32.2:
RMS Value in AC Circuit
An alternating current's half-cycles are equal and opposite in directions. So, the average value of an alternating current for one cycle is zero.
As a result, the average value of an alternating current is calculated for a half-cycle, and is the ratio of the enclosed area to the length of the base of the half-cycle.
Consider a small section of an alternating current waveform. Recall the value of instantaneous current, and by integrating it for the half-cycle, the area enclosed by the half-cycle can be determined. Dividing it by the base length of the half-cycle, the average value of an alternating current can be obtained.
Consider a small section of a squared alternating current waveform.
By integrating the area of the section, the area of the half-cycle of the squared wave can be determined. By dividing it by the base length, the average value of the squared alternating current waveform can be obtained.
The root-mean-square value is defined as the square root of the average of the square of the alternating current, which is greater than the average value.
32.2:
RMS Value in AC Circuit
The root mean square (RMS) value is a measure of the effective or average value of an alternating current (AC) waveform. In AC circuits, the voltage or current waveform constantly changes direction and magnitude, making it difficult to describe with a single value. The RMS value provides a convenient way to calculate the equivalent DC voltage or current that would produce the same heating effect in a resistor as the AC waveform.
Mathematically, the RMS value of an AC waveform is the square root of the mean of the square of the instantaneous values of the waveform over one cycle. The formula for calculating the RMS value of a sine wave is as follows:
where I0 is the peak current of the alternating current. For a waveform with a different shape, such as a square wave or a triangle wave, the RMS value can be calculated using different formulas.
In practical applications, the RMS value is an important parameter for determining the power consumption and efficiency of an AC circuit. For example, the power consumed by a resistor in an AC circuit can be calculated using the following formula:
where Vrms is the RMS voltage across the resistor, and R is the resistance of the resistor. This formula assumes that the voltage waveform is sinusoidal.
In addition to power calculations, the RMS value is used in many other aspects of AC circuit analysis, such as voltage regulation, transformer design, and motor control. It is also important for safety considerations, as the RMS value of the voltage or current can affect the performance and lifespan of the electrical device.
Overall, the RMS value is a key concept in AC circuit analysis that allows engineers and technicians to accurately describe and measure the behavior of AC waveforms. It provides a way to convert complex AC waveforms into simpler, equivalent DC values that can be easily understood and analyzed.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version], Pg 394 – 396. Retrieved from https://openstax.org/books/university-physics-volume-2@8ede2ba/pages/15-4-power-in-an-ac-circuit