Physics Lesson 16.15.2 - Equation of the Damped Oscillations in a RLC Circuit

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Welcome to our Physics lesson on Equation of the Damped Oscillations in a RLC Circuit, this is the second lesson of our suite of physics lessons covering the topic of Introduction to RLC Circuits, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Equation of the Damped Oscillations in a RLC Circuit

In the previous tutorial, we have explained that the total electromagnetic energy in a LC circuit is

W_tot = W_M + W_e = L ∙ i²/2 + Q²/2C

When a resistor is added in this circuit, the total electromagnetic energy will decrease at a rate of

dW_tot/dt = -i² ∙ R

because some of the energy in the system turns into thermal energy of resistor and is dissipated in the environment in the form of heat energy. The negative sign in the equation means that the total energy of the system decreases.

Differentiating the above equation with time, we obtain

dW_tot/dt = L ∙ i ∙ di/dt + Q/C ∙ dQ/dt = -i² ∙ R

Since i = dQ/dt, and di/dt = d² Q/dt² we obtain

L ∙ dQ/dt ∙ d² Q/dt² + Q/C ∙ dQ/dt = -R ∙ (dQ/dt)²
L ∙ dQ/dt ∙ d² Q/dt² + R ∙ (dQ/dt)² + Q/C ∙ dQ/dt = 0

Simplifying both sides of the last equation by dQ/dt, we obtain

L ∙ d² Q/dt² + R ∙ dQ/dt + 1/C ∙ Q = 0

This is the differential equation for the damped oscillations in a RLC circuit.

The charge decay in such a circuit is calculated through an expression, which is a combination of exponential and sinusoidal equation, as occurs in all types of damped oscillations. Thus, the charge left in a RLC circuit after a given time t of operation is found by:

Q(t) = Q₀ ∙ e^{-R ∙ t/2L} ∙ cos⁡(ω' ∙ t + φ)

where

ω' = √ω² - (R/2L)²

is the angular frequency of damped oscillations and

ω = 1/√L ∙ C

is the angular frequency of undamped oscillations.

In addition, you can see that the amplitude also contains an exponential decaying term e^{-R ∙ t/2L}. This means the amplitude of every successive oscillation is smaller than the previous one as the power of Euler's Number e here is negative.

Example 1

The potential difference between the plates of a 5μF capacitor connected in an alternating 50Hz RLC circuit shown in the figure is 20V and the resistance of resistor in the circuit has a value of 0.25Ω.

Physics Tutorials: This image provides visual information for the physics tutorial Introduction to RLC Circuits

Calculate:

The initial charge stored in the capacitor
The charge stored in the circuit 0.4 s after the switch turns on

Take the initial phase as zero.

Solution 1

The initial charge stored in the capacitor is
Q₀ = C ∙ ∆V
= (5 × 10^-6 F) ∙ (20V)
= 10^-4 C
First we must find the inductance and then calculate the charge stored in the circuit at the given time. Thus, since the circuit has a frequency of 50Hz, we have
ω = 2π ∙ f
= 2π ∙ 50Hz
= 100π rad/s
Hence, giving that
ω = 1/√L ∙ C
or
ω² = 1/LC
we obtain for the inductance L of the inductor:
L = 1/ω² ∙ C
= 1/(100π)² ∙ (10^-4 )
= 1/10⁴ ∙ π² ∙ 10^-4
= 1/π²
= 1/(3.14)²
= 0.1 H
The charge stored in the capacitor after 20s therefore is
Q(t) = Q₀ ∙ e^{-R ∙ t/2L} ∙ cos⁡(ω' ∙ t + φ)
Q(t) = Q₀ ∙ e^{-R ∙ t/2L} ∙ cos√ω² - (R/2L)² ∙ t
Q(20) = (10^-4 ) ∙ e^{-0.25 ∙ 0.4/2 ∙ 0.1} ∙ cos√(100π)² - (0.25/2 ∙ 0.1)² ∙ 0.4
Q(20) = (10^-4 ) ∙ e^-5 ∙ cos(√(100π)²-(1.2)² ∙ 0.4)
= (10^-4 ) ∙ e^-5 ∙ cos(100π ∙ 0.4)
= (10^-4 ) ∙ e^-5 ∙ cos(40π)
= (10^-4 ) ∙ e^-5 ∙ 1
= 10^-4 ∙ 6.7 × 10^-2 = 6.7 × 10^-6 C

From the above results, we draw the following conclusions:

For small values of resistance, ω' ≈ ω, so we can neglect the (R/2L)2 factor and focus only on the value of ω.

