Assuming a finite open-loop gain (A), a finite input resistance (Ri), and zero output

Derive an expression for \(v_{out}\) in terms of \(v_{in}\) for the circuit shown in Fig. 6.8.

Design a current source capable of providing \(500\ \mu A\) to a resistive load.

Assuming a finite open-loop gain (A), a finite input resistance (\(R_i\)), and zero output resistance (\(R_o\)), derive an expression for \(v_{out}\) in terms of \(v_{in}\) for the op amp circuit of Fig. 6.3.

Simulate the remaining op amp circuits described in this chapter, and compare the results to those predicted using the ideal op amp model.

Design a circuit that provides a 12 V output if a certain voltage (\(v_{signal}\)) exceeds 0 V, and a ?2 V output otherwise.

For the op amp circuit shown in Fig. 6.39, calculate \(v_{out}\) if (a) \(R_1 = R_2 = 100\ \Omega\) and \(v_{in} = 5\ V\); (b) \(R_2 = 200\ R_1\) and \(v_{in} = 1\ V\); (c) \(R_1 = 4.7\ k \Omega\), \(R_2 = 47\ k \Omega\), and \(v_{in} = 20\ sin\ 5t\ V\).

Determine the power dissipated by a \(100\ \Omega\) resistor connected between ground and the output pin of the op amp of Fig. 6.39 if \(v_{in} = 4\ V\) and (a) \(R_1 = 2R_2\); (b) \(R_1 = 1\ k \Omega\) and \(R_2 = 22\ k \Omega\); (c) \(R_1 = 100\ \Omega\) and \(R_2 = 101\ \Omega\).

Connect a \(1\ \Omega\) resistor between ground and the output terminal of the op amp of Fig. 6.39, and sketch \(v_{\mathrm{out}}(t)\) if (a) \(R_1 = R_2 = 10\ \Omega\) and \(v_{in} = 5\ sin\ 10t\ V\); (b) \(R_1 = 0.2R_2 = 1\ k \Omega\), and \(v_{in} = 5\ cos\ 10t\ V\); (c) \(R_1 = 10\ \Omega\), \(R_2 = 200\ \Omega\), and \(v_{in} = 1.5 + 5e^{?t}\ V\).

For the circuit of Fig. 6.40, calculate \(v_{out}\) if (a) \(R_1 = R_2 = 100\ k \Omega\), \(R_L = 100\ \Omega\), and \(v_{in} = 5\ V\); (b) \(R_1 = 0.1R_2\), \(R_L = \infty\), and \(v_{in} = 2\ V\); (c) \(R_1 = 1\ k \Omega\), \(R_2 = 0\), \(R_L = 1\ \Omega\), and \(v_{in} = 43.5\ V\).

(a) Design a circuit which converts a voltage \(v_1(t) = 9\ cos\ 5t\ V\) into 9 sin 5t V. (b) Verify your design by analyzing the final circuit.

A certain load resistor requires a constant 5 V dc supply. Unfortunately, its resistance value changes with temperature. Design a circuit which supplies the requisite voltage if only 9 V batteries and standard 10% tolerance resistor values are available.

For the circuit of Fig. 6.40, \(R_1 = R_L = 50\ \Omega\). Calculate the value of \(R_2\) required to deliver 5 W to \(R_L\) if \(V_{in}\) equals (a) 5 V; (b) 1.5 V. (c) Repeat parts (a) and (b) if \(R_L\) is reduced to \(22\ \Omega\).

Calculate \(v_{out}\) as labeled in the schematic of Fig. 6.41 if (a) \(i_{in} = 1\ mA\), \(R_p = 2.2\ k \Omega\), and \(R_3 = 1\ k \Omega\); (b) \(i_{in} = 2\ A\), \(R_p = 1.1\ \Omega\), and \(R_3 = 8.5\ \Omega\). (c) For each case, state whether the circuit is wired as a noninverting or an inverting amplifier. Explain your reasoning.

(a) Design a circuit using only a single op amp which adds two voltages \(v_1\) and \(v_2\) and provides an output voltage twice their sum (i.e., \(v_{\mathrm {out}}=2 v_{1}+2 v_{2}\)). (b) Verify your design by analyzing the final circuit.

