Convert a time-domain sinusoid such as 10 cos(ωt + 30°) to phasor form in polar and rectangular coordinates, or convert a phasor back to the time domain.
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Time Domain To Phasor Formula
The following formula is used to convert a time domain signal to its phasor representation:
Variables:
- P is the phasor representation (A∠° or V∠°)
- M is the magnitude of the sinusoid (A or V) in the convention you use: peak, or RMS = peak / √2.
- θ is the cosine-reference phase in degrees; subtract 90° from a sine-reference phase.
To convert a time domain signal to its phasor representation, use the magnitude and phase angle to create a polar form representation of the signal. The frequency is not used in the conversion to phasor form but is important for understanding the signal's behavior over time.
| Frequency (Hz) | Period (ms) | Angular Frequency (rad/s) |
|---|---|---|
| 1 | 1000.000 | 6.283 |
| 10 | 100.000 | 62.832 |
| 50 | 20.000 | 314.159 |
| 60 | 16.667 | 376.991 |
| 100 | 10.000 | 628.319 |
| 200 | 5.000 | 1256.637 |
| 400 | 2.500 | 2513.274 |
| 440 | 2.273 | 2764.602 |
| 500 | 2.000 | 3141.593 |
| 1000 | 1.000 | 6283.185 |
| 2000 | 0.500 | 12566.371 |
| 5000 | 0.200 | 31415.927 |
| 10000 | 0.100 | 62831.853 |
| 12000 | 0.083 | 75398.224 |
| 20000 | 0.050 | 125663.706 |
| 32768 | 0.031 | 205887.416 |
| 44100 | 0.023 | 277088.472 |
| 48000 | 0.021 | 301592.895 |
| 50000 | 0.020 | 314159.265 |
| 100000 | 0.010 | 628318.531 |
| Formulas: T (ms) = 1000 / f (Hz); ω (rad/s) = 2π f. Rounded to 3 decimals. | ||
What is a Phasor?
A phasor is a complex number used to represent the magnitude and phase of a sinusoidal function in the frequency domain. It simplifies the analysis of linear electrical circuits that operate in steady state. Phasors are used extensively in electrical engineering, especially in the context of alternating current (AC) circuits.
How to Calculate a Phasor from Time Domain?
The following steps outline how to convert a time domain signal to its phasor representation:
- First, determine the sinusoid’s peak or RMS coefficient and select that convention.
- Next, enter its phase angle in degrees or radians and select cosine or sine.
- Use the formula P = M∠θ° to find the phasor representation.
- Finally, enter the magnitude and phase angle into the calculator to get the phasor.
Example Problem :
Use the following variables as an example problem to test your knowledge.
RMS magnitude (M) = 5 A, corresponding to a peak coefficient of 5√2 A for a cosine waveform.
Frequency (f) = 60 Hz (not used in phasor calculation)
Cosine-reference phase angle (θ) = 30 degrees; the RMS phasor is 5∠30° A.
