Download RLC Circuits

RL and LC Circuits
Capacitor and Inductors in Series
Resistors and Inductors in Series
RL Circuits
As 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 Circuits
The EMF across the inductor is negative
which reflects the decrease in electric
potential that occurs in going across the
inductor.
RL Circuits
After 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 Circuits
The 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 Circuits
Once current is flowing it is hard to stop
Current decays
I = Io e-t/τ
I = (εo/R) e-t/τ
LC Circuits
If 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 Circuits
When 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 Circuits
The 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 Circuits
The 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 Oscillations
The 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