An alternating current is given by i = 14.14 sin(377t). Find the RMS value of current, frequency, and instantaneous value of current when t = 3 ms, and also find the time taken by... An alternating current is given by i = 14.14 sin(377t). Find the RMS value of current, frequency, and instantaneous value of current when t = 3 ms, and also find the time taken by current to reach 10 A first time after passing through zero and increasing positively.
Understand the Problem
The question asks to analyze an alternating current formula to find the root mean square (RMS) value of the current, its frequency, and to determine the time taken for the current to reach a specific value of 10 A the first time it increases positively after passing through zero.
Answer
The RMS value is $10 \, A$, frequency is $60 \, Hz$, and time to reach $10 \, A$ is $2.08 \, ms$.
Answer for screen readers
The RMS value of the current is $10 , A$, the frequency is approximately $60 , Hz$, and the time taken for the current to reach $10 , A$ is approximately $2.08 , ms$.
Steps to Solve
- Identify the Given Function
The alternating current is given by the equation: $$ i(t) = 14.14 \sin(377t) $$
- Calculate the RMS Value of Current
The root mean square (RMS) value for a sinusoidal function can be calculated using the formula: $$ I_{rms} = \frac{I_{m}}{\sqrt{2}} $$ where ( I_m = 14.14 ).
Substituting the value: $$ I_{rms} = \frac{14.14}{\sqrt{2}} = 10 , \text{A} $$
- Determine the Frequency
The angular frequency ( \omega ) is given as ( 377 , \text{rad/s} ). The relationship between angular frequency and frequency is: $$ f = \frac{\omega}{2\pi} $$
Thus, $$ f = \frac{377}{2\pi} \approx 60 , \text{Hz} $$
- Find the Instantaneous Value When ( t = 3 , ms )
Substitute ( t = 0.003 , s ) into the equation: $$ i(0.003) = 14.14 \sin(377 \times 0.003) $$
Calculate the value: $$ i(0.003) \approx 14.14 \sin(1.131) \approx 14.14 \times 0.8988 \approx 12.7 , \text{A} $$
- Determine Time to Reach 10 A
Set the equation equal to 10 A: $$ 10 = 14.14 \sin(377t) $$
Rearranging gives: $$ \sin(377t) = \frac{10}{14.14} $$
Calculate ( \sin^{-1} \left( \frac{10}{14.14} \right) ): $$ 377t = \sin^{-1}(0.7071) $$
Solving for ( t ): $$ t = \frac{\sin^{-1}(0.7071)}{377} $$
Calculate ( t ): $$ t \approx \frac{0.7854}{377} \approx 0.00208 , s = 2.08 , ms $$
The RMS value of the current is $10 , A$, the frequency is approximately $60 , Hz$, and the time taken for the current to reach $10 , A$ is approximately $2.08 , ms$.
More Information
The RMS value is useful for understanding the effective power of AC currents, frequency indicates how fast the current cycles, and the time to reach a specific value is essential for phase analysis in AC circuits.
Tips
- Miscalculating the sine inverse or forgetting to account for the amplitude when solving for specific current values.
- Confusing the angular frequency with frequency, leading to errors in calculations.
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