lab #3: operational amplifiers - iu bphysics.indiana.edu/~courses/p309/exp-procedure/03_op...lab #3:...
TRANSCRIPT
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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 +ππ and at the βββ input βππ. The open-loop voltage output of the output of amplifier is: ππππππππ = πΊπΊπΊπΊπΊπΊπΊπΊ Γ (+ππ β βππ). (eq. 1) For a normal amplifier, like a stereo amplifier, πΊπΊπΊπΊπΊπΊπΊπΊ is adjustable and we operate the amplifier with the output completely separate from the inputs. Operational amplifiers have a very high gain, πΊπΊπΊπΊπΊπΊπΊπΊ~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 πΊπΊπΊπΊπΊπΊπΊπΊ~β, in which case, ππππππππ~ Β± β, unless +ππ= βππ. Negative feedback between output and input (i.e. where a bigger ππππππππ reduces ππππππ) 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 simplest such
Figure 1: Amplifier in open-circuit mode, showing +, β and ππππππππ connections.
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
2
configuration. As in all stable circuits using op-amps, the amplifier will set ππππππππ to be whatever is necessary to make +ππ= βππ. 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: β’ No current flows in or out of either of the two inputs to the op-amp. β’ ππππππππ in any negative-feedback configuration strives to make the voltage difference
between the two inputs zero, i.e., +ππ= βππ. 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 ππππππππ > 15V, then the actual ππππππππ = 15V, and if equation 1 predicts ππππππππ < β15V, then the actual ππππππππ = 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 πΊπΊπΊπΊπΊπΊπΊπΊ. 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. The op-amp gain is given by
Figure 2. Inverting amplifier circuit. The figure shows the two power supply pins to the op-amp, ππ+ and ππβ. 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 ππ+ and ππβ power supply pins.
-
+
_
R
V+
V-
DIP, top view
V
R
V = - V R / R
1
out 12inin
2
Function Generator ππππππ~
Oscilloscope
Channel 2 Channel 1
ππππππππ
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
3
πΊπΊπΊπΊπΊπΊπΊπΊ =1
2
RR
VV
i n
o u t β= . (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 πΊπΊπΊπΊπΊπΊπΊπΊ~ β 10. Use a DMM to measure the actual resistance of the resistors and calculate the expected value for πΊπΊπΊπΊπΊπΊπΊπΊ. 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 ππππππ. Use the oscilloscope to measure ππππππ and ππππππππ simultaneously. Determine the gain πΊπΊπΊπΊπΊπΊπΊπΊ = ππππππππ
ππππππ .
Questions: Compare your measured πΊπΊπΊπΊπΊπΊπΊπΊ to the theoretical value πΊπΊπΊπΊπΊπΊπΊπΊππβππππππππππππππππππ = βπ π 2π 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 πΊπΊπΊπΊπΊπΊπΊπΊ independent of the input voltage (i.e. is the amplifier linear)? Clipping Increase the signal generator amplitude until you observe clipping of ππππππππ. At what output voltage do you see clipping? Change the power supply voltages to the op-amp (first ππ+, then ππβ. What happens to the output? Sketch what you observe and label the graph of ππππππππ vs. π‘π‘ with respect to ππ+ and ππβ. 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 ππππ
ππππ
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 ππππππππ 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, ππππππ = ππππβππ
2sin (πππ‘π‘). If you
increase the frequency, the output signal will change from a sine wave to a triangle wave. Why? Calculate the theoretical ππππππππ of the op-amp circuit as a function of the πΊπΊπΊπΊπΊπΊπΊπΊ, the slew rate, ππππβππ and ππ. You should find that for high frequencies the op-amp can only
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
4
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 ππππβππ οΏ½ππππβππ
5,ππππβππ πΊπΊπΊπΊππ 5 ππππβπποΏ½. For
each ππππβππ 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, ππππππππ = 0V if +ππ= βππ. A real op-amp, will have ππππππππ = a small offset voltage ππππππ, when +ππ= βππ. 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 πΊπΊπΊπΊπΊπΊπΊπΊ~ β 1000. As usual, measure both π π 1 and π π 2 to calculate πΊπΊπΊπΊπΊπΊπΊπΊππβππππππππππππππππππ. Set ππππππ = 0V by connecting the input of the resistor to ground. Now measure ππππππππwith the oscilloscope and also with a DMM. Question: Consider R1 and R2 as a voltage divider. What is ππβ? Compare the measured offset voltage with ππππππ 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 πΊπΊπΊπΊπΊπΊπΊπΊ~ β 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:
Figure 3. Measurement of offset voltage by grounding the input voltage. You will need to use π π 1 = 10Ξ©,π 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 ππ+ and ππβ power supply pins.
