1. Complex Numbers for AC Circuits 📊

Essential for phasor analysis and impedance calculations!

Rectangular Form

Z = R + jX

R = Real (resistance)

X = Imaginary (reactance)

j = √(-1) (engineers use j, mathematicians use i)

Polar Form

Z = |Z|∠θ

|Z| = √(R² + X²) (magnitude)

θ = tan⁻¹(X/R) (angle)

Euler: Z = |Z|e^(jθ)

Conversions

Rectangular → Polar: |Z| = √(R² + X²), θ = tan⁻¹(X/R)

Polar → Rectangular: R = |Z|cos(θ), X = |Z|sin(θ)

2. Differential Equations 📈

Crucial for transient analysis in circuits!

First Order (RC/RL Circuits)

τ(dx/dt) + x = x_final

Solution: x(t) = x_final + (x_initial - x_final)e^(-t/τ)

RC circuit: τ = RC
RL circuit: τ = L/R

Second Order (RLC Circuits)

LC(d²x/dt²) + RC(dx/dt) + x = 0

Overdamped: α > ω₀ (two real roots)
Critically damped: α = ω₀ (repeated root)
Underdamped: α < ω₀ (oscillatory)

3. Laplace Transforms 🔄

f(t)	F(s)	Application
1 (unit step)	1/s	DC source
e^(-at)	1/(s+a)	Exponential decay
sin(ωt)	ω/(s²+ω²)	AC source
cos(ωt)	s/(s²+ω²)	AC source
t·e^(-at)	1/(s+a)²	Critical damping

4. Fourier Analysis 🌊

Decompose periodic signals into sinusoidal components

f(t) = a₀ + Σ[aₙcos(nωt) + bₙsin(nωt)]

Coefficients:

a₀ = (1/T)∫f(t)dt (DC component)

aₙ = (2/T)∫f(t)cos(nωt)dt

bₙ = (2/T)∫f(t)sin(nωt)dt

Applications:

Signal spectrum analysis
Filter design
Harmonic distortion

5. Matrices & Linear Algebra 📐

Circuit Applications

Nodal analysis: [Y][V] = [I]
Mesh analysis: [Z][I] = [V]
Two-port networks: [ABCD] parameters

Key Operations

Determinant (Cramer's rule)
Matrix inversion
Eigenvalues (stability analysis)

6. Practice Questions 📚

Common Board Exam Questions

Q1: Convert 3 + j4 to polar form.

A: |Z| = √(9+16) = 5, θ = tan⁻¹(4/3) = 53.13°. Answer: 5∠53.13°

Q2: An RC circuit has R=1kΩ, C=1μF. Find the time constant.

A: τ = RC = 1000 × 10⁻⁶ = 0.001s = 1ms

Q3: Find L{e⁻³ᵗ·sin(4t)}.

A: Using s-shift: 4/[(s+3)² + 16] = 4/(s² + 6s + 25)

🔥 ECE Challenge 🔥

Master the math! Complex numbers, Laplace, Fourier - the tools for analyzing any circuit!

Math is the language of electronics!

Mathematics