Linearity for energy storage devices

[Altairs]

Homework Statement

I am supposed to prove that linearity holds for energy storage elements.

Homework Equations

So far I have got to know that linearity can be proved by superposition.

The Attempt at a Solution

I attempted to make a circuit with a resistor, capacitor, a voltage source and a current source. Suppresed the sources one by one and solved the first order differential equations but did not get to the linearity. Equations didnt satisfy the superposition principle.

What is the correct way to prove it ?

 

[KataKoniK]

it is important to approach any problem or question with a clear and systematic method. In order to prove that linearity holds for energy storage elements, we must first define what we mean by linearity in this context. Linearity refers to the property of a system where the output is directly proportional to the input. In other words, if we double the input, the output will also double.

To prove linearity in energy storage elements, we must show that the response of the element to a given input is directly proportional to the input. This can be done by using the principle of superposition, which states that the total response of a linear system to multiple inputs is equal to the sum of the individual responses to each input.

To begin, let's consider a simple circuit with a resistor, capacitor, and voltage source. We can represent this circuit using Kirchhoff's laws and the equations for the resistor and capacitor:

Kirchhoff's Voltage Law: V = IR + Q/C

Ohm's Law: V = IR

Capacitor Equation: I = C(dV/dt)

Now, let's apply a voltage input V1 to the circuit and solve for the resulting current I1 using the above equations. We can then suppress the voltage source and apply a different voltage input V2 to the circuit and solve for the resulting current I2. Finally, we can combine these two solutions to find the total current response to the inputs V1 and V2.

I_total = I1 + I2 = (V1/R + C(dV1/dt)) + (V2/R + C(dV2/dt))

= (1/R)(V1 + V2) + C(dV1/dt + dV2/dt)

= (1/R)(V1 + V2) + C(d(V1 + V2)/dt)

= (1/R)(V1 + V2) + C(dV_total/dt)

From this, we can see that the total current response is equal to the sum of the individual current responses, and it is also directly proportional to the total voltage input V_total. This satisfies the principle of superposition and shows that the response of the circuit is linear.

This same approach can be applied to other energy storage elements, such as inductors, by using the relevant equations and applying the principle of superposition. Therefore, we can conclude that linearity holds for energy storage elements.

 

[Mene]

it is important to approach this problem with a systematic and analytical mindset. To prove linearity for energy storage devices, we must first understand what linearity means in this context. Linearity refers to the property of a system where the output is directly proportional to the input. In other words, the response of the system to a sum of inputs is equal to the sum of the individual responses to each input.

To prove linearity for energy storage devices, we can use the superposition principle as mentioned. This principle states that the response of a linear system to a sum of inputs is equal to the sum of the individual responses to each input. In other words, we can break down the inputs into smaller, simpler components and analyze their individual effects on the system.

To apply this principle to energy storage devices, we can consider a circuit with a resistor, capacitor, and a voltage or current source. We can then apply the superposition principle by individually turning off one source at a time and analyzing the response of the system. By doing so, we can determine the individual contributions of each source and then sum them up to see if they equal the total response of the system when all sources are active.

If the individual responses do indeed add up to the total response, then we can conclude that linearity holds for energy storage devices. If the responses do not add up, then there may be non-linear elements present in the system.

In conclusion, to prove linearity for energy storage devices, we can use the superposition principle to analyze the individual effects of each input and determine if they add up to the total response. This approach allows for a systematic and rigorous method of proving linearity in a scientific manner.