Am Pl Ific Adores

242 )-------The O p e ra t io n a l A m p l i f ie r

P R O B L E M S

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.

Answer: —4 V

20 kft

10 kQ

Figure P 6.3-1

P 6.3-2 Find v0 and i0 for the circuit of Figure P 6.3-2.

3 kQ 4 kQ

*0

Figure P 6.3-2

P 6.3-3 Find vG and iQ for the circuit of Figure P 6.3-3.

Answer: v0 = -3 0 V and iG = 3.5 mA

4 kQ 8 kQ

Figure P 6.3-3

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

r

Figure P 6.3-4

P 6.3-5 Find vQ and iQ for the circuit of Figure P 6.3-5.

Answer: vc = -1 5 V and i0 = 7.5 mA

3 kQ

4 kQ— V W

12V( + ) 2mA

t

c m

Q)

Figure P 6.3-5

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

Answer: 7.5 V

Figure P 6.3-6

P 6.3-7 Find vQ and i0 for the circuit of Figure P 6.3-7.

Rx R2

P r o b l e m s --------( 243

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

Answer: i0 = 2.5 mA

Figure P 6.3-11

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

5 kQ 20 kQ

Figure P 6.3-8

P 6.3-9 Determine the voltage v0 for the circuit shown in Figure P 6.3-9.

Answer: vQ = - 8 V

Figure P 6.3-12

P 6.3-13 The output of the circuit shown in Figure P 6.3-13 is v0 = 3.5 V. Determine the value of (a) the resistance R, (b) the power supplied be each independent source, and (c) the power, Poa = *oa x v0 supplied by the op amp.

Figure P 6.3-9

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

the input. Design the circuit so that the gain is ^ = 20 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 6.3-10

P 6,3-11 The circuit shown in Figure P 6.3-11 has one input, vs, and one output, v0. Show that the output is proportional to the input. Design the circuit so that vQ = 5 vs.


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.

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.

Answer: va = 2 V, vb = -0.25 V, vc = - 5 V, vd = -2 .5 V, and ve = -0.25 V

Figure P 6.4-1

P 6.4-2 Find v0 and iQ for the circuit of Figure P 6.4-2.

Answer: v0 = - 4 V and iQ = 1.33 mA

Figure P 6.4-2

P 6.4-3 If R | =4.8 kfl and R2~ R 4 = 30kll, find v0/vs for the circuit shown in Figure P 6.4-3 when /?3 = 1 kll.

Answer: vG/v s = —200

R2 r4

P 6.4-4 The output of the circuit shown in Figure P 6.4-4 is vG. The inputs are Vj and v2. Express the output as a function of the inputs and the resistor resistances.

Figure P 6.4-4

P 6.4-5 The outputs of the circuit shown in Figure P 6.4-5 are vG and i0. The inputs are vj and v2. Express the outputs as functions of the inputs and the resistor resistances.

P r o b l e m s --------( 245

P 6 . 4 - 6 Determine the node voltages for the circuit shown in Answer: va = - 1 2 V, vb = - 4 V , vc - - 4 V, vd - - 4 V

Figure P 6.4-6.Answer: va = -0.75 V, = 0 V, and vc = -0.9375 V

Figure P 6.4-6

P 6.4-7

15 kQ

= -3 .2 V, vf = -4 .8 V, and vg = -3 .2 V

Figure P 6.4-7

Answer: vG =Ro A R

R0 + R\ Ro

R — Rn + A R

Figure P 6.4-10 A strain gauge circuit.

P 6.4-8 Find vQ and iQ for the circuit shown in Figure P 6.4-8. P 6.4-11 Find v0 for the circuit shown in Figure P 6.4-11.

10 kQ 2C|— VW —| \

10 kQ 20 kQ-AAA/— —°

10 kQ

Figure P 6.4-11

Figure P 6.4-8 P 6.4-12 The circuit shown in Figure P 6.4-12 has one output,v0, and two inputs, v, and v2. Show that when ^ the

P 6.4-9 Determine the node voltages for the circuit shown in output is proportional to the difference of the inputs, v\ — V2.Figure P 6.4-9. Specify resistance values to cause v0 = 5 (vi - v2).

