lab #3: operational amplifiers - department of physics ...courses/p309/exp-procedure/03...p309...

P309 Intermediate Lab, Indiana University Dept. of Physics

Last revised 01/2017

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Lab #3: Operational Amplifiers

Goal: So far we have looked at passive circuits composed of resistors, capacitors and

inductors. The problem with passive circuits is that the real part of the impedance always

decreases the amplitude of voltage and current in the circuit. Often we wish to take a

small voltage or current and amplify it, so that we can measure it with greater precision.

We might also want to add, subtract, integrate or differentiate two or more voltage or

current amplitudes. Amplifiers allow us to perform all of these linear mathematical

operations and more on an AC or DC voltage or current. The operational amplifier (op-

amp) is a type of integrated circuit amplifier with properties that makes implementing

these functions particularly simple. In this laboratory, you will learn the basic properties

of an ideal op-amp, how to use operational amplifiers with various types of feedback

control to perform simple transformations of an input signal and also some of the

limitations of real op-amps. You will also apply the integrator circuit to measure the

amplitude and direction of earth’s magnetic field in the laboratory. For a good primer on

op-amps, see Wikipedia (https://en.wikipedia.org/wiki/Operational_amplifier).

Equipment: OP07 op-amp, proto-board, assorted resistors and capacitors, DMM,

oscilloscope, large inductor coil.

1 Introduction:

A classical amplifier has two inputs: a ‘non-inverting’ input labeled “+,” and an

‘inverting’ input labeled “–.” Call the voltage at the “+” input +𝑉 and at the “−“ input

−𝑉. The open-loop voltage output of the output of amplifier is:

𝑉𝑜𝑢𝑡 = 𝐺𝑎𝑖𝑛 × (+𝑉 − −𝑉). (eq. 1)

For a normal amplifier, like a stereo amplifier, 𝐺𝑎𝑖𝑛

is adjustable and we operate the amplifier with the

output completely separate from the inputs.

Operational amplifiers have a very high gain,

𝐺𝑎𝑖𝑛~106, which is not too useful in an open-loop

configuration, unless you are looking at an input

voltage in the micro-Volt range. Indeed, in an ideal

op-amp, we assume that 𝐺𝑎𝑖𝑛~∞, in which case,

𝑉𝑜𝑢𝑡~±∞, unless +𝑉= −𝑉. Negative feedback

between output and input (i.e. where a bigger 𝑉𝑜𝑢𝑡 reduces 𝑉𝑖𝑛) allows many practical op-

amp applications, where the amplifier has linear response over more conditions than an

open-loop amplifier (e.g. we can design the feedback so that the gain does not change

despite changes in temperature). In most useful op-amp circuits, we determine the

negative feedback by connecting the output of the op-amp to one or both inputs via

appropriate passive components (resistors, capacitors, inductors,…). Figure 2 shows the

Figure 1: Amplifier in open-

circuit mode, showing +, −

and 𝑉𝑜𝑢𝑡 connections.

https://en.wikipedia.org/wiki/Operational_amplifier



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simplest such configuration. As in all stable circuits using op-amps, the amplifier will set

𝑉𝑜𝑢𝑡 to be whatever is necessary to make +𝑉= −𝑉. The arrangement of the feedback

determines the function of the op-amp circuit. Negative feedback is an important and

somewhat counter-intuitive concept. Please review it at:

https://en.wikipedia.org/wiki/Negative_feedback

We can determine the function of an ideal op-amp circuit from two ‘golden’ rules:

No current flows in or out of either of the two inputs to the op-amp.

𝑉𝑜𝑢𝑡 in any negative-feedback configuration strives to make the voltage difference

between the two inputs zero, i.e., +𝑉= −𝑉.

Our op-amp is an OP07, an integrated circuit with dozens of transistors, packaged in an

8-pin plastic DIP (Dual In-Line Package). You will find a data sheet for the OP07 at the

end of this document. Unlike the other components you have studied so far, the op-amp is

an active device: it requires a power supply to operate. The OP07 op-amp requires

power-supply voltages of ±15 V. If the output wants to exceed the supply voltage, the

signal is ‘clipped,’ i.e., if equation 1 predicts 𝑉𝑜𝑢𝑡 > 15V, then the actual 𝑉𝑜𝑢𝑡 = 15V,

and if equation 1 predicts 𝑉𝑜𝑢𝑡 < −15V, then the actual 𝑉𝑜𝑢𝑡 = −15V. Clipping is one of

the differences between a real and an ideal op-amp.

