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What is a radian?

Consider a circle of radius r as shown to the right. Now, mark off a portion of the circumference such that the selected distance around the circumference is equal to r. This portion of the circumference defines an angle, which we have labelled in the figure with the lower-case Greek letter phi (φ, pronounced "fee"). Because the portion of the circumference that is enclosed by this angle is equal in length to the radius of the circle, the angle itself is designated as one radian.

We already know that the entire circumference, C, of the circle is give by the expression C = 2πr, where π is the lower-case Greek letter pi, and represents a value of approximately 3.14159265. We also know that the entire circle encompasses a total of 360°. Therefore we can calculate the angle of one radian in terms of degrees as follows:

Where Do We Use Radians?

In electronics, we commonly use radians in two ways:

1. To express frequency, in radians per second. 

As we have already noted when looking at What is Alternating Current?, most people are used to referring to frequency in cycles per second, or hertz (Hz). In the US and Canada, for example, we know our household power lines carry AC at a frequency of 60 Hz. In Europe, the frequency is 50 Hz, but the general technology involved is still the same.

When analyzing or designing electronic circuits, however, it is often more convenient — and easier — to work with frequency in radians per second.

We normally use the letter f to represent frequency in hertz. To avoid confusion as to which representation we're seeing, we use the Greek letter omega (ω) to represent frequency in radians per second. Mathematically, ω = 2πf.

2. To express phase angle, in radians. 

Phase relationships can become important for many reasons. We may need to keep track of the phase relationship between voltage and current in a circuit, or we may want to be able to control that phase relationship.

We may also need to know how a circuit (such as a filter) that contains a reactance causes the signal voltage or current to shift phase as it passes through that circuit. Some oscillator circuits make use of such a phase shift, while some circuits that aren't supposed to oscillate may do so if proper consideration isn't given to these phase shifts.

In most cases, we don't need to calculate the numerical phase relationship as such; we can simply describe it in terms of π. Thus, you might see a reference to a phase angle of π/2 or π/4 radians, to represent angles of 90° or 45°, respectively.

Why Use Radians?

While it is certainly possible to make all of our calculations for electronic circuits in terms of hertz rather than radians, there is one extremely practical reason to stick with radians except at the very beginning and the very end of your procedure, and maybe avoid hertz altogether: use of radians simplifies the mathematical calculations.

This is because the factor 2π appears very often in calculations involving hertz, but is eliminated from those same calculations involving radians. By avoiding that factor throughout a circuit analysis or design process, we can reduce the chances of mathematical errors. Also, since π is a transcendental number — a number with no final resolution — using its approximate value many times in your calculations can lead to accumulating roundoff errors and a gradual drift away from accurate numbers.

By performing our calculations in radians rather than cycles or degrees, and then converting back and forth only at the very start and end of the procedure, we bypass these issues and maintain greater precision in our computations, and a more accurate final result.

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