# 14.8 Problems

## Review Questions

- What is the basic function of an integrator?
- What is the basic function of a differentiator?
- What is the function of the capacitor in the basic integrator and differentiator?
- Why are capacitors used in favor of inductors?
- What practical modifications need to be done to the basic integrator, and why?
- What practical modifications need to be done to the basic differentiator, and why?
- What are the negative side effects of the practical versus basic integrator and differentiator?
- What is an analog computer, and what is it used for?
- What are some of the advantages and disadvantages of the analog computer versus the digital computer?
- How might integrators and differentiators be used as wave-shaping circuits?

## Problems

### Analysis Problems

- Sketch the output waveform for the circuit of Figure 10.8.1 if the input is a 4 V peak square wave at 1 kHz.

- Repeat Problem 1 if πππ(π‘)=5sin2π318π‘ .
- Repeat Problem 1 if πππ(π‘)=20cos2π10000π‘ .
- Using the circuit of Problem 1, if the input is a ramp with a slope of 10 V/s, find the output after 1 ms, 10 ms, and 4 s. Sketch the resulting wave.
- Determine the low frequency gain for the circuit of Figure 10.8.2 .

- If πΆ=33 nF in Figure 10.8.2 , determine π
_{ππ’π‘}if π_{ππ}is a 200 mV peak sine wave at 50 kHz. - Repeat Problem 6 using a 200 mV peak square wave at 50 kHz.
- Sketch the output of the circuit of Figure 10.2.7 with the input signal given in Figure 10.8.3 .
- Assume that the input to the circuit of Figure 10.2.7 is 100 mV DC. Sketch the output waveform, and determine if it goes to saturation.

- Sketch the output waveform for Figure 10.2.7, given an input of: πππ(π‘)=0.5cos2π9000π‘ .
- Given the circuit of Figure 10.8.4 , sketch the output waveform if the input is a 100 Hz, 1 V peak triangle wave.

- Repeat Problem 11 if the input is a 2 V peak square wave at 500 Hz. Assume that the rise and fall times are 1 π s and are linear (vs. exponential).
- Repeat Problem 11 for the following input: πππ(π‘)=3cos2π60π‘ .
- Repeat Problem 11 for the following input: πππ(π‘)=0.5sin2π1000π‘ .
- Repeat Problem 11 for the following input: πππ(π‘)=10π‘2 .

- Given the input shown in Figure 10.8.5 , sketch the output of the circuit of Figure 10.19.
- Given the input shown in Figure 10.8.5 , sketch the output of the circuit of Figure 10.19.

- Sketch the output waveform for Figure 10.3.9, given an input of πππ(π‘)=0.5cos2π4000π‘ .
- Sketch the output waveform for Figure 10.3.9, if π
_{ππ}is a 300 mV peak triangle wave at 2500 Hz.

### Design Problems

- Given the circuit of Figure 10.8.1 ,
- Determine the value of πΆ required in order to yield an integration constant of β2000.
- Determine the required of value of π for minimum integration offset error.
- Determine ππππ€ .
- Determine the 99% accuracy point.

- Given the differentiator of Figure 10.8.7 , determine the value of π
that will set the differentiation constant to β103 .

- Determine the value of π π such that the maximum gain is 20 for the circuit of Figure 10.8.7 . (Use the values of Problem 21.)
- Determine the value of πΆπ in Problem 22 such that noise above 5 kHz is attenuated.
- Design an integrator to meet the following specifications: integration constant of β4500, ππππ€ no greater than 300 Hz, πππ at least 6 k Ξ© , and DC gain no more than 32 dB.
- Design a differentiator to meet the following specifications: differentiation constant of β1.2πΈβ4 , πβππβ at least 100 kHz, and a minimum πππ of 50 Ξ© .

### Challenge Problems

- Sketch the output waveform for the circuit of Figure 10.2.7 if the input is the following damped sinusoid: πππ(π‘)=3πβ200π‘sin2π1000π‘ .
- A 1.57 V peak, 500 Hz square wave may be written as the following infinite series: πππ(π‘)=2ββπ=112πβ1sin2π500(2πβ1)π‘

Using this as the input signal, determine the infinite series output Equation for the circuit of Figure 10.2.7. - Sketch the output for the preceding problem. Do this by graphically adding the first few terms of the output.
- Remembering that the voltage across an inductor is equal to the inductance times the rate of change of current, determine the output Equation for the circuit of Figure 10.8.8 .

- Repeat the preceding Problem for the circuit of Figure 10.8.9 .

- A 2.47 V peak, 500 Hz triangle wave may be written as the following infinite series: πππ(π‘)=2ββπ=11(2πβ1)2cos2π500(2πβ1)π‘

Using this as the input signal, determine the infinite series output Equation for the circuit of Figure 10.3.9. - Sketch the output for the preceding problem. Do this by graphically adding the first few terms of the output.

### Computer Simulation Problems

- Simulate Problem 1 and determine the steady-state response.
- Model Problem 8 using a simulator. Determine both the steady-state and initial outputs.
- Simulate Problem 14. Determine the steady-state output.
- Model Problem 22 using a simulator and determine the output signal.
- Compare the resulting waveform produced by the circuit of Problem 8 using both the 741 op amp and the medium-speed LF411. Does the choice of op amp make a discernible difference in this application?