(a) Design a circuit that provides a current i which is equal in magnitude to the sum of three input voltages \(v_1\), \(v_2\), and \(v_3\). (Compare volts to amperes.) (b) Verify your design by analyzing the final circuit.

(a) Design a circuit that provides a voltage \(v_{out}\) which is equal to the difference between two voltages \(v_2\) and \(v_1\) (i.e., \(v_{out} = v_2 ? v_1\)), if you have only the following resistors from which to choose: two \(1.5\ k \Omega\) resistors, four \(6\ k \Omega\) resistors, or three \(500\ \Omega\) resistors. (b) Verify your design by analyzing the final circuit.

Analyze the circuit of Fig. 6.42 and determine a value for \(V_1\), which is referenced to ground.

Derive an expression for \(v_{out}\) as a function of \(v_1\) and \(v_2\) for the circuit represented in Fig. 6.43.

Explain what is wrong with each diagram in Fig. 6.44 if the two op amps are known to be perfectly ideal.

For the circuit depicted in Fig. 6.45, calculate \(v_{out}\) if \(I_s = 2\ mA\), \(R_Y = 4.7\ k \Omega\), \(R_X = 1\ k \Omega\), and \(R_f = 500\ \Omega\).

Consider the amplifier circuit shown in Fig. 6.45. What value of \(R_f\) will yield \(v_{out} = 2\ V\) when \(I_s = 10\ mA\) and \(R_Y = 2R_X = 500\ \Omega\)?

With respect to the circuit shown in Fig. 6.46, calculate \(v_{out}\) if \(v_s\) equals (a) 2 cos 100t mV; (b) \(2\ sin(4t + 19^{\circ})\ V\).

Calculate \(v_{out}\) as labeled in the circuit of Fig. 6.47 if \(R_x = 1\ k \Omega\).

For the circuit of Fig. 6.47, determine the value of \(R_x\) that will result in a value of \(v_{out} = 10\ V\).

Referring to Fig. 6.48, sketch \(v_{out}\) as a function of (a) \(v_{in}\) over the range of \(?2\ V\ \leq\ v_{in}\ \leq\ + 2\ V\), if \(R_4 = 2\ k \Omega\); (b) \(R_4\) over the range of \(1\mathrm{\ k}\Omega\le R_4\le10\mathrm{\ k}\Omega\), if \(v_{in} = 300\ mV\).

The 1.5 V source of Fig. 6.49 is disconnected, and the output of the circuit shown in Fig. 6.48 is connected to the left-hand terminal of the \(500\ \Omega\) resistor instead. Calculate \(v_{out}\) if \(R_4 = 2\ k \Omega\) and (a) \(v_{in} = 2\ V\), \(v_1 = 1\ V\); (b) \(v_{in} = 1\ V\), \(v_1 = 0\); (c) \(v_{in} = 1\ V\), \(v)_1 = ?1\ V\).

For the circuit shown in Fig. 6.50, compute \(v_{out}\) if (a) \(v_1 = 2v_2 = 0.5v_3 = 2.2\ V\) and \(R_1 = R_2 = R_3 = 50\ k \Omega\); (b) \(v_1 = 0\), \(v_2 = ?8\ V\), \(v_3 = 9\ V\), and \(R_1 = 0.5R_2 = 0.4R_3 = 100\ k \Omega\).

(a) Design a circuit which will add the voltages produced by three separate pressure sensors, each in the range of \(0\ \leq\ v_{sensor}\ \leq\ 5\ V\), and produce a positive voltage \(v_{out}\) linearly correlated to the voltage sum such that \(v_{out} = 0\) when all three voltages are zero, and \(v_{out} = 2\ V\) when all three voltages are at their maximum. (b) Verify your design by analyzing the final circuit.

(a) Design a circuit which produces an output voltage \(v_{out}\) proportional to the difference of two positive voltages \(v_1\) and \(v_2\) such that \(v_{out} = 0\) when both voltages are equal, and \(v_{out} = 10\ V\) when \(v_1 ? v_2 = 1\ V\). (b) Verify your design by analyzing the final circuit.

