Alternating Current: AC Circuits, LCR Resonance & Power
Study sinusoidal AC, RMS values, reactance and impedance, LCR series resonance, power factor, wattless current and transformer for JEE Main Physics.
The central idea of this chapter
Alternating current varies sinusoidally with time. This chapter connects voltage, current, phase, reactance and power through AC circuits, LCR resonance and transformers.
What should you understand first?
Alternating current is used in homes and industries. Once you understand RMS, phase difference, reactance and resonance, most problems become systematic applications of formulas.
Core concepts
- RMS value I₀/√2, V₀/√2
- Reactance XL, XC
- Impedance Z = √(R² + (XL−XC)²)
- LCR resonance XL = XC
- Power factor cos φ = R/Z
Useful building blocks
- Inductive reactance XL = ωL
- Capacitive reactance XC = 1/ωC
- Resonant frequency ω₀ = 1/√LC
- Average power P = VrmsIrmscosφ
- Transformer Vs/Vp = Ns/Np
Alternating Current formula sheet
Use RMS values for power calculations. Apply resonance condition and phase relations consistently in LCR circuits.
| Topic | Formula / Relation | Meaning or use |
|---|---|---|
| Instantaneous AC | V = V₀ sinωt, I = I₀ sin(ωt − φ) | Sinusoidal voltage and current with phase difference φ |
| RMS value | Irms = I₀/√2, Vrms = V₀/√2 | Equivalent DC value for same heating effect |
| Inductive reactance | XL = ωL = 2πfL | Opposition by inductor in AC |
| Capacitive reactance | XC = 1/ωC = 1/2πfC | Opposition by capacitor in AC |
| Impedance (series LCR) | Z = √[R² + (XL−XC)²] | Total opposition in LCR circuit |
| Phase difference | tanφ = (XL−XC)/R | Angle between V and I |
| Resonance condition | XL = XC | At resonance, Z = R, current maximum |
| Resonant frequency | ω₀ = 1/√(LC), f₀ = 1/(2π√LC) | Frequency at which resonance occurs |
| Quality factor | Q = ω₀L/R = (1/R)√(L/C) | Sharpness of resonance |
| Average power | P = VrmsIrmscosφ | Real power consumed in AC circuit |
| Power factor | cosφ = R/Z | Fraction of total power actually used |
| Transformer ratio | Vs/Vp = Ns/Np | Ideal transformer, s = secondary, p = primary |
How to approach AC problems
First identify circuit type (R, L, C, LR, RC, LCR), then find reactance, impedance and phase. Use resonance and power formulas for LCR and power-related questions.
AC circuits
- Only R? V & I in phase
- Only L? I lags V by 90°
- Only C? I leads V by 90°
- LCR series? Use Z & φ formulas
- Resonance? XL = XC
Power & transformer
- Average power? P = VI cosφ
- Wattless current? cosφ = 0
- Power factor? cosφ = R/Z
- Transformer? V ratio = N ratio
- Step-up/down? Step-up: Ns > Np; Step-down: Ns < Np
How to prepare Alternating Current
Start with RMS values and basic AC circuits, then learn LCR series resonance and power. Finish with transformer and wattless current concepts.
What to do
- Learn RMS value and sinusoidal AC representation
- Practise reactance and impedance for R, L, C, LR, RC, LCR
- Master LCR series resonance condition and resonant frequency
- Understand average power, power factor and wattless current
- Revise transformer ratio and step-up/step-down conditions
Common mistakes
- Using peak values instead of RMS in power formulas
- Confusing XL and XC formulas (ωL vs 1/ωC)
- Forgetting phase difference sign in LCR circuits
- Not checking resonance condition (XL = XC)
- Ignoring ideal transformer assumptions in problems
Ready to test Alternating Current?
Revise the formula sheet, then solve mixed JEE Main problems on AC circuits, LCR resonance and power.
Alternating Current FAQ
Short answers to frequently tested ideas in this chapter.
RMS (root mean square) value of AC is the equivalent DC value that produces the same heating effect. For sinusoidal AC, Irms = I₀/√2 and Vrms = V₀/√2.
Resonance occurs when inductive reactance equals capacitive reactance (XL = XC). At resonance, impedance is minimum (Z = R) and current is maximum.
Power factor is the cosine of phase difference between voltage and current (cos φ = R/Z). It represents the fraction of total power that is actually used.