lab #3: operational amplifiers - department of physics ...courses/p309/exp-procedure/03...p309...
TRANSCRIPT
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
1
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 earthβs 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 βnon-invertingβ input labeled β+,β and an
βinvertingβ input labeled ββ.β Call the voltage at the β+β input +π and at the βββ input
βπ. The open-loop voltage output of the output of amplifier is:
πππ’π‘ = πΊπππ Γ (+π β βπ). (eq. 1)
For a normal amplifier, like a stereo amplifier, πΊπππ
is adjustable and we operate the amplifier with the
output completely separate from the inputs.
Operational amplifiers have a very high gain,
πΊπππ~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 πΊπππ~β, in which case,
πππ’π‘~Β±β, unless +π= βπ. Negative feedback
between output and input (i.e. where a bigger πππ’π‘ reduces πππ) 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,β¦). Figure 2 shows the
Figure 1: Amplifier in open-
circuit mode, showing +, β
and πππ’π‘ connections.
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
2
simplest such configuration. As in all stable circuits using op-amps, the amplifier will set
πππ’π‘ to be whatever is necessary to make +π= βπ. 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 βgoldenβ rules:
No current flows in or out of either of the two inputs to the op-amp.
πππ’π‘ in any negative-feedback configuration strives to make the voltage difference
between the two inputs zero, i.e., +π= βπ.
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 Β±15 V. If the output wants to exceed the supply voltage, the
signal is βclipped,β i.e., if equation 1 predicts πππ’π‘ > 15V, then the actual πππ’π‘ = 15V,
and if equation 1 predicts πππ’π‘ < β15V, then the actual πππ’π‘ = β15V. 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 πΊπππ. 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 βββ and also connect the
feedback resistor R2 to the inverting input βββ. Connect the non-inverting input β+β to
ground.
Figure 2. Inverting amplifier circuit. The figure shows the two power supply pins to
the op-amp, π+ and πβ. 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 β pins are not the same as the π+ and πβ power supply
pins.
-
+
_
R
V+
V-
DIP, top view
V
R
V = - V R / R
1
out 12inin
2
Function
Generator πππ~
Oscilloscope
Channel 2
Channel 1
πππ’π‘
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
3
The op-amp gain is given by
πΊπππ =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 1kΞ© and 10kΞ© so that πΊπππ~ β 10. Use a DMM to
measure the actual resistance of the resistors and calculate the expected value for πΊπππ.
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 πππ. Use the oscilloscope to measure πππ and πππ’π‘ simultaneously. Determine the gain
πΊπππ =πππ’π‘
πππ .
Questions: Compare your measured πΊπππ to the theoretical value πΊππππ‘βπππππ‘ππππ = βπ 2
π 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 πΊπππ independent of
the input voltage (i.e. is the amplifier linear)?
Clipping
Increase the signal generator amplitude until you observe clipping of πππ’π‘. At what output
voltage do you see clipping? Change the power supply voltages to the op-amp (first π+, then πβ. What happens to the output? Sketch what you observe and label the graph of
πππ’π‘ vs. π‘ with respect to π+ and πβ.
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 ππ
ππ‘ 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 πππ’π‘ 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,
πππ =ππβπ
2sin (ππ‘). If you increase the frequency, the output signal will change from a
sine wave to a triangle wave. Why? Calculate the theoretical πππ’π‘ of the op-amp circuit
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
4
as a function of the πΊπππ, the slew rate, ππβπ and π. 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 ππβπ
(ππβπ
5, ππβπ πππ 5 ππβπ). For each ππβπ 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, πππ’π‘ = 0V if +π= βπ. A
real op-amp, will have πππ’π‘ = a small offset voltage πππ, when +π= βπ. Measure the
offset voltage of the OP07. Use the circuit in Figure 3, and change R1 and R2 to have
nominal resistances of 10Ξ© and 10kΞ© so that πΊπππ~ β 1000. As usual, measure both π 1
and π 2 to calculate πΊππππ‘βπππππ‘ππππ. Set πππ = 0V by connecting the input of the resistor
to ground. Now measure πππ’π‘with the oscilloscope and also with a DMM.
Question: Consider R1 and R2 as a voltage divider. What is πβ? Compare the measured
offset voltage with πππ specified in the OP07 data sheet.