(a) Three pressure-sensitive sensors are used to double-check the weight readings obtained from the suspension systems of a long-range jet airplane. Each sensor is calibrated such that \(10\ \mu V\) corresponds to 1 kg. Design a circuit which adds the three voltage signals to produce an output voltage calibrated such that 10 V corresponds to 400,000 kg, the maximum takeoff weight of the aircraft. (b) Verify your design by analyzing the final circuit.

(a) The oxygen supply to a particular bathysphere consists of four separate tanks, each equipped with a pressure sensor capable of measuring between 0 (corresponding to 0 V output) and 500 bar (corresponding to 5 V output). Design a circuit which produces a voltage proportional to the total pressure in all tanks, such that 1.5 V corresponds to 0 bar and 3 V corresponds to 2000 bar. (b) Verify your design by analyzing the final circuit.

For the circuit shown in Fig. 6.51, let \(v_{in} = 8\ V\), and select values for \(R_1\), \(R_2\), and \(R_3\) to ensure an output voltage \(v_{out} = 4\ V\).

For the circuit of Fig. 6.52, derive an expression for \(v_{out}\) in terms of \(v_{in}\).

Construct a circuit based on the 1N4740 diode which provides a reference voltage of 10 V if only 9 V batteries are available. Note that the breakdown voltage of this diode is equal to 10 V at a current of 25 mA.

Employ a 1N4733 Zener diode to construct a circuit which provides a 4 V reference voltage to a \(1\ k \Omega\) load, if only 9 V batteries are available as sources. Note that the Zener breakdown voltage of this diode is 5.1 V at a current of 76 mA.

(a) Design a circuit which provides a 5 V dc reference voltage to an unknown (nonzero resistance) load, if only a 9 V battery is available as a supply. (b) Verify your design with an appropriate simulation. As part of this, determine the acceptable range for the load resistor.

A particular passive network can be represented by a Thévenin equivalent resistance between \(10\ \Omega\) and \(125\ \Omega\) depending on the operating temperature. (a) Design a circuit which provides a constant 2.2 V to this network regardless of temperature. (b) Verify your design with an appropriate simulation (resistance can be varied from within a single simulation, as described in Chap. 8).

Calculate the voltage \(V_1\) as labeled in the circuit of Fig. 6.53 if the battery is rated at \(V_{batt}\) equal to (a) 9 V; (b) 12 V. (c) Verify your solutions with appropriate simulations, commenting on the possible origin of any discrepancies.

(a) Design a current source based on the 1N750 diode which is capable of providing a dc current of \(750\ \mu A\) to a load \(R_L\), such that \(1\ k \Omega\ <\ R_L\ <\ 50\ k \Omega\). (b) Verify your design with an appropriate simulation (note that resistance can be varied within a single simulation, as described in Chap. 8).

(a) Design a current source able to provide a dc current of 10 mA to an unspecified load. Use a 1N4747 diode (\(V_{br} = 20\ V\) at 12.5 mA). (b) Use an appropriate simulation to determine the permissible range of load resistance for your design.

The circuit depicted in Fig. 6.54 is known as a Howland current source. Derive expressions for \(v_{out}\) and \(I_L\), respectively as a function of \(V_1\) and \(V_2\).

For the circuit depicted in Fig. 6.54, known as a Howland current source, set \(V_2 = 0\), \(R_1 = R_3\), and \(R_2 = R_4\); then solve for the current \(I_L\) when \(R_1 = 2R_2 = 1\ k \Omega\) and \(R_L = 100\ \Omega\).

(a) Employ the parameters listed in Table 6.3 for the \(\mu A741\) op amp to analyze the circuit of Fig. 6.55 and compute a value for \(v_{out}\). (b) Compare your result to what is predicted using the ideal op amp model.

(a) Employ the parameters listed in Table 6.3 for the \(\mu A741\) op amp to analyze the circuit of Fig. 6.10 if \(R = 1.5\ k \Omega\), \(v_1 = 2\ V\), and \(v_2 = 5\ V\). (b) Compare your solution to what is predicted using the ideal op amp model.

Define the following terms, and explain when and how each can impact the performance of an op amp circuit: (a) common-mode rejection ratio; (b) slew rate; (c) saturation; (d) feedback.

