Definition of a phasor

Consider a field quantity, such as the electric field $\underline{\mathcal{ E}}(\underline{r},t)$, that varies sinusoidally with angular frequency $\omega$:

$$\underline{\mathcal{E}}(\underline{r},t)=\underline{E}_0(\underline{r}) \cos \left[ \omega t + \varphi(\underline{r}) \right] \space ,$$ (2.51)

where $\underline{E}_0(\underline{r})$ is the vector amplitude of the field,

$$\omega=2 \pi f$$ (2.52)

is measured in rad/s, $f$ is the frequency in Hz (or cycle/s) and $\varphi(\underline{r})$ is the initial phase, i.e. the argument of the $\cos$ function at the initial time t=0.

By using Euler's formula

$$e^{j\alpha}= \cos \alpha + j \sin \alpha$$ (2.53)

the instantaneous electric field of (A.51) can be rewritten as

$$\underline{\mathcal{E}}(\underline{r},t)=\mathrm{Re}\left\{ \unde... ...e^{j \omega t}+ \underline{E}^* (\underline{r})e^{-j \omega t} \right] \space ,$$ (2.54)

where the asterisk * indicates the complex conjugate,

$$\underline{E}(\underline{r})=\underline{E}_0 (\underline{r}) e^{j \varphi(\underline{r})}$$ (2.55)

is the phasor electric field, and $j=\sqrt{-1}$ is the imaginary unit. The phasor of an instantaneous quantity (i.e., a quantity evaluated in the time domain) is a complex quantity independent of time, whose amplitude and phase are the amplitude and initial phase of the instantaneous quantity. In the most general case, the components $\mathcal{E}_x$, $\mathcal{E}_y$, $\mathcal{E}_z$ of $\underline{\mathcal{E}}$ may have different initial phases, so that the instantaneous electric field

$$\underline{\mathcal{E}}(\underline{r},t)=\hat{x} E_{0x}(\underlin... ...y(\underline{r})] + \hat{z} E_{0z} \cos [ \omega t + \varphi_z (\underline{r})]$$ (2.56)

has the phasor

$$\underline{E}(\underline{r})=\hat{x} E_{0x}(\underline{r}) e^{j \... ...line{r})}+\hat{z} E_{0z}(\underline{r}) e^{j \varphi_z(\underline{r})} \space .$$ (2.57)

Note that $\vert\underline{E}\vert$ is the peak value of $\underline{\mathcal{E}}$; some authors prefer to introduce a factor $\sqrt{2}$ in front of $\mathrm{Re}\{...\}$ in (A.54), so that the phasor's amplitude becomes the root-mean-square (rms) values of $\underline{\mathcal{E}}$. The instantaneous electric field $\underline{\mathcal{E}}$ is obtained from the phasor electric field $\underline{E}$ by multiplying $\underline{E}$ times the time-dependence factor $\exp(j\omega t)$ and taking the real part of the product.

The assumption that quantities such as the electric field vary sinusoidally at a given fixed frequency is true for power systems, but not for transmission of information which requires a nonzero bandwidth. For narrow-band signals propagating through a communication system whose characteristics may be taken to be independent of frequency over the signal bandwidth, the behavior of the system can be described with sufficient accuracy by studying it at the center frequency. However, if the signal is wide-band (or broadband), then the communication channel is usually dispersive, i.e. its properties vary significantly over the frequency band of operation. In such a case, the behavior of the system can be studied at a number of different frequencies within the operational bandwidth, and special techniques can then be used to describe the system behavior in the time domain; such techniques are outside the scope of this textbook.