Alternating Current

1. AC Voltage Applied to Elements

Alternating Current (AC): Current that changes direction periodically. I = I_m sin(ωt).

Pure Resistor

Voltage and Current are in Same Phase.
V = V_m sin(ωt)
I = I_m sin(ωt)

Pure Inductor

Current lags behind Voltage by π/2 (90°).
Inductive Reactance: X_L = ωL = 2πfL
(X_L ∝ f)

Pure Capacitor

Current leads Voltage by π/2 (90°).
Capacitive Reactance: X_C = 1/(ωC) = 1/(2πfC)
(X_C ∝ 1/f)

(Asked in NEET 2016, 2019)

2. Series LCR Circuit

Inductor (L), Capacitor (C), and Resistor (R) connected in series across AC source.

Impedance (Z):
Z = √[ R² + (X_L - X_C)² ]
Phase Angle (φ):
tanφ = (X_L - X_C) / R

Resonance

Condition where Current is Maximum (Impedance Minimum).

Condition: X_L = X_C
Resonant Frequency: ω₀ = 1 / √(LC)
At resonance: Z = R (Purely Resistive), Power Factor = 1.

(Asked in NEET 2018, 2021, 2023)

3. Power in AC Circuits

Average power dissipated over a complete cycle.

P = V_rms I_rms cosφ

cosφ = R / Z (Power Factor)

Resistive Circuit: φ=0, cosφ=1 → Max Power.
Pure Inductive/Capacitive: φ=90°, cosφ=0 → Zero Power (Wattless Current).
Wattless Current: Component of current (I sinφ) which consumes no power.

(Asked in NEET 2020)

4. LC Oscillations

Exchange of energy between Capacitor and Inductor.

Total Energy U = U_E + U_B = Constant
U = q²/2C + ½Li²
Frequency ν = 1 / 2π√(LC)

5. Transformers

Device to change AC voltage (works on Mutual Induction).

V_s / V_p = N_s / N_p = I_p / I_s = k

Types & Losses

Step-up: N_s > N_p (Voltage increases, Current decreases).
Step-down: N_s < N_p (Voltage decreases, Current increases).
Energy Losses: Flux Leakage, Resistance (Copper loss), Eddy Currents (Iron loss), Hysteresis.
Laminated Core: Reduces Eddy current loss.

(Asked in NEET 2017, 2019, 2022)

Numericals: Alternating Current

Q1. The equation of an AC voltage is V = 200 sin(100πt). Find the RMS voltage and frequency.

Solution:
Compare with V = V_m sin(ωt).
V_m = 200 V. ω = 100π.
V_rms = V_m / √2 = 200 / 1.414 = 141.4 V.
ω = 2πf = 100π → 2f = 100 → f = 50 Hz.

Q2. A 40 mH inductor is connected to a 220 V, 50 Hz AC supply. Find the rms current in the coil.

Solution:
L = 40 mH = 40 × 10^-3 H.
X_L = 2πfL = 2 × 3.14 × 50 × 0.04
X_L = 314 × 0.04 = 12.56 Ω.
I_rms = V_rms / X_L = 220 / 12.56 ≈ 17.5 A.

Q3. A 60 μF capacitor is connected to a 110 V, 60 Hz ac supply. Determine the rms value of the current in the circuit.

Solution:
X_C = 1 / (2πfC)
X_C = 1 / (2 × 3.14 × 60 × 60 × 10^-6)
X_C = 1 / (0.0226) ≈ 44.2 Ω.
I_rms = V / X_C = 110 / 44.2 ≈ 2.49 A.

Q4. In a series LCR circuit, R = 40 Ω, X_L = 80 Ω and X_C = 50 Ω. Find the Impedance.

Solution:
Z = √[ R² + (X_L - X_C)² ]
Z = √[ 40² + (80 - 50)² ]
Z = √[ 1600 + 30² ]
Z = √[ 1600 + 900 ] = √2500 = 50 Ω.

Q5. Calculate resonant frequency for a circuit with L = 20 mH and C = 20 μF.

Solution:
ω₀ = 1 / √(LC)
ω₀ = 1 / √(20×10^-3 × 20×10^-6)
ω₀ = 1 / √(400 × 10^-9) = 1 / √(4 × 10^-7) = 1 / (6.32 × 10^-4) ≈ 1581 rad/s.
f₀ = ω₀ / 2π ≈ 1581 / 6.28 ≈ 251 Hz.