For the circuit of Fig. 6.56, replace the \(470\ \Omega\) resistor with a short circuit, and compute \(v_{out}\) using (a) the ideal op amp model; (b) the parameters listed in Table 6.3 for the \(\mu A741\) op amp; (c) an appropriate PSpice simulation. (d) Compare the values obtained in parts (a) to (c) and comment on the possible origin of any discrepancies.

If the circuit of Fig. 6.55 is analyzed using the detailed model of an op amp (as opposed to the ideal op amp model), calculate the value of open-loop gain A required to achieve a closed-loop gain within 2% of its ideal value.

Replace the 2 V source in Fig. 6.56 with a sinusoidal voltage source having a magnitude of 3 V and radian frequency \(\omega = 2 \pi f\). (a) Which device, a \(\mu A741\) op amp or an LF411 op amp, will track the source frequency better over the range \(1\ Hz\ <\ f\ <\ 10\ MHz\)? Explain. (b) Compare the frequency performance of the circuit over the range \(1\ Hz\ <\ f\ <\ 10\ MHz\) using appropriate PSpice simulations, and compare the results to your prediction in part (a).

(a) For the circuit of Fig. 6.56, if the op amp (assume LF411) is powered by matched 9 V supplies, estimate the maximum value to which the \(470\ \Omega\) resistor can be increased before saturation effects become apparent. (b) Verify your prediction with an appropriate simulation.

For the circuit of Fig. 6.55, calculate the differential input voltage and the input bias current if the op amp is a(n) (a) \(\mu A741\); (b) LF411; (c) AD549K; (d) OPA690.

Human skin, especially when damp, is a reasonable conductor of electricity. If we assume a resistance of less than \(10\ M \Omega\) for a fingertip pressed across two terminals, design a circuit which provides a +1 V output if this nonmechanical switch is “closed” and ?1 V if it is “open.”

Design a circuit which provides an output voltage \(v_{out}\) based on the behavior of another voltage \(v_{in}\), such that \(v_{\text {out }}=\left\{\begin{aligned} +2.5 \mathrm{~V} &\ \ \ \ \ \ \ \ \ v_{\text {in }}>1 \mathrm{~V} \\ 1.2 \mathrm{~V} &\ \ \ \ \ \ \ \ \text { otherwise } \end{aligned}\right.\)

For the instrumentation amplifier shown in Fig. 6.38a, assume that the three internal op amps are ideal, and determine the CMRR of the circuit if (a) \(R_1 = R_3\) and \(R_2 = R_4\); (b) all four resistors have different values.

For the circuit depicted in Fig. 6.57, sketch the expected output voltage \(v_{out}\) as a function of \(v_{active}\) for \(?5\ V\ \leq\ v_{active}\ \leq\ + 5\ V\), if \(v_{ref}\) is equal to (a) ?3 V; (b) +3 V.

For the circuit depicted in Fig. 6.58, (a) sketch the expected output voltage \(v_{out}\) as a function of \(v_1\) for \(?5\ V\ \leq\ v_1\ \leq\ + 5\ V\), if \(v_2 = +2\ V\); (b) sketch the expected output voltage \(v_{out}\) as a function of \(v_2\) for \(?5\ V\ \leq\ v_2\ \leq\ + 5\ V\), if \(v_1 = +2\ V\).

For the circuit depicted in Fig. 6.59, sketch the expected output voltage \(v_{out}\) as a function of \(v_{active}\), if \(?2\ V\ \leq\ v_{active}\ \leq\ + 2\ V\). Verify your solution using a \(\mu A741\) (although it is slow compared to op amps designed specifically for use as comparators, its PSpice model works well, and as this is a dc application speed is not an issue). Submit a properly labeled schematic with your results.

In digital logic applications, a +5 V signal represents a logic “1” state, and a 0 V signal represents a logic “0” state. In order to process real-world information using a digital computer, some type of interface is required, which typically includes an analog-to-digital (A/D) converter—a device that converts analog signals into digital signals. Design a circuit that acts as a simple 1-bit A/D, with any signal less than 1.5 V resulting in a logic “0” and any signal greater than 1.5 V resulting in a logic “1.”

