Transfer Function Calculation for \(\frac{{dc\left( t \right)}}{{dt}} + 2c\left( t \right) = r\left( t \right)\)

The transfer function of a linear, time-invariant (LTI) system is defined as the ratio of the Laplace transform of the output signal to the Laplace transform of the input signal, assuming all initial conditions are zero. For a system described by a differential equation, we can find the transfer function by taking the Laplace transform of the entire equation.

The given differential equation is:

\(\frac{{dc\left( t \right)}}{{dt}} + 2c\left( t \right) = r\left( t \right)\)

Here, \(c(t)\) is the output and \(r(t)\) is the input. We apply the Laplace transform to both sides of the equation. The Laplace transform of a derivative \(\frac{dc(t)}{dt}\) is \(sC(s) - c(0)\), where \(C(s)\) is the Laplace transform of \(c(t)\) and \(c(0)\) is the initial condition of \(c(t)\). The Laplace transform of \(2c(t)\) is \(2C(s)\). The Laplace transform of \(r(t)\) is \(R(s)\).

Taking the Laplace transform of the equation:

\(\mathcal{L}\left\{ \frac{{dc\left( t \right)}}{{dt}} \right\} + \mathcal{L}\left\{ 2c\left( t \right) \right\} = \mathcal{L}\left\{ r\left( t \right) \right\}\)

This becomes:

\(sC(s) - c(0) + 2C(s) = R(s)\)

To find the transfer function, we assume that all initial conditions are zero. So, we set \(c(0) = 0\).

\(sC(s) - 0 + 2C(s) = R(s)\)

\(sC(s) + 2C(s) = R(s)\)

Now, we factor out \(C(s)\) from the terms on the left side:

\(C(s)(s + 2) = R(s)\)

The transfer function \(G(s)\) is defined as the ratio of the Laplace transform of the output \(C(s)\) to the Laplace transform of the input \(R(s)\), i.e., \(G(s) = \frac{C(s)}{R(s)}\). We rearrange the equation to find this ratio:

\(\frac{C(s)}{R(s)} = \frac{1}{s + 2}\)

Therefore, the transfer function of the system is:

\(G(s) = \frac{1}{s + 2}\)

Comparing this result with the given options:

• Option 1: \(G\left( s \right) = \frac{s}{{s - 2}}\)
• Option 2: \(G\left( s \right) = \frac{1}{{s - 2}}\)
• Option 3: \(G\left( s \right) = \frac{s}{{s + 2}}\)
• Option 4: \(G\left( s \right) = \frac{1}{{s + 2}}\)

The calculated transfer function matches Option 4.

The information about \(r(t)\) being a unit impulse signal is relevant for finding the system's impulse response \(c(t)\) if \(r(t) = \delta(t)\), but it is not needed for finding the general transfer function \(G(s)\).