Electrical Engineering ⇒ Topic : Superposition Theorem

SUPERPOSITION THEOREM

It can be stated as

When a number of voltage and current sources are acting in a linear network simultaneously, the resultant current in any branch of the circuit is the algebraic sum of currents flowing through it taking one source at a time while deactivating the other sources. The voltage source is replaced by its internal resistance while the current source is replaced by open circuit.

Explanation

In Figure 1 (a), let us apply Superposition theorem to find the current in the resistor r₃. Let us first deactivate the source V₂. To deactivate V₂, its internal resistance must replace it. The new circuit configuration is shown in Figure 1(b).

                                                                                      figure (1)

From Figure 1 (b), we can write

                           ................ (1)

Next, let us deactivate the source V₁ to get the circuit configuration illustrated in Figure 1 (c). From Figure 1(c), we get

                           ................. (2)

Therefore, net current in r₃ is given by

Similarly, the current through r₁ and r₂ can be obtained. To apply superposition theorem, the direction of currents calculated for each source must be taken care of.

The superposition theorem for an a.c. circuit is the same as that for a d.c. circuit except that phase angle of all quantities (impedances, voltages and currents) must be taken into consideration.Therefore, superposition theorem for a.c. circuits can be stated as under :

In an a.c. network containing more than one source of voltage or current, the total current or voltage in any branch of the network is the phasor sum of currents or voltages produced in that branch by each source acting independently.

The procedure for using this theorem is as under

1. Select one source and replace all other sources with their *internal impedances.
2. Determine current through or voltage across the desired branch as a result of the single source acting alone.
3. Repeat steps (i) and (ii) using each source in turn until the branch current/voltage components have been calculated for all sources.
4. Determine the phasor sum of current/voltage components to obtain the actual current through or voltage across that branch.

The superposition theorem for an a.c. circuit is the same as that for a d.c. circuit except that phase angle of all quantities (impedances, voltages and currents) must be taken into consideration.Therefore, superposition theorem for a.c. circuits can be stated as under :

In an a.c. network containing more than one source of voltage or current, the total current or voltage in any branch of the network is the phasor sum of currents or voltages produced in that branch by each source acting independently.

The procedure for using this theorem is as under

1. Select one source and replace all other sources with their *internal impedances.
2. Determine current through or voltage across the desired branch as a result of the single source acting alone.
3. Repeat steps (i) and (ii) using each source in turn until the branch current/voltage components have been calculated for all sources.
4. Determine the phasor sum of current/voltage components to obtain the actual current through or voltage across that branch.

Superposition Theorem

The superposition theorem is applied to simplify complicated networks when two or more sources of emf are present. The theorem is stated as follows.In any network containing more than one source of emf, the resultant current in any branch is the algebraic sum of the currents which should be produced by each emf acting alone, all other sources of emf being replaced by their respective internal resistances (or impedances in case of ac circuits).

This may be explained from the following analysis. Consider a circuit as shown in Fig. (a). Let the currents due to V₁ alone be I'₁ and I'₂. Similarly, let the currents due to V₂ alone be I'1 and I'2, where V₁ and V₂ are the two independent sources of emf.

                        FIGURE (a) Circuit to illustrate superposition theorem