The Superposition Principle states that in any linear network composed of resistors and voltage sources, the voltage across a resistor is simply the sum of all the voltages individually applied to it by each voltage source in the circuit.  The superposition principle is a useful method for finding the voltage across any resistor in a network where more than one voltage source is present.  A voltage source in this context may refer to any source of voltage or voltage signal, e.g., power supplies, batteries, oscillators, signal generators, etc.

To apply the superposition principle, one has to calculate the individual 'voltage contribution' of each voltage source to the resistor of interest.  To determine the voltage contribution of a source, simply 'zero out' all the other sources by replacing them with a short circuit, or with their internal resistances if these are known.  Kirchoff's Voltage Law and Ohm's Law are then applied to calculate the voltage across the resistor with this single source present.  Once all the voltage contributions have been determined, the voltage across the resistor is simply the sum of all the individual voltages contributed by the voltage sources.

                 

Figure 1.  A circuit with two voltage sources; find the voltage across the 3K resistor

Figure 2.  Contribution of the 5V source to the voltage across the 3K resistor

Figure 3.   Contribution of the 3V source to the voltage across the 3K resistor

To illustrate the Superposition Principle, consider the circuit in Figure 1.  The voltage across the 3K resistor may be calculated by getting the individual voltages applied by the 5V and 4V sources across it.  Figure 2 shows the equivalent circuit wherein the 4V source is 'zeroed out', i.e., replaced by a short circuit.  This circuit gives the voltage V1 applied by the 5V source across the 3K resistor, wherein V1 = 5V (0.75/2.75) = 1.364 V.  Figure 3 shows the equivalent circuit wherein the 5V source is replaced by a short circuit.  This circuit gives the voltage V2 applied by the 4V source across the 3K resistor, wherein V2 = 4V (1.2/2.2) = 2.182 V.   Thus, from the Superposition Principle, the voltage across the 3K resistor in Figure 1 is 1.364V + 2.182V = 3.546V.

Similarly, the voltage across the 2K resistor is -5V(2/2.75) + 4V(1.2/2.2) = -3.636 + 2.1818 = -1.454V and the voltage across the 1 K resistor is 5V(.75/2.75) - 4V(1/2.2) = 1.364 - 1.818 = -0.454.