Problem 2 - University Physics with Modern Physics

Question

A sinusoidal current \(i=I \cos \omega t\) has an rms value \(I_{\mathrm{rms}}=\)2.10 A. (a) What is the current amplitude? (b) The current is passed through a full-wave rectifier circuit. What is the rectified average current? (c) Which is larger: \(I_{\text { ms or }} I_{\text { rav }} ?\) Explain, using graphs of \(i^{2}\) and of the rectified current.

Step-by-Step Solution

Answer

(a) 2.97 A, (b) 1.89 A, (c) RMS is larger due to squaring effect.

Step 1: Relationship Between RMS and Amplitude

The root mean square (rms) value of an alternating current is related to its amplitude by the formula \( I_{\text{rms}} = \frac{I}{\sqrt{2}} \). To find the amplitude \( I \), we rearrange this equation as follows:\[ I = I_{\text{rms}} \times \sqrt{2} \]

Step 2: Calculate Current Amplitude

Given \( I_{\mathrm{rms}} = 2.10 \) A, substitute this value into the rearranged equation to find the amplitude:\[ I = 2.10 \times \sqrt{2} \approx 2.97 \text{ A} \]. So, the current amplitude is 2.97 A.

Step 3: Average Current of Full-Wave Rectifier

The average current \( I_{\text{rav}} \) of a full-wave rectified signal is \( \frac{2I}{\pi} \), where \( I \) is the amplitude of the original sinusoidal current. Substitute the amplitude found earlier:\[ I_{\text{rav}} = \frac{2 \times 2.97}{\pi} \approx 1.89 \text{ A} \].

Step 4: Compare RMS and Average Rectified Current

The rms value of the current, 2.10 A, is the effective value, and the average value through the full-wave rectifier, 1.89 A, concerns the net positive flow. Comparatively, the rms value is always larger because rms considers the magnitude square of the waveform, effectively emphasizing larger amplitudes, while the average rectified current only considers the absolute waveform.

Key Concepts

RMS Value

Understanding the root mean square (RMS) value is fundamental when dealing with alternating current (AC). RMS value of an AC current, denoted as \( I_{\text{rms}} \), is essentially the DC equivalent value that delivers the same power to a resistor as the original AC current. It smooths out the sinusoidal fluctuations to present a more consistent measure.
For a sinusoidal current \( i = I \cos \omega t \), the RMS value is given by the formula \( I_{\text{rms}} = \frac{I}{\sqrt{2}} \), where \( I \) is the peak current or current amplitude.

• This formula arises because RMS involves squaring the current, finding the mean over one cycle, and then taking the square root.
• The RMS value is crucial as it provides a meaningful representation of an AC's ability to do work.

Learning this helps in comparing AC and DC powers effectively.

Current Amplitude

The current amplitude, indicated as \( I \), represents the peak or maximum value reached by an alternating sinusoidal current. It is visible as the highest point on the AC waveform graph which repetitively oscillates.
To calculate the amplitude from a known RMS value, use the formula \( I = I_{\text{rms}} \times \sqrt{2} \). For example, with an RMS value of 2.10 A, the amplitude would be approximately 2.97 A.

• This calculation is important in scenarios requiring peak currents, like in the sizing of electrical components.
• Understanding amplitude aids in designing systems to manage peak loads without damage.

The amplitude provides insight into the maximum potential current in a circuit at any given instant.

Full-Wave Rectifier

A full-wave rectifier is an electronic device that converts the entire AC input waveform into a unidirectional current. This is achieved by flipping the negative half-cycles of the AC input to become positive, essentially making the waveform entirely above the zero line on a graph.
The output is not constant but pulsates, unlike a smooth DC. The rectified waveform, however, has twice the frequency of the input waveform. For a sinusoidal input, the average or mean value of this full-wave rectified output, \( I_{\text{rav}} \), is calculated using the formula \( I_{\text{rav}} = \frac{2I}{\pi} \), where \( I \) is the current amplitude.

• Full-wave rectifiers are used in power supply circuits to convert AC to DC.
• This transformation is crucial for powering loads that require constant positive voltage.

Understanding rectifiers is essential for plenty of applications, including battery charging and DC motor operations.

Average Current

Average current in the context of rectified AC circuits refers to the mean value of the amplitude over one complete cycle. For full-wave rectification, this is found as \( I_{\text{rav}} = \frac{2 \times I}{\pi} \), where \( I \) is the current amplitude. This concept allows you to understand how much net current flows over time.

• The average current expresses the effective net current flow and contrasts with the RMS value, which accounts for the entire waveform including direction reversals.
• In your exercise, calculating shows an average current of approximately 1.89 A when rectified, against the higher RMS value of 2.10 A.

Typically, the RMS value is larger than the rectified average because RMS accounts for the squares of the waveform's amplitude variations, giving it a higher value. In electricity, both concepts highlight different performance aspects of AC signals.