LC Circuits

Introduction to LC Circuits

An LC circuit consists of a capacitor (with initial charge ), an inductor, and a switch. When the switch is closed after being open for a long time, the circuit exhibits oscillatory behavior analogous to a mass-spring system.

• Capacitor: Stores electric charge and energy.

• Inductor: Stores energy in its magnetic field.

• Oscillation: The charge and current oscillate sinusoidally.

Stepwise Behavior of LC Circuits

1. Step A: Capacitor is fully charged (), current . Analogous to a fully stretched spring (velocity ).

2. Step B: Capacitor discharges, current reaches maximum (, ). Analogous to maximum speed of a block on a spring.

3. Step C: Capacitor is recharged with opposite polarity (, ). Analogous to spring compressed in the opposite direction.

4. Step D: Current continues until the initial charge is restored (, ).

Mathematical Description

• Charge on the capacitor:

• Current through the inductor:

• Angular frequency:

Applications of LC Circuits

• Used in radio transmitters and receivers to establish transmission frequency.

• Example: Cell phones use high-frequency LC circuits (e.g., 1000 MHz).

Example Calculation

Given an inductor mH, to achieve a frequency kHz:

• Use to solve for .

LR Circuits

Introduction to LR Circuits

An LR circuit consists of an inductor and a resistor in series, driven by an external battery. When the switch is closed, the current decays exponentially.

• Time constant:

• Exponential decay:

Graphical Representation

• At , has decreased to 37% of its initial value.

• At , has decreased to 13% of its initial value.

QuickCheck Examples

• Immediately after closing the switch, the current is determined by the inductor's opposition to change.

• After a long time, the current is determined by Ohm's law: .

General Principles of Electromagnetic Induction

Lenz's Law

Lenz's Law states that an induced current in a closed loop occurs only if the magnetic flux through the loop is changing. The direction of the induced current creates a magnetic field that opposes the change in flux.

Faraday's Law

• An emf is induced around a closed loop if the magnetic flux changes.

• Magnitude:

• Direction: As given by Lenz's law.

Using Electromagnetic Induction

• Model: Make simplifying assumptions.

• Visualize: Use Lenz's law for direction.

• Solve: (multiply by for -turn coil)

• Induced current:

Important Concepts

Magnetic Flux

Magnetic flux () measures the amount of magnetic field passing through a surface:

Changing Magnetic Flux

• Moving a loop into/out of a magnetic field

• Changing the loop's area or orientation

• Changing the magnetic field strength through the loop

Creating Induced Current

• Motional emf: (due to moving charge carriers in a magnetic field)

• Induced electric field: (due to changing magnetic field)

Applications

Inductors

• Solenoid inductance:

• Potential difference:

• Energy stored:

• Magnetic energy density:

LC and LR Circuit Summary Table

Additional info:

• LC circuits are foundational in radio, communications, and signal processing.

• LR circuits are important in transient analysis and filtering applications.