Phasors

Quickly

Phasors are a mathematical construct used to represent sinusoidal waves of some constant frequency with complex numbers.

In other words,

• It is another way of looking at waves that is sometimes more convenient
• The waves must be both sinusoidal and of a fixed frequency (and we know from Fourier that we can basically represent any wave as a superposition of waves with the aforementioned properties)
• It uses complex numbers

A generic, planar wave propagating along $\hat k$ with ray direction along $\hat n$ will have the form

\[\tilde{E}(\vec r) = \hat nE_0e^{-j\vec k\cdot \vec r}\]

Breakdown

• The tilde ~ above denotes it is a complex quantity
• $\hat n$ is the ray direction

• $\vec k$ is the wave vector. Its direction $\hat k$ is the direction the wave propagates. Its magnitude $

  k

  $ is the angular wave number which is related to wavelength with $k = \pi/\lambda$

• $\vec r$ is the position we are evaluating at
• $E_0$ is the amplitude

Converting Between E and H

The exact conversion depends on the medium the wave is going through.

In free space, generically,

\[\tilde{E}(\vec r) = \hat nE_0e^{-j\vec k\cdot \vec r} \Rightarrow \tilde{H}(\vec r) = (\hat k\times\hat n)\frac{E_0}{\eta}e^{-j\vec k\cdot \vec r}\]

For a wave propagating along $\hat z$ with $\vec E$ along $\hat x$,

\[\tilde{E}(\vec r) = \hat xE_0e^{-jkz} \Rightarrow \tilde{H}(\vec r) = \hat y\frac{E_0}{\eta}e^{-jkz}\]

Converting With Time-Domain

For $\tilde E$ and $\tilde H$, to get the time-domain equations, you need to multiply by $e^{j\omega t}$ then get the real component.

\[\vec E(\vec r, t) = \Re\left\{\tilde E(\vec r)e^{j\omega t}\right\}\] \[\vec H(\vec r, t) = \Re\left\{\tilde H(\vec r)e^{j\omega t}\right\}\]