Consider a discharging RL circuit. Obtain expressions as functions of time for t

Question

Consider a discharging RL circuit. Obtain expressions as functions of time for the voltage across the resistor, VR (t), and the voltage across the inductor, VL(t). Obtain an expression for the energy stored in the inductor as a function of time, UL(t) and PR(t). Repeat part (a) for a discharging RC circuit, but in the prompts exchange the capacitance for the inductance. Now consider a freely oscillating LC circuit where R = 0. Obtain expressions for VL(t), VC (T), UL(t), and UC(t). What is observed about the functions UL and UC?

Explanation / Answer

a)

Voltage across resistor in a discharging RL circuit is given by:

VR(t) = Vo*e^(-t/(L/R))

where Vo = initial voltage across inductor

and across inductor, VL(t) = Vo*e^(-t/(L/R))

energy stored = UL(t) = 0.5*L*(Vo^2/R^2)*e^(-2t/(L/R))

Power, PR(t) = (Vo^2/R)*e^(2t/(L/R))

So, UL(t) = (0.5*L/R)*PR(t)

b)

For RC circuit,

VR(t) = VC(t) = Vo*(e^(-t/RC))

UC(t) = 0.5*C*Vo^2*e^(-2t/RC)

PR(t) = (Vo^2/R)*e^(-t/RC)

So, UC(t) = (0.5*C*R)*PR(t)

c)

VL(t) = Vo*sin((1/sqrt(LC))*t) = VC(t)

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