5.9 Op Amp Circuit Analysis with PSpice

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5.9 Op Amp Circuit Analysis with PSpice

PSpice for Windows does not have a model for an ideal op amp, although one may create one as a subcircuit using the Create Subcircuit line in the Tools menu. Rather than creating an ideal op amp, we will use one of the four nonideal, commercially a vailable op amps supplied in the PSpice library eval.slb. The op amp models have the part names LF411, LM111, LM324, and uA741, as shown in Fig. 5.34. Each of them can be obtained from Draw/Get New Part/libraries … /eval.lib or by simply selecting Draw/Get New Part and typing the part name in the PartName dialog box, as usual. Note that each of them requires dc supplies, without which the op amp will not work. The dc supplies should be connected as shown in Fig. 5.3.

(a) JFET–input op amp subcircuit

(b) Op amp subcircuit

(c) Five– connection op amp subcircuit

(d) Five–connection op amp subcircuit

Figure 5.34

Nonideal op amp model available in PSpice.

Use PSpice to solve the op amp circuit for Example 5.1.

Solution:

Using Schematics, we draw the circuit in Fig. 5.6(a) as shown in Fig.�5.35. Notice that the positive terminal of the voltage source vs is connected to the inverting terminal (pin 2) via the 10-kΩ resistor, while the noninverting terminal (pin 3) is grounded as required in Fig. 5.6(a). Also, notice how the op amp is powered; the positive power supply terminal V+ (pin 7) is connected to a 15 -V dc voltage source, while the negative power supply terminal V− (pin 4) is connected to −15 V. Pins 1 and 5 are left floating because they are used for offset null adjustment, which does not concern us in this chapter. Besides adding the dc power supplies to the original circuit in Fig. 5.6(a), we have also added pseudocomponents VIEWPOINT and IPROBE to respectively measure the output voltage vo at pin 6 and the required current i through the 20-kΩ resistor.

Schematic for Example 5.11.

After sa ving the schematic, we simulate the circuit by selecting Analysis/Simulate and ha ve the results displayed on VIEWPOINT and IPROBE. From the results, the closed-loop gain is

$$\frac{v_o}{v_s} = \frac{-3.9983}{2} = -1.99915$$

and i = 0.1999 mA, in agreement with the results obtained analytically in Example 5.1.

Answer: 9.0027, 650.2 μA.

Rework Practice Prob. 5.1 using PSpice. Practice Problem 5.11

5.10 † Applications

The op amp is a fundamental b uilding block in modern electronic instrumentation. It is used e xtensively in man y de vices, along with resistors and other passi ve elements. Its numerous practical applica tions include instrumentation amplifiers, digital-to-analog converters, analog computers, level shifters, filters, calibration circuits, inverters, summers, integrators, differentiators, subtractors, log arithmic amplifiers, comparators, gyrators, oscillators, rectifiers, regulators, voltageto- current con verters, current -to-voltage converters, and clippers. Some of these we have already considered. We will consider two more applications here: the digital-to-analog converter and the instrumentation amplifier.

5.10.1 Digital-to-Analog Converter

The digital-to-analog converter (DAC) transforms digital signals into analog form. A typical e xample of a four -bit DAC is illustrated in Fig. 5.36(a). The four-bit DAC can be realized in many ways. A simple realization is the binary weighted ladder, shown in Fig. 5.36(b). The bits are weights according to the magnitude of their place v alue, by descending value of Rf∕Rn so that each lesser bit has half the weight of the ne xt higher. This is ob viously an in verting summing amplifier. The output is related to the inputs as sho wn in Eq. (5.15). Thus,

$$-V_o = \frac{R_f}{R_1}V_1 + \frac{R_f}{R_2}V_2 + \frac{R_f}{R_3}V_3 + \frac{R_f}{R_4}V_4$$

(5.23)

Input V1 is called the most significant bit (MSB), while input V4 is the least significant bit (LSB). Each of the four binary inputs V1, … , V4 can assume only two voltage levels: 0 or 1 V. By using the proper input and feedback resistor values, the DAC provides a single output that is pro portional to the inputs.

