📘 AC Circuits – Phasor Analysis & Impedance (Complete Theory + Worked Examples)
🔹 1. Introduction to AC Circuits
In AC circuits, voltages and currents vary sinusoidally with time. Unlike DC circuits, resistance alone does not define opposition to current. We introduce a new concept called Impedance (Z).
🔹 2. Sinusoidal Signal
v(t) = Vm sin(ωt + φ)
- Vm → Maximum value
- ω = 2Ï€f → Angular frequency
- φ → Phase angle
🔹 3. Phasor Representation
Phasor converts time-domain sinusoid into frequency-domain complex form.
Vm∠φ
Example:10 sin(ωt + 30°) → 10∠30°
🔹 4. Impedance (Z)
- Resistor → Z = R
- Inductor → Z = jωL
- Capacitor → Z = 1 / (jωC)
🔹 5. Worked Example 1 – Pure RLC
Given: R = 4Ω L = 0.1H C = 100µF f = 50Hz
Step 1: Find ω
ω = 2Ï€f = 2Ï€×50 = 314 rad/s
Step 2: Inductive Reactance
XL = ωL = 314×0.1 = 31.4Ω
Step 3: Capacitive Reactance
XC = 1/(ωC) = 1/(314×100×10⁻⁶) = 31.8Ω
Step 4: Total Impedance
Z = R + j(XL − XC) Z = 4 + j(-0.4)
Magnitude:
|Z| = √(4² + 0.4²) |Z| ≈ 4.02Ω
🔹 6. Worked Example 2 – Current Calculation
If voltage applied = 100∠0° V, Find current.
I = V/Z I = 100 / 4.02 I = 24.88A
🔹 7. Power in AC Circuit
Real Power:
P = VI cosφ
Reactive Power:
Q = VI sinφ
Apparent Power:
S = VI
🎯 Exam Insights
- Impedance calculation is common
- Resonance condition frequently asked
- Power factor questions are very important
- Phasor conversion errors are common traps
AC Circuits Form the Backbone of Power Systems & Control
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