Lab 2 – Introduction to
Op Amps
Lab Performed on September 17, 2008 by Nicole Kato, Ryan Carmichael, and Ti Wu
E11 Laboratory Report – Submitted October 1, 2008
Department of Engineering, Swarthmore College
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Abstract:
In this laboratory, an operational amplifier was connected with resistors two circuits, an
inverting op amp circuit and a non-inverting programmable-gain circuit with two switches. Tests
were then performed to determine the input and output voltages (Vin and Vout) of the inverting op
amp configuration and the four non-inverting configurations. Vout was then divided by Vin,
determining the gain of each configuration. This value was then compared to the theoretical gain
determined by a function of resistors specific to each configuration. Our results are summarized in
Table 1 below.
Configuration
Sb
Sa
Theoretical Gain
Measured Gain
(actual resistors)
(without distortions)
Inverting
Non-inverting
Non-inverting
Non-inverting
Non-inverting
-open
open
closed
closed
-open
closed
open
closed
-1.993
1.000
2.00
2.99
3.99
-2.02
0.945
2.04
2.97
3.98
Percent
Error (%)
1.355
5.50
2.00
0.667
0.251
Table 1: Summary of Gains
Introduction:
An operational amplifier, or op amp
(shown in Figure 1), is an electronic circuit that
can control both the voltage and current of
electrical circuits, often used in thermostats, strain
gages, and accelerometers, among other things. In
our lab, we used the LM741 operational amplifier,
Figure 1: A Circuit Symbol for an Op Amp
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commonly produced by National Semiconductor. A full understanding of op amps requires an
understanding of the circuit’s components, including diodes and transistors. However, these
complicated circuits can be treated as black box devices by addressing its terminal behavior as a
function, rather than the circuit’s actual composition. When used in a circuit with resistors and a
necessary dependent source, the op amp can be used to sum, scale, subtract, and perform other
useful functions. When used with inductors and capacitors, it is used in integrating and
differentiating circuits.
Theory:
The LM741 Op amps have eight terminals, but we will only examine five of them: the VS+ ,
VS-, inverting input (V- ), non-inverting input (V+), and output (Vout ). The VS+ and VS- are the
positive and negative power supplies. The output voltage cannot be greater than the positive power
supply nor can it be less than the negative power supply. In an ideal op amp, no current flows into
the inverting and non-inverting inputs, although current can flow out of the input-controlled output.
There is also no voltage difference between the inputs. One of these inputs cannot have its voltage
change without the other input either increasing or decreasing in voltage to match. This ensures
linear operation and keeps the output at the same voltage. This equilibrium is maintained by
negative feedback through a feedback loop. Feedback loops direct the output voltage back to the
inverting input. This negative feedback drives the circuit back to equilibrium in a manner specific to
the circuit configuration.
For non-ideal op amp there is a voltage difference between the inputs. Because the inputs
are not equal, the output cannot return to its original value and the op amp leaves the linear region.
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Here negative feedback, in the circuit of which the op amp is a part of, limits the differences
between the two inputs, allowing us to assume linear operation to a good approximation.
In this lab, op amps are used in two different configurations: inverting and non-inverting.
With the inverting configuration shown below in Figure 2, the input voltage is connected to the
inverting input and the non-inverting
input is connected to ground. This
causes gain, or the ratio of the output
to input, to be negative, which in turn
causes the sine waves of the output
and input to oscillate at an offset of
Figure 2: Inverting Op Amp Configuration
pi. The output voltage then appears to
be an inversion of the input voltage.
Also, for this configuration, the feedback loop sends output voltage back through Rf into the
inverting input. Were Vin to rise, it would increase the voltage at the input directly as well as
decrease the output voltage. The output voltage would cause the feedback loop to decrease the
voltage at the inverting input returning the
system to equilibrium.
Kirchoff’s current law states that the
current going into a node must be equal to the
Node A
Σ I = 0 = Vin - (R1 * I)
Vin = (R1 * I)
current going out of the node. Using this fact
Node B
Σ I = 0 = Vout + (Rf * I)
Vout = -(Rf * I)
in our analysis, we can determine the current
Vout/Vin = Gain = -(Rf * I)/(R1 * I) = -Rf / R1
into and out of nodes A and B and, in the math
right, find that the gain is -Rf / R1. The negative sign indicates that this configuration is inverting.
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In the non-inverting configuration below in Figure 3, the voltage source is connected to the
non-inverting input, and the inverting input is connected to the output and to ground through
resistors. This causes the gain to be positive. When we look at the sine waves of the output and
input verses time we can see
that they oscillate with no
offset and so the output
does not appear inverted. In
this configuration, the
feedback loop sends the
output voltage through Rf
into the inverting input.
Figure 3: Non-Inverting Op Amp Configuration
Should Vin increase the
output would also increase. The feedback loop would then send the output voltage through Rf and
into the non-inverting terminal, which would decrease the output voltage, returning the circuit to
equilibrium.
