Alternating Current (AC) 

Focus: AC basics, RMS and mean values, reactance, impedance, LCR circuits, power, resonance, choke coil (JEE Main).

1) Alternating Current & Voltage

• Instantaneous current:

I = I₀ sin(ωt)
  

• Instantaneous voltage:

E = E₀ sin(ωt)
  

• Angular frequency: ω = 2πf
• Time period: T = 2π / ω
• Frequency of AC in India = 50 Hz

2) Mean Value of AC

• Mean value over a full cycle is zero.
• Mean value over half cycle:

Imean = (2I₀) / π
  

3) RMS (Virtual) Value

• RMS value is the DC equivalent producing same heating effect.
• Measured by AC ammeter and voltmeter.

Irms = I₀ / √2
Vrms = V₀ / √2
  

Form factor:

Form factor = Irms / Imean = 1.11
  

4) AC Through Pure Resistor (R)

• Current and voltage are in phase.

I = (E₀ / R) sin(ωt)
Irms = Vrms / R
  

5) AC Through Pure Inductor (L)

• Inductor opposes change in current.
• Current lags voltage by π/2.

XL = ωL
I = (E₀ / XL) sin(ωt − π/2)
Irms = Vrms / (ωL)
  

6) AC Through Pure Capacitor (C)

• Current leads voltage by π/2.

XC = 1 / (ωC)
I = (E₀ / XC) sin(ωt + π/2)
Irms = Vrms / XC
  

7) AC Through RL Circuit

Z = √(R² + XL²)
tanφ = XL / R
I = (E₀ / Z) sin(ωt − φ)
  

8) AC Through RC Circuit

Z = √(R² + XC²)
tanφ = XC / R
I = (E₀ / Z) sin(ωt + φ)
  

9) AC Through Series LCR Circuit

Z = √[ R² + (ωL − 1/ωC)² ]
Irms = Vrms / Z
  

Phase angle:

tanφ = (ωL − 1/ωC) / R
  

Voltage relation:

V = √[ VR² + (VL − VC)² ]
  

10) Power in AC Circuit

Pavg = Vrms Irms cosφ
  

• cosφ is called power factor.
• Wattless current = I_rms cosφ

11) Resonance in Series LCR Circuit

• Impedance is minimum.
• Current is maximum.
• Phase angle φ = 0 → power factor = 1

ω₀ = 1 / √(LC)
f₀ = 1 / (2π√LC)
  

At resonance:

Z = R
Imax = V / R
VL = VC
  

12) Quality Factor (Q)

Q = ω₀L / R = 1 / (ω₀CR)
  

13) Choke Coil

• High inductance and negligible resistance.
• Controls AC without significant power loss.
• Used in fluorescent lamps.