The given alternating current is represented by the equation:
\[ I(t) = 160 \sin(314{,}159 \, t) \]
We will determine each of the requested values step by step.
### A) The Frequency of this Current
The general form of a sinusoidal current is:
\[ I(t) = I_{\text{max}} \sin(\omega t + \phi) \]
where:
- \( I_{\text{max}} \) is the peak current (160 A in this case),
- \( \omega \) is the angular frequency,
- \( t \) is time,
- \( \phi \) is the phase angle (which is 0 in this case).
The angular frequency \(\omega\) is related to the actual frequency \(f\) by:
\[ \omega = 2\pi f \]
Given that \(\omega = 314{,}159 \, \text{rad/s}\), we can solve for \(f\):
\[ 314{,}159 = 2\pi f \]
\[ f = \frac{314{,}159}{2\pi} \]
\[ f \approx 50{,}000 \, \text{Hz} \]
### B) The RMS (Root Mean Square) Value
The RMS value (also known as the virtual value) of a sinusoidal current is given by:
\[ I_{\text{RMS}} = \frac{I_{\text{max}}}{\sqrt{2}} \]
For this current:
\[ I_{\text{RMS}} = \frac{160}{\sqrt{2}} \]
\[ I_{\text{RMS}} \approx 113.14 \, \text{A} \]
### C) The Average Value of this Current
The average value over a full cycle of a pure sinusoidal waveform is zero because the positive and negative halves cancel each other out. However, if we need the average of just the positive half-cycle, it can be computed as:
\[ I_{\text{avg}} = \frac{2 I_{\text{max}}}{\pi} \]
For this current:
\[ I_{\text{avg}} = \frac{2 \cdot 160}{\pi} \]
\[ I_{\text{avg}} \approx 101.86 \, \text{A} \]
### D) The Form Factor of this Waveform
The form factor is the ratio of the RMS value to the average value of the absolute value of the current:
\[ \text{Form Factor} = \frac{I_{\text{RMS}}}{I_{\text{avg (positive half-cycle)}}} \]
So:
\[ \text{Form Factor} = \frac{113.14}{101.86} \]
\[ \text{Form Factor} \approx 1.11 \]
### E) The Crest Factor of this Waveform
The crest factor is the ratio of the peak value to the RMS value:
\[ \text{Crest Factor} = \frac{I_{\text{max}}}{I_{\text{RMS}}} \]
So:
\[ \text{Crest Factor} = \frac{160}{113.14} \]
\[ \text{Crest Factor} \approx 1.414 \, (\sqrt{2}) \]
### F) The Instantaneous Value 0.003 Seconds After Zero
The instantaneous value of the current at \( t = 0.003 \, \text{s} \) can be found by substituting \( t \) in the original equation:
\[ I(0.003) = 160 \sin(314{,}159 \cdot 0.003) \]
Calculate the argument of the sine function:
\[ 314{,}159 \times 0.003 \approx 942.477 \, \text{rad} \]
Since \( 942.477 \, \text{rad} = 150 \cdot 2\pi + \pi \), we are essentially at the angle \( \pi \) on the unit circle. Therefore,
\[ \sin(942.477) = \sin(\pi) = 0 \]
So,
\[ I(0.003) = 160 \sin(942.477) = 160 \cdot 0 = 0 \, \text{A} \]
The instantaneous value of the current 0.003 seconds after zero is 0 A.
An alternating current is represented by 160sin(314,159t)amperes .
Determine:
A) The frequency of this current
B) The virtual value of this value
C)THe average value of this current
D)The form factor of this waveform
E) The crest factor of this waveform
F) The instantaneous value 0,003 seconds after zero
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