Problems 6 circuit wiley

5/17/13 Problems

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Problems

Section 6.3 The Ideal Operational Amplifier

P 6.3-1 Determine the value of voltage measured by the voltmeter in Figure P 6.3-1.

FIGURE P 6.3-1

Answer:

P 6.3-2 Find and for the circuit of Figure P 6.3-2.

FIGURE P 6.3-2


FIGURE P 6.3-3

Kelvin SiuSticky Note


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Answer:

and

P 6.3-4 Find v and i for the circuit of Figure P 6.3-4.

FIGURE P 6.3-4


FIGURE P 6.3-5

Answer:

and

P 6.3-6 Determine the value of voltage measured by the voltmeter in Figure P 6.3-6.

FIGURE P 6.3-6

Answer:

7.5 V




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FIGURE P 6.3-7

P 6.3-8 Determine the current for the circuit shown in Figure P 6.3-8.

FIGURE P 6.3-8

Answer:

P 6.3-9 Determine the voltage for the circuit shown in Figure P 6.3-9.

FIGURE P 6.3-9

Answer:

P 6.3-10 The circuit shown in Figure P 6.3-10 has one input, , and one output, . Show that the output is

proportional to the input. Design the circuit so that the gain is .


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FIGURE P 6.3-10

P 6.3-11 The circuit shown in Figure P 6.3-11 has one input, , and one output, . Show that the outputis proportional to the input. Design the circuit so that .

FIGURE P 6.3-11

P 6.3-12 The input to the circuit shown in Figure P 6.3-12 is the voltage . The output is the voltage .The output is related to the input by the equation where m and b are constants.Determine the values of m and b.

FIGURE P 6.3-12

P 6.3-13 The output of the circuit shown in Figure P 6.3-13 is . Determine the value of (a) theresistance R, (b) the power supplied be each independent source, and (c) the power,

supplied by the op amp.

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FIGURE P 6.3-13

P 6.3-14 Determine the node voltages at nodes a, b, c, and d of the circuit shown in Figure P 6.3-14.

FIGURE P 6.3-14

P 6.3-15 Determine the node voltages at nodes a, b, c, and d of the circuit shown in Figure P 6.3-15.

FIGURE P 6.3-15

Section 6.4 Nodal Analysis of Circuits Containing Ideal Operational Amplifiers

P 6.4-1 Determine the node voltages for the circuit shown in Figure P 6.4-1.

FIGURE P 6.4-1

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Answer:

, , , , and


FIGURE P 6.4-2

Answer:

and

P 6.4-3 If and , find / for the circuit shown in Figure P 6.4-3 when .

FIGURE P 6.4-3

Answer:

P 6.4-4 The output of the circuit shown in Figure P 6.4-4 is . The inputs are and . Expressthe output as a function of the inputs and the resistor resistances.

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FIGURE P 6.4-4

P 6.4-5 The outputs of the circuit shown in Figure P 6.4-5 are and . The inputs are and .Express the outputs as functions of the inputs and the resistor resistances.

FIGURE P 6.4-5


FIGURE P 6.4-6

Answer:

, , and

P 6.4-7 Find and for the circuit shown in Figure P 6.4-7.

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FIGURE P 6.4-7

P 6.4-8 Find and for the circuit shown in Figure P 6.4-8.

FIGURE P 6.4-8


FIGURE P 6.4-9

Answer:

, , , , , ,and

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P 6.4-10 The circuit shown in Figure P 6.4-10 includes a simple strain gauge. The resistor R changes itsvalue by R when it is twisted or bent. Derive a relation for the voltage gain / and show that itis proportional to the fractional change in R, namely, R/ .

FIGURE P 6.4-10 A strain gauge circuit.

Answer:

P 6.4-11 Find for the circuit shown in Figure P 6.4-11.

FIGURE P 6.4-11

P 6.4-12 The circuit shown in Figure P 6.4-12 has one output, , and two inputs, and . Show that

when , the output is proportional to the difference of the inputs, . Specify

resistance values to cause .

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FIGURE P 6.4-12

P 6.4-13 The circuit shown in Figure P 6.4-13 has one output, , and one input, . Show that the output isproportional to the input. Specify resistance values to cause .

