2 The Superposition Theorem
Thursday, 12 July 2018
8:11 AM
To superimpose something means to lay one thing on top of another. The superposition theorem is a means by which we can solve circuits that have more than one independent source. Each source is taken, one at a time, as if it were the only source in the circuit. All other sources are set to 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 circuit. When all sources have been taken, the overall circuit can be solved. The algebraic sum of the superimposed currents and voltages is computed.
Currents that are in the same direction are added; those that are in opposing directions are subtracted with the sign of the larger applied to the result. Voltages are treated in a like manner.
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 a short circuit, you may completely short out a resistor in the
circuit. If this occurs, there will be no current in that resistor for this part of the calculation. The final sum will still have the correct current.
The Superposition Theorem
Measure the resistors with listed values in Table 4-6.
Table 4-6
Component

Listed Value
Measured Value
R1
3.9 kΩ
R2
5.6 kΩ
R3
10.0 kΩ
Construct the circuit shown in Figure 4-5. This circuit has two voltage sources connected to a common 0 V reference.
Figure 4-5
Remove the 10 V source and place a jumper between the points labelled C and D, as shown in Figure 4-6. This jumper effectively sets the 𝑉𝑆2 source to zero.
Figure 4-6
Compute the total resistance, 𝑅𝑇 seen by the +5.0 V source. Then temporarily remove the + 5.0 V source and measure the resistance between points A and B to confirm your calculation. Record the computed and measured values in Table 4-7.
Table 4-7
Quantity
Computed
Measured
Step 4
RT  (VS1 operating alone)
Step 7
RT  (VS2 operating alone)
Use the source voltage, 𝑉𝑆1 and the total resistance to compute the total current, 𝐼𝑇, from the +5.0 V source. This current is through 𝑅1 so record it as 𝐼1 in Table 4-8. Use the current divider rule to determine the currents in 𝑅2 and 𝑅3. Record all three currents as positive in Table 4-8. This will be the assigned direction of current. Mark the magnitude and direction of the current in Figure 4-6.
Table 4-8
Computed Current
I1
I2
I3
Computed Voltage
V1
V2
V3
Measured Voltage
V1
V2
V3
Step 5
Step 6
Step 8
Step 9
Step 10
(totals)
Use the currents computed in step 5 and the measured resistances to calculate the expected voltage across each resistor of Figure 4-6. Then connect the +5.0 V power supply and measure the actual voltages present in this circuit. Record the computed and measured voltages in Table 4-8. Since all currents in step 5 were considered positive, all voltages in this step are also positive.
Remove the +5.0 V source from the circuit and move the jumper from between points 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 4-7.
Compute the current through each resistor in Figure 4-7. Note that this time the total current is through  𝑅2 and divides between 𝑅1 and 𝑅3. Mark the magnitude and direction of the current on Figure 4-7.
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 direction as in step 5.
Figure 4-7
Record the computed currents in Table 4-8.
Use the currents computed in step 8 and the measured resistances to compute the voltage drops across each resistor. If the current through a resistor was a positive current, record the resistor's voltage as a positive voltage. If a current was a negative current, record the voltage as a negative voltage. Then connect the + 10 V source as illustrated in Figure 4-7. Measure and record the voltages in Table 4-8. The measured voltages should confirm your calculation.
Compute the algebraic sum of the currents and voltages listed in Table 4-8. Enter the computed sums in Table 4-8. Then replace the jumper between A and B with the +5.0 V source, as shown in the original circuit in Figure 4-5. Measure the voltage across each resistor in this circuit. The measured voltages should agree with the algebraic sums. Record the measured results in Table 4-8.