(a) You’re given an electronic switch which requires 5 V at 1 mA in order to close; it is open with no voltage present at its input. If the only microphone available produces a peak voltage of 250 mV, design a circuit which will energize the switch when someone speaks into the microphone. Note that the audio level of a general voice may not correspond to the peak voltage of the microphone. (b) Discuss any issues that may need to be addressed if your circuit were to be implemented.

You’ve formed a band, despite advice to the contrary. Actually, the band is pretty good except for the fact that the lead singer (who owns the drum set, the microphones, and the garage where you practice) is a bit tone-deaf. Design a circuit that takes the output from each of the five microphones your band uses, and adds the voltages to create a single voltage signal which is fed to the amplifier. Except not all voltages should be equally amplified. One microphone output should be attenuated such that its peak voltage is 10% of any other microphone’s peak voltage.

Cadmium sulfide (CdS) is commonly used to fabricate resistors whose value depends on the intensity of light shining on the surface. In Fig. 6.61 a CdS “photocell” is used as the feedback resistor \(R_f\). In total darkness, it has a resistance of \(100\ k \Omega\), and a resistance of \(10\ k \Omega\) under a light intensity of 6 candela. \(R_L\) represents a circuit that is activated when a voltage of 1.5 V or less is applied to its terminals. Choose \(R_1\) and \(V_s\) so that the circuit represented by \(R_L\) is activated by a light of 2 candela or brighter.

A fountain outside a certain office building is designed to reach a maximum height of 5 meters at a flow rate of 100 l/s. A variable position valve in line with the water supply to the fountain can be controlled electrically, such that 0 V applied results in the valve being fully open, and 5 V results in the valve being closed. In adverse wind conditions the maximum height of the fountain needs to be adjusted; if the wind velocity exceeds 50 km/h, the height cannot exceed 2 meters. A wind velocity sensor is available which provides a voltage calibrated such that 1 V corresponds to a wind velocity of 25 km/h. Design a circuit which uses the velocity sensor to control the fountain according to specifications.

Derive an expression for \(v_{out}\) in terms of \(v_1\) and \(v_2\) for the circuit shown in Fig. 6.10, also known as a difference amplifier.

An historic bridge is showing signs of deterioration. Until renovations can be performed, it is decided that only cars weighing less than 1600 kg will be allowed across. To monitor this, a four-pad weighing system is designed. There are four independent voltage signals, one from each wheel pad, with 1 mV = 1 kg. Design a circuit to provide a positive voltage signal to be displayed on a DMM (digital multimeter) that represents the total weight of a vehicle, such that 1 mV = 1 kg.You may assume there is no need to buffer the wheel pad voltage signals.

Design a circuit to provide a reference voltage of 6 V using a 1N750 Zener diode and a noninverting amplifier.

Obtain an expression for \(v_{out}\) as labeled in the circuit of Fig. 6.49 if \(v_1\) equals (a) 0 V; (b) 1 V; (c) ?5 V; (d) 2 sin 100t V.

(a) Design a current source able to provide a dc current of 50 mA to an unspecified load. Use a 1N4733 diode (\(V_{br} = 5.1\ V\) at 76 mA). (b) Use an appropriate simulation to determine the permissible range of load resistance for your design.

Calculate the common-mode gain for each device listed in Table 6.3. Express your answer in units of V/V, not dB.

A common application for instrumentation amplifiers is to measure voltages in resistive strain gauge circuits. These strain sensors work by exploiting the changes in resistance that result from geometric distortions, as in Eq. [6] of Chap. 2. They are often part of a bridge circuit, as shown in Fig. 6.60a, where the strain gauge is identified as \(R_G\). (a) Show that \(V_{\text {out }}=V_{\text {in }}\left[\frac{R_{2}}{R_{1}+R_{2}}-\frac{R_{3}}{R_{3}+R_{\mathrm{Gaugc}}}\right]\). (b) Verify that \(V_{out} = 0\) when the three fixed-value resistors \(R_1\), \(R_2\), and \(R_3\) are all chosen to be equal to the unstrained gauge resistance \(R_{Gauge}\). (c) For the intended application, the gauge selected has an unstrained resistance of \(5\ k \Omega\), and a maximum resistance increase of \(50\ m \Omega\) is expected. Only \(\pm 12\ V\) supplies are available. Using the instrumentation amplifier of Fig. 6.60b, design a circuit that will provide a voltage signal of +1 V when the strain gauge is at its maximum loading. AD622 Specifications Amplifier gain G can be varied from 2 to 1000 by connecting a resistor between pins 1 and 8 with a value calculated by \(R = \frac{50.5}{G-1}\ k \Omega\).