Q6. In a series LCR circuit, V = 230 V, I = 4 A, and phase angle φ = 60°. Calculate power dissipated.

Solution:
P = VI cosφ
P = 230 × 4 × cos(60°)
P = 920 × 0.5
P = 460 Watts.

Q7. A transformer steps down 2200 V to 220 V. Number of turns in primary is 5000. Find number of turns in secondary.

Solution:
V_s / V_p = N_s / N_p
220 / 2200 = N_s / 5000
1 / 10 = N_s / 5000
N_s = 500 turns.

Q8. Find Q-factor of a series LCR circuit with L = 2 H, C = 32 μF and R = 10 Ω.

Solution:
Q = (1/R) √(L/C)
Q = (1/10) √(2 / 32×10^-6)
Q = 0.1 √(1/16 × 10⁶)
Q = 0.1 × (1000 / 4) = 0.1 × 250 = 25.

Q9. At resonance, what is the peak current if V_rms = 220 V and R = 100 Ω?

Solution:
At resonance, Z = R = 100 Ω.
I_rms = V_rms / R = 220 / 100 = 2.2 A.
Peak Current I_m = I_rms √2
I_m = 2.2 × 1.414 = 3.11 A.

Q10. In a circuit with voltage 220 V and current 5 A, the phase difference is π/2. What is power consumed?

Solution:
P = VI cosφ
Here φ = π/2 = 90°.
cos(90°) = 0.
P = 220 × 5 × 0 = 0 Watts.
(This is Wattless Current).

Important Formulae

1. Peak & RMS Value

V = V_m sinωt

V_rms = V_m / √2 = 0.707 V_m
V_avg = 2V_m / π = 0.637 V_m (Half cycle)

2. Reactance & Impedance

Inductive Reactance:

X_L = ωL = 2πfL

Capacitive Reactance:

X_C = 1/ωC = 1/2πfC

LCR Impedance:

Z = √(R² + (X_L - X_C)²)

3. Resonance

X_L = X_C

ω₀ = 1 / √(LC)
f₀ = 1 / 2π√(LC)

Quality Factor Q = (1/R) √(L/C)

4. Power & Transformer

Average Power:

P = V_rms I_rms cosφ

Transformer Ratio:

V_s/V_p = N_s/N_p = I_p/I_s

20 NEET Golden Facts

1. Voltage Preference: AC is preferred over DC because it can be stepped up/down (Transformer) and transmitted with lower loss.
2. Mean Value: Average value of AC over complete cycle is Zero.
3. RMS Meter: Hot wire ammeters measure RMS value. Ordinary DC ammeters measure average value (hence read 0 for AC).
4. Household Supply: 220V, 50Hz. This 220V is RMS value. Peak voltage = 220√2 ≈ 311V.
5. Capacitor Block: Capacitor blocks DC (f=0, X_C=∞) but allows AC to pass.
6. Inductor Choke: Inductor opposes change in current. Used as Choke coil to reduce current without power loss (Wattless).
7. Phase Diff: Pure R (φ=0), Pure L (V leads I by 90°), Pure C (I leads V by 90°).
8. Resonance Use: Used in radio/TV tuning to select specific frequency signals.
9. Power Factor: Max value 1 (Resonance), Min value 0 (Pure L or C).
10. Transformer DC: Does NOT work on DC. Will burn if DC applied (Resistance is low).
11. Sharpness: High Q-factor → Sharper resonance → Better selectivity.
12. Wattless Current: In pure L or C, power consumed is zero even if current flows.
13. Skin Effect: AC flows on surface of conductor. Thick wires are made of multiple thin strands to reduce resistance.
14. Energy in LC: Oscillates between Electric (Capacitor) and Magnetic (Inductor). Like Spring-Mass system.
15. Eddy Currents: Circulating currents in metal core. Minimized by Lamination.
16. Step Up: Increases Voltage, Decreases Current. Power remains constant (ideal).
17. Power Transmission: Done at High Voltage to minimize I²R losses.
18. 110V vs 220V: 220V is more dangerous than 110V (higher peak). 110V requires thicker wires (more current for same power).
19. Soft Iron: Used in Transformer core due to high permeability and low hysteresis loss.
20. Frequency Change: Transformer does NOT change frequency. Reactance changes with frequency.

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