Superposition Theorem
Network Analysis Techniques
Don’t just sit there! Build something!! 
Learning to mathematically analyze circuits requires much study and practice. Typically, students practice by working through lots of sample problems and checking their answers against those provided by the textbook or the instructor. While this is good, there is a much better way.
You will learn much more by actually building and analyzing real circuits, letting your test equipment provide the “answers” instead of a book or another person. For successful circuitbuilding exercises, follow these steps:
 Carefully measure and record all component values prior to circuit construction.
 Draw the schematic diagram for the circuit to be analyzed.
 Carefully build this circuit on a breadboard or other convenient medium.
 Check the accuracy of the circuit’s construction, following each wire to each connection point, and verifying these elements onebyone on the diagram.
 Mathematically analyze the circuit, solving for all values of voltage, current, etc.
 Carefully measure those quantities, to verify the accuracy of your analysis.
 If there are any substantial errors (greater than a few percent), carefully check your circuit’s construction against the diagram, then carefully recalculate the values and remeasure.
Avoid very high and very low resistor values, to avoid measurement errors caused by meter “loading”. I recommend resistors between 1 kΩ and 100 kΩ, unless, of course, the purpose of the circuit is to illustrate the effects of meter loading!
One way you can save time and reduce the possibility of error is to begin with a very simple circuit and incrementally add components to increase its complexity after each analysis, rather than building a whole new circuit for each practice problem. Another timesaving technique is to reuse the same components in a variety of different circuit configurations. This way, you won’t have to measure any component’s value more than once.
Suppose we have a single resistor powered by two seriesconnected voltage sources. Each of the voltage sources is “ideal,” possessing no internal resistance:

Calculate the resistor’s voltage drop and current in this circuit.
Now, suppose we were to remove one voltage source from the circuit, replacing it with its internal resistance (0 Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

Now, suppose we were to remove the other voltage source from the circuit, replacing it with its internal resistance (0 Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

One last exercise: “superimpose” (add) the resistor voltages and superimpose (add) the resistor currents in the last two circuit examples, and compare these voltage and current figures with the calculated values of the original circuit. What do you notice?
Suppose we have a single resistor powered by two parallelconnected current sources. Each of the current sources is “ideal,” possessing infinite internal resistance:

Calculate the resistor’s voltage drop and current in this circuit.
Now, suppose we were to remove one current source from the circuit, replacing it with its internal resistance (∞ Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

Now, suppose we were to remove the other current source from the circuit, replacing it with its internal resistance (∞ Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

One last exercise: “superimpose” (add) the resistor voltages and superimpose (add) the resistor currents in the last two circuit examples, and compare these voltage and current figures with the calculated values of the original circuit. What do you notice?
Suppose a DC generator is powering an electric motor, which we model as a 100 Ω resistor:

Calculate the amount of current this generator will supply to the motor and the voltage measured across the motor’s terminals, taking into account all the resistances shown (generator internal resistance r_{gen}, wiring resistances R_{wire}, and the motor’s equivalent resistance).
Now suppose we connect an identical generator in parallel with the first, using connecting wire so short that we may safely discount its additional resistance:

Use the Superposition Theorem to recalculate the motor current and motor terminal voltage, commenting on how these figures compare with the first calculation (using only one generator).
Suppose we have a single resistor powered by two seriesconnected voltage sources. Each of the voltage sources is “ideal,” possessing no internal resistance:

Calculate the resistor’s voltage drop and current in this circuit.
Now, suppose we were to remove one voltage source from the circuit, replacing it with its internal resistance (0 Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

Now, suppose we were to remove the other voltage source from the circuit, replacing it with its internal resistance (0 Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

One last exercise: “superimpose” (add) the resistor voltages and superimpose (add) the resistor currents in the last two circuit examples, and compare these voltage and current figures with the calculated values of the original circuit. What do you notice?
Suppose we have a single resistor powered by two parallelconnected current sources. Each of the current sources is “ideal,” possessing infinite internal resistance:

Calculate the resistor’s voltage drop and current in this circuit.
Now, suppose we were to remove one current source from the circuit, replacing it with its internal resistance (∞ Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

Now, suppose we were to remove the other current source from the circuit, replacing it with its internal resistance (∞ Ω). Recalculate the resistor’s voltage drop and current in the resulting circuit:

One last exercise: “superimpose” (add) the resistor voltages and superimpose (add) the resistor currents in the last two circuit examples, and compare these voltage and current figures with the calculated values of the original circuit. What do you notice?
A windmillpowered generator and a battery work together to supply DC power to a light bulb. Calculate the amount of current through each of these three components, given the values shown in the schematic diagram. Assume internal resistances of the generator and battery to be negligible:

I_{batt} = I_{bulb} = I_{gen} =
A windmillpowered generator and a battery work together to supply DC power to a light bulb. Calculate the amount of current through each of these three components, given the values shown in the schematic diagram. Assume internal resistances of the generator and battery to be negligible:

I_{batt} = I_{bulb} = I_{gen} =
Electrical signals are frequently used in industrial control applications to communicate information from one device to another. An example of this is motor speed control, where a computer outputs a speed command signal to a motor “drive” circuit, which then provides metered power to an electric motor:

Two common standards for analog control signals are 15 volts DC and 420 mA DC. In either case, the motor will spin faster when this signal from the computer grows in magnitude (1 volt = motor stopped, 5 volts = motor runs at full speed; or 4 mA = motor stopped, 20 mA = motor runs at full speed).
At first, it would seem as though the choice between 15 volts and 420 mA as control signal standards is arbitrary. However, one of these standards exhibits much greater immunity to induced noise along the twowire cable than the other. Shown here are two equivalent schematics for these signal standards, complete with an AC voltage source in series to represent the “noise” voltage picked up along the cable’s length:

Use the superposition theorem to qualitatively determine which signal standard drops the greatest amount of noise voltage across the motor drive input’s resistance, thereby most affecting the motor speed control.
Published under the terms and conditions of the Creative Commons Attribution License