Lecture 2 example, LC circuit (ENGR5001)

Lecture 2 example, LC circuitTutorialCurrent spec2:210 min

ENGR5001 Lecture 2 worked example 2, LC circuit (slides 9-14)

For the LC circuit in the figure (an inductor $L$ in series with a capacitor $C$ , input voltage $V_i$ applied across the pair, output voltage $V_o$ taken across the capacitor, loop current $i$ ):

LC circuit: inductor L in series with capacitor C, input V_i, output V_o across C, loop current i.

Formulas you may need

Inductor $V_L = L\dfrac{di}{dt}$ ; capacitor $V_C = \dfrac{q}{C} = \dfrac{1}{C}\displaystyle\int i\,dt$ ; Kirchhoff's voltage law (learn this)
Generalised second order form $\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t)$ (learn this)

(a)
State the assumptions.
(b)
Develop the linear model relating $V_i$ to $V_o$ .
(c)
Convert the model into the generalised form and identify the input, output, $K$ , natural frequency, damping ratio and steady state gain. Which mechanical model has the same form?