# Lab #3 Linearity, Proportionality, and Superposition ```ECE 170
Lab #3 Linearity, Proportionality, and Superposition
Lab #3 Linearity, Proportionality, and Superposition
This lab experiment will focus on three concepts. Those concepts are linearity,
proportionality, and superposition. Linearity and proportionality are like twins; they look
similar at first glance, but you can find differences on closer inspection. Linearity is a
system property that is very, very desirable. Consider an airplane. You would like to
think that if it is 6 months old, or 25 years old, that it is pretty much the same condition it
was when it left manufacturing plant. You would also like to think that if you were
flying 100mph or 1000mph that the airplane would react the same way (effects of drag,
etc. scaled of course). That is a linear system. It exhibits the same constant linear
properties over any time frame and input. In reality, an airplane is a highly complex
grouping of many linear and non-linear systems. The fact that it can take you from point
A to point B is the property of linearity focused upon in this example. A circuit that
contains only resistors and sources is a special type of linear circuit. It is a linear
algebraic circuit because all the elements can be considered constants. A true linear
system can have time varying quantities (speed, mass) as long as the plant (airplane),
behaves like a constant.
Proportionality is a way to relate two quantities together. In a linear system, this means
that if you supply more input, you get more output that is proportional to you input. In
other words, when you crank the volume on your stereo to 10, it takes 10x units of input
to get that output, but only 1x unit of input to get a volume of 1.
Consider the circuit given in Figure 3.1.
+
Vin
Linear
Resistive
Circuit
R Vout
-
Figure 3.1: Linearity and Proportionality.
In the linear circuit shown in Figure 3.1, we can find the output voltage relative to the
input voltage as
Vout  k  Vin
Where the k is called the proportionality constant of the circuit. You should note that you
could get a different proportionality constant for each set of quantities you would like to
relate. This is a fundamental property of a linear circuit.
Superposition is another way to solve a linear electrical circuit. Let us first state the
superposition theorem.
3-1
ECE 170
Lab #3 Linearity, Proportionality, and Superposition
Superposition Theorem: In any linear electrical circuit, any voltage or current
value can be obtained by taking the individual contributions to that voltage or
current as a result of each source taken alone and summing them together.
What this means is that if we have a circuit with two sources and need to find the output
voltage, we can first determine the output voltage as a result of source 1 (getting rid of
source 2) and then adding it to the output voltage resulting from source 2 (getting rid of
source 1). Consider the circuit with two sources shown in Figure 3.2.
R1
R2
+
V1
Vout
R3
V2
Figure 3.2: Circuit to Illustrate Superposition.
If we would like to solve for Vout in Figure 3.2, we can use the superposition theorem to
break this circuit into two sub circuits. These are shown in Figures 3.3 and 3.4.
R1
R2
R1
+
V1
Vout1
R2
+
R3
Vout2
R3
V2
-
-
Figure 3.4: Superposition Circuit 2.
Figure 3.3: Superposition Circuit 1.
By the superposition theorem, Vout as shown in Figure 3.2 is the sum of Vout1 and Vout2 as
shown in Figures 3.3 and 3.4. In mathematical form,
Vout  Vout1  Vout 2
It is very important to note how you eliminate sources when applying the superposition
theorem. Voltage sources are replaced with short circuits and current sources are simply
opened, or left disconnected.
3-2
ECE 170
Lab #3 Linearity, Proportionality, and Superposition
Instructional Objectives
3.1
3.2
3.3
Measure circuit parameters to determine if they are linearly related.
Measure circuit parameters to determine proportionality constants.
Verify the superposition theorem.
Procedure
1.
10k 
10k 
+
Vin
10k 
10k 
10k 
Vout
-
Figure 3.5: Linear Circuit to Study Proportionality.
2.
At 5 different voltages of Vin between 0 and 15V, measure the output voltage,
Vout. Calculate the proportionality constant that relates the output voltage to the
input voltage. Record your data in Table 3.1.
Input Voltage
Vin (V)
Output Voltage
Vout (V)
Table 3.1: Data for Figure 3.5.
3-3
Proportionality Constant
K = Vout/Vin
ECE 170
3.
Lab #3 Linearity, Proportionality, and Superposition
We will now verify the superposition circuit. Since we will be constructing three
circuits, it is advised that each lab partner builds one circuit and takes the
measurements for that circuit. Construct the circuit shown in Figure 3.6 on your
4k
5k
+
5V
Vout
6.2k 
15V
Figure 3.6: Circuit to Verify Superposition.
Quantity
Measured Voltage
(V)
Vout
V5V
V15V
Table 3.2: Data to Verify Superposition.
4.
Now remove the 15V source by replacing it with a short circuit. This is shown in
Figure 3.7. Measure the voltage across the 6.2k resistor.
4k
5k
+
5V
V5V
6.2k 
Figure 3.7: Superposition Circuit with One Source Removed.
5.
Reconnect the 15V source and replace the 5V source with a short circuit and
measure the voltage across the 6.2k resistor. This circuit is shown in Figure 3.8.
3-4
ECE 170
Lab #3 Linearity, Proportionality, and Superposition
4k
5k
+
V15V
6.2k 
15V
Figure 3.8: Superposition Circuit Missing A Voltage Source.
Post Lab Questions
1.
Plot the data in Table 3.1 on the same plot along with your data using the
proportionality constant obtained in the pre-lab. Connect your measured data
with a best-fit line and be sure to distinguish the actual measured data points. The
theoretical data may be plotted with a smooth line.
2.
For the circuit of Figure 3.5, how did the measurements made in the lab compare
to the predicted values calculated using proportionality? Explain any differences.
3.
For each of the three circuits you built for the superposition portion of this lab,
how well did the calculated value from the pre-lab compare to the measured
outputs? Explain any differences.
4.
If each of the voltage supplies were independently increased in magnitude (not
polarity) by 1V, which one would have a bigger effect on the change in the output
5.
If the input in Figure 3.5 were a 5V peak to peak sine wave instead of a constant
DC voltage, plot Vin and Vout. Is there still a linear relationship between Vin and
Vout?
3-5
ECE 170
Lab #3 Linearity, Proportionality, and Superposition
Name: ____________________
Section: ____________________
Pre-Lab #4: Thevenin’s Theorem
1.
Solve for Vload in Figure 4.0.
5k
4k
+