Problem $1$ An LVDT has the ideal element circuit shown below. The primary voltage $e_p(t)=E_{p}sint$ has $E_p$ and $$ fixed. The core displacement $x(t)$ causes a linear change in the secondary inductances and a mutual coupling between primary and secondary coils. Excitation frequency $$ is usually made as large as possible so that $x(t)$ does not change much over one cycle of $e_p(t).$ However if it is too large, the output signal will be small. To show this, answer the following questions.

$($a$)$ Find the transfer function relating $E_o(s)$ to $E_p(s)$ for fixed $x.$

$($b$)$ If $4K_m^2{x}^{2}2L_p{L}_o,$ draw the Bode plot of the transfer function from $($a$).$

$($c$)$ From the Bode plot in $($b$),$ what can you say about the relationship between the amplitude of $e_o(t)$ and $x$ and the phase angle between $e_o(t)$ and $e_p(t)?$ How is it affected by frequency?

$L_{S_1}=L_0-K_{S}x$

$L_{S_2}=L_0+K_{S}x$

$L_0=$ constant

$K_S=$ constant

$x=$ core displacement

$1$