Example 5.12

In the op amp circuit of Fig. 5.36(b), let Rf = 10 k Ω, R¹ = 10 k Ω, R² = 20 kΩ, R³ = 40 kΩ, and R⁴ = 80 kΩ. Obtain the analog output for binary inputs [0000], [0001], [0010], … , [1111].

Solution:

Substituting the given values of the input and feedback resistors in Eq. (5.23) gives

$$-V_o = \frac{R_f}{R_1}V_1 + \frac{R_f}{R_2}V_2 + \frac{R_f}{R_3}V_3 + \frac{R_f}{R_4}V_4$$ $$= V_1 + 0.5V_2 + 0.25V_3 + 0.125V_3$$

Using this equation, a digital input [ V1V2V3V4] = [0000] produces an analog output of −Vo = 0 V; [V1V2V3V4] = [0001] gives −Vo = 0.125 V.

(b) binary weighted ladder type.

In practice, the voltage levels may be typically 0 and ± 5 V.

Similarly,

$$[V_1 V_2 V_3 V_4] = [0010] \Rightarrow -V_o = 0.25 \text{ V}$$

\n

$$[V_1 V_2 V_3 V_4] = [0011] \Rightarrow -V_o = 0.25 + 0.125 = 0.375 \text{ V}$$

\n

$$[V_1 V_2 V_3 V_4] = [0100] \Rightarrow -V_o = 0.5 \text{ V}$$

\n:
\n

$$[V_1 V_2 V_3 V_4] = [1111] \Rightarrow -V_o = 1 + 0.5 + 0.25 + 0.125$$

\n

$$= 1.875 \text{ V}$$

Table 5.2 summarizes the result of the digital -to-analog conversion. Note that we have assumed that each bit has a value of 0.125 V. Thus, in this system, we cannot represent a voltage between 1.000 and 1.125, for example. This lack of resolution is a major limitation of digital-to-analog conversions. For greater accuracy, a word representation with a greater number of bits is required. Even then a digital representation of an ana log voltage is never exact. In spite of this inexact representation, digital representation has been used to accomplish remarkable things such as audio CDs and digital photography.

TABLE 5.2

Input and output values of the four-bit DAC.

Binary input		Output
[V1V2V3V4]	Decimal value	−Vo
0000	0	0
0001	1	0.125
0010	2	0.25
0011	3	0.375
0100	4	0.5
0101	5	0.625
0110	6	0.75
0111	7	0.875
1000	8	1.0
1001	9	1.125
1010	10	1.25
1011	11	1.375
1100	12	1.5
1101	13	1.625
1110	14	1.75
1111	15	1.875

A three-bit DAC is shown in Fig. 5.37.

• (a) Determine |Vo| for [V1V2V3] = [010].
• (b) Find |Vo| if [V1V2V3] = [110].
• (c) If |Vo| = 1.25 V is desired, what should be [V1V2V3]?
• (d) To get |Vo| = 1.75 V, what should be [V1V2V3]?

Answer: 0.5 V, 1.5 V, [101], [111].

Three-bit DAC; for Practice Prob. 5.12.

5.10.2 Instrumentation Amplifiers

One of the most useful and v ersatile op amp circuits for precision mea surement and process control is the instrumentation amplifier (IA), so called because of its widespread use in measurement systems. Typical applications of IAs include isolation amplifiers, thermocouple amplifiers, and data acquisition systems.

The instrumentation amplifier is an extension of the difference amplifier in that it amplifies the difference between its input signals. As shown in Fig. 5.26 (see Example 5.8), an instrumentation amplifier typically consists of three op amps and seven resistors. For convenience, the amplifier is shown again in Fig. 5.38(a), where the resistors are made equal except for the external gain-setting resistor RG, connected between the gain set terminals. Figure 5.38(b) shows its schematic symbol. Example 5.8 showed that

$$v_o = A_v(v_2 - v_1)$$

(5.24)

where the voltage gain is

$$A_v = 1 + \frac{2R}{R_G} \tag{5.25}$$

As shown in Fig. 5.39, the instrumentation amplifier amplifies small differential signal v oltages superimposed on lar ger common -mode

Instrumentation amplifier Amplified dierential signal,

no common-mode signal

Figure 5.39

The IA rejects common voltages but amplifies small signal voltages.

voltages. Since the common-mode voltages are equal, they cancel each other.