Question: What is the open-loop gain of the OP07 op-amp (look at the data sheet at the

end of this write-up)?

2 Inverting Amplifier

We will first build a circuit to multiply the input signal by a fixed negative 𝐺𝑎𝑖𝑛. Follow

Figure 2 to build this circuit. In this op amp configuration, connect the input signal

through the series input resistor R1 to the inverting input ‘−‘ and also connect the

feedback resistor R2 to the inverting input ‘−‘. Connect the non-inverting input ‘+’ to

ground.

Figure 2. Inverting amplifier circuit. The figure shows the two power supply pins to

the op-amp, 𝑉+ and 𝑉−. Most op-amp schematics do not show these pins, but you

always must connect the power supply to these pins for the op-amp to function.

Remember that the + and − pins are not the same as the 𝑉+ and 𝑉− power supply

pins.

-

+

_

R

V+

V-

DIP, top view

V

R

V = - V R / R

1

out 12inin

2

Function

Generator 𝑉𝑖𝑛~

Oscilloscope

Channel 2

Channel 1

𝑉𝑜𝑢𝑡

https://en.wikipedia.org/wiki/Negative_feedback



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The op-amp gain is given by

𝐺𝑎𝑖𝑛 =1

2

R

R

V

V

in

out . (eq. 2)

Question: derive equation 2 for this circuit starting with the two golden rules.

Using the Proto Board, build the inverting amplifier as shown in Figure 2. Pick R1 and R2

to have nominal resistances of 1kΩ and 10kΩ so that 𝐺𝑎𝑖𝑛~ − 10. Use a DMM to

measure the actual resistance of the resistors and calculate the expected value for 𝐺𝑎𝑖𝑛.

Refer to the photo in Figure 10 to see what your configuration will look like. Use a

simple color scheme to help you remember the function of the different wires on the

breadboard; e.g., red for power, green for ground, white or blue for signals. Use a signal

generator to produce a 1kHz sine wave of 1V peak-to-peak amplitude with no DC offset

for 𝑉𝑖𝑛. Use the oscilloscope to measure 𝑉𝑖𝑛 and 𝑉𝑜𝑢𝑡 simultaneously. Determine the gain

𝐺𝑎𝑖𝑛 =𝑉𝑜𝑢𝑡

𝑉𝑖𝑛 .

Questions: Compare your measured 𝐺𝑎𝑖𝑛 to the theoretical value 𝐺𝑎𝑖𝑛𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = −𝑅2

𝑅1.

Change the frequency of the function generator to 100Hz and 10kHz and measure the

gain again. Is the gain independent of frequency? Change the input peak-to-peak voltage

to 0.1V, 0.2V, 0.5V and 1.5V. To get a small voltage on the function generator, pull out

the amplitude knob, which reduces the voltage by a factor of 10. Is 𝐺𝑎𝑖𝑛 independent of

the input voltage (i.e. is the amplifier linear)?

Clipping

Increase the signal generator amplitude until you observe clipping of 𝑉𝑜𝑢𝑡. At what output

voltage do you see clipping? Change the power supply voltages to the op-amp (first 𝑉+, then 𝑉−. What happens to the output? Sketch what you observe and label the graph of

𝑉𝑜𝑢𝑡 vs. 𝑡 with respect to 𝑉+ and 𝑉−.

Slew Rate

An ideal op-amp has an output voltage that changes instantly as the input voltage

changes. A real op-amp has a maximum change in output voltage/second called the slew

rate. Estimate the slew rate of your op-amp by setting the function generator to produce a

square wave signal. Display both the square wave input voltage and the output voltage on

the oscilloscope. Increase the frequency of the signal until the shapes of the waves in the

two traces are clearly different. Now sketch or record the traces and measure the

maximum 𝑑𝑉

𝑑𝑡 for the op-amp. Compare this result to the slew-rate quoted in the data sheet

for the op-amp.

