THE RLC CIRCUIT
THE LOSSLESS LC CIRCUIT
The RLC Circuit Step Response
In the circuit of Fig. 8.23, find i(t) and iR(t) for t > 0.
➢ For t < 0 the switch is open, and the circuit is partitioned into two
independent subcircuits
➢ The 4-A current flows through the inductor, so that i(0-t) = 4 A
➢ Since 30u(-t) = 30 V, and capacitor acts as an open circuit the initial
voltage across capacitor is v(0-) = 15 V
The RLC Circuit Step Response
➢ For t > 0 the switch is closed, and we have a parallel RLC circuit with a current
source
➢ The voltage source is zero which means it acts like a short-circuit
➢ The two resistors are now in parallel and can be combined to give;
R = 10 Ω, and α =
1
2×10×8×10−3
= 6.2 s-1 and 𝜔0 =
➢ That gives s1 = -11.978 s-1 and s2 = - 0.5218 s-1
➢ Since α > 𝜔0 , we have the overdamped case
➢ Hence
1
20×8×10−3
= 2.5 rad/s,
4
The Lossless LC Circuit
➢ When we considered the source-free RLC circuit, it became
apparent that the resistor served to dissipate any initial energy
stored in the circuit
➢ At some point it might occur to us to ask: what would happen if
we could remove the resistor?
➢ If the value of the resistance in a parallel RLC circuit becomes
infinite, or zero in the case of a series RLC circuit, we have a
simple LC loop in which an oscillatory response can be
maintained forever!
➢ Let us look briefly at an example of such a circuit, and then
discuss another means of obtaining an identical response without
the need of supplying any inductance
5
The Lossless LC Circuit
➢ Consider the source-free circuit of Fig. 9.35, in which the large values
1
F are used so that the calculations will be simple
36
1
A and v(0) = 0 and find that α = 0 and 𝜔02 = 9 s−2, so
6
L = 4 H and C =
➢ We let i(0) = −
that ωd = 3 rad/s
➢ In the absence of exponential damping, the voltage v is simply
v = A cos 3t + B sin 3t
➢ Since v(0) = 0, we see that A = 0
➢ Next,
➢ But i(0) = −
1
6
𝑑𝑣
|
𝑑𝑡 𝑡=0
= 3𝐵 =
𝑖(0)
1/36
A, and
𝑑𝑣
| =
𝑑𝑡 𝑡=0
6 V/s,
therefore 𝐵 = 2 V and
v(t) =2 sin 3t V
➢ Which is an undamped sinusoidal response; in other words, our
voltage response does not decay
6
The Lossless LC Circuit Ex 9.11
Determine i(t) for t > 0 for the circuit shown.
➢ For t < 0, we see that the capacitor will initially charge to a
voltage of v(0) = 9 V, and the inductor will have a current i(0) = 0
➢ For t > 0, the capacitor disconnects from the voltage source in a
charged state and connects instantaneously to the inductor
7
The Lossless LC Circuit Ex 9.11 Solution
➢ For the LC circuit configuration, we find that
α = 0, and
1
𝜔0 =
40
3×10−3 × 3 ×10−6
= 5000 rad/s,
ωd = 5000 rad/s,
so that
i = A cos (5000t) + B sin (5000t)
➢ Since i(0) = 0, A = 0
➢ Applying initial conditions to the derivative of current at t = 0+
thus
𝑑𝑖
| +
𝑑𝑡 𝑡=0
➢ therefore 𝐵 =
= 5000𝐵 =
3000
5000
𝑣(0+ )
𝐿
= 0.6 A and
i(t) = 0.6 sin(5000t) A
=
9
3×10−3
= 3000 A/s