Superposition Theorem 

The superposition theorem depends on the idea of linearity between the reaction and excitation of an electrical circuit. It expresses that the reaction in a specific branch of a direct circuit when different free sources are acting in the meantime is equal to the entirety of the reactions because of every autonomous source acting at once. 

In this strategy, we will consider just a single autonomous source at any given moment. In this way, we need to wipe out the staying free sources from the circuit. We can take out the voltage sources by shorting their two terminals and also, the present sources by opening their two terminals. 

Statement of Superposition Theorem

The Superposition Theorem states that for any linear system, the response (output) caused by any combination of input signals is the same as the sum of the individual responses caused by each input signal acting alone.

In mathematical terms, the Superposition Theorem states that for a linear system with input signals x1(t), x2(t), ..., xn(t) and corresponding output signals y1(t), y2(t), ..., yn(t), the overall output signal y(t) is given by:

y(t) = y1(t) + y2(t) + ... + yn(t)

This applies to both time-domain and frequency-domain analysis, for both continuous-time and discrete-time systems.

The Superposition Theorem is a fundamental principle in the study of linear systems and it is widely used in fields such as electrical engineering, mechanical engineering, and control systems. It allows simplifies the analysis of complex systems by breaking them down into simpler components and studying each component separately. This principle is useful in circuit analysis and control systems design.

It is important to note that the Superposition Theorem only applies to linear systems, meaning systems that obey the principle of superposition. A non-linear system would not obey this principle.

Superposition Theorem Formula

The Superposition Theorem states that for a linear system with input signals x1(t), x2(t), ..., xn(t) and corresponding output signals y1(t), y2(t), ..., yn(t), the overall output signal y(t) is given by the linear combination of the individual output signals:

y(t) = y1(t) + y2(t) + ... + yn(t)

where the inputs are applied one at a time.

It can be also written as:

y(t)= Σ(from i = 1 to n) yi(t) = Σ(from i = 1 to n) [h(t) * xi(t)],

where h(t) is the impulse response of the system, xi(t) is the input signal and yi(t) is the output corresponding to the input signal xi(t).

The Superposition Theorem applies to both time-domain and frequency-domain analysis, for both continuous-time and discrete-time systems. In the frequency domain, the superposition theorem is known as the linearity property of the Fourier transforms and it states that the Fourier transform of a linear combination of signals is equal to the linear combination of the Fourier transforms of the individual signals.

Superposition Theorem Steps

The steps to apply the Superposition Theorem to a linear system are as follows:

1. Identify the input signals: Determine the input signals x1(t), x2(t), ..., xn(t) that will be applied to the system.
2. Determine the impulse response: Obtain the impulse response h(t) of the system, which is the output of the system when the input is a delta function.
3. Determine the output for each input: For each input signal xi(t), calculate the corresponding output signal yi(t) by convolving the impulse response h(t) with the input signal xi(t): yi(t) = h(t) * xi(t).
4. Sum the individual output signals: Sum the individual output signals y1(t), y2(t), ..., yn(t) to obtain the overall output signal y(t): y(t) = y1(t) + y2(t) + ... + yn(t).
5. Check the system's linearity: It is important to confirm that the system is linear, otherwise, the superposition theorem can not be applied.

In the frequency domain, the Superposition Theorem is known as the linearity property of the Fourier transform. The steps are similar, but instead of convolving the impulse response with the input signal, the Fourier transform of the impulse response and the Fourier transform of the input signal is multiplied.

Superposition Theorem Proof

The Superposition Theorem can be proven mathematically by showing that the overall output signal y(t) of a linear system is linear with respect to the input signals. This means that if we have two input signals x1(t) and x2(t) and corresponding output signals y1(t) and y2(t), then the overall output signal y(t) caused by the combination of x1(t) and x2(t) is equal to the sum of y1(t) and y2(t).

Here is a proof of the Superposition Theorem for a continuous-time linear system:

Let h(t) be the impulse response of the system, and let x1(t) and x2(t) be two input signals. Then, according to the convolution theorem, the output signals y1(t) and y2(t) caused by x1(t) and x2(t) respectively are given by:

y1(t) = h(t) * x1(t)

y2(t) = h(t) * x2(t)

Now, let x(t) = x1(t) + x2(t) be the overall input signal. Then, the overall output signal y(t) is given by:

y(t) = h(t) * x(t)

Using the distributive property of convolution and the fact that h(t) is linear, we can expand the product:

y(t) = h(t) * (x1(t) + x2(t)) = h(t) * x1(t) + h(t) * x2(t)

Comparing this to the previous expressions for y1(t) and y2(t), we can see that:

y(t) = y1(t) + y2(t)

This shows that the overall output signal y(t) is equal to the sum of the individual output signals y1(t) and y2(t), which is the definition of the Superposition Theorem. This proof can be easily generalized to any number of input signals xi(t) and corresponding output signals yi(t) and it can be used to prove the theorem for the time domain and frequency domain.