3 Non-inverting Amplifier
What if we donβt 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 πΊπππ~ β 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 π 1 = 10Ξ©,π 2 = 10kΞ©. Set the trigger mode of the oscilloscope to
βLineβ so you can measure the DC offset voltage. Remember to connect the power
supply to the π+ and πβ power supply pins.
-
+
_
R
V+
V-
DIP, top view
V
R
V = - V R / R
1
out 12inin
2
Oscilloscope
Channel 2
and
DMM
πππ’π‘
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
5
Question: Using the golden rules for op-amps show that the theoretical value for the gain
of this circuit is:
πΊπππ =πππ’π‘πππ
= 1 +π 2
π 1. (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 πΊπππ
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 (π‘ = 0).
Question: Use the two golden rules, to show that for a time-dependent input voltage,
πππ’π‘(π‘) = β1
π πΆβ«πππ(π‘β²)ππ‘
β².
π‘
0
(eq. 4)
Figure 4. Non-inverting amplifier circuit. The figure does not show the two power
supply pins to the op-amp, π+ and πβ, but you always must connect the power supply
to these pins for the op-amp to function. Note that the wire to π 1 does not connect to
the wire from πππ. 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
πππ’π‘ = πππ 1 +π 2
π 1
No connection here
Function
Generator
Oscilloscope
Channel 2
Channel 1
πππ’π‘
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
6
Drift
First reset the integrator by briefly pressing the switch on the 2ΞΌF capacitor. Connect
the input of the resistor to ground. Since the voltage on the βββ input of the op-amp is
0V, πππ’π‘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.
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 βRun/Stopβ
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 20kΞ©
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 βRun/Stopβ button to make it scan slowly across the
screen.
Switc
Oscilloscope
Channel 2
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
7
balance of the two inputs. Build the circuit in Figure 6, by installing the offset trim,
connecting a blue precision 20kΞ© 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 π0 to the input. In this case, equation 4 tells us that
the output voltage is a linear function of the time. Use the 10kΞ© potentiometer on the
Proto-Board to make a voltage divider to generate a small π0~10mV, so πππ’π‘ takes about
30s to increase from 0V to 15V. Select the divider resistors accordingly.
Question: Why should π 0 be much less than π ?
Measure the rate of increase of πππ’π‘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 π 1 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 πππππβππππ = 2V as πππ. Calculate the expected output signal
πππ’π‘ from equation 3 and compare to your experimental πππ’π‘. 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 (π 1 and π 2). 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
10kΞ© potentiomete
Oscilloscope
Channel 2
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
8
periodically reset the integrator by pushing the discharge button on the capacitor because
the average voltage from the function generator is not exactly 0π 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 π₯π¦ 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
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
9
A large many-turn inductor coil is an excellent transducer for magnetic-field
measurements because of Faradays law, which states that an electromotive force ν is
induced in the coil when the magnetic field flux changes. When the coil is flipped by
180Β°, 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:
ππππππ = β1
π πΆβ« ν(π‘β²)ππ‘β²
π‘
0
= β2π΄ππ΅
π πΆ, (4)
where π΅ is the component of the magnetic field in the direction of the coil axis, π is the
number of turns of the coil and π΄ the effective coil area. The average area of a multi-layer
coil, whose mean radius is π and whose maximum and minimum radii are π Β± πΏ , is:
π΄ = π (π2 +1
3πΏ2). (5)
Choose the input resistor π such that a single flip of the coil causes a πππ’π‘ that you can
measure with at least 10% accuracy with the oscilloscope. Note that any drift in the
integrator is faster when π is smaller. You need not completely eliminate the drift; just
make it small compared to the final value for πππ’π‘. Make a series of measurements
flipping the coil by 180Β°perpendicular to its axis. Repeat your measurement three times
to measure: π΅π§ with the coil axis vertical, π΅π₯ with N-S horizontal coil axis (along the lab
room), and π΅π¦ 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 (π 1 and π 2). Remember to connect
the power supply to the op-amp.
Switc
Large Inductor Coil
Oscilloscope
Channel 2
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised 01/2017
10
which orientation of the coil measures which axis of the earthβs 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 π΅ (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 π΅ of the field.
Question: Compare your measurement of the earthβs 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 20kΞ© 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.