Question: How can the finite slew rate of an op-amp affect its function? You should

notice that once 𝑉𝑜𝑢𝑡 is limited by the slew rate, the output voltage is no longer

proportional to the input voltage and the shape of the output waveform is no longer the

same as the shape of the input waveform. Describe what happens instead? Suppose you

connect a sine-wave input signal to the op-amp of a fixed peak-to-peak amplitude,

𝑉𝑖𝑛 =𝑉𝑝−𝑝

2sin (𝜔𝑡). If you increase the frequency, the output signal will change from a

sine wave to a triangle wave. Why? Calculate the theoretical 𝑉𝑜𝑢𝑡 of the op-amp circuit



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as a function of the 𝐺𝑎𝑖𝑛, the slew rate, 𝑉𝑝−𝑝 and 𝜔. You should find that for high

frequencies the op-amp can only amplify small amplitude signals and for large

amplitudes it can only amplify lower frequencies. Derive the relationship between the

maximum amplitude and maximum frequency at which the op-amp linearly amplifies the

input signal. Now, repeat your experiment with a sine-wave input for three different 𝑉𝑝−𝑝

(𝑉𝑝−𝑝

5, 𝑉𝑝−𝑝 𝑎𝑛𝑑 5 𝑉𝑝−𝑝). For each 𝑉𝑝−𝑝 sweep the frequency in powers of 100 and

measure the output peak-to-peak voltage and the wave shape. Compare your results to

your theoretical calculation.

Offset Voltage

Connect the circuit shown in Figure 3. For an ideal op-amp, 𝑉𝑜𝑢𝑡 = 0V if +𝑉= −𝑉. A

real op-amp, will have 𝑉𝑜𝑢𝑡 = a small offset voltage 𝑉𝑂𝑆, when +𝑉= −𝑉. Measure the

offset voltage of the OP07. Use the circuit in Figure 3, and change R1 and R2 to have

nominal resistances of 10Ω and 10kΩ so that 𝐺𝑎𝑖𝑛~ − 1000. As usual, measure both 𝑅1

and 𝑅2 to calculate 𝐺𝑎𝑖𝑛𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙. Set 𝑉𝑖𝑛 = 0V by connecting the input of the resistor

to ground. Now measure 𝑉𝑜𝑢𝑡with the oscilloscope and also with a DMM.

Question: Consider R1 and R2 as a voltage divider. What is 𝑉−? Compare the measured

offset voltage with 𝑉𝑂𝑆 specified in the OP07 data sheet.

3 Non-inverting Amplifier

What if we don’t want to have the output voltage inverted with respect to the input

voltage? Consider the non-inverting linear amplifier circuit in Figure 4. Here the input

voltage connects to the non-inverting input and the voltage divider returns a fraction of

the output voltage to the inverting input. Use the same resistors that you used in Section 2

for a nominal 𝐺𝑎𝑖𝑛~ − 10 to construct the circuit. Measure Vin and Vout, determine the

actual gain.

Figure 3. Measurement of offset voltage by grounding the input voltage. You will

need to use 𝑅1 = 10Ω,𝑅2 = 10kΩ. Set the trigger mode of the oscilloscope to

“Line” so you can measure the DC offset voltage. Remember to connect the power

supply to the 𝑉+ and 𝑉− power supply pins.

-

+

_

R

V+

V-

DIP, top view

V

R

V = - V R / R

1

out 12inin

2

Oscilloscope

Channel 2

and

DMM

𝑉𝑜𝑢𝑡



5

Question: Using the golden rules for op-amps show that the theoretical value for the gain

of this circuit is:

𝐺𝑎𝑖𝑛 =𝑉𝑜𝑢𝑡𝑉𝑖𝑛

= 1 +𝑅2

𝑅1. (eq. 3)

Compare your experimental and theoretical results. Change the frequency of the function

generator to 100Hz and 10kHz and measure the gain again. Is the gain independent of

frequency? Change the input peak-to-peak voltage to 0.1V, 0.2V, 0.5V and 1.5V. Is 𝐺𝑎𝑖𝑛

independent of the input voltage (i.e. is the amplifier linear)?

4 Integrator

Op-amps can be used to construct a circuit that integrates an electrical signal over time

(Figure 5). A capacitor serves as the memory of the integrator. To clear the memory, we

simply short circuit the capacitor by closing a switch. When we open the switch, the

integration starts (𝑡 = 0).

Question: Use the two golden rules, to show that for a time-dependent input voltage,

𝑉𝑜𝑢𝑡(𝑡) = −1

𝑅𝐶∫𝑉𝑖𝑛(𝑡′)𝑑𝑡

′.