FIGURE P 6.4-13

P 6.4-14 The circuit shown in Figure P 6.4-14 has one input, , and one output, . Show that the outputis proportional to the input. Design the circuit so that .

FIGURE P 6.4-14

P 6.4-15 The circuit shown in Figure P 6.4-15 has one input, , and one output, . The circuit contains

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seven resistors having equal resistance, R. Express the gain of the circuit, / , in terms of theresistance R.

FIGURE P 6.4-15

P 6.4-16 The circuit shown in Figure P 6.4-16 has one input, , and one output, . Express the gain, /

, in terms of the resistances , , , , and . Design the circuit so that .

FIGURE P 6.4-16

P 6.4-17 The circuit shown in Figure P 6.4-17 has one input, , and one output, . Express the gain of thecircuit, , in terms of the resistances , , , , , and . Design the circuit so that

.

FIGURE P 6.4-17

P 6.4-18 The circuit shown in Figure P 6.4-18 has one input, , and one output, . Express the gain of thecircuit, , in terms of the resistances , , , and . (This circuit contains a pair ofresistors having resistance and another pair having resistance .) Design the circuit so that

.

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FIGURE P 6.4-18

P 6.4-19 The circuit shown in Figure P 6.4-19 has one input, , and one output, . The circuit containsone unspecified resistance, R.

(a) Express the gain of the circuit, , in terms of the resistance R.

(b) Determine the range of values of the gain that can be obtained by specifying a value for theresistance R.

(c) Design the circuit so that .

FIGURE P 6.4-19

P 6.4-20 The circuit shown in Figure P 6.4-20 has one input, , and one output, . The circuit containsone unspecified resistance, R.

(a) Express the gain of the circuit, / , in terms of the resistance R.

(b) Determine the range of values of the gain that can be obtained by specifying a value for the

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resistance R.

(c) Design the circuit so that .

FIGURE P 6.4-20

P 6.4-21 The circuit shown in Figure P 6.4-21 has three inputs: , , and . The output of the circuit is

. The output is related to the inputs by

where a, b, and c are constants. Determine the values of a, b, and c.

FIGURE P 6.4-21

P 6.4-22 The circuit shown in Figure P 6.4-22 has two inputs: and . The output of the circuit is .

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The output is related to the inputs by

where a and b are constants. Determine the values of a and b.

FIGURE P 6.4-22

P 6.4-23 The input to the circuit shown in Figure P 6.4-23 is the voltage source voltage . The output is the

node voltage The output is related to the input by the equation where is

called the gain of the circuit. Determine the value of the gain k.

FIGURE P 6.4-23

P 6.4-24 The input to the circuit shown in Figure P 6.4-24 is the voltage source voltage . The output is thenode voltage The output is related to the input by the equation where m and b areconstants. Determine the values of m and b.

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FIGURE P 6.4-24

P 6.4-25

The input to the circuit shown in Figure P 6.4-25 is the node voltage The output is the node

voltage . The output is related to the input by the equation where is called the

gain of the circuit. Determine the value of the gain k.

FIGURE P 6.4-25

P 6.4-26 The values of the node voltages , and in Figure P 6.4-26 are , ,and , Determine the value of the resistances , and .

FIGURE P 6.4-26

P 6.4-27 The input to the circuit shown in Figure P 6.4-27 is the voltage source voltage, . The output is the

node voltage, . The output is related to the input by the equation where is

called the gain of the circuit. Determine the value of the gain k.

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FIGURE P 6.4-27

Section 6.5 Design Using Operational Amplifiers

P 6.5-1 Design the operational amplifier circuit in Figure P 6.5-1 so that

where

FIGURE P 6.5-1


where

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FIGURE P 6.5-2


FIGURE P 6.5-3



P 6.5-6 The voltage divider shown in Figure P 6.5-6 has a gain of

Design an operational amplifier circuit to implement the resistor.

FIGURE P 6.5-6 A circuit with a negative resistor.


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FIGURE P 6.5-7

P 6.5-8 Design an operational amplifier circuit with output , where and areinput voltages.

P 6.5-9 Determine the voltage for the circuit shown in Figure P 6.5-9.

FIGURE P 6.5-9

Hint: Use superposition.

