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In an A.C. circuit, the current is given by i = 5 sin (100t - π/2) ampere and the A.C. potential is V = 200 sin (100t) volt. Then the power consumption is
- a)20 watts
- b)40 watts
- c)1000 watts
- d)0 watt
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
In an A.C. circuit, the current is given by i = 5 sin (100t - π/2) amp...
AC Circuit Power Consumption
Explanation:
To find the power consumption in an AC circuit, we need to use the formula: P = VI cos(theta), where P is the power in watts, V is the voltage in volts, I is the current in amperes, and theta is the phase angle between the voltage and current.
Given:
Current, i = 5 sin (100t - π/2) A
AC potential, V = 200 sin (100t) V
Step 1: Calculate the instantaneous power
Instantaneous power, p = vi
p = (5 sin (100t - π/2)) (200 sin (100t))
p = 1000 sin (100t) sin (100t - π/2)
Step 2: Simplify the expression
Using the identity sin (A - B) = sin A cos B - cos A sin B, we get:
p = 1000 sin (100t) [sin (100t) cos (π/2) - cos (100t) sin (π/2)]
p = -1000 sin^2 (100t)
Note that the power is negative because the phase angle between the voltage and current is π/2.
Step 3: Find the average power
The average power over one cycle is given by:
P = (1/T) ∫_0^T p dt
where T is the time period of the waveform (T = 2π/100 = π/50 seconds).
P = (1/π/50) ∫_0^π/50 -1000 sin^2 (100t) dt
P = -20 watts
Note that the average power is negative, which means that the circuit is absorbing power.
Step 4: Conclusion
The power consumption in the AC circuit is 0 watts because the average power is negative. This means that the circuit is not consuming any power, but rather it is returning power to the source. This is possible because the voltage and current are out of phase by π/2, which means that the circuit is purely reactive.
Explanation:
To find the power consumption in an AC circuit, we need to use the formula: P = VI cos(theta), where P is the power in watts, V is the voltage in volts, I is the current in amperes, and theta is the phase angle between the voltage and current.
Given:
Current, i = 5 sin (100t - π/2) A
AC potential, V = 200 sin (100t) V
Step 1: Calculate the instantaneous power
Instantaneous power, p = vi
p = (5 sin (100t - π/2)) (200 sin (100t))
p = 1000 sin (100t) sin (100t - π/2)
Step 2: Simplify the expression
Using the identity sin (A - B) = sin A cos B - cos A sin B, we get:
p = 1000 sin (100t) [sin (100t) cos (π/2) - cos (100t) sin (π/2)]
p = -1000 sin^2 (100t)
Note that the power is negative because the phase angle between the voltage and current is π/2.
Step 3: Find the average power
The average power over one cycle is given by:
P = (1/T) ∫_0^T p dt
where T is the time period of the waveform (T = 2π/100 = π/50 seconds).
P = (1/π/50) ∫_0^π/50 -1000 sin^2 (100t) dt
P = -20 watts
Note that the average power is negative, which means that the circuit is absorbing power.
Step 4: Conclusion
The power consumption in the AC circuit is 0 watts because the average power is negative. This means that the circuit is not consuming any power, but rather it is returning power to the source. This is possible because the voltage and current are out of phase by π/2, which means that the circuit is purely reactive.
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