10 kQAAV

P 6.4-10 The circuit shown in Figure P 6.4-10 includes a simple strain gauge. The resistor R changes its value by \R when it is twisted or bent. Derive a relation for the voltage gain vQ/v s and show that it is proportional to the fractional change in /?, namely, AR/R0.

10 kQ

Find vG and iQ for the circuit

30 kQ

shown in Figure P 6.4-7.

‘ 10 kQ

0 — The O p e ra t io n a l A m p l i f ie r

Figure P 6.4-12

Figure P 6.4-15

P 6.4-16 The circuit shown in Figure P 6.4-16 has one input, vs, and one output, v0. Express the gain, vQ/vs, in terms of the resistances R2, /?3, Ra* and R5. Design the circuit so thatVo = - 2 0 vs.

P 6.4-13 The circuit shown in Figure P 6.4-13 has one output, vc, and one input, Vj. Show that the output is proportional to the input. Specify resistance values to cause vD = 20vj.

Figure P 6.4-13

P 6.4-14 The circuit shown in Figure P 6.4-14 has one input, vs, and one output, vQ. Show that the output is proportional to the input. Design the circuit so that v0 = 20vs.

Figure P 6.4-16

P 6.4-17 The circuit shown in Figure P 6.4-17 has one input, vs, and one output, vc. Express the gain of the circuit, vG/vs, in terms of the resistances R \,R 2i R3, # 4, R5, and R*. Design the circuit so that v0 = —2 0vs.

P 6.4-15 The circuit shown in Figure P 6.4-15 has one input, vs, and one output, v0. The circuit contains seven resistors having equal resistance, R. Express the gain of the circuit, Vo/vs, in terms of the resistance R.

Figure P 6.4-17

P 6.4-18 The circuit shown in Figure P 6.4-18 has one input, vs, and one output, z'0. Express the gain of the circuit, 1'0/vs, in terms of the resistances R 1, R2, Ri, and R0. (This circuit contains a pair of resistors having resistance R\ and another pair having resistance R2.) Design the circuit so that ia = 0.02vs.

P r o b l e m s — ( 247

Figure P 6.4-18

P 6.4-19 The circuit shown in Figure P 6.4-19 has one input, vs, and one output, vG. The circuit contains one unspecified resistance, R.(a) Express the gain of the circuit, vG/vs, 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 resistance R.(c) Design the circuit so that v0= — 3vs.

P 6.4-20 The circuit shown in Figure P 6.4-20 has one input, vs, and one output, v0. The circuit contains one unspecified resistance, R.

(a) Express the gain of the circuit, vD/vs, 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 resistance R.

(c) Design the circuit so that vD = ~5vs.

P 6.4-21 The circuit shown in Figure P 6.4-21 has three inputs: vlf v2, and v3. The output of the circuit is vG. The output is related to the inputs by

v0 = tfvi + bv2 + cv 3

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

20 kfl 20 kft 40 kQ

P 6.4-22 The circuit shown in Figure P 6.4-22 has two inputs: V| and v2. The output of the circuit is vG. The output is related to the inputs by

v0 = av i + bv2

where a and b are constants. Determine the values o f a and b.

0 - T he O p e ra t io n a l A m p l i f ie r

20 kQ

50 kQ 25 kQ

5 0 kQ

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 vs. The output is the node voltage v0 The output is related to the input by the equation

v0 = kvs where k = — is called the gain of the circuit.Vs

Determine the value of the gain k.

Figure P 6.4-25

P 6.4-26 The values of the node voltages vb v2, and vG in Figure P 6.4-26 are = 6.25 V, v2 = 3.75 V, and v0 = -1 5 V. Determine the value of the resistances R\, R2, and R$.

20 kQ R,

Figure P 6.4-23

P 6.4-24 The input to the circuit shown in Figure P 6.4-24 is the current source current zs. The output is the node voltage vG. The output is related to the input by the equation vQ = mis + b where m and b are constants. Determine the values of m and b.

v2

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, v*, The output is the node voltage, v0. The output is related to the input by the equation vQ = kv j

where k = — is called the gain of the circuit. Determine theVj

value of the gain k.

