Why do linear components make linear circuits?

We can use the concept of linear circuits to simplify circuit analysis (e.g. superposition & Thevinin’s theorem.) In order to do this, we need a simple way to identify linear circuits.

I believe this is typically done by recognizing that a circuit is composed entirely of linear components. It makes intuitive sense that linear components would create linear circuits, but is there a more rigorous explanation as to why?

Even if all components are linear (obey additivity and homogeneity,) it’s not clear to me why every input/output relationship in that circuit has to share this property.

You can find the proof in elementary circuit theory texts such as Classical Circuit Theory By Omar Wing

Theorem 2.1

A circuit composed entirely of linear elements is a linear circuit. Proof Let {v(t),i(t),u(0)} be the solution of the circuit. Since the elements are linear by hypothesis, in the equations that define the elements, homogeneity and additivity hold for {v(t),i(t),u(0)}. Since KVL and KCL equations are homogeneous, {Av(t)} and {Ai(t)} satisfy the KVL equations and KCL equations, respectively. Since KVL and KCL are linear equations, additivity holds for {v(t)} and for {i(t)} in these equations. Therefore {v(t),i(t),u(0)} satisfies the homogeneity and additivity properties and by definition the circuit is linear.

• This is what I was looking for, thank you. Can you expand upon what u(0) means in this context? The author says that it represents initial conditions, but I don’t quite understand how the circuit isn’t fully defined by v(t) and i(t). Mar 3 at 12:22

I believe this is typically done by recognizing that a circuit is composed entirely of linear components.

This is the definition of linearity. Not the electronic component definition of linearity, but the mathematical definition of linearity from which the electronic component version of linearity is derived. Recall Fourier transforms which are linear and that for a linear system, the component frequencies in the output can only contain frequencies that were present in the input. No new frequencies can be introduced in a linear system. The concept of superposition is directly derived from this in that if your circuit only has linear components, you can analyze each source, whether it be DC, or AC of varying frequencies separately and then add them all up in the end. That's literally how the Fourier transform works.

One somewhat more rigourous but still intuitive way to look at it is to take two lines of different slopes that run over the same span of X coordinates. Then add them up and find what the slope of the new line is. It's the slope of the new line is just the sum of the slopes of the first two lines. Now instead of thinking of superimposing small segments of line, extend to derivatives which are is the slope at any point on a curve (or rather the slope of the tangent to any point on a curve).

That's why ideal capacitors and inductors are linear components:

$$\V_L = L\frac{di}{dt}\$$

$$\I_c = C\frac{dv}{dt}\$$

Even though they have that little derivative term.