The IA has three major characteristics:

• 1. The voltage gain is adjusted by one external resistor RG.
• 2. The input impedance of both inputs is v ery high and does not vary as the gain is adjusted.
• 3. The output vo depends on the dif ference between the inputs v1 and v2, not on the voltage common to them (common-mode voltage).

Due to the widespread use of IAs, manuf acturers have developed these amplifiers on single-package units. A typical e xample is the LH0036, developed by National Semiconductor . The gain can be v aried from 1 to 1,000 by an e xternal resistor whose v alue may vary from 100 to 10 kΩ.

In Fig. 5.38, let R = 10 kΩ, v¹ = 2.011 V, and v² = 2.017 V. If RG is Example 5.13 adjusted to 500 Ω, determine: (a) the voltage gain, (b) the output volt age vo.

Solution:

(a) The voltage gain is

$$A_v = 1 + \frac{2R}{R_G} = 1 + \frac{2 \times 10,000}{500} = 41$$

(b) The output voltage is

$$v_o = A_v(v_2 - v_1) = 41(2.017 - 2.011) = 41(6)$$

mV = 246 mV

Determine the value of the external gain-setting resistor RG required for the IA in Fig. 5.38 to produce a gain of 200 when R = 25 kΩ.

Practice Problem 5.13

Answer: 251.3 Ω.

5.11 Summary

• 1. The op amp is a high -gain amplifier that has high input resistance and low output resistance.
• 2. Table 5.3 summarizes the op amp circuits considered in this chapter. The expression for the gain of each amplifier circuit holds whether the inputs are dc, ac, or time-varying in general.

TABLE 5.3

Summary of basic op amp circuits.

• 3. An ideal op amp has an infinite input resistance, a zero output resistance, and an infinite gain.
• 4. For an ideal op amp, the current into each of its two input terminals is zero, and the voltage across its input terminals is negligibly small.
• 5. In an inverting amplifier, the output voltage is a negative multiple of the input.
• 6. In a noninverting amplifier, the output is a positive multiple of the input.
• 7. In a voltage follower, the output follows the input.
• 8. In a summing amplifier, the output is the weighted sum of the inputs.
• 9. In a difference amplifier, the output is proportional to the difference of the two inputs.
• 10. Op amp circuits may be cascaded without changing their inputoutput relationships.
• 11. PSpice can be used to analyze an op amp circuit.
• 12. Typical applications of the op amp considered in this chapter include the digital-to-analog converter and the instrumentation amplifier.

Review Questions

• 5.1 The two input terminals of an op amp are labeled as:
  • (a) high and low.
  • (b) positive and negative.
  • (c) inverting and noninverting.
  • (d) differential and nondifferential.
• 5.2 For an ideal op amp, which of the following statements are not true?
  • (a) The differential voltage across the input terminals is zero.
  • (b) The current into the input terminals is zero.
  • (c) The current from the output terminal is zero.
  • (d) The input resistance is zero.
  • (e) The output resistance is zero.
• 5.3 For the circuit in Fig. 5.40, voltage vo is:

(a) −6 V	(b) −5 V
(c) −1.2 V	(d) −0.2 V

Figure 5.40

For Review Questions 5.3 and 5.4.

5.4 For the circuit in Fig. 5.40, current ix is:

(a) 600 μA	(b) 500 μA
(c) 200 μA	(d) 1∕12 μA

5.5 If vs = 0 in the circuit of Fig. 5.41, current io is:

(a) −10 μA	(b) −2.5 μA
(c) 10∕12 μA	(d) 10∕14 μA

5.6 If vs = 8 mV in the circuit of Fig. 5.41, the output

Figure 5.41 For Review Questions 5.5, 5.6, and 5.7.

voltage is:

(a) −44 mV	(b) −8 mV
(c) 4 mV	(d) 7 mV

5.7 Refer to Fig. 5.41. If vs = 8 mV, voltage va is:

(a) −8 mV	(b) 0 mV
(c) 10∕3 mV	(d) 8 mV

5.8 The power absorbed by the 4-kΩ resistor in Fig. 5.42 is:

(a) 9 mW	(b) 4 mW
(c) 2 mW	(d) 1 mW

Figure 5.42

For Review Questions 5.8.

