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Introduction to the Operational Amplier

Dr. Sai KiranInstructor

Oakland University

Mohammed S. NazrullaLab TA

Oakland University

EE 326Macomb University Center

March 2004

1 Introduction

The operational amplier (henceforth referred to as op-amp) is a ubiquitous example of a general-purpose analogueintegrated circuit. When appropriately congured using simple external passive circuit elements such as resistorsand capacitors, it can be used to perform operations such as addition, subtraction, differentiation, integration etc., onarbitrary analogue input signals. Since it is an analogue device, its inputs and outputs must be treated as analoguewaveforms, though certain op-amp circuits can be built whose outputs (or inputs) can be interpreted as digital signals(e.g. digital to analogue converters, or DACs). Figure 1 shows a typical op-amp in an eight-pin dual inline package(DIP).

Figure 1: A 741 Operational Amplifier Package.

This gure shows the pin outs for the IC and the schematic (or black-box) representation of the op-amp as a right-pointing triangle with ve (or three if you dont count V CC ) terminals. The Null Offset pins are never shown inschematics; even the V CC supplies are often left out. We shall not be needing the null offset pins in this lab, whichcan be left oating when not in use. NC stands for no connection. Pin 2 (v ) is known as an inverting terminal,while pin 3 (v+ ) is called a non-inverting terminal .

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Op-amps are complicated devices that exhibit both linear and nonlinear behaviour. An op-amp is actually a multi-stage transistor amplier circuit comprising about a couple dozen transistors and some resistors and capacitors. Aschematic of the innards of a 741 op-amp is provided on page 894 of the text by Sedra and Smith [4]. Most op-ampinternal circuits typically have three stagesan input stage, an intermediate stage and an output stage. Additionaldetails can be found in the text and will be covered in more advanced courses.

For the purposes of our lab, we will only need to consider circuits that very closely approximate the behaviourpredicted by the simplest ideal op-amp model. This model is based on the golden rules of the ideal op-amp, statedbelow (please also refer to Figure 2).

Figure 2: Op-Amp Input Parameters.

Golden Rules of the Ideal Op-Amp

1. i+ = i = 0

2. v+ = v , i.e. v pn = 0

Op-amp circuits lend themselves to easy analysis via node equations in conjunction with the ideal constraintsstated above. The procedure for analysing op-amp circuits is as follows:

1. Identify the node voltages at all nonreference nodes, but do not formulate equations at the op-amp output nodes.

2. At the remaining nonreference nodes, formulate N 2 node equations (for an N -node circuit) and then use theideal op-amp input voltage constraint v+ = v to reduce the number of unknowns.

3. Solve the resulting system of equations for the desired unknown(s).

2 Mathematical Models of Ideal Op-Amp Circuits

2(a) Inverting Amplier

Consider the circuit shown in Figure 3.

The node equation at the inverting terminal is:

0 vinR in

+0 vout

R f = 0 . Thus,

vout = R f R in

vin (1)

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EE 326: Introduction to the Operational Amplier 3

Figure 3: Inverting Amplifier.

2(b) Non-Inverting Amplier

A non-inverting amplier circuit is shown in Figure 4.

Figure 4: Non-Inverting Amplifier.

It can be shown that the input-output relationship for this circuit is given by:

vout = 1 +R 1R 2

vin (2)

A special, limiting case of the non-inverting amplier occurs when R 1 = 0 and R 2 . This is known as avoltage follower, or a buffer; it is shown in Figure 5, and may be used to electrically isolate one part of a circuit fromanother, while keeping the voltages at either side of the node at which it is inserted the same.

From the gure, it is clear that vin = vout , hence the term voltage follower.

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Figure 5: Voltage Follower or Buffer.

2(c) Voltage Controlled Current Source (VCCS)

Figure 6 shows a possible implementation of a constant current source. This one happens to be voltage-controlled.

Figure 6: Voltage Controlled Current Source.

The node equation at the inverting input is as follows:

V inR

+V in V out

RL= 0

Furthermore, we note that V out V in = V L and I L = V LR L =V inR to get the following equations describing the

operation of this current-source (note that the load current is independent of the load resistance):

V out = ( R + RL )V inR

(3)

I L =V inR

(4)

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EE 326: Introduction to the Operational Amplier 5

2(d) Differentiator

The circuit in Figure 7 is a differentiator.

Figure 7: A Differentiator.

Writing a node equation at the inverting terminal gives:

C ddt

(0 vin ) +0 vout

R= 0 . Thus,

vout = RC dvindt

(5)

2(e) Integrator

Figure 8 shows a circuit whose output is proportional to the integral of the input.

Figure 8: An Integrator.

Analysis of this circuit yields the following input-output relationship:

vout = 1

RC vin dt (6)

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2(f) Subtractor

The circuit shown in Figure 9 is called a subtractor.

Figure 9: A Subtractor Circuit.

The output of this circuit is dependent on the two inputs as described by the equation below:

vout =(R1 + R 2 ) R4

(R3 + R 4 ) R1 vin 2

R 2

R 1 vin 1 (7)

3 Pre-Lab

In your lab report, please provide the following:

1. The derivation of the input-output relationship for the non-inverting amplier (Figure 4) that gives rise to equa-tion 2.