24 kQ

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 vs. The output is the node voltage vD. The output isrelated to the input by the equation vQ = kvs where k = — is

Vscalled the gain of the circuit. Determine the value of the gain k. Figure P 6.4-27

P r o b l e m s --------( 249

Section 6.5 Design Using O perational Am plifiers

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

wherevout — r ' l ir

r = 2 0 -mA

Operationalamplifiercircuit

20 kQ > vout

*out — g ' vin

where

g = 2 -

v0ut = 5 • vj - 2 • v2

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

P 6 .5 -5 Design the operational amplifier circuit in FigureP 6.5-3 so that

- 1 0 kfivout ___________________^ ~ - 5 kn + ( - io k n ) = 2

Design an operational amplifier circuit to implement the — 10-kfl resistor.

5 kQ

Figure P 6.5-1

P 6.5-2 Design the operational amplifier circuit in Figure P 6.5-2 so that Figure P 6.5-6 A circuit with a negative resistor.

P 6.5-7 Design the operational amplifier circuit in Figure P 6.5-7 so that

iin = 0 and vout = 3 • vin

| zout

Figure P 6.5-2


vw = 5 • vi + 2 • v2

P 6.5-8 Design an operational amplifier circuit with output v0 = 6 vi + 2 v2, where v, and v2 are input voltages.

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

Hint: Use superposition.

Answer: vG = (—3)(3) + (4)(-4) + ( 4)(8) = 7 V

8 kQ 24 kQ

Figure P 6.5-3


vout 35 5 • (vi — v2) Figure P 6.5-9

250 )------- The O p e ra t io n a l A m p l i f ie r

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 can be achieved by connecting the resistor terminals to either the input or the output voltage terminals.

Figure P 6.5-10 Resistances in kfl.

P 6.5-11 The circuit shown in Figure P 6.5-11 is called a Howland current source. It has one input, vin, and one output, zout. Show that when the resistances are chosen so that R2R3 = R\Ra, the output is related to the input by the equation

. _ Vi„'ou' ~ /?,

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 Howland current source. It has one input, it, and one output, vt. The output resistance, R0, is given by

Express the output resistance of the Howland current source in terms of the resistances R\, R2, R3, and R4.

P 6.5-13 The input to the circuit shown in Figure P 6.5-13a is the voltage vs. The output is the voltage vc. The voltage vb 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

v0 = av s + b

where a and b are constants that depend on R 1, R2, R ^ R4, R5, and vb.

(b) Design the circuit so that its input and output have the relationship specified by the graph shown in Figure P 6.5-136.

v0, V

Figure P 6.5-13

P 6.5-14 The input to the circuit shown in Figure P 6.5-\4a is the voltage vs. The output is the voltage vD. The voltage vb 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

v0 = avs + b

where a and b are constants that depend on R\< R2< R4,and vb.

(b) Design the circuit so that its input and output have the relationship specified by the graph shown in Figure P 6.5-I4h.

P r o b l e m s --------1 251

R- w v

O '

W V ---------f—

—w v —*—

R3-w v —

r2

(a)

Vn. V

J - uSf V

expression a/?p indicates the part of Rp that appears between potentiometer terminals y-w.

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

(b) Set Rx = R3 = R* = ^ p. Design the circuit so that the gain varies from -1 0 V to 10 V as the position of the potentiometer wiper is varied through its full range.

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 is called an active potentiometer (Graeme, 1982) because the equivalent resistance, Rcq, takes both positive and negative values as the position of the potentiometer wiper varies. Rp is the potentiometer resistance. The expressions aRp and (1 —a)Rp indicate the resistances that appear between potentiometer terminals y-w and x-w, respectively. Express the equivalent resistance of the active potentiometer source in terms of R, Rp, and a.

°—W VR

P 6.5-17 The input to the circuit shown in Figure P 6.5-17 is the voltage source voltage vs. The output is the node voltage v0. The output is related to the input by the equation v0 = kvs where k = ^ is called the gain of the circuit. (In Figure P 6.5-17, a and b are positive real constants, so the resistance aR and bR are a and b times as large as the resistances R). Derive an equation that shows how to pick values of a and b that cause the circuit to have a given gain k. Use this equation to design the circuit to have a gain k = 8 V/V using R = 20 kH.

