Why is the frequency of the response same as that of the forcing function in a linear circuit?
What's the case when the circuit isn't linear?
Electronic – Why is the frequency of the response same as that of the forcing function in a linear circuit
accircuit analysisfrequency responselinearitysuperposition
Best Answer
If a circuit (or any system) is linear, then the output is governed by a linear differential equation. This means that the input signal and all its derivatives are not cubed, squared or anything like that.
So if out input is \$x(t)\$ and the output is \$y(t)\$ we can write an equation of the form
$$ a_0 + a_1(t)y(t) + a_2(t)\frac{dy(t)}{dt}+a_3(t)\frac{d^2y(t)}{dt^2} + ... = b_0 + b_1(t)x(t) + b_2(t)\frac{dx(t)}{dt}+b_3(t)\frac{d^2x(t)}{dt^2} + ... $$
So if x(t) is \$sin(wt)\$ the RHS can contain only terms of sin or cosine to the first power. Thus the LHS can only contain terms of sin or cosine to the first power. These all have the same frequency.
If there was a squared term, or some higher power, then there would be a sine squared term which oscillates at twice the original frequency, or another multiple.