# Is there a formal proof for the superposition theorem?

I was just wondering whether there is a formal prove for the superposition theorem in electric circuits? I tried searching it online but couldn't find anything sufficient. Most of the sources assume it follows from the definition of linear systems, but then how do you prove linearity without using the superposition theorem? Any input will be much appreciated!

• Well, Wikipedia states: "The superposition theorem for electrical circuits states that for a linear system[..]", i.e. the fact "The circuits are linear systems" is assumed to prove the superposition theorem, so I do not understand the question. – ACuriousMind Feb 15 '15 at 1:44
• @user40908 you don't prove linearity using superposition theorem, you first check if your circuit is linear in the variables of interest to you. That is the same as checking if the superposition holds. – Sofia Feb 15 '15 at 1:58
• R. E. Scott, Linear Circuits, New York: Addison-Wesley, 1960. – Alfred Centauri Feb 15 '15 at 3:46
• "how do you prove the linearity without using superposition theorem" - the (differential) equation(s) constraining the circuit variables are linear. – Alfred Centauri Feb 15 '15 at 4:07
• Maxwell's equations are linear. As long as the response of the circuit elements is also linear then superposition works by construction. – DanielSank Feb 17 '15 at 18:41

There are a few key bits of physics and math to understand here, but one does need to be very careful to avoid producing a circular argument. The key, concept I think is that of a

• linear circuit element, for which the output is precisely proportional to the input.

The precise meaning of 'input' and 'output' depend on the precise device, but this does not matter so much. For a linear capacitor, if you double the charge on the plates, you double the potential difference across them. For a linear resistor, the potential drop between the terminals is proportional to the current through it. For a linear inductor, it is proportional to the rate of change of current.

It is important to note that not all circuit elements are linear. A diode, for example, will respond differently if you change its polarity. A light bulb will increase its resistance as the current increases. Iron-core inductors show hysteresis, so their inductance is different depending on whether their magnetization is increasing or decreasing. In general, most circuit elements will show some nonlinearity if you drive them hard enough (even if "hard enough" is "so hard that you fry it", which is also nonlinear behaviour).

The restriction to linear circuits, then, is part definition and part physics. You are explicitly ruling out, for your circuit, those elements which behave nonlinearly. If I hand you a circuit board with a complicated circuit printed on it, and you want to decide whether it is linear or not, you need to take it apart and measure the response curves of all its components. Are they all linear? Great! your circuit is linear.

Thus, when you start off your proof with "let $C$ be a linear circuit ...", you are assuming that this step (which is where most of the physics is) has already been completed. This is therefore a safe assumption to use in a proof, and it comes at the price of restricting the validity of the result to only those circuits that have been empirically checked to be linear.

This is essentially all you need. You know the (linear!) equations which connect the (charge / current / rate of change of current) in each element with the potential difference across it, and you can use Kirchhoff's laws (which embody charge conservation at each node and energy conservation along each loop, and therefore always hold) to link them up. This will naturally result in a linear system between your sources and your output. This linear system has the mathematical property that the final solution is the sum of what you'd get if each of the sources were turned on by itself in turn, which is what the superposition theorem, as stated in Wikipedia, states. If your system has the physical property corresponding to the mathematical linearity assumption, it will also have the physical property that corresponds to the mathematical result.

Superposition is essentially a mathematical concept. Inspection of a physical phenomenon and the choice of the mathematical model to represent it, define the mathematical relations that the phenomenon is suggested to be subject to, and superposition may be one of them.

For example, when we say that a capcitor follows the mathematical rule $I = C\frac{\mathrm{d}V}{\mathrm{d}t}$, and that an RC circuit follows the ODE $\frac{\mathrm{d}}{\mathrm{d}t}V(t)+\frac{1}{RC}V(t) = \frac{1}{R}V_{\text{in}}(t)$, then "automatically" we get the superposition property for $V_{\text{in}}(t)$ thanks to the mathematical nature of ODEs.

But, if you take into account that a capacitor is not linear (i.e. doesn't follow exactly $I = C\frac{\mathrm{d}V}{\mathrm{d}t}$) but rather a more complex non linear form, then superposition of $V_{\text{in}}(t)$ would not hold.

So, the "proof of superposition" is actually within the scope of mathematics, while the choice of the mathemtical model to represent a physical phenomenon is subject to theoretical and/or experimental studies.

There are many proofs, the problem is that they become too mathematical and hence end up distracting electrical engineering students from core electrical engineering.

Very rigorous proofs involve assuming your circuits can be represented as planar graphs. Additional assumptions including no short circuits, independent current and voltage sources only. Proofs involving dependent sources become complicated.

In any case, to do the full proof, you lay down KVL and KCL (node analysis, mesh analysis) and you end up with linear system of equations needed to solve the circuit.

Of course, you can just solve the linear equation using elementary linear algebra techniques, but if you pay careful attention you see that the equations are always of the form: linear combination of node voltages = linear combination of independent voltage sources and current sources

(OR, of course, linear combination of branch currents ) = ...

(pay attention to the fact that the coefficients for the voltage sources and current sources in the above equations are simply algebraic functions of resistors ONLY)

... from here it should be easy too see that we can solve circuits by simply flipping on one source at a time (and keeping all other sources set to zero), and then adding up the "effects"

so to sum it up: what we are doing is assuming a linear system of equations exist, that this is system has as many unknowns as there are equations (not over/under specified) -- but we are not going to write this system down and solve it, instead, we set all V's and I's to zero, and flip back one V or I at a time and solve for currents/node voltages at a time.