(Solved): #1: DC Motor Transfer Function Differential Equation Consider the following transfer function repr ...

#1: DC Motor Transfer Function Differential Equation Consider the following transfer function representing a DC motor system, \( \frac{V(s)}{E(s)}=G_{v}(s)=\frac{10}{s+1} \), where \( E(s) \) and \( V(s) \) are the Laplace transforms of the input voltage \( e(t) \) volts and the output velocity \( v(t) \mathrm{m} / \mathrm{s} \), respectively. (1a) What is the differential equation corresponding to the transfer function \( G_{v}(s) \) ? (1b) Assume \( v(0)=0 \) and \( e(t) \) is a unit step function. Find \( v(t) \) for \( t \geq 0 \) using the Laplace transform approach. (1c) Plot \( v(t) \) vs. time \( t \). In the plot, specify the final value \( v(\infty) \), the time constant \( \tau \), and the value of \( v(\tau) \). (1d) Assume \( e(t)=\sin t \). Compute the magnitude and the phase of \( G_{v}(j 1) \), and use them to determine the steady-state response in the form of \( v_{s s}(t)=A \sin (t+\theta) \). (1e) If the input is changed to \( e(t)=\sin 10 t \) volt, what will \( v_{s s}(t) \) be? Explain the physical meaning.