Unit – 9 Electrical circuits and Electronics | power in A.C. circuit

Power in A.C. Circuits | TRB Physics Notes

In alternating current (AC) circuits, power consumption is different from direct current (DC) circuits because current and voltage may not be in phase. Understanding power in AC circuits is very important for TRB, BRTE, and other competitive examinations.

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1. Instantaneous Power in AC Circuit

Instantaneous power is the power at any instant of time and is given by:

p = v × i

where
v = instantaneous voltage
i = instantaneous current

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2. Average Power in AC Circuit

Average power is the power actually consumed by the circuit over one complete cycle of AC.

Average power:

P = V_rms × I_rms × cos φ

where
V_rms = root mean square value of voltage
I_rms = root mean square value of current
φ = phase difference between voltage and current

The term cos φ is called the power factor.

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3. Power Factor

Power factor is defined as the cosine of the phase angle between voltage and current.

Power factor = cos φ

• Pure resistive circuit → cos φ = 1
• Pure inductive circuit → cos φ = 0
• Pure capacitive circuit → cos φ = 0

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4. Power in Different AC Circuits

(a) Pure Resistive Circuit

• Voltage and current are in phase
• Power factor = 1
• Maximum power is consumed

P = V_rms × I_rms

(b) Pure Inductive Circuit

• Current lags voltage by 90°
• Power factor = 0
• Average power consumed = 0

(c) Pure Capacitive Circuit

• Current leads voltage by 90°
• Power factor = 0
• Average power consumed = 0

(d) R–L–C Circuit

• Voltage and current have a phase difference
• Power factor lies between 0 and 1

P = V_rms × I_rms × cos φ

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5. Types of Power in AC Circuits

(a) Active (True) Power

Active power is the power actually consumed.

P = V_rms × I_rms × cos φ

Unit: Watt (W)

(b) Reactive Power

Reactive power is the power alternately stored and released by inductors and capacitors.

Q = V_rms × I_rms × sin φ

Unit: VAR (Volt Ampere Reactive)

(c) Apparent Power

Apparent power is the product of rms voltage and rms current.

S = V_rms × I_rms

Unit: VA (Volt Ampere)

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6. Power Triangle

The relation between active, reactive, and apparent power is represented by a right-angled triangle called the power triangle.

• S² = P² + Q²
• cos φ = P / S
• sin φ = Q / S

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7. Shortcut Tricks for TRB Exams

• Only resistance consumes power
• Pure L or C → Average power = 0
• Power factor decides real power
• cos φ = 1 → Maximum power
• At resonance → Power factor = 1
• P = VIcos φ (Most important formula)

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8. Comparison Table: Power in Different AC Circuits

Circuit	Phase Difference	Power Factor	Average Power
Pure R	0°	1	Maximum
Pure L	90° lag	0	Zero
Pure C	90° lead	0	Zero
RLC	Between 0° and 90°	0 < cos φ < 1	Finite

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9. TRB Level MCQs (10 Questions)

1. Average power in AC circuit is given by:
   (A) VI
   (B) VI sin φ
   (C) VI cos φ
   (D) V/I
   Answer: (C)
2. Power factor of pure inductive circuit is:
   (A) 1
   (B) 0
   (C) 0.5
   (D) Infinite
   Answer: (B)
3. Unit of reactive power is:
   (A) Watt
   (B) Joule
   (C) VA
   (D) VAR
   Answer: (D)
4. Which circuit consumes maximum power?
   (A) Pure L
   (B) Pure C
   (C) Pure R
   (D) LC
   Answer: (C)
5. At resonance, power factor is:
   (A) Zero
   (B) Less than 1
   (C) Greater than 1
   (D) Unity
   Answer: (D)
6. Apparent power is measured in:
   (A) W
   (B) VAR
   (C) VA
   (D) Joule
   Answer: (C)
7. Reactive power depends on:
   (A) Resistance
   (B) Phase difference
   (C) Voltage only
   (D) Current only
   Answer: (B)
8. In pure capacitive circuit, average power is:
   (A) Maximum
   (B) Minimum
   (C) Zero
   (D) Infinite
   Answer: (C)
9. cos φ is maximum when:
   (A) φ = 90°
   (B) φ = 45°
   (C) φ = 0°
   (D) φ = 180°
   Answer: (C)
10. Power factor improvement reduces:
    (A) Voltage
    (B) Current
    (C) Frequency
    (D) Resistance
    Answer: (B)

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Exam Tip: In TRB exams, power questions are mostly formula-based. Remember power factor concepts and power triangle relations.