Kirchoff’s voltage law states that the voltage of a closed loop must sum to zero. Using this
fact, we summed the voltages around loops 1 and 2 (shown above) and solved for Vout and Vin.
From this, in the math below, we found that the gain was 1 + (Rf/R1). The positive value indicates
that this configuration is non-inverting.
Loop 1
ΣV = 0 = Vout -(Rf * I) - (R1 * I)
Vout = I * (Rf + R1)
Loop 2
ΣV = 0 = Vin + (V+) - (V-) - R1I
0 = Vin - (R1 * I)
Vin = R1 * I
Vout/Vin = Gain = [I * (Rf + R1)] / [R1 * I] = 1 + (Rf/R1)
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Furthermore, in the noninverting programmable-gain op
amp configuration shown in Figure
4, we see that there are two
switches that can be either open or
Figure 4: Op Amp Configuration With Switches
closed, allowing us to adjust the
gain in equal increments by opening or closing the switches. Table 2 summarizes the formulas for
the theoretical gain for various switch positions.
Sb
Sa
Gain
open
open
1 + (Rf /∞)
open
closed
1 + (Rf /Ra)
closed
open
1 + (Rf /Rb)
closed
closed
1 + (Rf /(Ra || Rb)
Table 2: Theoretical Gain Formulas for Various Switch Positions
If Ra=20kΩ, Rb=10kΩ, and Rf=20kΩ, the numerical gains are adjusted by one through the opening
and closing of the switches. These gains are shown below in Table 3.
Sb
open
Sa
open
Gain
1.000
open
closed
2.00
closed
open
3.00
closed
closed
4.00
Table 3: Theoretical Gain for Various Switch Positions
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Procedure:
This section will be completed by Ti Wu with the assistance of Professor Cheever and handed in
later.
Results:
a. The Inverting Configuration:
Theoretically, for the inverting configuration shown in Figure 2, the gain should equal –Rf /
R1. Were the resistors Rf and R1 exactly 20 and 10 kΩ, the gain would be -2.00. However, we
measured the resistance of R1 as 9.88 kΩ and Rf as 19.69 kΩ, which yields a theoretical gain of
-1.993. We first determined the measured gain by taking the ratio of the peak-to-peak voltages of
Vout and Vin read directly off of the oscilloscope (the oscilloscope output is Figure 5 in Appendix
E). This method determined a gain of -1.941. This value, however, includes distortion: blips and
static in the oscilloscope graphs caused the oscilloscope to determine the Vout and Vin inaccurately.
To find a more accurate gain, we measured the difference in height between the Vout and Vin’s
maximum and minimum values using a ruler, and divided the values to determine our measured
gain with less distortion, which we determined to be -2.02.
In addition, we created a MultiSim model (Figure 6 below) of the circuit and used it to evaluate
the gain with 20 kΩ and 10 kΩ resistors (Figures 7,8 below), yielding a gain of -2.01. All of these
results are summarized in Table 3 below. It should also be noted that our supply voltages were
11.81 V and -11.90 V. If our output voltage was larger it could not have been greater than 11.81 V.
Similarly, the output voltage could not have been less than -11.90 V.
Vin
Vout
(with distortion)
(v)
(with distortion)
(v)
1.01
1.96
(actual resistors)
MultiSim
Gain
Percent
Error
-1.993
-2.01
1.355
Measured Gain
Measured Gain
Theoretical Gain
Theoretical Gain
(with distortion)
(without distortion)
(ideal resistors)
-1.941
-2.02
-2.00
(%)
Table 3: Gains for the Inverting Configuration
Note: Percent Error = | Measured Gain (without distortions)- Theoretical Gain (actual resistors)| * 100
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Theoretical Gain (actual resistors)
Figure 6: MultiSim Model of Inverting Configuration
Figure 7: MultiSim Simulation of Oscilloscope Reading
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Figure 8: MultiSim Graphical Determination of Gain (Slope)
b. The Non-Inverting Programmable-Gain Configuration:
For the non-inverting configuration shown in Figure 3, we calculated the gain for the four
possible permutations of switch positions (both switches open, Sb open with Sa closed, Sb closed
with Sa open, and both closed). Similar to the inverting configuration, we determined the gain with
distortion by taking direct readings from the oscilloscope (oscilloscope readouts are Figures 9-12 in
Appendix E) and then determined the gain without distortion by determining Vout and Vin with a
ruler. We also measured the values of our resistors as Ra = 19.69 kΩ, Rb=9.88 kΩ, Rf=19.69 kΩ.
We plugged these values into Table 2 to determine more accurate theoretical gains. The results of
these calculations are shown in Table 4 below.