• 5.9 Which of these amplifiers is used in a digital-toanalog converter?
  • (a) noninverter
  • (b) voltage follower
  • (c) summer
  • (d) difference amplifier
• 5.10 Difference amplifiers are used in (please check all that apply):
  • (a) instrumentation amplifiers
  • (b) voltage followers
  • (c) voltage regulators
  • (d) buffers
  • (e) summing amplifiers
  • (f ) subtracting amplifiers

Answers: 5.1c, 5.2c,d, 5.3b, 5.4b, 5.5a, 5.6c, 5.7d, 5.8b, 5.9c, 5.10a, f.

Problems

Section 5.2 Operational Amplifiers

• 5.1 The equivalent model of a certain op amp is shown in Fig. 5.43. Determine:
  • (a) the input resistance
  • (b) the output resistance
  • (c) the voltage gain in dB

Figure 5.43

For Prob. 5.1.

• 5.2 The open-loop gain of an op amp is 50,000. Calculate the output voltage when there are inputs of +10 μV on the inverting terminal and +20 μV on the noninverting terminal.
• 5.3 Determine the voltage input to the inverting terminal of an op amp when −40 μV is applied to the noninverting terminal and the output through an openloop gain of 150,000 is 15 V.
• 5.4 The output voltage of an op amp is −4 V when the noninverting input is 1 mV. If the open-loop gain of the op amp is 2 × 10⁶ , what is the inverting input?
• 5.5 For the op amp circuit of Fig. 5.44, the op amp has an open-loop gain of 100,000, an input resistance of 10 kΩ, and an output resistance of 100 Ω. Find the voltage gain vo∕vi using the nonideal model of the op amp.

5.6 Using the same parameters for the 741 op amp in Example 5.1, find vo in the op amp circuit of Fig. 5.45.

Figure 5.45

For Prob. 5.6.

5.7 The op amp in Fig. 5.46 has Ri = 100 kΩ, Ro = 100 Ω, A = 100,000. Find the differential voltage vd and the output voltage vo.

Figure 5.46 For Prob. 5.7.

Section 5.3 Ideal Op Amp

5.8 Obtain vo for each of the op amp circuits in Fig. 5.47.

For Prob. 5.8.

5.9 Determine vo for each of the op amp circuits in Fig. 5.48.

5.10 Find the gain vo∕vs of the circuit in Fig. 5.49.

For Prob. 5.10.

5.11 Using Fig. 5.50, design a problem to help other students better understand how ideal op amps work.

Figure 5.51

For Prob. 5.12.

Figure 5.52

For Prob. 5.13.

5.14 Determine the output voltage vo in the circuit of Fig. 5.53.

Figure 5.53

For Prob. 5.14.

Section 5.4 Inverting Amplifier

• 5.15 (a) Determine the ratio vo∕is in the op amp circuit of Fig. 5.54.
• (b) Evaluate the ratio for R¹ = 20 kΩ, R² = 25 kΩ, R³ = 40 kΩ.

5.16 Using Fig. 5.55, design a problem to help students better understand inverting op amps.

Figure 5.55

• 5.17 Calculate the gain vo∕vi when the switch in Fig. 5.56 is in:
  • (a) position 1 (b) position 2 (c) position 3.

For Prob. 5.17.

5.18 For the circuit shown in Figure 5.57, solve for the Thevenin equivalent circuit looking into terminals A and B. *

For Prob. 5.18.

* An asterisk indicates a challenging problem.

5.19 Determine io in the circuit of Fig. 5.58.

Figure 5.58

5.20 In the circuit of Fig. 5.59, calculate vo of v^s = 2 V.

Figure 5.59

For Prob. 5.20.

5.21 Calculate vo in the op amp circuit of Fig. 5.60.

Figure 5.60

For Prob. 5.21.

• 5.22 Design an inverting amplifier with a gain of −15.
• 5.23 For the op amp circuit in Fig. 5.61, find the voltage gain vo∕vs.

Figure 5.61 For Prob. 5.23.

5.24 In the circuit shown in Fig. 5.62, find k in the voltage transfer function v^o = kvs.

Figure 5.62

For Prob. 5.24.

