Question 131: AC fundamentals

Question

An ac current is represented as
$$i=5 \sqrt{2}+10 \cos \left(650 \pi t+\frac{\pi}{6}\right) A m p$$
The r.m.s value of the current is

Accepted Answer

To find the root mean square (RMS) value of the given alternating current, follow these steps:
The current is represented as:
$ i = 5\sqrt{2} + 10 \cos \left(650\pi t + \frac{\pi}{6}ight) \, 	ext{Amp} $
Here, the time-independent DC component is $5\sqrt{2}$ and the AC component is $10 \cos \left(650\pi t + \frac{\pi}{6}ight)$.
Calculate the square of the current, $i^2$:
$    i^2 = \left(5\sqrt{2}ight)^2 + \left(10 \cos \left(650\pi t + \frac{\pi}{6}ight)ight)^2 + 2 	imes 5\sqrt{2} 	imes 10 \cos \left(650\pi t + \frac{\pi}{6}ight)    $
Simplifying, we have:
$    i^2 = 50 + 100 \cos^2 \left(650\pi t + \frac{\pi}{6}ight) + 100\sqrt{2} \cos \left(650\pi t + \frac{\pi}{6}ight)    $
Find the average value $\langle i^2 angle$:
The average value of $\cos$ terms over a period is zero, simplifying our equation to:
$    \langle i^2 angle = 50 + \frac{100}{2} + 0    $
This simplifies to:
$    \langle i^2 angle = 50 + 50 = 100    $
Calculate the RMS current:
The RMS value is the square root of the mean of the squares of the current:
$    \langle i angle = \sqrt{100} = 10 \, 	ext{Amp}    $
Thus, the RMS value of the current is 10 Amps.

Answer

10 Amp

Answer

$5 \sqrt{2} \mathrm{~Amp}$

Answer

100 Amp

Answer

50 Amp

AC fundamentals

An ac current is represented as $i=5 \sqrt{2}+10 \cos \left(650 \pi t+\frac{\pi}{6}\right) A m p$ The r.m.s value of the current is

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