For the circuit of Fig. 6.43, let all resistor values equal \(5\ k \Omega\). Sketch \(v_{out}\) as a function of time if (a) \(v_1 = 5\ sin\ 5t\ V\) and \(v_2 = 5\ cos\ 5t\ V\); (b) \(v_1 = 4e^{?t}\ V\) and \(v_2 = 5e^{?2t}\ V\); (c) \(v_1 = 2\ V\) and \(v_2 = e^{?t}\ V\).

Derive an expression for \(v_{out}\) in terms of \(v_1\) and \(v_2\) for the circuit shown in Fig. 6.10, also known as a difference amplifier.

An historic bridge is showing signs of deterioration. Until renovations can be performed, it is decided that only cars weighing less than 1600 kg will be allowed across. To monitor this, a four-pad weighing system is designed. There are four independent voltage signals, one from each wheel pad, with 1 mV = 1 kg. Design a circuit to provide a positive voltage signal to be displayed on a DMM (digital multimeter) that represents the total weight of a vehicle, such that 1 mV = 1 kg.You may assume there is no need to buffer the wheel pad voltage signals.

Design a circuit to provide a reference voltage of 6 V using a 1N750 Zener diode and a noninverting amplifier.

Obtain an expression for \(v_{out}\) as labeled in the circuit of Fig. 6.49 if \(v_1\) equals (a) 0 V; (b) 1 V; (c) ?5 V; (d) 2 sin 100t V.

(a) Design a current source able to provide a dc current of 50 mA to an unspecified load. Use a 1N4733 diode (\(V_{br} = 5.1\ V\) at 76 mA). (b) Use an appropriate simulation to determine the permissible range of load resistance for your design.

Calculate the common-mode gain for each device listed in Table 6.3. Express your answer in units of V/V, not dB.

A common application for instrumentation amplifiers is to measure voltages in resistive strain gauge circuits. These strain sensors work by exploiting the changes in resistance that result from geometric distortions, as in Eq. [6] of Chap. 2. They are often part of a bridge circuit, as shown in Fig. 6.60a, where the strain gauge is identified as \(R_G\). (a) Show that \(V_{\text {out }}=V_{\text {in }}\left[\frac{R_{2}}{R_{1}+R_{2}}-\frac{R_{3}}{R_{3}+R_{\mathrm{Gaugc}}}\right]\). (b) Verify that \(V_{out} = 0\) when the three fixed-value resistors \(R_1\), \(R_2\), and \(R_3\) are all chosen to be equal to the unstrained gauge resistance \(R_{Gauge}\). (c) For the intended application, the gauge selected has an unstrained resistance of \(5\ k \Omega\), and a maximum resistance increase of \(50\ m \Omega\) is expected. Only \(\pm 12\ V\) supplies are available. Using the instrumentation amplifier of Fig. 6.60b, design a circuit that will provide a voltage signal of +1 V when the strain gauge is at its maximum loading. AD622 Specifications Amplifier gain G can be varied from 2 to 1000 by connecting a resistor between pins 1 and 8 with a value calculated by \(R = \frac{50.5}{G-1}\ k \Omega\).

For the circuit of Fig. 6.43, let all resistor values equal \(5\ k \Omega\). Sketch \(v_{out}\) as a function of time if (a) \(v_1 = 5\ sin\ 5t\ V\) and \(v_2 = 5\ cos\ 5t\ V\); (b) \(v_1 = 4e^{?t}\ V\) and \(v_2 = 5e^{?2t}\ V\); (c) \(v_1 = 2\ V\) and \(v_2 = e^{?t}\ V\).