Question

Provide an example of a non-linear circuit, for which the superposition theorem does not
work. Provide an explanation

Answer 1

Answer on Question #52497, Engineering / Other

Task: Provide an example of a non-linear circuit, for which the superposition theorem does not work. Provide an explanation

Answer:

the superposition theorem can be used to find out whether the circuit under consideration is a linear circuit. Any lumped circuit that is not a linear circuit is called a nonlinear circuit. The most important observation is that, in general, superposition does not apply to nonlinear circuits. A special class of nonlinear elements are the so-called, affine linear elements, which are characterized by a straight line not passing through the origin.

for example, an affine linear resistor is defined by: $F_{R} = \{(i(t),U(i)): U(t) - Ri(t) = a, \text{ with } a \neq 0\}$.

As an example, it is proven that the superposition theorem does not apply an affine linear time-invariant resistor. It is shown that if superposition is assumed a contradiction will follow:

If $u_{1}(t) = Ri_{1}(t) + a, (a \neq 0)$ and $u_{2}(t) = Ri_{2}(t) + a, (a \neq 0)$ then $u(t) = u_{1}(t) + u_{2}(t)$ must hold true if $i(t) = i_{1}(t) + i_{2}(t)$.

Starting with the currents, we obtain:

$$i_{1}(t) = \frac{1}{R} (u_{1}(t) - a), i_{2}(t) = \frac{1}{R} (u_{2}(t) - a)$$ $$i(t) = i_{1}(t) + i_{2}(t) = \frac{1}{R} (u_{1}(t) + u_{2}(t) - 2a)$$ $$u(t) = Ri(t) + a = \frac{R}{R} (u_{1}(t) + u_{2}(t) - 2a) + a = u_{1}(t) + u_{2}(t) - a \neq u_{1}(t) + u_{2}(t),$$

which completes the proof.