AC Circuit Analysis

🔌 AC Circuit Analysis: The Dance of Electricity

Imagine electricity as water flowing through pipes. Now imagine that water sloshing back and forth, like waves in a bathtub. That’s AC—Alternating Current!

🌊 The Big Picture

In AC circuits, electricity doesn’t just flow one way—it swings back and forth, like a swing in a playground. Special components called resistors ®, inductors (L), and capacitors © each react differently to this swinging electricity.

Think of it like a dance party:

• Resistors = The steady dancers who keep the same rhythm
• Inductors = The heavy dancers who take time to start moving
• Capacitors = The eager dancers who rush ahead of the music

🧲 LR Circuit: The Heavy Starter

What is it?

An LR circuit has a resistor ® and an inductor (L) connected together.

The Story

Imagine pushing a heavy shopping cart. When you first push, it’s hard to get moving. But once it’s rolling, it wants to keep going. That’s an inductor!

The inductor opposes sudden changes in current. It’s like a lazy giant—slow to wake up, slow to stop.

Key Formula

Time Constant: τ = L/R

This tells you how fast the circuit “wakes up.”

Simple Example

• L = 2 Henrys, R = 4 Ohms
• τ = 2/4 = 0.5 seconds
• After 0.5 seconds, current reaches about 63% of its final value

graph TD
    A["Battery ON"] --> B["Current starts slow"]
    B --> C["Inductor resists change"]
    C --> D["Current builds up"]
    D --> E["Full current after 5τ"]

⚡ RC Circuit: The Eager Charger

What is it?

An RC circuit has a resistor ® and a capacitor © connected together.

The Story

Imagine filling a water balloon. At first, water rushes in fast! But as the balloon fills up, it gets harder to push more water in. That’s a capacitor charging!

RC Time Constant

Time Constant: τ = R × C

This tells you how fast the capacitor fills up (or empties).

Simple Example

• R = 1000 Ohms (1 kΩ)
• C = 0.001 Farads (1 mF)
• τ = 1000 × 0.001 = 1 second

After 1 second, the capacitor is 63% charged. After 5 seconds (5τ), it’s basically full (99%).

Why This Matters

• Camera flash = Capacitor charges slowly, releases fast
• Touch screens = RC circuits detect your finger
• Audio filters = RC circuits shape sound

graph TD
    A["Start Charging"] --> B["Fast at first"]
    B --> C["Slows down"]
    C --> D["Nearly full at 5τ"]
    D --> E["Fully charged"]

🔄 LC Oscillations: The Energy Swing

What is it?

An LC circuit has only an inductor (L) and capacitor ©—no resistor!

The Story

Imagine a swing. You pull it back (store energy), let go, it swings forward (releases energy), swings back again. This keeps going back and forth!

In an LC circuit:

1. Capacitor stores energy in its electric field (like pulling the swing back)
2. Energy transfers to inductor (swing moves forward)
3. Inductor stores energy in its magnetic field (swing at other side)
4. Energy goes back to capacitor (swing returns)

The Magic Formula

Oscillation Frequency: f = 1/(2π√LC)

Simple Example

• L = 1 Henry, C = 1 Farad
• f = 1/(2π√1) = 0.159 Hz
• The circuit oscillates about once every 6 seconds

Real Life

• Radio tuning = LC circuits pick specific stations
• Metal detectors = LC oscillations detect metal

🎸 LCR Series Circuit: The Full Band

What is it?

An LCR circuit has all three: Inductor + Capacitor + Resistor in a line.

The Story

Now we have the full band playing:

• Inductor (L) = The bass player (heavy, slow response)
• Capacitor © = The lead guitarist (quick, eager response)
• Resistor ® = The drummer (keeps things steady, uses up energy)

Impedance: The Total Resistance

In AC circuits, we call the total “resistance” as impedance (Z).