Answer:

P 6.5-10 For the op amp circuit shown in Figure P 6.5-10, find and list all the possible voltage gains that canbe achieved by connecting the resistor terminals to either the input or the output voltage terminals.

FIGURE P 6.5-10 Resistances in k.

P 6.5-11

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The circuit shown in Figure P 6.5-11 is called a Howland current source. It has one input, , andone output, . Show that when the resistances are chosen so that , the output isrelated to the input by the equation

FIGURE P 6.5-11

P 6.5-12 The circuit shown in Figure P 6.5-12 is used to calculate the output resistance of the Howlandcurrent source. It has one input, , and one output, . The output resistance, , is given by

Express the output resistance of the Howland current source in terms of the resistances , , , and .

FIGURE P 6.5-12

P 6.5-13 The input to the circuit shown in Figure P 6.5-13a is the voltage . The output is the voltage .The voltage is used to adjust the relationship between the input and output.

(a) Show that the output of this circuit is related to the input by the equation

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where a and b are constants that depend on , , , , , and .

(b) Design the circuit so that its input and output have the relationship specified by the graphshown in Figure P 6.5-13b.

FIGURE P 6.5-13

P 6.5-14 The input to the circuit shown in Figure P 6.5-14a is the voltage . The output is the voltage .The voltage is used to adjust the relationship between the input and output.

(a) Show that the output of this circuit is related to the input by the equation

where a and b are constants that depend on , , , , and .

(b) Design the circuit so that its input and output have the relationship specified by the graphshown in Figure P 6.5-14b.

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FIGURE P 6.5-14

*P 6.5-15 The circuit shown in Figure P 6.5-15 contains both an op amp and a potentiometer. This circuit iscalled an active potentiometer (Graeme, 1982) because the equivalent resistance, , takes both

positive and negative values as the position of the potentiometer wiper varies. is the

potentiometer resistance. The expressions a and (1a) indicate the resistances that appear

between potentiometer terminals yw and xw, respectively. Express the equivalent resistance ofthe active potentiometer source in terms of R, , and a.

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FIGURE P 6.5-15

*P 6.5-16 The circuit shown in Figure P 6.5-16 contains both op amps and a potentiometer. This circuit hasan adjustable gain, / , that takes both positive and negative values as the position of thepotentiometer wiper varies (Albean, 1997). is the potentiometer resistance. The expression

indicates the part of that appears between potentiometer terminals yw.

(a) Express the gain in terms of the resistor resistances, and a.

(b) Set . Design the circuit so that the gain varies from to 10 V as

the position of the potentiometer wiper is varied through its full range.

FIGURE P 6.5-16

P 6.5-17 The input to the circuit shown in Figure P 6.5-17 is the voltage source voltage . The output is the

node voltage The output is related to the input by the equation is

called the gain of the circuit. (In Figure P 6.5-17, a and b are positive real constants, so theresistance aR and bR are a and b times as large as the resistances R). Derive an equation that showshow to pick values of a and b that cause the circuit to have a given gain k. Use this equation todesign the circuit to have a gain using .

FIGURE P 6.5-17

P 6.5-18 The input to the circuit shown in Figure P 6.5-18 is the current source current . The output is thenode voltage . The output is related to the input by the equation where m and bare constants. (In Figure P 6.5-18, c and d are positive real constants, so the resistance cR and dRare c and d times as large as the resistance R.) Derive an equation that shows how to pick values ofc and m that cause the circuit to have given values of i and b. Use this equation to design the circuit

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to have when

FIGURE P 6.5-18

P 6.5-19 The input to the circuit shown in Figure P 6.4-19 is the voltage source voltage The output is thenode voltage . The output is related to the input by the equation where m and bare constants, (a) Specify values of and that cause the output to be related to the input by theequation . (b) Determine the values of m and b when , and .

FIGURE P 6.5-19

P 6.5-20 The circuit shown in Figure P 6.5-20 uses a potentiometer to implement a variable resistor having aresistance R that varies over the range

The gain of this circuit is . Varying the resistance R over it's range causes the value of the

gain G to vary over the range

Determine the minimum and maximum values of the gains, and .