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 has an adjustable gain, v0/vj, that takes both positive and negative values as the position of the potentiometer wiper varies (Albean, 1997). Rp is the potentiometer resistance. The

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 is. The output is the node voltage vG. The output is related to the input by the equation vQ = m/s + b where m and b are constants. (In Figure P 6.5-18, c and d are positive real constants, so the resistance cR and dR are c and d times as large as the resistance R.) Derive an equation that shows how to pick values of c and d that cause the circuit to have given values of m and b. Use this equation to design the circuit to have m = -125 V/mA and b = 12 V when R = 25 kfl.

252 )-------T he O p e ra t io n a l A m p l i f ie r

cR dR

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 vs The output is the node voltage vG. The output is related to the input by the equation vc = mvs -I- b where m and b are constants, (a) Specify values of Rt, and va that cause the output to be related to the input by the equation v0 = 4vs + 7. (b) Determine the values of m and b when R$ = 20 kO, and va = 2.5V.

10 kQ 30 kQ 20 kQ

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 a resistance R that varies over the range

0 < R < 2 0 0 kO The gain of this circuit is G = *■. Varying the resistance R over it’s range causes the value of the gain G to vary over the range

Gmin < < GmaxVS

Determine the minimum and maximum values of the gains,^min and Gmax.

! F igure P 6 .5-20

P 6.5-21 The input to the circuit shown in Figure P 6.5-2la is the voltage, vs. The output is the voltage v0. The voltage vb is used to adjust the relationship between the input and output. Determine values of R4 and vb that cause the circuit input and output to have the relationship specified by the graph shown in Figure P 6.5-21 b.

Answer: v*, = 1.62 V and R4 =62.5 kO

Figure P 6.5-21

Section 6.6 Operational Am plifier Circuits and Linear Algebraic Equations

P 6.6-1 Design a circuit to implement the equation

JCZ = 4w + - - 3v

4

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

P 6.6-2 Design a circuit to implement the equation

0 = 4w + jc+10 — (6y -f 2z)

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 typical OP-07E (Table 6.7-1). Use the offsets model of the operational amplifier to calculate the output offset voltage. (Recall that the input, vin, is set to zero when calculating the output offset voltage.)

+ 6

P r o b l e m s --------( 253

10 kQ 100 kQ

Answer: 0.45 mV

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

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

Vo _ _________ Rm(R0 - AR2)_________vin (^1 + în)(ô + ^ 2) + ^1 în( 1 + A)

Rl R2

P 6.7-4 Consider the inverting amplifier shown in Figure P 6.7-3. Suppose the operational amplifier is ideal, /?, = 5 kfl, and /?2 = 50kfl. The gain of the inverting amplifier will be

Vin

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

(a) The operational amplifier is ideal, but 2 percent resistors are used and R\ = 5.1 kfl and R2 — 49 kfl.

(b) The operational amplifier is represented using the finite gain model with A = 200,000, Rx = 2 Mfl, and R0 = 75 fl; /?, = 5 kfl and R2 = 5 0 kfl.

(c) The operational amplifier is represented using the finite gain model with A = 200,000, Ri = 2 Mfl, and R0 = 75 fl; Ry = 5.1 kfl and R2 = 4 9 kfl.

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

^4 / \Vo = -^ (V n + Vp)

when

R1 = R1 Ri Ri

R i Ra

Section 6.10 How Can We Check . . . ?

P 6.10-1 Analysis of the circuit in Figure P 6.10-1 show s that /0 = — 1 mA and vc = 7 V. Is this analysis correct?

254 )------- The O p e ra t io n a l A m p l i f ie r

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

6 kQ 4 kft

Figure P 6.10-1

P 6.10-2 Your lab partner measured the output voltage of the circuit shown in Figure P 6.10-2 to be vQ = 9.6 V. Is this the correct output voltage for this circuit?

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

4 kft 10 kft 12 kft

P 6.10-3 Nodal analysis of the circuit shown in Figure P 6.10-3 indicates that vc = —12 V. Is this analysis correct?

