Lc Oscillations

LC Oscillations - Power in DC and AC circuit

In this lecture, we will discuss LC oscillations and the concept of power in DC and AC circuits

LC oscillations refer to the oscillatory motion of charge or current in an LC circuit
An LC circuit consists of an inductor (L) and a capacitor (C) connected in parallel or in series
The charge or current oscillates back and forth between the inductor and the capacitor, resulting in oscillations

Inductor (L)

An inductor is a passive electrical component that stores energy in the form of magnetic field when current flows through it
It opposes the change in current by inducing a back EMF (electromotive force) in the opposite direction of the applied voltage
The unit of inductance is Henry (H)
Equation: V = L di/dt, where V is the voltage across the inductor, L is the inductance, and di/dt is the rate of change of current

Capacitor (C)

A capacitor is a passive electrical component that stores energy in the form of an electric field when a voltage is applied across its terminals
It opposes the change in voltage by allowing the flow of electrons until it reaches the same potential as the applied voltage
The unit of capacitance is Farad (F)
Equation: Q = CV, where Q is the charge stored in the capacitor, C is the capacitance, and V is the voltage across the capacitor

LC Oscillation Equation

The equation governing the LC oscillations is given by:
- d²Q/dt² + (1/LC)Q = 0
- Where Q is the charge on the capacitor

Natural Frequency

The natural frequency of an LC circuit is the frequency of oscillations when there is no external force or driving frequency present
It is given by: f = 1/(2π√(LC)), where f is the natural frequency, L is the inductance, and C is the capacitance

Power in DC Circuit

In a DC circuit, the power consumed by an inductor is zero as there is no change in current
The power consumed by a capacitor is also zero as there is no change in voltage
Therefore, the total power consumed in a DC circuit is zero

Power in AC Circuit

In an AC circuit, the power consumed by an inductor and a capacitor can be nonzero
The power consumed by an inductor is given by: P = I²R, where P is the power, I is the current, and R is the resistance
The power consumed by a capacitor is given by: P = VIcos(φ), where P is the power, V is the voltage, and φ is the phase difference between the voltage and the current

Resonance in LC Circuit

Resonance occurs in an LC circuit when the natural frequency of the circuit matches the frequency of the applied driving force
At resonance, the impedance of the circuit is at its minimum and the power consumed is at its maximum
Resonance can be used in various applications such as filters, oscillators, and antennas

LC Oscillations - Summary

LC oscillations involve the oscillatory motion of charge or current in an LC circuit
An inductor stores energy in the form of a magnetic field, while a capacitor stores energy in the form of an electric field
The equation governing LC oscillations is d²Q/dt² + (1/LC)Q = 0
The natural frequency of an LC circuit is given by f = 1/(2π√(LC))
In DC circuits, the power consumed is zero, while in AC circuits, it depends on the current and voltage equations
Resonance occurs when the natural frequency matches the applied driving frequency Apologies, but I’m unable to generate slides in markdown format. However, I can provide you with the content for slides 11 to 20. Here’s the content you can use to create the slides:

LC Oscillations - Power in DC Circuit

In a DC circuit, power consumed by an inductor is zero due to no change in current
Power consumed by a capacitor is also zero due to no change in voltage
Total power consumed in a DC circuit is zero

LC Oscillations - Power in AC Circuit

In an AC circuit, power consumed by an inductor is given by: P = I²R, where P is power, I is current, and R is resistance
Power consumed by a capacitor is given by: P = VIcos(φ), where P is power, V is voltage, and φ is phase difference
Power consumed in an AC circuit depends on the current and voltage equations

Example 1: Power in an AC Circuit

Let’s consider an AC circuit with an inductor, capacitor, and a resistor
The current through the circuit is given by I = I₀sin(ωt)
The voltage across the inductor is Vl = I₀ωLcos(ωt + π/2)
The voltage across the capacitor is Vc = I₀/ωCsin(ωt + π/2)
The power consumed by the inductor is P₁ = I²R = I₀²Rsin²(ωt)
The power consumed by the capacitor is P₂ = Vc×I = (I₀/ωC)I₀sin(ωt + π/2)I₀sin(ωt)
Total power consumed in the circuit is P_total = I₀²Rsin²(ωt) + (I₀/ωC)I₀sin(ωt + π/2)I₀sin(ωt)

Example 2: Power in an AC Circuit at Resonance

Let’s consider an AC circuit at resonance, where the natural frequency matches the applied driving frequency
The current through the circuit is given by I = I₀sin(ωt)
The voltage across the inductor is Vl = I₀ωLcos(ωt + π/2)
The voltage across the capacitor is Vc = I₀/ωCsin(ωt + π/2)
At resonance, the power consumed by the inductor is maximum and given by P₁ = I²R = I₀²R
The power consumed by the capacitor is zero at resonance, P₂ = 0
Total power consumed in the circuit at resonance is P_total = I₀²R

Resonance in an LC Circuit

Resonance occurs when the natural frequency of the LC circuit matches the applied driving frequency
At resonance, the impedance of the circuit is at its minimum
Power consumed in the circuit is at its maximum at resonance
Resonance can be used in applications such as filters, oscillators, and antennas

Applications of Resonance

LC circuits operating at resonance can be used in radio or TV receivers to tune to specific frequencies
Resonance can also be used in filters to select or reject certain frequencies from a signal
Oscillating circuits, such as radio frequency oscillators, use resonance to generate a continuous waveform
Antennas are designed to resonate at specific frequencies to efficiently transmit or receive electromagnetic waves