In absence of an external source, the charge in the circuits drops very fast (in only 0.4 s it has dropped about 15 times, from 10^-4C to 6.7 × 10^-6 C). This means that if the emf source is not active, the RLC circuit discharges almost immediately due to the electromagnetic-to-heat energy conversion, which takes place in the resistor.

We can use a similar approach when calculating the energy decay in a RLC circuit operating without an external energy supply. For this, we observe what happens to the electric energy in the capacitor. Since

W_e = Q²/2C

we can write this expression as a function of time, i.e.

W_e (t) = [Q(t)]²/2C
= [Q₀ ∙ e^{-R ∙ t/2L} ∙ cos⁡(ω' ∙ t + φ) ]²/2C
= Q²₀/2C ∙ e^{- R ∙ t/L} ∙ cos²⁡(ω' ∙ t + φ)

This means the energy of the electric field in a RLC circuit oscillates in a cos2 fashion while the amplitude decreases exponentially with time.

Example 2

A 50Hz RLC circuit contains a 10Ω resistor, a 0.4H inductor and a 2nF capacitor connected in series. The capacitor initially stores a charge of 5μC. Calculate:

The initial electric energy stored in the capacitor plates
The energy left in the capacitor plates 2 s after the switch turns OFF.

Take the initial phase equal to zero.

Solution 2

Clues:

R = 10 Ω
L = 0.4 H
C = 2nF = 2 × 10^-9 F
Q₀ = 5μC = 5 × 10^-6 C
f = 50 Hz
a) W_{e initial} = ?
b) W(2) = ?

The initial electric energy stored in the capacitor plates is
W_{e initial} = Q²₀/2C
= (5 × 10^-6 C)²/2 ∙ (2 × 10^-9 F)
= 6.25 × 10^-3 J
The energy left in the capacitor plates after 2 s of supply interruption is
W_e (t) = Q²₀/2C ∙ e^{- R ∙ t/L} ∙ cos²⁡(ω' ∙ t + φ)
W_e (t) = Q²₀/2C ∙ e^{- R ∙ t/L} ∙ cos²⁡(√ω²-(R/2L)² ∙ t)
W_e (2) = (6.25 × 10^-3 J) ∙ e^{-10 ∙ 2/0.4} ∙ cos²√(2 ∙ π ∙ 50)²-(10/2 ∙ 0.4)² ∙ 2
= (6.25 × 10^-3 J) ∙ e^-50 ∙ cos²(√98596 - 156.25 ∙ 2)
= (6.25 × 10^-3 J) ∙ e^-50 ∙ cos²(√98439.75 ∙ 2)
= (6.25 × 10^-3 J) ∙ e^-50 ∙ cos²(627.5 rad)
= (6.25 × 10^-3 J) ∙ (1.93 × 10^-22 ) ∙ (0.6833)²
= 5.63 × 10^-25 J
This value is very small, close to zero. Hence, we say the flow of energy through a RLC circuit, practically stops immediately after the switch turns off. Again, here you can see how necessary a sustainable external source such as a battery or an AC power supply is, in order to maintain constant the electricity flow through a RLC circuit.

You have reached the end of Physics lesson 16.15.2 Equation of the Damped Oscillations in a RLC Circuit. There are 4 lessons in this physics tutorial covering Introduction to RLC Circuits, you can access all the lessons from this tutorial below.

More Introduction to RLC Circuits Lessons and Learning Resources

Magnetism Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
16.15	Introduction to RLC Circuits
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
16.15.1	What is a RLC Circuit? Damped Oscillations in a RLC Circuit
16.15.2	Equation of the Damped Oscillations in a RLC Circuit
16.15.3	Forced Oscillations. Alternating Current and Emf in a RLC Circuit caused by Forced Oscillations
16.15.4	Resistive, Inductive and Capacitive Load

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