11-The Superposition
Theorem
Objectives:
1. Apply the superposition theorem to linear circuits with more
than one voltage source.
2. Construct a circuit with two voltage sources, solve for the
currents and voltages throughout the circuit, and verify your
computation by measurement.
Summary of Theory:
The superposition theorem is a means by which we can solve circuits
that have more than one independent voltage source. Each source is
taken, one at a time, as if it were the only source in the circuit. All
other sources are replaced with their internal resistance (the internal
resistance of a dc power supply or battery can be considered to be
zero). The currents and voltages for the first source are computed.
The results are marked on the schematic, and the process is
repeated for each source in the circuits. When all sources have been
taken the overall circuit can be solved. The superposition theorem
will work for any number of sources as long as you are consistent in
accounting for the direction of currents and the polarity of voltages.
One way to keep the accounting straightforward is to assign a
polarity, right or wrong, to each component. Tabulate any current
which is in the same direction as the assignment as a positive current
and any current which opposes the assigned direction as a negative
current. When the final algebraic sum is completed, positive currents
are in the assigned direction, negative currents are in the opposite
direction of the assignment. In the process of replacing a voltage
source with its zero internal resistance.
Materials Needed:
Resistors: one 10 kΩ, one 2.2 kΩ, one 22 kΩ.
Procedure:
 Obtain the resistors listed in Table 11-1. Measure and resistors
and record the measured value in table 11-1.
 Construct the circuit shown in figure 11-1. This circuit has two
voltages sources connected to a common reference ground.
Table 11-1
Listed
Measured
R1
R2
R3
value
01kΩ
2.2 kΩ
22 kΩ
value
07..1kΩ
7700. kΩ
21.840 kΩ
 Compute the total resistance, RT, seen by the +5 v source. Then
remove the +5 v source and measure the resistance between
point A and B to confirm calculation. Record the computed and
measured values in Table 11-2.
 Use the source voltage, Vs1, and the total resistance to compute
the total current, IT, from the +5v source. This current flows
through R1 so record it as I1 in Table 11-2. Use the current divider
rule to determine the currents in R2 and R3. Record all three
currents as positive values in Table 11-2.
R2 ‖ R3
R23 = R2 R3/ R2 + R3 = 2.197 * 21.840 /(2.197+21.840)= 1.996KΩ
RT = 1.996 + 9.770 = 11.766 KΩ
I1 = IT = Vs/RT = 5/11.766 = 0.424 mA
I2 = IT * R3 / (R2 + R3 ) = 0.424 * 21.840/(2.197 + 21.840) = 0.385mA
I3 = IT * R2 / (R2 + R3 ) = 0.424 * 2.197/(2.197 + 21.840) = 0.038mA
 Use the currents computed in step 5 and the measured
resistances to calculate the expected voltage across each resistor
of figure 11-2. The connect the 5v power supply and measure the
actual voltages present in this circuit. Record the computed and
measured voltages in Table 11-2.
V1 = R1 * I1 = 9.770 * 0.424 = 4.142 V
V2 = R2 * I2 = 2.197 * 0.385 = 0.845 V
V3 = R3 * I3 = 21.840 * 0.038 = 0.829 V
 Remove the +5 v sources from the circuit and move the jumper from
between point C and D to between points A and B. compute the
total resistance between points C and D. Measure the resistance to
confirm your calculation. Record the computed and measured
resistance in Table 11-2.
Table 11-2.
Step 4
Step 5
Step 6
Step 7
Step 8
Step 9
Step10
Computed
Resistance
11.766KΩ
Measured
Resistance
11.764KΩ
Computed
Current
I1 I2 I3
0.424mA 0.385mA
Computed
Voltages
V1 V2 V3
V1
V2
V3
0.038mA
4.142V
8.947KΩ
Measured voltages
0.845V
0.829V
4.109V
0.841V
0.841V
7.532V 2.454V 7.534V
1.60V 8.363V
3.39V
7.528V
-3.41V
2.451V 7.528V
1.6V
8.37V
8.941 KΩ
0.771mA
Total
0.347mA
1.117mA
1.502mA
0.345mA
0.307
 Compute the current through each resistor in Figure 11-3. Note
that the total current flows through R2 and divides between R1
and R3. Mark the magnitude and direction of the current on figure
11-3. Important: record the current as a positive current if it is in
the same direction as recorded in step 5 and as a negative current
if it is in the opposite directions as in step 4.
 Use the currents computed in step 8 and the measured resistance
to compute the voltages drops across each resistors. Record the
voltage as a negative voltage. Then connect the 10 v source as
illustrated in figure 11-3 and measure the voltages
 Compute the algebraic sum of the currents and voltages listed in
Table 11-2. Enter the computed sums in Table 11-2. Then replace
the jumper between A and B with the 5 v sources, as shown in the
original circuit in figure 11-1. Measure the voltage across each
resistor in this circuit. The measured voltages should agree with
the algebraic sums. Record the measured results n Table 11-2.
Conclusion:
The superposition uses to find the currents through keep of a single
source in the circuit and then find a total current. Then remove the
source and put the next source and so on.