I've been learning about RLC networks, superposition, two-port circuits etc. and a lot of derivations and equations implicitly assume the fact that the circuits in question are linear, no matter the weird and wonderful arrangements of the resistors, inductors and capacitors.
This sort of makes sense to me intuitively, but is there any proof (or any way to prove) that all RLC passive networks (except exceptions such as with non-ideal components) behave linearly?
For a resistor, inductor and capacitor there is a linear relationship between voltage and current.
For a resistor: \$ V_R=RI\$
For an inductor: \$V_L=Z_LI=j\omega L\cdot I \$
For a capacitor: \$V_C=Z_CI= \frac{1}{j\omega C}\cdot I \$
As you can see, the voltage across the various components is proportional to the current flowing through it, with the proportional constant being either \$R, \: j\omega L \$ or \$\frac{1}{j\omega C} \$.
Hence, any combination of a resistor, inductor and capacitor will also result in a linear relationship between voltage and current. So the entire system is linear.
there any proof (or any way to prove) that all RLC passive networks (except exceptions such as with non-ideal components) behave linearly?
Yes , Let's say a complex circuit of N- meshes having R,L,C elements(with no initial conditions), then you can always write mesh equation for each mesh and you'll get N integro -differential equation which on differentiation gives you a linear constant cofficient differential equation , so now you have N variables and N differential equation , and by substitution method, you can always get a unique linear constant cofficient differential equation with desired input output variables .
And hence, we can say that no matter how complex R,L,C circuit is you'll always get a LCCDE between input and output ,which is certainly a linear relation .
