x and y in Fig. P15.64(b) are always at the same potential. c) Using the information in (a) and (b), show that the circuit in Fig. 15.33 can be drawn as shown in Fig. P15.64(c). d) Show that the circuit in Fig. P15.64(c) is in the form of the circuit in Fig. 15.2, where Zf = R1 + aR2 + R1R2C1s 1 + R2C1s . Zi = R1 + (1 - a)R2 + R1R2C1s 1 + R2C1s ,

An engineering project manager has received a proposal from a subordinate who claims the circuit shown in Fig. P15.65 could be used as a treble volume control circuit if The subordinate further claims that the voltage transfer function for the circuit is where Fortunately the project engineer has an electrical engineering undergraduate student as an intern and therefore asks the student to check the subordinates claim. The student is asked to check the behavior of the transfer function as as and the behavior when and varies between 0 and 1. Based on your testing of the transfer function do you think the circuit could be used as a treble volume control? Explain.

In the circuit of Fig. P15.65 the component values are and a) Calculate the maximum boost in decibels. b) Calculate the maximum cut in decibels. c) Is significantly greater than ? d) When what is the boost in decibels when ? e) When what is the cut in decibels when ? f) Based on the results obtained in (d) and (e), what is the significance of the frequency when ?

Using the component values given in Problem 15.66, plot the maximum gain in decibels versus when is inifinite. Let vary from 0 to 1 in increments of 0.1.