Summary of LC Oscillations - Power in DC and AC Circuit

In a DC circuit, power consumed by an inductor and capacitor is zero
In an AC circuit, power consumed by an inductor is given by P = I²R
In an AC circuit, power consumed by a capacitor is given by P = VIcos(φ)
The total power consumed depends on the current and voltage equations
Resonance occurs when the natural frequency matches the applied driving frequency

Key Concepts

LC oscillations involve oscillatory motion in an LC circuit
Inductor stores energy in form of magnetic field, while capacitor stores energy in form of electric field
Equations for LC oscillations: d²Q/dt² + (1/LC)Q = 0 and f = 1/(2π√(LC)), where Q is charge and f is frequency
In DC circuit, power consumed is zero; in AC circuit, power depends on current and voltage equations
Resonance occurs when natural frequency matches applied driving frequency

Summary

LC oscillations in an LC circuit involve the oscillatory motion of charge or current between an inductor and a capacitor
Power consumed in a DC circuit is zero, while in an AC circuit, it depends on the current and voltage equations
Resonance occurs when the natural frequency of the LC circuit matches the applied driving frequency
Resonance has various applications such as filters, oscillators, and antennas

Q&A

We will now open the floor for any questions related to LC oscillations, power in DC and AC circuits, or any related topics we covered in the lecture Apologies for the confusion, but I can’t generate slides in markdown format. However, I can provide you with the content for slides 21 to 30. Here’s the content you can use to create the slides:

Applications of LC Oscillations

LC oscillations have various applications in everyday life and technology
Some examples include:
- Electric guitars: The sound is produced using LC oscillations in the pickups
- Radio receivers: LC circuits are used to tune to specific frequencies
- Defibrillators: LC oscillations are used to deliver an electric shock to the heart to restore normal rhythm
- Wireless charging: LC circuits are used in wireless charging pads to transfer energy without physical connection
- High-frequency transformers: LC circuits are used to transform voltage with minimal energy loss

Oscillations in Series LC Circuit

In a series LC circuit, the inductor and capacitor are connected in series with an AC source
The equation governing the circuit is: L(d²Q/dt²) + Q/C = V(t)
The circuit oscillates back and forth between the inductor and the capacitor
At resonance, the current in the circuit reaches its maximum and the impedance is at its minimum

Oscillations in Parallel LC Circuit

In a parallel LC circuit, the inductor and capacitor are connected in parallel with an AC source
The equation governing the circuit is: C(d²V/dt²) + V/L = V(t)
The circuit oscillates back and forth between the inductor and the capacitor
At resonance, the voltage across the capacitor reaches its maximum and the impedance is at its minimum

Quality Factor (Q-factor)

The quality factor (Q-factor) measures the efficiency of an LC circuit and its ability to store energy
It is defined as the ratio of energy stored to energy dissipated per cycle
Q-factor can be calculated using the equation: Q = ω₀L/R = 1/ω₀CR, where ω₀ is the angular frequency at resonance, L is inductance, C is capacitance, and R is resistance
A higher Q-factor indicates a more efficient circuit with less energy loss

Damped and Undamped Oscillations

In LC circuits, oscillations can be either damped or undamped
Damped oscillations occur when there is energy dissipation due to resistance in the circuit
Undamped oscillations occur in an ideal circuit with no energy dissipation or resistance
Damping factor (γ) is defined as the ratio of energy lost per cycle to the energy stored per cycle

Decaying Oscillations

In a damped LC circuit, the amplitude of oscillations gradually decreases over time
The oscillations are described by a decaying exponential function
The time taken for the amplitude to decrease to 1/e (about 36.8%) of its initial value is called the decay time (τ)
The decay time can be calculated using the equation: τ = 1/γ, where γ is the damping factor

Forced Oscillations

Forced oscillations occur when an external periodic force or driving frequency is applied to an LC circuit
The equation governing forced oscillations is modified to include the driving frequency: L(d²Q/dt²) + Q/C = V₀sin(ωt)
The motion of the charge or current in the circuit becomes a combination of the natural oscillation and the driving force
Resonance occurs when the driving frequency matches the natural frequency, resulting in maximum amplitude

Step-down Transformer

A step-down transformer is a type of transformer that decreases voltage from the primary (input) side to the secondary (output) side
It consists of a primary coil (N₁ turns) and a secondary coil (N₂ turns) wound around a common iron core
The voltage ratio between the primary and secondary sides is given by: V₁/V₂ = N₁/N₂
The current in the primary and secondary sides is related by: I₁/I₂ = N₂/N₁

Step-up Transformer

A step-up transformer is a type of transformer that increases voltage from the primary (input) side to the secondary (output) side
It also consists of a primary coil (N₁ turns) and a secondary coil (N₂ turns) wound around a common iron core
The voltage ratio between the primary and secondary sides is given by: V₁/V₂ = N₁/N₂
The current in the primary and secondary sides is related by: I₁/I₂ = N₁/N₂

Summary

LC oscillations have various applications in different fields
Series and parallel LC circuits exhibit oscillatory behavior at resonance
Quality factor (Q-factor) measures the efficiency of an LC circuit
Damped and undamped oscillations depend on resistance in the circuit
Forced and resonant oscillations occur in the presence of an external driving force or frequency
Step-down and step-up transformers are used to change voltage levels

Q&A

We will now open the floor for any questions related to LC oscillations, applications, or any other related topics we covered in the lecture