Alternating Current: CBSE Class 12 Physics Chapter 7 Notes

Key Terms in Alternating Current

Alternating Current (AC)

An electric current that periodically reverses its direction and changes its magnitude continuously with time. Represented by I = I₀ sin(ωt) or I = I₀ cos(ωt).

RMS Value (Root Mean Square)

The effective value of an AC. It is the value of steady DC that would produce the same amount of heat in a given resistor in a given time. I_rms = I₀/√2 ≈ 0.707 I₀.

Reactance (X)

The opposition offered by an inductor (Inductive Reactance, X_L = ωL) or a capacitor (Capacitive Reactance, X_C = 1/ωC) to the flow of alternating current.

Impedance (Z)

The total opposition offered by an AC circuit (containing R, L, C) to the flow of current. It's the vector sum of resistance and reactance. Z = √(R² + (X_L - X_C)²).

Power Factor (cos φ)

The cosine of the phase angle between voltage and current in an AC circuit. It determines the true power consumed. cos φ = R/Z.

Resonance

A condition in a series LCR circuit where the inductive reactance equals the capacitive reactance (X_L = X_C), leading to maximum current and minimum impedance (Z = R).

Quality Factor (Q-factor)

A measure of the sharpness of resonance in an LCR circuit. It is defined as Q = ω₀L/R or Q = 1/(ω₀CR). A higher Q-factor indicates a sharper, more selective resonance peak.

Wattless Current

The component of current in an AC circuit that does not contribute to the average power consumption. This occurs in purely inductive or capacitive circuits where the phase difference is π/2.

Understanding the Series LCR Circuit

The series LCR circuit is a cornerstone of this chapter and a frequent topic in board exams. It consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series to an AC voltage source, V = V₀ sin(ωt). The current in the circuit is I = I₀ sin(ωt + φ), where φ is the phase difference between the voltage and current. The behavior of the circuit is determined by the interplay between inductive reactance (X_L = ωL), which causes voltage to lead current by 90°, and capacitive reactance (X_C = 1/ωC), which causes voltage to lag current by 90°. The resistance (R) has voltage and current in phase.

The total opposition to the current is called impedance (Z), calculated using the formula: Z = √[R² + (X_L - X_C)²]. The phase angle φ is given by tan φ = (X_L - X_C)/R. The relationship between these quantities is visualized using a phasor diagram, where voltages and current are represented as rotating vectors. The circuit exhibits three behaviors based on the comparison of reactances:

1. Inductive Dominant (X_L > X_C): The circuit is inductive, voltage leads current (φ > 0).
2. Capacitive Dominant (X_L < X_C): The circuit is capacitive, voltage lags current (φ < 0).
3. Resonance (X_L = X_C): The reactances cancel out, impedance is minimum (Z = R), current is maximum, and the circuit behaves as a purely resistive circuit (φ = 0).

Must-Remember Formulas & Concepts

• AC Voltage & Current: V = V₀ sin(ωt), I = I₀ sin(ωt), where ω = 2πf = 2π/T.
• RMS Values: V_rms = V₀/√2 and I_rms = I₀/√2. Standard domestic supply (e.g., 220V) refers to the RMS value.
• Inductive Reactance: X_L = ωL = 2πfL. Proportional to frequency.
• Capacitive Reactance: X_C = 1/(ωC) = 1/(2πfC). Inversely proportional to frequency.
• Series LCR Impedance: Z = √[R² + (X_L - X_C)²].
• Resonant Frequency (f₀): The frequency at which X_L = X_C. f₀ = 1 / (2π√LC). At resonance, impedance is minimum (Z=R) and current is maximum.
• Average Power: P_avg = V_rms I_rms cos(φ). Here, cos(φ) = R/Z is the Power Factor.
• Quality Factor (Q-factor): Q = ω₀L/R = 1/(R√(L/C)). Measures the sharpness of resonance.
• Transformer Ratio: V_s/V_p = N_s/N_p = I_p/I_s (for an ideal transformer). Step-up: N_s > N_p. Step-down: N_s < N_p.
• LC Oscillations: The natural frequency of oscillation is f = 1 / (2π√LC). In an ideal LC circuit, total energy remains constant.

AC Circuits: Resistor vs. Inductor vs. Capacitor

Worked Mini-Examples

• {"title":"Calculating Impedance and Current","bodyMarkdown":"Problem: A series circuit has R=10 Ω, L=0.1 H, C=100 μF, connected to a 220V, 50 Hz AC source. Find the impedance (Z) and RMS current (I_rms).\n\nSolution:\n1. Angular Frequency: ω = 2πf = 2 3.14 50 = 314 rad/s.\n2. Reactances: \n X_L = ωL = 314 0.1 = 31.4 Ω.\n X_C = 1/(ωC) = 1 / (314 100 * 10⁻⁶) = 31.85 Ω.\n3. Impedance: Z = √[R² + (X_L - X_C)²] = √[10² + (31.4 - 31.85)²] = √[100 + (-0.45)²] ≈ √100.2 ≈ 10.01 Ω.\n4. RMS Current: I_rms = V_rms / Z = 220 / 10.01 ≈ 21.98 A."}
• {"title":"Finding Resonant Frequency","bodyMarkdown":"Problem: An LCR circuit has L = 2.0 H, C = 32 μF, and R = 10 Ω. Find its resonant frequency (ω₀).\n\nSolution:\nThe resonant angular frequency is given by:\nω₀ = 1/√LC = 1 / √(2.0 32 10⁻⁶) = 1 / √(64 10⁻⁶) = 1 / (8 10⁻³) = 1000 / 8 = 125 rad/s."}

Board Exam Traps & Tips

Examiners often test subtle points in the AC chapter.

• RMS vs. Peak Value: Always check if the question gives V_rms or V₀ (peak voltage). If a voltage is given as '220V', it is the RMS value unless specified otherwise. Use V_rms = V₀/√2 correctly.
• Phasor Diagrams: Practice drawing neat, labeled phasor diagrams for all circuit types, especially the series LCR circuit under different conditions (X_L > X_C, X_L < X_C, and X_L = X_C). Arrows are mandatory on vectors.
• Power Factor: A power factor close to 1 is desirable for efficiency. Questions might ask 'how to improve the power factor'. The answer usually involves adding a capacitor in parallel to an inductive load.
• Transformer: Remember that transformers work only on AC, not DC. Also, distinguish between power loss causes (flux leakage, eddy currents, hysteresis, copper loss) as they are frequently asked.

Practice Questions with Solutions

• What is the power factor of a purely inductive circuit? What does it signify? The power factor (cos φ) is 0. This signifies that the average power consumed over a full cycle is zero. The current is 'wattless'.
• Why is an AC ammeter or voltmeter calibrated to read the RMS value? Because the RMS value corresponds to the equivalent DC value that would produce the same heating effect, making it the 'effective' value for power calculations and measurements.
• What happens to the impedance of a series LCR circuit at resonance? At resonance (X_L = X_C), the impedance becomes minimum and is equal to the resistance of the circuit (Z = R).
• How does a step-up transformer affect voltage and current? A step-up transformer increases the voltage (V_s > V_p) and, to conserve power, decreases the current (I_s < I_p).

Frequently Asked Questions