𝑡

0

(eq. 4)

Figure 4. Non-inverting amplifier circuit. The figure does not show the two power

supply pins to the op-amp, 𝑉+ and 𝑉−, but you always must connect the power supply

to these pins for the op-amp to function. Note that the wire to 𝑅1 does not connect to

the wire from 𝑉𝑖𝑛. Connect Channel 1 of the oscilloscope to the Function generator

directly as in Figure 2.

-

+

_

V

R

R

V = V (1 + R / R )out in 12

1

2

in

𝑉𝑜𝑢𝑡 = 𝑉𝑖𝑛 1 +𝑅2

𝑅1

No connection here

Function

Generator

Oscilloscope

Channel 2

Channel 1

𝑉𝑜𝑢𝑡



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Drift

First reset the integrator by briefly pressing the switch on the 2μF capacitor. Connect

the input of the resistor to ground. Since the voltage on the “−“ input of the op-amp is

0V, 𝑉𝑜𝑢𝑡should remain zero for an ideal op-amp. Usually, however, the output will drift

because the golden rules are not exactly true. Measure the drift rate in Volts/second from

your oscilloscope trace.

To reduce this drift, the OP07 provides an offset trim that allows you to adjust the

-

+

_

VV

in

out

R

C

pin 1 pin 8

20k pot.

+V

V0

+V

R

R0

1

Figure 5. Basic voltage integrator circuit. Remember to connect the power supply to

the op-amp. Set the oscilloscope to a very slow scan time and use the “Run/Stop”

button to make it scan slowly across the screen.

Switc

Oscilloscope

Channel 2

-

+

_

VV

in

out

R

C

pin 1 pin 8

20k pot.

+V

V0

+V

R

R0

1

Figure 6. Voltage integrator circuit with drift control. Attach a blue precision 20kΩ

potentiometer connected to the +15V power supply to pins 1 and 8 of the op-amp.

Remember to connect the power supply to the op-amp. Set the oscilloscope to a very

slow scan time and use the “Run/Stop” button to make it scan slowly across the

screen.

Switc

Oscilloscope

Channel 2



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balance of the two inputs. Build the circuit in Figure 6, by installing the offset trim,

connecting a blue precision 20kΩ potentiometer (variable resistor) between pins 1 and 8

of the op-amp. Connect the adjustable contact of the potentiometer to the +15V supply.

Adjust the potentiometer until the drift of the integrator is as near zero as possible. Use

the adjusting tool (a miniature screwdriver) to rotate the potentiometer. Determine the

residual drift rate in Volts/second (you will need this result in Section 5).

To show that the circuit integrates the input voltage as in equation 4, build the circuit in

Figure 7 and apply a constant voltage 𝑉0 to the input. In this case, equation 4 tells us that

the output voltage is a linear function of the time. Use the 10kΩ potentiometer on the

Proto-Board to make a voltage divider to generate a small 𝑉0~10mV, so 𝑉𝑜𝑢𝑡 takes about

30s to increase from 0V to 15V. Select the divider resistors accordingly.

Question: Why should 𝑅0 be much less than 𝑅?

Measure the rate of increase of 𝑉𝑜𝑢𝑡from the oscilloscope trace (set the oscilloscope for a

very slow sweep and use manual triggering. Compare with the rate calculated from the

values of the resistors and the capacitor in the circuit. Change 𝑅1 and repeat your

measurement. Do the two results agree with equation 4?

Questions: As shown in Figure 8, use the function generator to apply a square-wave of

frequency = 1kHz and 𝑉𝑝𝑒𝑎𝑘−𝑝𝑒𝑎𝑘 = 2V as 𝑉𝑖𝑛. Calculate the expected output signal

𝑉𝑜𝑢𝑡 from equation 3 and compare to your experimental 𝑉𝑜𝑢𝑡. You will need to

-

+

_

VV

in

out

R

C

pin 1 pin 8

20k pot.

+V

V0

+V

R

R0

1

Figure 7. Voltage integrator circuit with drift control and small voltage applied to the

input via a voltage divider (𝑅1 and 𝑅2). Remember to connect the power supply to

the op-amp.

Switc

-

+

_

VV

in

out

R

C

pin 1 pin 8

20k pot.

