Question

A standard North American wall socket plug is labeled \(110 \mathrm{~V}\). This label indicates the _______ value of the voltage. a) average b) maximum c) root-mean-square (rms) d) instantaneous

Short answer

Answer: The voltage value represented by the label on a standard North American wall socket plug is the root-mean-square (rms) voltage.

Step-by-step solution

Understand the basic definitions

Let's briefly explain the types of voltage values mentioned in the question: a) Average voltage: The average value is the value obtained when we sum the total voltage at each point in a complete cycle and divide by the number of points. Average Voltage cannot be directly related to power delivered. b) Maximum voltage (or peak voltage): It is the highest voltage value in one cycle of an alternating current. c) Root-mean-square (rms) voltage: RMS voltage is a measurement that calculates the effective value of an alternating current voltage. It is used to convert the alternating current (AC) power to direct current (DC) equivalent power. This is the most important value for practical applications and calculation of power. d) Instantaneous voltage: The value of voltage at any specific point of time in a cycle of an alternating current.

Analyze the relevance of each value to the problem

We need a value that can calculate the power delivered by the wall socket plug, as it represents the useful energy provided. Average voltage cannot be directly related to power delivered. Maximum or instantaneous voltage only represents the highest point in a cycle, but power delivery depends on the effectiveness of voltage throughout the cycle.

Determine the correct value of the voltage

Root-mean-square (rms) voltage represents the effectiveness of an alternating current's voltage and is commonly used in power calculations. It's the most important value for practical applications as it helps in determining the effective power delivered by the wall socket plug. Therefore, the correct answer is: c) root-mean-square (rms)

Key concepts

Root-Mean-Square (RMS) Voltage

Understanding the concept of Root-Mean-Square (RMS) Voltage is essential when working with alternating current (AC) circuits. RMS voltage represents the equivalent direct current (DC) value which would deliver the same power as the AC voltage over one cycle. It's the square root of the average of the squares of all instantaneous voltages in the cycle.

For example, a North American wall socket labeled as '110 V' is providing an RMS voltage. This means that the effective voltage that can do work, like powering your electronics, is 110 V just as if 110 V of DC was applied. RMS voltage is used because AC current varies over time, and this measurement helps in equating AC to a constant DC value for practical use.

An RMS voltage is calculated using the equation: \( V_{rms} = \frac{1}{\sqrt{2}} \times V_{max} \), where \( V_{max} \) is the maximum voltage. This equation is derived from the waveform of a sinusoidal AC voltage because the power delivered by AC is not constant and must be averaged over a cycle.

Average Voltage

The average voltage of an AC signal is somewhat different from the RMS voltage. It is the arithmetic mean of all instantaneous voltages throughout a full cycle of the AC waveform. However, because AC waveforms typically have a symmetric shape, centered around zero voltage, the average voltage over a complete cycle equals zero for a pure sinusoidal wave.

This doesn't mean an AC signal delivers no power; rather, power delivery is not effectively conveyed by average voltage due to the constant polarity change. Average voltage can be more useful in other contexts, such as when analyzing rectified waveforms not yet filtered to a DC level. The calculation generally involves integrating the voltage over a cycle and then dividing by the cycle's duration.

Maximum Voltage

Maximum voltage, or peak voltage, of an AC signal represents the highest positive value the voltage reaches during a cycle. For sinusoidal AC waveforms, which are common in household power supply and many electronics, the voltage fluctuates between a positive peak and a negative peak.

This peak value is especially relevant in the context of voltage insulation and safety standards, as any equipment used must be able to withstand voltage spikes up to the peak value. However, while maximum voltage is necessary for certain design considerations, it is not directly useful for everyday power calculations because it does not represent the average effectiveness of the voltage over time.

Instantaneous Voltage

Instantaneous voltage is the value of an AC voltage at any specific moment in time. Unlike RMS voltage, which is a sort of average, instantaneous voltage is ever-changing in an AC system as the cycle progresses from zero to peak, down to zero, to negative peak, and back to zero again.

To get a sense of how instantaneous voltage works, think of taking snapshots of the voltage level at countless points in time throughout the AC cycle; that's your instantaneous voltage at each point. It gives you a 'real-time' reading but does not directly inform about the power-delivering capability of the AC source over time. Engineers may use instantaneous voltage readings to analyze dynamic behavior of circuits, such as during transient conditions or waveform distortion analysis.