Download 03_Op_Amps-JAGv7

P309 Intermediate Lab, Indiana University Dept. of Physics
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 opamp, 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 openloop 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 Figure 1: Amplifier in openvoltage in the micro-Volt range. Indeed, in an ideal circuit mode, showing +, −
op-amp, we assume that 𝐺𝑎𝑖𝑛~∞, in which case, and 𝑉𝑜𝑢𝑡 connections.
𝑉𝑜𝑢𝑡 ~ ± ∞, unless +𝑉 = −𝑉 . Negative feedback
between output and input (i.e. where a bigger 𝑉𝑜𝑢𝑡 reduces 𝑉𝑖𝑛 ) allows many practical opamp 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 simplest such
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
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 counterintuitive concept. Please review it at:
https://en.wikipedia.org/wiki/Negative_feedback
R
2
DIP, top view
_
R
V+
1
-
Function
V
in
Generator
𝑉𝑖𝑛 ~
Oscilloscope
+
V-
𝑉𝑜𝑢𝑡
Channel
Vout
= - V R2 / R
in
2
1
Channel 1
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.
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 8pin 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 powersupply 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.
The op-amp gain is given by
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
Vout
R2
𝐺𝑎𝑖𝑛 = V R .
in
1
Question: derive equation 2 for this circuit starting with the two golden rules.
(eq. 2)
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
𝑉
the oscilloscope to measure 𝑉𝑖𝑛 and 𝑉𝑜𝑢𝑡 simultaneously. Determine the gain 𝐺𝑎𝑖𝑛 = 𝑉𝑜𝑢𝑡 .
𝑖𝑛
𝑅
Questions: Compare your measured 𝐺𝑎𝑖𝑛 to the theoretical value 𝐺𝑎𝑖𝑛𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = − 𝑅2.
1
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
𝑑𝑉
traces are clearly different. Now sketch or record the traces and measure the maximum 𝑑𝑡
for the op-amp. Compare this result to the slew-rate quoted in the data sheet for the opamp.
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
𝑉𝑝−𝑝
input signal to the op-amp of a fixed peak-to-peak amplitude, 𝑉𝑖𝑛 = 2 sin(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 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
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P309 Intermediate Lab, Indiana University Dept. of Physics
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
𝑉𝑝−𝑝
experiment with a sine-wave input for three different 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.
R
2
DIP, top view
_
R
V+
1
-
V
in
Oscilloscope
Channel 2
+
V-
𝑉𝑜𝑢𝑡
Vout= - V R / R
and
DMM
in
2
1
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.
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 𝑉𝑜𝑢𝑡 with 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.
Question: Using the golden rules for op-amps show that the theoretical value for the gain
of this circuit is:
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
𝐺𝑎𝑖𝑛 =
𝑉𝑜𝑢𝑡
𝑅2
= 1+ .
𝑉𝑖𝑛
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
No connection here
R
2
𝑉𝑜𝑢𝑡 = 𝑉𝑖𝑛 1 +
_
R
𝑅2
𝑅1
1
-
Function V
Generator in
Oscilloscope
𝑉𝑜𝑢𝑡
+
Channel
Vout
= V (1 + 2
R /R)
in
2
1
Channel 1
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.
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,
𝑡
1
𝑉𝑜𝑢𝑡 (𝑡) = −
∫ 𝑉𝑖𝑛 (𝑡′)𝑑𝑡 ′ .
𝑅𝐶
(eq. 4)
0
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
Switch
C
_
R
+V
Vin
Oscilloscope
Channel 2
+
Vout
R1
pin 1
R
pin 8
V
Figure
5. Basic voltage integrator circuit. Remember to connect the power supply to
0
the op-amp. Set the oscilloscope to a very slow scan time
and use the “Run/Stop”
20k pot.
button to make it scan slowly across the screen.
0
+V
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, 𝑉𝑜𝑢𝑡 should 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.
Switch
C
_
R
+V
Vin
Oscilloscope
Channel 2
+
Vout
R1
pin 1
V
R
pin 8
0
20k pot.
0
+V
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.
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
To reduce this drift, the OP07 provides an offset trim that allows you to adjust the 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 white
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 3, build the circuit in
Figure 7 and apply a constant voltage 𝑉0 to the input. In this case, equation 3 tells us that
the output voltage is a linear function of the time. Use the 10kΩ potentiometer on the ProtoBoard to make a voltage divider to generate a small 𝑉0 ~10mV, so 𝑉𝑜𝑢𝑡 takes about 30s
C to
increase from 0V to 15V. Select the divider resistors accordingly. R
_
-
Switch
+V
Vin
+
R1
Vout
C
_
R
pin 1
V
+V
0
Vin
R
Oscilloscope
Channel 2
+
0
pin 8
20k pot.
Vout
R1
+V
10kΩ
potentiometer
V
R
pin 1
pin 8
0
20k pot.
0
+V
Figure 7. Voltage integrator circuit with drift control and small voltage applied to the
input via a voltage divider (𝑅1 and 𝑅2 ) For simplicity, use the 10kΩ potentiometer on
your Proto-Board. Remember to connect the power supply to the op-amp.
Question: Why should 𝑅0 be less than 𝑅?
Measure the rate of increase of 𝑉𝑜𝑢𝑡 from 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 3?
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
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
from equation 3 and compare to your experimental 𝑉𝑜𝑢𝑡 . You will need to 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.
Switch
C
_
R
-
Function V
in
Generator
+V
R1
+
pin 1
V
R
Oscilloscope
Channel 2
Vout
pin 8
0
20k pot.
0
+V
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.
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.
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
A large many-turn inductor coil is an excellent transducer for magnetic-field measurements
because of Faradays law, which states that an electromotive force 𝜀 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 𝑡 ′
𝐴𝑁𝐵
=−
∫ 𝜀(𝑡 )𝑑𝑡 ′ = −2
,
𝑅𝐶 0
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:
1
𝐴 = 𝜋 (𝑟 2 + 3 𝛿 2 ).
(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 which orientation
Switch
C
_
R
-
Large Inductor
Coil
+V
R1
Vin
Oscilloscope
Channel 2
+
Vout
pin 1
V
R
pin 8
0
20k pot.
0
+V
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.
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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P309 Intermediate Lab, Indiana University Dept. of Physics
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 = 104 G.
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: http://ngdc.noaa.gov/seg/geomag/jsp/struts/calcPointIGRF.
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.
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P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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Last Revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15