+V

V0

+V

R

R0

1

10kΩ potentiomete

Oscilloscope

Channel 2



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periodically reset the integrator by pushing the discharge button on the capacitor because

the average voltage from the function generator is not exactly 0𝑉 and the drift

compensation on your op-amp is not perfect. Repeat the derivation and comparison for a

square-wave and a triangle wave at your three frequencies. You may either save the

oscilloscope outputs to a file or take pictures with your cell phone. If you have time,

repeat for a sine-wave input.

5 The Magnetic Field of the Earth

We will now use the integrator in Section 3 to measure the magnetic field of the earth.

The magnetic field of the earth varies in amplitude and direction with geographical

position. A classical compass measures only the field direction in the 𝑥𝑦 direction. We

will measure the full magnetic field vector in the laboratory. Build the circuit shown in

Figures 9 and 10.

-

+

_

VV

in

out

R

C

pin 1 pin 8

20k pot.

+V

V0

+V

R

R0

1

Figure 8. Voltage integrator circuit with drift control and alternating voltage applied

to the input. Remember to connect the power supply to the op-amp.

Switc

Function

Generator

Oscilloscope

Channel 2



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A large many-turn inductor coil is an excellent transducer for magnetic-field

measurements because of Faradays law, which states that an electromotive force 휀 is

induced in the coil when the magnetic field flux changes. When the coil is flipped by

180°, in a fixed magnetic field, the flux changes by twice the starting value. Thus,

integrating the change in voltage suffices to determine the flux, according to:

𝑉𝑓𝑖𝑛𝑎𝑙 = −1

𝑅𝐶∫ 휀(𝑡′)𝑑𝑡′

𝑡

0

= −2𝐴𝑁𝐵

𝑅𝐶, (4)

where 𝐵 is the component of the magnetic field in the direction of the coil axis, 𝑁 is the

number of turns of the coil and 𝐴 the effective coil area. The average area of a multi-layer

coil, whose mean radius is 𝑟 and whose maximum and minimum radii are 𝑟 ± 𝛿 , is:

𝐴 = 𝜋 (𝑟2 +1

3𝛿2). (5)

Choose the input resistor 𝑅 such that a single flip of the coil causes a 𝑉𝑜𝑢𝑡 that you can

measure with at least 10% accuracy with the oscilloscope. Note that any drift in the

integrator is faster when 𝑅 is smaller. You need not completely eliminate the drift; just

make it small compared to the final value for 𝑉𝑜𝑢𝑡. Make a series of measurements

flipping the coil by 180°perpendicular to its axis. Repeat your measurement three times

to measure: 𝐵𝑧 with the coil axis vertical, 𝐵𝑥 with N-S horizontal coil axis (along the lab

room), and 𝐵𝑦 with horizontal E-W axis (perpendicular to both). Think carefully about

-

+

_

VV

in

out

R

C

pin 1 pin 8

20k pot.

+V

V0

+V

R

R0

1

Figure 9. Inductor connected to voltage integrator circuit with drift control and small

voltage applied to the input via a voltage divider (𝑅1 and 𝑅2). Remember to connect

the power supply to the op-amp.

Switc

Large Inductor Coil

Oscilloscope

Channel 2



10

which orientation of the coil measures which axis of the earth’s magnetic field, which

way you need to flip it, and include a sketch of the orientations and the rotations you

performed in your lab book. For each orientation determine the amount of drift during the

measurement and subtract it from your final values. Determine the component of the field

in each direction. Combine the three components to get the orientation and magnitude of

the B vector.

The S.I. unit for 𝐵 (appropriate for equation 4) is 1T (Tesla). 1T = 104G.

Questions: List possible sources for uncertainties. Evaluate the error of the three

individual field measurements. Combine the errors to get the uncertainty of the

magnitude 𝐵 of the field.

Question: Compare your measurement of the earth’s magnetic field with the accepted

value: https://www.ngdc.noaa.gov/geomag-web/#igrfwmm .

Figure 10. Photo of the apparatus for the magnetic field measurement. The large coil

is at the upper left. Most of them are mounted on gimbals to make them easier to

rotate. Use the white plastic adjustment tool to set the 20kΩ potentiometer to

minimize the drift in the integrator. Using a color scheme for the wires can help you

keep track of the wiring. The 2uF capacitor has a push-button reset switch attached.

https://www.ngdc.noaa.gov/geomag-web/#igrfwmm



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Last Revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15

lab #3: operational amplifiers - department of physics ...courses/p309/exp-procedure/03...p309...

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