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FIGURE P 6.5-20

P 6.5-21 The input to the circuit shown in Figure P 6.5-21a is the voltage, . The output is the voltage .The Voltage is used to adjust the relationship between the input and output. Determine values of

and that cause the circuit input and output to have the relationship specified by the graphshown in Figure P 6.5-21b.

FIGURE P 6.5-21

Answer:

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Section 6.6 Operational Amplifier Circuits and Linear Algebraic Equations

P 6.6-1 Design a circuit to implement the equation

The circuit should have one output, corresponding to z, and three inputs, corresponding to w,x, and y.

P 6.6-2 Design a circuit to implement the equation

The output of the circuit should correspond to z.

Section 6.7 Characteristics of Practical Operational Amplifiers

P 6.7-1 Consider the inverting amplifier shown in Figure P 6.7-1. The operational amplifier is a typicalOP-07E (Table 6.7-1). Use the offsets model of the operational amplifier to calculate theoutput offset voltage. (Recall that the input, , is set to zero when calculating the outputoffset voltage.)

FIGURE P 6.7-1

Answer:

0.45 mV

P 6.7-2 Consider the noninverting amplifier shown in Figure P 6.7-2. The operational amplifier is atypical LF351 (Table 6.7-1). Use the offsets model of the operational amplifier to calculate theoutput offset voltage. (Recall that the input, , is set to zero when calculating the outputoffset voltage.)

FIGURE P 6.7-2



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P 6.7-3 Consider the inverting amplifier shown in Figure P 6.7-3. Use the finite gain model of theoperational amplifier (Figure 6.7-1c) to calculate the gain of the inverting amplifier. Show that

FIGURE P 6.7-3

P 6.7-4 Consider the inverting amplifier shown in Figure P 6.7-3. Suppose the operational amplifier isideal, , and . The gain of the inverting amplifier will be

Use the results of Problem P 6.7-3 to find the gain of the inverting amplifier in each of thefollowing cases:

(a) The operational amplifier is ideal, but 2 percent resistors are used and and .

(b) The operational amplifier is represented using the finite gain model with ,

, and ; and .

(c) The operational amplifier is represented using the finite gain model with ,

, and ; and .

P 6.7-5 The circuit in Figure P 6.7-5 is called a difference amplifier and is used for instrumentationcircuits. The output of a measuring element is represented by the common mode signal and the differential signal ( + ). Using an ideal operational amplifier, show that

when

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FIGURE P 6.7-5

Section 6.10 How Can We Check ?

P 6.10-1 Analysis of the circuit in Figure P 6.10-1 shows that and . Is this analysiscorrect?

FIGURE P 6.10-1

Hint: Is KCL satisfied at the output node of the op amp?

P 6.10-2 Your lab partner measured the output voltage of the circuit shown in Figure P 6.10-2 to be

. Is this the correct output voltage for this circuit?

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FIGURE P 6.10-2

Hint: Ask your lab partner to check the polarity of the voltage that he or she measured.

P 6.10-3 Nodal analysis of the circuit shown in Figure P 6.10-3 indicates that . Is this analysiscorrect?

FIGURE P 6.10-3

Hint: Redraw the circuit to identify an inverting amplifier and a noninverting amplifier.

P 6.10-4 Computer analysis of the circuit in Figure P 6.10-4 indicates that the node voltages are , , , , , , and . Is this analysis correct?

Justify your answer. Assume that the operational amplifier is ideal.

FIGURE P 6.10-4

Hint: Verify that the resistor currents indicated by these node voltages satisfy KCL at nodes b, c, d,

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and f.

P 6.10-5 Computer analysis of the noninverting summing amplifier shown in Figure P 6.10-5 indicates thatthe node voltages are , , , , and

.

(a) Is this analysis correct?

(b) Does this analysis verify that the circuit is a noninverting summing amplifier? Justify youranswers. Assume that the operational amplifier is ideal.

FIGURE P 6.10-5

Hint: 1st Hint: Verify that the resistor currents indicated by these node voltages satisfy KCL at

nodes b and e.2nd Hint: Compare to Figure 6.5-1e to see that and . Determine ,

, and from the resistance values. Verify that .

Copyright 2010 John Wiley & Sons, Inc. All rights reserved.

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