Question

What is the maximum value of the AC voltage whose root-mean-square value is (a) \(110 \mathrm{~V}\) or (b) \(220 \mathrm{~V} ?\)

Short answer

Answer: (a) The maximum value of the AC voltage with an RMS value of 110 V is approximately 155.56 V. (b) The maximum value of the AC voltage with an RMS value of 220 V is approximately 311.13 V.

Step-by-step solution

Write down the given RMS values

We are given the RMS values of two AC voltages as: (a) \(V_\mathrm{rms} = 110 \mathrm{~V}\) (b) \(V_\mathrm{rms} = 220 \mathrm{~V}\)

Calculate the maximum value of the AC voltage for part (a)

Using the relationship between the maximum value and the RMS value of an AC voltage, we can find the maximum value for part (a): \(V_\mathrm{max} = \sqrt{2} \times V_\mathrm{rms}\) For part (a), \(V_\mathrm{rms} = 110 \mathrm{~V}\), so we have: \(V_\mathrm{max} = \sqrt{2} \times 110 \mathrm{~V} \approx 155.56 \mathrm{~V}\)

Calculate the maximum value of the AC voltage for part (b)

Similarly, we can find the maximum value for part (b). For part (b), \(V_\mathrm{rms} = 220 \mathrm{~V}\), so we have: \(V_\mathrm{max} = \sqrt{2} \times 220 \mathrm{~V} \approx 311.13 \mathrm{~V}\)

Write down the final answers

The student can now write the final answers for the maximum value of the AC voltage for both given RMS values: (a) The maximum value of the AC voltage with an RMS value of \(110 \mathrm{~V}\) is approximately \(155.56 \mathrm{~V}\). (b) The maximum value of the AC voltage with an RMS value of \(220 \mathrm{~V}\) is approximately \(311.13 \mathrm{~V}\).

Key concepts

Root-Mean-Square Voltage

Root-mean-square voltage (RMS) is an important concept in alternating current (AC) circuits. Unlike direct current (DC), where the voltage remains constant, AC voltage fluctuates over time. This makes it tricky to measure its effective power. RMS provides a solution by measuring the equivalent DC value that would deliver the same power to a load.
Because of this, RMS is essential in everyday electrical devices. It tells us how much usable voltage there is in terms of power delivery.
In mathematics, RMS voltage is calculated using the maximum voltage, also known as peak voltage.

• The formula to convert RMS to maximum voltage is: \( V_\mathrm{max} = \sqrt{2} \times V_\mathrm{rms} \).
• This equation derives from the fact that the peak value of a sine wave is \( \sqrt{2} \approx 1.414 \) times its RMS value.

Understanding this relationship helps electricians, engineers, and students to predict safely how appliances operate with AC power. It ensures that devices receive the correct amount of power, avoiding overloading or underutilization.

Maximum Voltage

Maximum voltage (often called peak voltage) is the highest voltage level in an AC waveform. While RMS voltage provides an average value of voltage for power calculation, maximum voltage indicates the extreme of voltage at any given moment.
This peak value is crucial for designing circuits and insulation systems since they must tolerate the highest potential voltages.
To calculate maximum voltage from RMS, you simply multiply the RMS value by approximately 1.414.

• For example, given an RMS voltage of \(110 \mathrm{~V}\), the maximum voltage is \(V_\mathrm{max} = \sqrt{2} \times 110 \approx 155.56 \mathrm{~V}\).
• Similarly, for an RMS voltage of \(220 \mathrm{~V}\), it becomes \(V_\mathrm{max} = \sqrt{2} \times 220 \approx 311.13 \mathrm{~V}\).

By grasping the concept of maximum voltage, one develops a better understanding of how AC systems operate under the most extreme conditions, aiding in the safer and more efficient design and troubleshooting of electrical circuits.

AC Circuit Calculations

AC circuit calculations are essential for accurately understanding and designing AC power systems. Because AC voltage and current continuously vary over time, computations require particular calculations for both root-mean-square and maximum values.
This allows one to determine parameters like power consumption and system stability.

• One key equation is the conversion from RMS to maximum voltage: \( V_\mathrm{max} = \sqrt{2} \times V_\mathrm{rms} \).
• Knowing the maximum voltage helps in ensuring that components like capacitors, resistors, and transformers are adequately rated to handle the peaks in voltage.
• Accurate calculations prevent overloads, which can damage circuits or cause safety hazards.

Because AC circuits are ubiquitous in household and industrial applications, knowing how to perform these calculations is indispensable for anyone working with electric power. Whether you're a new student or a seasoned engineer, these skills ensure that your electrical systems operate smoothly, efficiently, and safely.