Lectures on Circuits from the Arthur and Mei Mei School YouTube Channel

• Voltage Divider Circuit
• Passive RC Filter Circuits
• LC Circuit

Voltage Divider Circuit

A voltage divider circuit consists of two resistors (R1 and R2) connected in series between the input voltage and ground, with the output taken between the resistors.

Key Points

• The circuit follows Ohm's Law, which establishes a ratio between the output and input voltage
• When tested with different waveforms, each with 4 volts peak-to-peak input:
  • Sine wave: 2 volts peak-to-peak output
  • Square wave: 2 volts peak-to-peak output
  • Saw tooth wave: 2 volts peak-to-peak output

Conclusion

The voltage divider consistently reduces the amplitude by half while preserving the shape of the waveform, regardless of the input waveform type.

Passive RC Filter Circuits

Resistive Voltage Divider

• A voltage divider circuit consists of two resistors (R1 and R2) in series between input and ground
• The output is taken across R2 (between R2 and ground)
• Using Ohm's law, the output voltage is proportional to the input voltage by the ratio of resistances
• In the demonstration, a 4V peak-to-peak input sine wave produced a 2V peak-to-peak output
• With resistors, various input waveforms (sine, square, sawtooth) maintain their shape but have reduced amplitude
• The circuit reduces amplitude while preserving waveform shape

Capacitive Voltage Divider (RC Circuit)

• When replacing a resistor with a capacitor, the behavior changes
• A capacitor's resistance (impedance) varies based on its charge state
• The impedance of a capacitor with a sine wave input can be calculated using complex notation:
  • $Z=\frac{1}{i\omega C}$ where:
    • i is the imaginary unit
    • ω (omega) is the angular frequency
    • C is the capacitance
  • The units of impedance are ohms

RC Circuit Frequency Response

• With a 5V, 600Hz square wave input, the output resembles "shark fins"
• At lower frequencies (200Hz), the output nearly resembles a square wave as the capacitor has time to charge/discharge
• At higher frequencies (17,000Hz), the output takes on a sawtooth appearance
• Unlike the resistive divider, the RC circuit changes both the amplitude and shape of the waveform depending on the frequency

The circuit demonstrates fundamental principles of electrical engineering including voltage division, impedance, and frequency response.

LC Circuit

Basic Components

• Inductor: A coil of wire that induces a magnetic field when current flows through it
• Capacitor: A component used to store electric charge

Key Physical Principles

• Faraday's Law: A changing magnetic field causes current
  • Demonstrated with a magnet and meter: when the magnet moves, current is detected
  • When stationary, no current flows

Mathematical Analysis

• The general solution for charge (q) in an LC circuit is:
  • $q=A\cos (\omega t)+B\sin (\omega t)$
  • Alternatively: $q=C\sin (\omega t+\varphi )$
  • All solutions are wave-like, where:
    • Changing phase moves the wave left / right
    • Changing amplitude makes the wave bigger / smaller

Differential Equations

• The differential equation for an LC circuit: $\ddot{q}+q=0$
• This shows both sine and cosine functions are solutions
• Similar to the equation governing a spring system

Energy and Dampening

• In a real circuit, oscillations decay over time
• Energy is not perfectly conserved due to resistance
• All components and wires have some resistance
• Current flow produces heat, leaking energy to the environment

LRC Circuit

• The governing equation: $L\ddot{q}+R\dot{q}+\frac{q}{C}=0$
• Similar to a bouncing ball that loses energy with each bounce

Thermodynamics Connection

• No system is perfectly insulated
• All systems leak energy (Second Law of Thermodynamics)

How the Circuit Works

1. When a capacitor discharges, current in the inductor creates a magnetic field (Ampere's Law)
2. When fully discharged, no current flows and the magnetic field collapses
3. The collapsing field causes a back current (Faraday's Law)
4. This recharges the capacitor in the opposite direction
5. This cycle continues, creating oscillations