2. Steps showing how equation 6 for the integrator (Figure 8) arises.

3. A step-by-step derivation for the subtractor circuit of Figure 9, yielding equation 7. What does this equationsimplify to when R 3 = R1 and R 4 = R2 ?

4 Lab Experiments

NOTE: For all the circuits to be implemented, the supply voltages must be set to V CC = 15 V.

4(a) Amplier

We rst need to create a subcircuit that provides an adjustable DC input to the amplier. We accomplish this by con-structing a potentiometer using the op-amp V CC supplies, as shown in Figure 10.

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EE 326: Introduction to the Operational Amplier 7

Figure 10: Amplifier Input Generator.

The vin output of this circuit is connected to the vin input of the amplier circuit.

1. Select one of the following amplier circuits to implement:

(a) The inverting amplier of Figure 3, with R f = 33 k and R in = 10 k , OR:(b) The non-inverting amplier of Figure 4, with R 1 = 22 k and R2 = 10 k .

2. Connect the input generator subcircuit (Figure 10) to the amplier circuit and connect two voltmeters to measurevin and vout simultaneously.

3. Calculate the expected gain, i.e. vout /v in , for your chosen circuit.

4. Adjust the potentiometer and record vin and vout as you vary vin from -6 V to 6 V in 1 V steps.

5. Using your expected gain value, calculate the theoretical values of vout for comparison with the actual values.

4(b) Current Source1. Construct the current source circuit as shown in Figure 6, with R = 1 k , V in = 5 V and a variable resistor for

RL .

2. Step RL from 300 to 3k in 300 steps. Record V L and calculate I L at each step.

3. Calculate the theoretical I L predicted by equation 4.

4(c) Differentiator/Integrator

1. With R = 22 k and C = 0 .1 F, set up either a differentiator (Figure 7) or an integrator (Figure 8) circuit.

2. Provide a 1 V peak-to-peak square-wave input to your circuit with T = 5 ms, then plot vin and vout on acommon time axis.

3. Repeat the above step with a triangle wave input of the same specications.

4. Calculate the theoretical relationship between vin and vout based on the above parameter values for comparisonwith the actual results.

5 Analysis and Discussion

In your lab report, please provide the analysis results requested below and also answer the questions posed herein:

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5(a) Amplier Circuit

1. Provide two plots of vout vs. vin an experimental plot using your data and a theoretical one based on theequation obtained from the ideal op-amp model.

2. Plot Gv vs. vin Gv = vout /v in , vin = 0 again provide a theoretical and an experimental plot.

3. Are the experimental plots what you expected? Comment (also compare the theoretical ideal with the actualresultsdo they always agree closely?).

4. Over the input range that provides close agreement, how does the actual gain compare with the theoreticallycalculated value (please quantify the error)?

5. What happens when the output voltage approaches the supply rail (i.e. within about 1 V or so of V CC )?

6. The input generator circuit alters the load resistance at the input of the amplier. Does this mean that youhave to modify the theoretical formula accordingly, or does the original formula still work? Explain.

7. Comment on the limitations of the amplier you built, if any.

5(b) Current Source

1. Plot I L vs. RL theoretical and actual.

2. Comment on the experimental plotdoes it look like something you would expect from a constant currentsource? If not, does it at least behave as a constant source over some range of load resistances? What is thisrange, as indicated by the plot?

3. Calculate the percent difference between theory and lab results when the load current is more or less constant(use a representative value for the experimental load current).

4. Do you suppose the behaviour of the load current for various load resistances has something to do with the V outoutput voltage of equation 3? Explain.

5. Compare equations 2 and 3 (also Figures 4 and 6). Are there any similarities? Do you notice any differences(other than notational)? Comment.

5(c) Integrator or Differentiator

1. Provide scope printouts for each set of inputs and outputs on common time axes (they may be vertically sepa-rated, but should be on a common grid).

2. Quantitatively compare the following parameters between theory and actual resultspeak-to-peak voltages (for

all output waveformsignore abnormal spikes, if any) and slopes (triangle wave output only). Qualitativelyconrm whether the output waveforms are the correct function types for the given inputs.

3. Based on your knowledge (gained from one of the previous circuits) of the op-amp behaviour when the outputvoltage swing approaches the supply rail, what is the maximum input signal amplitude that would yield theexpected output without any distortion?

4. Which of the twothe integrator or the differentiatorwould you expect to provide betterlooking outputgiven any arbitrary, continuous (in time) input signal? Explain (you can think about and answer this questioneven though you havent implemented both circuits).

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References

[1] Richard Johnston. Circuits II Lab . Electrical Engineering, Lawrence Technological University, 1999

[2] Roland E. Thomas, Albert J. Rosa. The Analysis and Design of Linear Circuits . John Wiley & Sons, Inc., 2001

[3] John C. Getty. The Analysis and Design of Linear Circuits: Lab Manual . John Wiley & Sons, Inc., 1999

[4] Adel S. Sedra, Kenneth C. Smith. Microelectronic Circuits . Oxford University Press, Inc., New York, New York,1998

[5] Richard C. Dorf, James A. Svoboda. Introduction to Electric Circuits . John Wiley & Sons, Inc., 1999

[6] Hitachi Datasheet. HA17741/PS General-Purpose Operational Amplier (Frequency Compensated) . Hitachi, Ltd.,Tokyo, Japan, December 2000