The sine function describes the ratio between the radius of a unit circle (or a circle in the Cartesian plane with unit radius) and the y-axis position of a point on the circle. The complementary function is the cosine, which describes the same ratio but for the x-axis position.

The power of a sine wave refers to an alternating current, in which the current, and therefore voltage, varies with time as a sine wave. Sometimes it is important to calculate average quantities for periodic (or repetitive) signals such as alternating current, while designing or building circuits.

## What is a Sine Function

It will be beneficial to define the sine function, in order to understand it's properties, and therefore how to calculate an average sine value.

In general, the sine function as it is defined, always has unit amplitude, 2π period and no phase offset. As mentioned, it is a ratio between the radius, *R*, and the y-axis position, *y*, of a point on the circle of radius *R*. For that reason, the amplitude is defined for a unit circle, but can be scaled by *R* as needed.

A phase offset would describe some angle away from the x-axis, where the new "starting point" of the circle has been shifted to. While this may be useful for some problems, it does not adjust the average amplitude, or power of a sine function.

## Calculating an Average Value

Remember that for a circuit the equation for power is, **P = I V,** where *V* is the voltage and *I* is the current. Because **V = I R**, for a circuit with resistance *R*, we now know that

First, consider a time-varying current *I(t)* of the form

The current has amplitude *I _{0}*, and period 2π/ω. If the resistance in the circuit is known to be

*R*, then the power as a function of time is

To calculate the average power, it is necessary to follow the general procedure for averaging: the total power at each instant in the period of interest, divided by the time period, T.

Therefore, the second step is to integrate P(t) over a full period.

The integral of I_{0}^{2}Rsin^{2}(ωt) over a period T is given by:

Then the average is the integral, or total power, divided by the period T:

It may be useful to know that the **average value of the sine function squared over its period** is always 1/2. Remembering this fact can help with calculating quick estimates.

## How to Calculate Root Mean Square Power

Just like the procedure for calculating the average value, **root mean square** is another useful quantity. It is calculated (almost) exactly as it is named: Take the quantity of interest, square it, calculate the mean (or average) and then take the square root. This quantity is often abbreviated as RMS.

So what is the RMS value of a sine wave? Just as done before, we know that the average value of a sine wave squared is 1/2. If we take the square root of 1/2, we can determine that the RMS value of a sine wave is approximately 0.707.

Often in circuit design, the RMS current or voltage is needed as well as the average. The fastest way to determine these is to determine the peak current or voltage (or the maximum value of the wave), and then multiply the peak value by 1/2 if you need the average, or 0.707 if you need the RMS value.

References

About the Author

Lipi Gupta is currently pursuing her Ph. D. in physics at the University of Chicago. She earned her Bachelor of Arts in physics with a minor in mathematics at Cornell University in 2015, where she was a tutor for engineering students, and was a resident advisor in a first-year dorm for three years. With this experience, when not working on her Ph. D. research, Gupta participates in STEM outreach activities to promote young women and minorities to pursue science careers.