The problem provides information about a sinusoidal waveform: a peak value of 20 A and a frequency of 50 Hz. It then asks four questions: (I) Write the mathematical expression for that sin wave. (II) Find the value of the current at time $t = 0.0075$ s after starting. (III) Find the time from the starting when the current value is $17.15$ A. (IV) Define the term RMS value for a sin wave, and write the expression.

1. Problem Description

The problem provides information about a sinusoidal waveform: a peak value of 20 A and a frequency of 50 Hz. It then asks four questions:

(I) Write the mathematical expression for that sin wave.

(II) Find the value of the current at time $t = 0.0075$ s after starting.

(III) Find the time from the starting when the current value is $17.15$ A.

(IV) Define the term RMS value for a sin wave, and write the expression.

2. Solution Steps

(I)

The general form of a sinusoidal current is:

$i(t) = I_m \sin(\omega t)$

where $I_m$ is the peak value and $\omega$ is the angular frequency.

We are given $I_m = 20$ A and $f = 50$ Hz. The angular frequency $\omega$ is related to the frequency $f$ by the formula:

$\omega = 2\pi f$

$\omega = 2\pi (50) = 100\pi$ rad/s

Therefore, the mathematical expression for the sinusoidal current is:

$i(t) = 20 \sin(100\pi t)$

(II)

To find the current at $t = 0.0075$ s, we substitute $t = 0.0075$ into the equation:

$i(0.0075) = 20 \sin(100\pi (0.0075))$

$i(0.0075) = 20 \sin(0.75\pi)$

$i(0.0075) = 20 \sin(\frac{3\pi}{4})$

$i(0.0075) = 20 \sin(135^\circ)$

$i(0.0075) = 20 \cdot \frac{\sqrt{2}}{2} = 10\sqrt{2}$ A

$i(0.0075) \approx 14.14$ A

(III)

We need to find the time $t$ when $i(t) = 17.15$ A.

$17.15 = 20 \sin(100\pi t)$

$\sin(100\pi t) = \frac{17.15}{20} = 0.8575$

$100\pi t = \arcsin(0.8575)$

$100\pi t = 1.0306$ radians (approximately $59.05^\circ$)

$t = \frac{1.0306}{100\pi} \approx 0.00328$ s

There will be multiple times where the current equals $17.15$ A, so we should consider the solution in the second quadrant as well. The reference angle is approximately $59.05^\circ$, thus the other angle is $180^\circ - 59.05^\circ = 120.95^\circ$, or $2.11$ radians.

$100\pi t = 2.11$

$t = \frac{2.11}{100\pi} \approx 0.00672$ s

(IV)

The RMS value (Root Mean Square) of a sinusoidal waveform is the effective value of the waveform. For current, it's the DC current that would produce the same heating effect as the AC current.

For a sinusoidal waveform $i(t) = I_m \sin(\omega t)$, the RMS value is given by:

$I_{rms} = \frac{I_m}{\sqrt{2}}$

$I_{rms} = \frac{20}{\sqrt{2}} = 10\sqrt{2} \approx 14.14$ A

3. Final Answer

(I) $i(t) = 20 \sin(100\pi t)$

(II) $i(0.0075) \approx 14.14$ A

(III) $t \approx 0.00328$ s and $t \approx 0.00672$ s.

(IV) The RMS value of a sinusoidal waveform is the effective value of the waveform. $I_{rms} = \frac{I_m}{\sqrt{2}}$.