Phasor Diagrams
Phasors
Sidebands
Rotation
What is a Phasor?
Recall our discussion on vectors: they are quantities with magnitude and direction.
A phasor is another useful mathematical concept, and in like fashion as with vectors, it has magnitude and…rotation!
Why do we need them??
The Famous Rotating Vector
Look at the figure below:
We have the white-pointed line rotating CC, generating two projections on the x and y axes. These are the cosine and sine components respectively.
The rotating line is our phasor: a vector that has a variable phase due to rotation.
Why is this Useful?
In sound synthesis projects, we need to generate sounds with multiple frequencies simultaneously. It is useful to view these are a set of rotating phasors, and see how they add up vectorially. This vector sum may be easier to understand and may lead to better understanding of the physics involved. Some phasors may rotate CC and others may rotate CCW, as we will see.
ImportantFrames of Reference under Rotation
Sometimes, when we have several components represented as phasors, it is useful to freeze one of the important ones, and let the others rotate relative to this frozen component. See for example the figure below:
In a phasor diagram representation, we would “freeze” the main red component, and then observe the others rotate around it. It should become obvious that keeping the red component steady is tantamount to reducing all frequencies by the red component’s frequency! This means that components at frequencies lower than the red component will effectively have negative frequencies and will rotate CW, with the others will rotate CCW. (Angle increases CCW)
Kaplan, Daniel T., Randall Pruim, and Nicholas J. Horton. 2024. mosaicCalc: R-Language Based Calculus Operations for Teaching. https://doi.org/10.32614/CRAN.package.mosaicCalc.
Mao, Renfei. 2024. gm: Create Music with Ease. https://doi.org/10.32614/CRAN.package.gm.
Pedersen, Thomas Lin, and Jordan Peck. 2025. ambient: A Generator of Multidimensional Noise. https://doi.org/10.32614/CRAN.package.ambient.
Soetaert, Karline. 2025. plot3D: Plotting Multi-Dimensional Data. https://doi.org/10.32614/CRAN.package.plot3D.
Citation
BibTeX citation:
@online{2026,
author = {},
title = {Phasor {Diagrams}},
date = {2026-02-20},
url = {https://mathforcoders.netlify.app/content/courses/MathModelsDesign/Modules/20-Systems/15-PhasorDiagrams/},
langid = {en}
}
For attribution, please cite this work as:
“Phasor Diagrams.” 2026. February 20, 2026. https://mathforcoders.netlify.app/content/courses/MathModelsDesign/Modules/20-Systems/15-PhasorDiagrams/.