A resistor-inductor-capacitor (RLC) network is depicted in Figure 2. v(t) + L oooo i(t) R vc...

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A resistor-inductor-capacitor (RLC) network is depicted in Figure 2. v(t) + L oooo i(t) R vc (t) Figure 2: A resistor-inductor-capacitor (RLC) network. a) Find the differential equation relating the capacitor voltage, vc(t), to the input voltage, v(t). [5 Marks] Hint: The voltage-current relationships of capacitor, resistor, and inductor are vc(t) = ½ fői(t)dt, vr(t) = Ri(t), and v₁(t) = Li(t). dt b) Find the transfer function relating the capacitor voltage, Vc(s), to the input volt- age, V (s). [5 Marks] c) Determine the characteristic equation of the control system, if R = 22, L = 1H, C = 1F. [5 Marks] d) Find the damping ratio (5) of the closed-loop system with the parameters given in c). [10 Marks] A resistor-inductor-capacitor (RLC) network is depicted in Figure 2. v(t) + L oooo i(t) R vc (t) Figure 2: A resistor-inductor-capacitor (RLC) network. a) Find the differential equation relating the capacitor voltage, vc(t), to the input voltage, v(t). [5 Marks] Hint: The voltage-current relationships of capacitor, resistor, and inductor are vc(t) = ½ fői(t)dt, vr(t) = Ri(t), and v₁(t) = Li(t). dt b) Find the transfer function relating the capacitor voltage, Vc(s), to the input volt- age, V (s). [5 Marks] c) Determine the characteristic equation of the control system, if R = 22, L = 1H, C = 1F. [5 Marks] d) Find the damping ratio (5) of the closed-loop system with the parameters given in c). [10 Marks]