Understanding Sinusoidal Waveforms in Electrical Engineering

A sinusoidal waveform is a type of alternating current (AC) waveform that is commonly encountered in electrical circuits. It varies smoothly and periodically, taking the shape of a sine or cosine function. Key characteristics of a sinusoidal waveform include its maximum value (also known as peak value or amplitude) and its Root Mean Square (RMS) value.

Sinusoidal Waveform: What are RMS and Maximum Values?

• Maximum Value ($I_{max}$ or $V_{max}$): This is the peak amplitude of the waveform, representing the highest instantaneous value reached during a cycle.
• RMS Value ($I_{rms}$ or $V_{rms}$): The Root Mean Square value is a measure of the effective value of an AC quantity. It is equivalent to the DC value that would produce the same amount of heat in a resistive load. For a sinusoidal waveform, the RMS value is related to the maximum value by a specific constant factor.

Calculating RMS Value from Maximum Value for a Sinusoidal Current

For a pure sinusoidal current waveform, the relationship between the RMS value ($I_{rms}$) and the maximum value ($I_{max}$) is given by the formula:

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

This formula shows that the RMS value is the maximum value divided by the square root of 2.

To find out what fraction or multiple the RMS value is of the maximum value, we can rearrange the formula:

$$\frac{I_{rms}}{I_{max}} = \frac{1}{\sqrt{2}}$$

Now, let's calculate the numerical value of $\frac{1}{\sqrt{2}}$:

$$\sqrt{2} \approx 1.41421$$

So,

$$\frac{1}{\sqrt{2}} \approx \frac{1}{1.41421} \approx 0.707$$

Therefore, for a sinusoidal waveform, the RMS value of current is approximately 0.707 times the maximum value of current.

Checking the Options

Let's compare our calculated value with the given options:

The ratio of the RMS value to the maximum value for a sinusoidal waveform is $1/\sqrt{2}$, which is approximately 0.707.

Final Answer Determination

Based on the relationship $I_{rms} = \frac{I_{max}}{\sqrt{2}}$, the RMS value is $\frac{1}{\sqrt{2}}$ times the maximum value. Calculating $\frac{1}{\sqrt{2}}$ gives approximately 0.707. This matches option 4.