Variable capacitor. Tuned circuit.

The oscillating circuit of a radio has an inductor L and a rotary variable capacitor with a rotor with $n$ plates having the shape of a semicircular crown of radius r, respectively R, the distance between fixed and mobile plates of the capacitor being $d$. The rotor axis is located in the center of the semicircular crown.

a) Plot the capacity variation as a function of the angle made by the rotor plates inserted into the stator plates, knowing that when the plates are completely out, the capacitor has the capacity $C_0$.

b) Plot the frequency that can be received by the radio as a function of the angle made by the mobile (rotor) plates with the stator plates.

The variable capacitor having $n$ plates is equivalent to $2n$ individual capacitors wired in parallel. The maximum capacitance is obtained when the mobile plates are completely inserted into the stator.

$C_{max} = 2n frac {epsilon_0*S}{d} =2nfrac{epsilon_0}{d} frac{pi}{2} (R^2-r^2)$

For this type of variable capacitors the dependence of the capacitance on the insertion angle $alpha$ is linear:

$C =malpha+n$ where the constants $m$ and $n$  are found from conditions at the ends:

$alpha =0$ means $C(0) =C_0 =n$

$alpha = pi$ means $C(pi) =C_{max}$ so that $m =(C_{max}-C_0)/pi$

Therefore

$C(alpha) = frac {C_max-C_0}{pi}*alpha+C_0$.

The dependence of the electromagnetic oscillations frequency on the angle $alpha$ is given by:

$nu = 1/[2pisqrt{LC}] =1/left [2pisqrt{L left (frac{C_{max}-C_0}{pi}alpha +C_0 right )}right]$

The values of the frequency at the ends are:

for $alpha=0$, $/nu_{max} =1/[2pisqrt{LC_0}]$

for $alpha=pi$, $nu_{min} =1/[2pisqrt{LC_{max}}]$

Both graphs of the capacitance and frequency are below. the values taken for the components are $C_0 =5 pF$, $C_max =20 pF$ and $L =1 mu H$.