Z = √(R² + (XL - XC)²)

Where:

• XL = 2πfL (inductor’s opposition)
• XC = 1/(2πfC) (capacitor’s opposition)

Simple Example

At 60 Hz with R = 10Ω, L = 0.1H, C = 100µF:

• XL = 2π × 60 × 0.1 = 37.7Ω
• XC = 1/(2π × 60 × 0.0001) = 26.5Ω
• Z = √(100 + (37.7-26.5)²) = √(100 + 125) = 15Ω

🎯 LCR Resonance: The Sweet Spot

What is it?

Resonance happens when XL = XC. The inductor and capacitor cancel each other out!

The Story

Remember pushing someone on a swing? If you push at exactly the right moment (right frequency), tiny pushes create huge swings. That’s resonance!

Resonant Frequency

f₀ = 1/(2π√LC)

At this special frequency:

• Impedance is minimum (just R)
• Current is maximum
• The circuit “likes” this frequency best

Simple Example

• L = 10 mH, C = 100 nF
• f₀ = 1/(2π√(0.01 × 0.0000001))
• f₀ = 5,033 Hz ≈ 5 kHz

At exactly 5 kHz, this circuit lets the most current through!

graph TD
    A["LCR Circuit"] --> B{Frequency?}
    B -->|Below f₀| C["XC > XL"]
    B -->|At f₀| D["XL = XC ✨"]
    B -->|Above f₀| E["XL > XC"]
    D --> F["Maximum Current!"]

⭐ Quality Factor: How Sharp is the Peak?

What is it?

The Quality Factor (Q) tells you how “picky” a circuit is about its resonant frequency.

The Story

Imagine two singers:

• Low Q singer: Can hit many notes (wide range, not picky)
• High Q singer: Only hits one perfect note (very picky, sharp)

The Formula

Q = f₀/Bandwidth = (1/R)√(L/C)

What Q Means

• High Q (100+): Very sharp peak, very selective
• Low Q (< 10): Broad peak, accepts many frequencies

Simple Example

• f₀ = 1000 Hz
• Bandwidth = 100 Hz
• Q = 1000/100 = 10

This circuit is moderately selective.

📊 Bandwidth and Selectivity

What is Bandwidth?

Bandwidth = The range of frequencies the circuit responds to well.

Bandwidth = f₀/Q

The Story

Think of a radio station. Bandwidth is like how “wide” the channel is:

• Narrow bandwidth = Only that exact station, no neighbors
• Wide bandwidth = Picks up nearby stations too (interference!)

Selectivity

Selectivity = How well a circuit picks ONE frequency and rejects others.

• High Q = High Selectivity = Sharp tuning
• Low Q = Low Selectivity = Broad tuning

Simple Example

Radio tuner circuit:

• f₀ = 100 MHz (your favorite station)
• Q = 100
• Bandwidth = 100/100 = 1 MHz

This means frequencies from 99.5 MHz to 100.5 MHz get through.

⏱️ RC Time Constant: The Speed Dial

Deep Dive

We mentioned τ = RC earlier. Let’s understand it better!

The Story

The time constant is like a speed setting:

• Small τ = Fast charging/discharging (quick camera flash)
• Large τ = Slow charging/discharging (gradual fade)

Charging Curve

After each time constant:

Practical Example

Nightlight with photoresistor:

• In dark: R = 1 MΩ, C = 10 µF
• τ = 1,000,000 × 0.00001 = 10 seconds
• Takes about 50 seconds to fully respond to darkness

🎓 Quick Summary

*In real life, some energy is always lost!

🚀 Why This Matters

These circuits are everywhere:

• 📻 Radio = LC resonance picks stations
• 📱 Phone charger = Smooth power delivery
• 🔊 Speakers = Filter circuits shape sound
• 🏥 MRI machines = Precise frequency control
• ⚡ Power grid = AC transmission everywhere

You now understand how electricity dances! 💃🕺

“Electricity is really just organized lightning.” — Now you know how to organize it with LCR circuits!