P$1.(20$ marks$)$ The Magnetic Levitation $($Maglev$)$ train $($see the figure to the left$)$ uses a propulsion system that is essentially a moving coil transducer; the actual system is much more complicated, but lets look at a simplified version. Here we will only look at the forward propulsion system, not the levitation system. Assume that the track is one long permanent magnet $($not true, but lets run with this idea$)$ as shown in the figure to the right; this permanent magnet creates a magnetic field of magnetic field density $\backslash ($ B $\backslash ).$ Assume that the train is of mass $\backslash ($ m $\backslash )$ and has losses $($wind resistance, friction, etc.$)$ with damping coefficient $\backslash ($ c $\backslash ).$ The train mass is driven by an electric coil that is wrapped around the south pole of the permanent magnet with a coil radius of $\backslash ($ r $\backslash )$ and has $\backslash ($ N $\backslash )$ coils. Also assume that this coil has an inductance of $\backslash ($ L $\backslash )$ and a resistance of $\backslash ($ R $\backslash ).$ Voltage is supplied to the coil by voltage source $\backslash ($ v$($t$)\backslash ).$ The velocity of the train mass $\backslash ($ m $\backslash )$ is represented by $\backslash ($ u$($t$)\backslash ).$

a$)$ Find the transfer function between the train velocity $\backslash ($ u$($t$)\backslash )$ and the voltage $\backslash ($ v$($t$)\backslash ).$

b$)$ Find the steady$-$state velocity $\backslash ($ u$\_\{$s s$\}($t$)\backslash )$ if $\backslash ($ v$($t$)=100\backslash$mathrm$\{$~V$\}\backslash )$ for $\backslash ($ t $\backslash$geq $0\backslash ).$ Your answer should be in symbolic form.