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Question about angular frequency

  1. Apr 30, 2012 #1

    I've been studying AC circuit theory for a while now and there's always been something that's been bothering me. When using the complex impedance method to determine phase differences between current and voltage (and vice versa) we calculate the angular frequency in radians/seconds (omega = 2*pi*f) but we use degrees when we write the phase differences in the equations. The problem is that I'd assume that we'd have to use radians since the angular frequency is measured (in this case) in radians/second but apparently that's not the case?

    Could anyone explain this to me?

  2. jcsd
  3. May 1, 2012 #2


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    Maybe that is just easier to recognize (45° is easier to imagine than 0.79 if you are not used to the second one). You can use any angular measure everywhere, as long as you use it consistently and transform values given in other measures, if necessary.
  4. May 1, 2012 #3
    Sorry, I might not have been clear with what I was asking. Let me rephrase it:

    Why is it possible to use degrees when you specify phase difference when using radians/second for the angular frequency?

    For example: 5*sin(ωt - 10°)

    Where ω = 2*∏*f <- (obviously measured in radians/second).
  5. May 1, 2012 #4
    You would have to convert the 10 degrees to radians to get a value for that expression. But when you add expressions with the same angular frequency but different phase shifts, superposition applies, so you only need to add the phase angles. You don't need to convert them since they are in the same units already; its only when you want to find the total value of the expression that you have to convert.

    Like mfb said, the phase is usually kept in degrees so that it is more easily read and for most people it is more intuitive to work in units of degrees than radians (when you say two things are perpendicular, is it more natural and convenient to say they are 90 degrees different in orientation than to say they are 1.571 radians different in orientation?).
    Last edited: May 1, 2012
  6. May 1, 2012 #5
  7. May 1, 2012 #6
    Yes, but I should clarify that the phase angles don't necessarily just "add", but you can add the individual sine expressions with superposition, and you then have to do some trig if the sine expressions are of different magnitudes. But the point is that in all of the calculations for the final phase, you can remain in units of degrees.
  8. May 1, 2012 #7


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    The reason why Radians are used in formulae containing trigonometric functions is that, when you differentiate the function with ω (angular frequency in radians per second) in it, you keep your ω's.* When you use f (cycles per second) or degrees, you keep getting spurious and annoying 2π's all over the place.

    We are more familiar with degree measurement so we often present results in degrees (except when the result is a neat and familiar multiple or fraction of π).

    *When you first learn to differentiate trig functions this is pointed out to you (or should have been!) and you may be given exercises to show what odd results you can get when not doing the right thing.
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