-
+
_
R
V+
V-
DIP, top view
V
R
V = - V R / R
1
out 12inin
2
Oscilloscope Channel 2 and DMM
ππππππππ
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
5
πΊπΊπΊπΊπΊπΊπΊπΊ =ππππππππππππππ
= 1 +π π 2π 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 πΊπΊπΊπΊπΊπΊπΊπΊ
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‘π‘ = 0). Question: Use the two golden rules, to show that for a time-dependent input voltage,
ππππππππ(π‘π‘) = β1π π π π
οΏ½ππππππ(π‘π‘β²)πππ‘π‘β².ππ
0
(eq. 4)
Figure 4. Non-inverting amplifier circuit. The figure does not show the two power supply pins to the op-amp, ππ+ and ππβ, but you always must connect the power supply to these pins for the op-amp to function. Note that the wire to π π 1 does not connect to the wire from ππππππ. 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
ππππππππ = ππππππ οΏ½1 +π π 2π π 1οΏ½
No connection here
Function Generator
Oscilloscope
Channel 2 Channel 1
ππππππππ
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
6
Drift First reset the integrator by briefly pressing the switch on the 2Β΅F capacitor. Connect the input of the resistor to ground. Since the voltage on the βββ input of the op-amp is 0V, ππππππππshould 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.
-
+
_
VV
inout
R
C
pin 1 pin 8
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.
Switch
Oscilloscope Channel 2
-
+
_
VV
inout
R
C
pin 1 pin 8
20k pot.
+V
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.
Switch
Oscilloscope Channel 2
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
7
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 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 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 ππ0 to the input. In this case, equation 3 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 ππ0~10mV, so ππππππππ takes about 30s to increase from 0V to 15V. Select the divider resistors accordingly.
Question: Why should π π 0 be less than π π ? Measure the rate of increase of ππππππππfrom 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 π 1 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 ππππππππππβππππππππ = 2V as ππππππ. Calculate the expected output signal ππππππππ
-
+
_
VV
inout
R
C
pin 1 pin 8
20k pot.
+V
Figure 7. Voltage integrator circuit with drift control and small voltage applied to the input via a voltage divider (π π 1 and π π 2) For simplicity, use the 10kΞ© potentiometer on your Proto-Board. Remember to connect the power supply to the op-amp.
Switch
+V
R
R0
1
10kΞ© potentiometer
Oscilloscope Channel 2
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
8
from equation 3 and compare to your experimental ππππππππ. 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 0ππ 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₯π₯π₯ direction. We will measure the full magnetic field vector in the laboratory. Build the circuit shown in Figures 9 and 10.
-
+
_
Vout
R
C
pin 1 pin 8
20k pot.
+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.
Switch
Function Generator
Oscilloscope Channel 2
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
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 ππ 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:
ππππππππππππ = β1π π π π
οΏ½ ππ(π‘π‘β²)πππ‘π‘β²ππ
0= β2
π΄π΄π΄π΄π΄π΄π π π π
, (4)
where π΄π΄ is the component of the magnetic field in the direction of the coil axis, π΄π΄ is the number of turns of the coil and π΄π΄ the effective coil area. The average area of a multi-layer coil, whose mean radius is ππ and whose maximum and minimum radii are ππ Β± πΏπΏ , is:
π΄π΄ = ππ οΏ½ππ2 + 13πΏπΏ2οΏ½. (5)
Choose the input resistor π π such that a single flip of the coil causes a ππππππππ that you can measure with at least 10% accuracy with the oscilloscope. Note that any drift in the integrator is faster when π π is smaller. You need not completely eliminate the drift; just make it small compared to the final value for ππππππππ. Make a series of measurements flipping the coil by 180Β°perpendicular to its axis. Repeat your measurement three times to measure: π΄π΄π§π§ with the coil axis vertical, π΄π΄π₯π₯ with N-S horizontal coil axis (along the lab room), and π΄π΄π¦π¦ with horizontal E-W axis (perpendicular to both). Think carefully about which orientation
-
+
_
VV
inout
R
C
pin 1 pin 8
20k pot.
+V
Figure 9. Inductor connected to voltage integrator circuit with drift control and small voltage applied to the input via a voltage divider (π π 1 and π π 2). Remember to connect the power supply to the op-amp.
Switch
Large Inductor Coil
Oscilloscope Channel 2
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
10
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΄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΄π΄ of the field. Question: Compare your measurement of the earthβs 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 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.
P309 Intermediate Lab, Indiana University Dept. of Physics
Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15
11