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

4 kft

P 6.10-4 Computer analysis of the circuit in Figure P 6.10-4 indicates that the node voltages are va = - 5 V, Vb = 0 V, vc = 2V, vd = 5 V, ve = 2V, vf = 2 V, and vg = 11 V. Is this analysis correct? Justify your answer. Assume that the operational amplifier is ideal.

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

Figure P 6.10-4

P 6.10-5 Computer analysis of the noninverting summing amplifier shown in Figure P 6.10-5 indicates that the node voltages are va = 2V, vb — —0.25 V, vc = - 5 V, vd = -2 .5 V, and ve = -0 .25 V.

(a) Is this analysis correct?(b) Does this analysis verify that the circuit is a noninverting

summing amplifier? Justify your answers. Assume that the operational amplifier is ideal.

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-le to see that Ra = lOkfl and Rb = 1 kft. Determine K u K2, and K4 from the resistance values. Verify that vd = AT4(AT1va + AT2vc).

Figure P 6.10-3 Figure P 6 .10-5

P S p ic e P r o b l e m s --------( 255

PSpice ProblemsSP 6-1 The circuit in Figure SP 6-1 has three inputs: vw, vx, and vy. The circuit has one output, vz. The equation

j —V ^ r - p60 kQ

20 kQAAA

20 kn i—o—WV—11

(T) 20 kQ ^

Figure SP 6-1

expresses the output as a function of the inputs. The coefficients a, b, and c are real constants.

(a) Use PSpice and the principle of superposition to determine the values of a, b, and c.

(b) Suppose vw = 2 V, vx = *, vy= y and we want the output to be vz = z. Express z as a function of x and y.

Hint: The output is given by v2 = a when vw = 1 V, vx = 0 V, and vy = 0 V.Answer: (a) vz = vw + 4 vx- 5 vy (b) z = 4 x - 5 v + 2SP 6-2 The input to the circuit in Figure SP 6-2 is vs, and the output is v0. (a) Use superposition to express vG as a function of v8. (b) Use the DC Sweep feature of PSpice to plot v0 as a function of vs. (c) Verify that the results of parts (a) and (b) agree with each other.

25 ki2 80 kQ

10 kQ

30 kQ

30 kQ > v0

Figure SP 6-3 Bridge circuit.

SP 6-4 Use PSpice to analyze the VCCS shown in Figure SP 6-4. Consider two cases:

(a) The operational amplifier is ideal.(b) The operational amplifier is a typical /a A741 represented

by the offsets and finite gain model.

Figure SP 6-2

SP 6 3 A circuit with its nodes identified is shown in Figure SP 6-3. Determine v34, v23, v50, and i0.

Figure SP 6-4 A VCCS.


Design ProblemsDP 6-1 Design the operational amplifier circuit in Figure DP 6-1 so that j

*out ~ ^ ' *in

Hint: A constant input is required. Assume that a 5-V source is available.

DP 6-4 Design a circuit having three inputs, vj, v2, v3, and two outputs, va, vb, that are related by the equation

Vb _12 - 2

0

Vl " 2V2 +

- 4

Figure DP 6-1

DP 6-2 Figure DP 6-2a shows a circuit that has one input, vl5 and one output, vc. Figure DP 6-2b shows a graph that specifies a relationship between v0 and v,. Design a circuit having input, vj, and output, vG, that have the relationship specified by the graph in Figure DP 6-2b.


*o.V


DP 6-5 A microphone has an unloaded voltage vs = 20 mV, as shown in Figure DP 6-5a. An op amp is available as shown in Figure DP 6-5b. It is desired to provide an output voltage of 4 V. Design an inverting circuit and a noninverting circuit and contrast the input resistance at terminals x-y seen by the microphone. Which configuration would you recommend to achieve good performance in spite of changes in the microphone resistance Rs?

Hint: We plan to connect terminal a to terminal x and terminal b to terminal y or vice versa.

(a)

(a)

Figure DP 6-2

DP 6-3 Design a circuit having input, v*, and output, vD, that are related by the equations (a) vG= \2vl + 6 , (b) vG= 12v j-6 ,(c) v0 = - 12vi + 6 , and (d) v0 = - 12vj - 6 .

Figure DP 6-5 Microphone and op amp circuit.

(b)

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