Vin (with Vout (with
distortion) distortion)
(v)
(v)
Sb
Sa
(with distortion)
(without distortion)
(ideal resistors)
(actual resistors)
Percent
Error (%)
Measured Gain Measured Gain Theoretical Gain Theoretical Gain
1.03
1.32
open
open
1.282
0.945
1.000
1.000
5.50
1.02
2.20
open closed
2.16
2.04
2.00
2.00
2.00
1.02
3.14
closed open
3.08
2.97
3.00
2.99
0.667
1.03
4.14
closed closed
4.02
3.98
4.00
3.99
0.251
Table 4: Theoretical and Actual Gains for Various Switch Configurations
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Discussion:
For both the inverting configuration and the non-inverting programmable-gain
configuration, our measured gains were very close to the theoretical gains. With the exception of
one measurement, all of our measured gains were within 2.00% of the theoretical gain. The
exception, the non-inverting configuration with both switches open, logically has the highest
percent error because it produces the smallest gain, and thus slight errors result in larger percentage
errors. In addition to having the most oscilloscope distortion, the measured value only differed from
the theoretical value by 0.05. For comparison, the non-inverting configuration with Sb open and Sa
closed (2.00% error) had measured and theoretical values that differed by 0.04. When working with
such small ratios, every 0.01 of voltage, resistance, and gain makes a significant impact on the
percent error of the experiment. Such small error comes both from machine and human error. The
resistors were accurate to 5%, the oscilloscope produced blips and static that distorted gain by as
much as 0.3, in lab we noted that the Vin and Vout had an error of about 0.02 V, and manuallydetermined gain from the oscilloscope output is only accurate to about 0.02. Furthermore, rounding
a value such as 0.005 to 0.01 produces an error significant enough to be noted.
Sources of error considered, all of the measured values fall reasonably within error of the
theoretical values. This fact provides evidence that the op amps used in the E11 lab can be
considered ideal op amps to a good degree of accuracy, providing the other equipment used are of
similar accuracy to that of the equipment used in the E11 “Introduction to Op Amps” lab. If more
accurate equipment were used and the theoretical value still differed from the measured value, then
the error could not be feasibly attributed to the equipment and procedures used, revealing error due
to a non-ideal op amp. However, for Engineering 11 we can consider the op amps used ideal.
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Conclusions and Further Work:
In conclusion, we obtained the following data for our lab:
Configuration
Sb
Sa
Theoretical Gain
Measured Gain
(actual resistors)
(without distortions)
Inverting
Non-inverting
Non-inverting
Non-inverting
Non-inverting
-open
open
closed
closed
-open
closed
open
closed
-1.993
1.000
2.00
2.99
3.99
-2.02
0.945
2.04
2.97
3.98
Percent
Error (%)
1.355
5.50
2.00
0.667
0.251
Table 1: Summary of Gains
The measured results obtained are within error of the theoretical values for a given configuration.
This suggests that the op amps used can be considered ideal in the configurations we used. To
further check this conclusion, the experiment could be redone using a more accurate oscilloscope,
more accurate resistors, and a more consistent voltage source. However, in the scope of Engineering
11, this is unnecessary.
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Acknowledgements:
We would like to acknowledge the following sources for their aid in this lab report:
Cheever, Erik. "E11 Formal Lab Report Format." Swarthmore College. 28 Sep. 2008
<http://www.swarthmore.edu/NatSci/echeeve1/Class/e11/Formal_report.html>.
Cheever, Erik. "E11 Lab #2: Lab Procedure." Swarthmore College. 28 Sep. 2008
<http://www.swarthmore.edu/NatSci/echeeve1/Class/e11/E11L2/Lab2(LabProcedure).html
>.
Cheever, Erik. "E11 Lab #2: Prelab." Swarthmore College. 28 Sep. 2008
<http://www.swarthmore.edu/NatSci/echeeve1/Class/e11/E11L2/Lab2(PreLab).html>.
Nilsson, James W, and Susan Riedel. Electric Circuits (8th Edition). Alexandria, VA: Prentice
Hall, 2007.
In addition, we would like to thank Anne Krikorian for proof reading our grammar, punctuation,
etc.
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Appendices A-D: Assigned Questions
These sections will be completed by Ti Wu with the assistance of Professor Cheever and
handed in later.
Appendix E: Oscilloscope Readouts for All Five Configurations
Voltage vs. Time: Inverting Configuration
(measured gain without distortion: -2.02)
Vout = light line
Vin = dark line
Figure 5: Oscilloscope Output for the Inverting Configuration
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Voltage vs. Time: Sb=Closed, Sa=Closed
(measured gain without distortion: 4.02)
Vout = light line
Vin = dark line
Figure 9: Oscilloscope Output for Sb=Closed, Sa=Closed
Voltage vs. Time: Sb=Closed, Sa=Open
(measured gain without distortion: 3.08)
Vout = light line
Vin = dark line
Figure 10: Oscilloscope Output for Sb=Closed, Sa=Open
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Voltage vs. Time: Sb=Open, Sa=Closed
(measured gain without distortion: 2.16)
Vout = light line
Vin = dark line
Figure 11: Oscilloscope Output for Sb=Open, Sa=Closed
Voltage vs. Time: Sb=Open, Sa=Open
(measured gain without distortion: 0.945)
Vout = light line
Vin = dark line
Figure 12: Oscilloscope Output for Sb=Open, Sa=Open
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