Superposition theorem

What is the superposition theorem?

In a linear time-invariant circuit with multiple voltage and current sources, the superposition theorem conveniently obtains the node voltage and branch current.

The superposition theorem states, ” The total current in any part of a linear circuit equals the algebraic sum of the currents produced by each source separately. “

To find the current from Rload, we may apply the superposition theorem. The method is more intuitive and less complex than using KCL and KVL.

superposition_theorem-1
Fig 1: A network with multiple sources.

Keep only one power source "Turn on". "Turn off" all other power sources.

To make sure that only one power source is “on” at a time and the rest are turned off:

  1. We are replacing all other independent voltage sources with a short circuit (using a zero resistance wire).
  2. Replacing all other independent current sources with an open circuit (circuit break having infinite resistance).
superposition theorem. creating an open where current source was present
Fig 2: I1 and V2 are turned off. Only V1 is turned-on.
Superposition theorem. Creating shorts where voltage sources were present
Fig 3: V1 and V2 are turned off. Only I1 is turned-on.
superposition_theorem_V2-1
Fig 4: I1 and V2 are turned off. Only V2 is turned-on.

Calculate the current and voltage (with direction) because of all the sources

Current from Rload because of V1 from terminal A to terminal B (Fig. 2),

$$I_{V1}=\cfrac{V_1}{R_1+R_{load}||(R_2+R_4)}\cfrac{R_2+R_4}{R_{load}+R_2+R_4}$$

Current from Rload because of I1 from terminal A to terminal B (Fig. 3),

$$I_{I1}=-I_1\cfrac{R_4}{R_2+R_1||R_{load}}\cfrac{R_1}{R_1+R_{load}}$$

Current from Rload because of V2 from terminal A to terminal B (Fig. 4),

$$I_{V2}=\cfrac{V_2}{R_2+R_4+R_{load}||R_1}\cfrac{R_1}{R_{load}+R_1}$$

Superimpose the calculated currents and voltages together.

The current through Rload is (simply adding all the current obtained above with their sign) :

$$I_{load}=I_{V1}+I_{I1}+I_{V2}$$

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