Section 5.5 Noninverting Amplifier

Figure 5.63 For Prob. 5.25.

5.26 Using Fig. 5.64, design a problem to help other students better understand noninverting op amps.

Figure 5.64 For Prob. 5.26.

5.27 Find vo in the op amp circuit of Fig. 5.65.

Figure 5.65 For Prob. 5.27.

5.28 Find io in the op amp circuit of Fig. 5.66.

Figure 5.66

For Prob. 5.28.

5.29 Determine the voltage gain vo∕vi of the op amp circuit in Fig. 5.67.

Figure 5.67 For Prob. 5.29.

5.30 In the circuit shown in Fig. 5.68, find ix and the power absorbed by the 20-kΩ resistor.

Figure 5.68

For Prob. 5.30.

5.31 For the circuit in Fig. 5.69, find ix.

Figure 5.69 For Prob. 5.31.

5.32 Calculate ix and vo in the circuit of Fig. 5.70. Find the power dissipated by the 60-kΩ resistor.

5.33 Refer to the op amp circuit in Fig. 5.71. Calculate ix and the power absorbed by the 3-kΩ resistor.

Figure 5.71

For Prob. 5.33.

5.34 Given the op amp circuit shown in Fig. 5.72, express vo in terms of v1 and v2.

5.35 Design a noninverting amplifier with a gain of 7.5.

5.36 For the circuit shown in Fig. 5.73, find the Thevenin equivalent at terminals a-b. (Hint: To find RTh, apply a current source io and calculate vo.)

Figure 5.73

For Prob. 5.36.

Section 5.6 Summing Amplifier

5.37 Determine the output of the summing amplifier in Fig. 5.74.

Figure 5.74

For Prob. 5.37.

5.38 Using Fig. 5.75, design a problem to help other students better understand summing amplifiers.

Figure 5.75

For Prob. 5.38.

5.39 For the op amp circuit in Fig. 5.76, determine the value of v2 in order to make v^o = −16.5 V.

5.40 Referring to the circuit shown in Fig. 5.77, determine Vo in terms of V1 and V2.

Figure 5.77 For Prob. 5.40.

5.41 An averaging amplifier is a summer that provides an output equal to the average of the inputs. By using proper input and feedback resistor values, one can get

$$-v_{\text{out}} = \frac{1}{4}(v_1 + v_2 + v_3 + v_4)$$

Using a feedback resistor of 10 kΩ, design an averaging amplifier with four inputs.

• 5.42 The feedback resistor of a three-input averaging summing amplifier is 50 kΩ. What are the values of R1, R2, and R3?
• 5.43 The feedback resistor of a five-input averaging summing amplifier is 40 kΩ. What are the values of R1, R2, R3, R4, and R5?
• 5.44 Show that the output voltage vo of the circuit in Fig. 5.78 is

$$v_o = \frac{(R_3 + R_4)}{R_3(R_1 + R_2)}(R_2v_1 + R_1v_2)$$

Figure 5.78 For Prob. 5.44.

5.45 Design an op amp circuit to perform the following operation:

$$v_o = 3.5v_1 - 2.5v_2$$

All resistances must be ≤ 100 kΩ.

5.46 Using only two op amps, design a circuit to solve

$$-v_{\text{out}} = \frac{v_3 - v_1}{5} + \frac{v_1 - v_2}{2}$$

Section 5.7 Difference Amplifier

5.47 The circuit in Fig. 5.79 is for a difference amplifier. Find vo given that v¹ = 1 V and v² = 2 V.

5.48 The circuit in Fig. 5.80 is a differential amplifier driven by a bridge. Find vo.

5.49 Design a difference amplifier to have a gain of 4 and a common-mode input resistance of 20 kΩ at each input.

5.50 Design a circuit to amplify the difference between two inputs by 2.5.

(a) Use only one op amp.

(b) Use two op amps.

5.52 Design an op amp circuit such that *

$$v_o = 4v_1 + 6v_2 - 3v_3 - 5v_4$$

Let all the resistors be in the range of 20 to 200 kΩ.

