CR · DC analysis

Superposition

Learn to split a linear circuit driven by several sources into easy one-source problems and add them up. Confirm for yourself that the partial responses sum to the full response.

Turn on the sources one at a time and add

Toggle the two sources and watch the current in the middle branch. Do the currents from each source alone add up to the current with both on?

Current I in the middle branch

I = 1.5 A

I = 1.0 + 0.5 = 1.5 A

Both · sum = total ·

Linearity is what lets you add

In a linear circuit, doubling the input exactly doubles the output, and the sum of two inputs is answered by the sum of the two outputs. This proportionality and additivity let you solve for several sources separately and add the results. The moment a nonlinear element like a diode enters, the theorem breaks.

Keep one source, zero the rest

To see one source’s contribution, set the others to zero. Zeroing a voltage source means a short across its ends; zeroing a current source means an open break. Any internal resistance the source has stays in the circuit. Here, turning V₂ off makes its place a short, revealing the current V₁ produces alone.

Add the partial responses, signs and all

Find the current each source produces alone in the same branch, with the same reference direction. Then add them with their signs, and you have the full current. If one source pushes current backward through that branch, it subtracts. Here both sources push the same way, so 1.0 A and 0.5 A simply add.

ObserveI = I₁ + I₂

The total is the sum of the partial responses.

ChooseI₁ = ?

The current with V₁ alone.

Fill inI₂ = ?

The current with V₂ alone.

On your ownI = 1.0 + 0.5 = ?

Add the two parts for the total.

Back to the first screen

With only one source on, the branch carried 1.0 A or 0.5 A; with both on, it carried exactly their sum, 1.5 A. There was no need to solve both sources at once — just solve the easy one-source circuit twice and add. As long as the circuit is linear, any tangled multi-source problem can always be split this way.

The superposition theorem says any current or voltage in a linear circuit equals the sum of the responses each independent source produces alone. Keep one source and zero the rest: short a voltage source, open a current source, leaving internal resistances in place. It applies only to linear circuits, where proportionality and additivity hold.