Discussion
What is the transfer function of the given open loop system with zero initial condition?
Answer: A
The circuit is a series resistor with a capacitor across the output, the first order RC low pass circuit. With zero initial conditions, in the Laplace domain the capacitor has the impedance \(1/(Cs)\), so the output is a potential divider.
\[ \begin{aligned} \frac{E_o(s)}{E_i(s)} = \frac{1/(Cs)}{R + 1/(Cs)} \\ &= \frac{1}{RCs + 1} \end{aligned} \]
The pole is at \(s = -1/RC\) and the dc gain is 1. The option \(1/(RCs)\) is the transfer function of an integrator, and \(RCs + 1\) and \(RCs - 1\) are not transfer functions of this circuit, since they would increase with frequency. The answer is option A.
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