5.53 The ordinary difference amplifier for fixed-gain operation is shown in Fig. 5.81(a). It is simple and reliable unless gain is made variable. One way of providing gain adjustment without losing simplicity and accuracy is to use the circuit in Fig. 5.81(b). Another way is to use the circuit in Fig. 5.81(c). Show that: *

(a) for the circuit in Fig. 5.81(a),

$$\frac{v_o}{v_i} = \frac{R_2}{R_1}$$

(b) for the circuit in Fig. 5.81(b),

$$\frac{v_o}{v_i} = \frac{R_2}{R_1} \frac{1}{1 + \frac{R_1}{2R_G}}$$

(c) for the circuit in Fig. 5.81(c),

$$\frac{v_o}{v_i} = \frac{R_2}{R_1} \left( 1 + \frac{R_2}{2R_G} \right)$$

Figure 5.81

For Prob. 5.53.

Section 5.8 Cascaded Op Amp Circuits

5.54 Determine the voltage transfer ratio vo∕vs in the op amp circuit of Fig. 5.82, where R = 10 kΩ.

Figure 5.82

For Prob. 5.54.

• 5.55 In a certain electronic device, a three-stage amplifier is desired, whose overall voltage gain is 42 dB. The individual voltage gains of the first two stages are to be equal, while the gain of the third is to be onefourth of each of the first two. Calculate the voltage gain of each.

5.56 Using Fig. 5.83, design a problem to help other students better understand cascaded op amps.

5.57 Find vo in the op amp circuit of Fig. 5.84.

Figure 5.84

For Prob. 5.57.

5.58 Calculate io in the op amp circuit of Fig. 5.85.

Figure 5.85

• For Prob. 5.58.
  • 5.59 In the op amp circuit of Fig. 5.86, determine the voltage gain vo∕vs. Take R = 10 kΩ.

Figure 5.86 For Prob. 5.59.

5.60 Calculate vo∕vi in the op amp circuit of Fig. 5.87.

For Prob. 5.60.

5.61 Determine vo in the circuit of Fig. 5.88.

Figure 5.88

For Prob. 5.61.

5.62 Obtain the closed-loop voltage gain vo∕vi of the circuit in Fig. 5.89.

Figure 5.89 For Prob. 5.62.

5.63 Determine the gain vo∕vi of the circuit in Fig. 5.90.

Figure 5.90 For Prob. 5.63.

5.64 For the op amp circuit shown in Fig. 5.91, find vo∕vs.

Figure 5.91 For Prob. 5.64.

5.65 Find vo in the op amp circuit of Fig. 5.92.

For Prob. 5.65.

5.66 For the circuit in Fig. 5.93, find vo.

Figure 5.93 For Prob. 5.66.

5.67 Obtain the output vo in the circuit of Fig. 5.94.

Figure 5.96 For Prob. 5.70.

5.68 Find vo in the circuit of Fig. 5.95, assuming that Rf = ∞ (open circuit).

Figure 5.95 For Probs. 5.68 and 5.69.

• 5.69 Repeat the previous problem if Rf = 10 kΩ.
• 5.70 Determine vo in the op amp circuit of Fig. 5.96.

30 kΩ

40 kΩ

5.71 Determine vo in the op amp circuit of Fig. 5.97.

• 5.72 Find the load voltage vL in the circuit of Fig. 5.98.

5.73 Determine the load voltage vL in the circuit of Fig. 5.99.

5.74 Find io in the op amp circuit of Fig. 5.100.

Figure 5.100 For Prob. 5.74.

Section 5.9 Op Amp Circuit Analysis with PSpice

• 5.75 Rework Example 5.11 using the nonideal op amp LM324 instead of uA741.
• 5.76 Solve Prob. 5.19 using PSpice or MultiSim and op amp uA741.
• 5.77 Solve Prob. 5.48 using PSpice or MultiSim and op amp LM324.
• 5.78 Use PSpice or MultiSim to obtain vo in the circuit of Fig. 5.101.

Figure 5.101 For Prob. 5.78.

5.79 Determine vo in the op amp circuit of Fig. 5.102, using PSpice or MultiSim.

Figure 5.102 For Prob. 5.79.

• 5.80 Use PSpice or MultiSim to solve Prob. 5.70.
• 5.81 Use PSpice or MultiSim to verify the results in Example 5.9. Assume nonideal op amps LM324.