RL and LC Circuits

Capacitor and Inductors in Series

Resistors and Inductors in Series

RL CircuitsAs the switch is thrown closed in an RL circuit, the current in the circuit begins to increase and a back EMF that opposes the increasing current is induced in the inductor.

The back EMF is εL = -L(dI/dt)Because the current is increasing dI/dt is positive.

RL CircuitsThe EMF across the inductor is negative which reflects the decrease in electric potential that occurs in going across the inductor.

RL CircuitsAfter the switch is closed there is a large back EMF that opposes current flow

EMF –L(dI/dt)

So not much current flows

I = εo/R

Using Kirchoff’s loop rule we find

εo –I/R - L(dI/dt)=0

RL CircuitsThe current does not increase instantly, but increases as an RC circuit does.It increases to its final equilibrium value when the switch is closed but instead increases according to an exponential function.

I(t) = (εo/R)(1-e-t/τ)Where τ = L/R

RL CircuitsOnce current is flowing it is hard to stop

Current decays

I = Io e-t/τ

I = (εo/R) e-t/τ

LC CircuitsIf the capacitor is initially charged and the switch is then closed, both the current in the circuit and the charge on the capacitor oscillate between maximum positive and negative values.

When the capacitor is fully charged, the energy U in the circuit is stored in the electric field of the capacitor and is

Qmax2 /2C

LC Circuits

LC CircuitsWhen the switch in the circuit is thrown then the capacitor discharges, this is providing a current in the circuit and the energy stored in the electric field of the capacitor now becomes stored in the magnetic field of the inductor

When the capacitor is fully discharged, it stores no energy. At this time the current reaches its maximum value and all the energy is stored in the inductor.

LC CircuitsThe current continues in the same direction, decreasing in magnitude, with the capacitor becoming fully charged again but with the polarity of its plates now opposite its initial polarity. This is followed by another discharge until the circuit returns to its original state of maximum charge Qmax.

LC CircuitsThe capacitor initially carries a charge Qo.

When the switch is closed:

-L dI/dt = Q/C

L (d2Q/dt2) + Q/C = 0

Where I = dQ/dt

Charge and Current in LC Circuits

I = ωQo

ω is the angular frequency

Q(t) = Qo cos ωt

I(t) = ωQo sin ωt

ω = 1/√(LC)

Current OscillationsThe current oscillates periodically and the stored energy is

U = ½ Q2/C + ½ LI2 = a constant

This is the total energy in an LC circuit