# Analytical proof that L and C are linear devices?

• xopek
In summary, the conversation discusses the linearity of inductors and capacitors and how they obey the ohm's law. It also explains that taking derivatives is a linear operation and that inductors and capacitors use a transfer function to convert input signals to output signals. The transfer function is linear and can be applied to any signal, not just sine or cosine signals.
xopek
I mean I know they are linear since they obey the ohms law. But I don't quite understand the reasoning that since, say, V=Ldi/dt and taking a derivative is a linear operation therefore it is a linear device?? I can verify that sin'(x) = cos(x) or sin(x+90) so the signal is time shifted but its form remains the same. That sounds logical to me but that has nothing to do with the fact that differentiation is a linear operation (which I believe is related to limits and differentials dy=m*dx etc). But what if we feed some sort of a high order polynomial instead of sin into L? Then taking derivative would distort the signal and would make the transformation nonlinear. So is it only linear when the signal is sin or cos?

They are linear in the sense that if ##f(.)## is the transfer function of the inductor or capacitor, then given two signals ##x_1(t)## and ##x_2(t)##, $$f(a_1x_1(t) +a_2x_2(t)) = a_1f(x_1(t)+a_2f(x_2(t))$$ That is ##f(.)## is a linear operator on functions of time. That means that you can use Fourier analysis to decompose a signal into a sum of ##\sin(.) ##and ##\cos(.)## signals, apply the transfer function to each one, and then sum them up to get the result of applying the transfer function to the original signal. So inductors and capacitors do not preserve phase relationships among frequency components.

anorlunda
Thanks. So just to clarify f(.) is a derivative? What is a transfer function in this context? I/V, i.e. impedance?
If so, then for a capacitor, I(t)=CdV/dt, and V(t)=cos(wt), so the transfer function, if we can call it that, is Z(t) or simply Xc?

The transfer function converts an input signal to an output signal. So in this case, if the input signal is ##V(t)## and the output signal is ##I(t) = C\frac{dV}{dt}##, the transfer function is ##f(.) = C\frac{d}{dt}##. You can easily show that it is linear in the sense that I defined it in post #2.

xopek said:
So is it only linear when the signal is sin or cos?
The term 'linear' implies that the same transfer function applies for any circuit with just RLC in it, irrespective of the amplitude of the signal. I think you are over thinking this. Differentiation still scales with the value of a constant multiplier.

tnich

## 1. What is the definition of a linear device?

A linear device is a type of electronic component or system that follows the principle of superposition, meaning that the output is directly proportional to the input. In other words, the response of the device is a straight line when plotted on a graph.

## 2. How can we prove that L and C are linear devices?

We can prove that L (inductance) and C (capacitance) are linear devices by applying the principle of superposition. This means that we can analyze the behavior of the device by considering each individual input (voltage or current) separately and then adding the results together to determine the overall response.

## 3. What is the analytical proof that L and C are linear devices?

The analytical proof of linearity for L and C devices involves using mathematical equations to show that the output is directly proportional to the input. For inductors, this can be shown through the equation V=L(di/dt), where V is the voltage, L is the inductance and di/dt is the rate of change of current. For capacitors, the equation is I=C(dv/dt), where I is the current, C is the capacitance, and dv/dt is the rate of change of voltage.

## 4. What are some real-life examples of L and C devices?

Inductors and capacitors can be found in a variety of electronic devices, such as radios, televisions, and computers. They are also commonly used in power supplies, filters, and oscillators. In everyday life, inductors can be found in electric motors and transformers, while capacitors can be found in household appliances like air conditioners and refrigerators.

## 5. Are there any limitations to the linearity of L and C devices?

While L and C devices are generally considered linear, there are some limitations to their linearity. These devices may exhibit non-linear behavior at high frequencies or high voltages, and their behavior may also be affected by temperature changes. Additionally, the materials used to make these devices can affect their linearity, so careful design and selection of components is important for achieving optimal performance.

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