Introduction to Phasors William L. Wilson wlw@rice.edu William L. Wilson wlw@rice.edu Jared L. Singer singer@rice.edu blah blah We will not always be dealing with transmission lines excited with a pulse. Although this is a good model for digital circuitry,it will not always apply. When we go to analog signals (rf, high frequency analog etc.) we will need more tools than are available to us at this point. In the not-too-distant-past, the material we will next consider was starting to be considered pass. The rf spectrum was more or less filled up, and the watchword was "digital". Now, in the new age of wireless communication, cell phones, and rf Local Area Networks, demand for engineers who understand ac behavior on transmission lines and who can design systems which work well with rf signals are very much in demand. Pay heed to what we say here, and you might well find yourself with many lucrative job offers in the future. To begin, we want to consider a transmission line which is being excited with an oscillating source.

Sinusoidal excitation of a loaded transmission line Sinusoidal excitation of a loaded transmission line

The usual set-up includes a source, with a sinusoidal output, a source impedance Zg a transmission line with impedance Z0 , L meters long, and a load of impedance ZL at the end.

Excitation waveform Excitation waveform

Let's look at the source first. We can describe the output waveform from the generator as V t Vg ω t θ Which when plotted looks like . The oscillating waveform has a period T and its angular frequency ω is given as ω 2 T 2 f The angle, θ , which specifies how much the wave is leading a cosine function with zero off-set is given by θ 2 τ T What we do not want to do, is carry a bunch of sine and cosine functions around with us everywhere. Once we start multiplying and dividing, the trig turns into a big mess, and gets in the way of our understanding of what is going on. The way we deal with this, as every good 242 student knows, is to introduce phasors. Since we know from Euhler's Identity Vg e j ω t θ Vg ω t θ j ω t θ If we take the real part of Vg e j ω t θ we will extract the voltage waveform we desire. We will re-write this function as Vg e j ω t θ Vg e j θ e j ω T and then define V˜g as the phasor voltage where V˜g Vg e j θ Note that V˜g is a complex quantity, with both a magnitude Vg and a phase angle θ . In order to retrieve a real voltage signal from a phasor, we have to multiply the phasor by e j ω t and then take the real part. Note that this is the same thing as plotting the phasor on the complex plane, and then observing the projection of the phasor on the real axis, as the phasor rotates around at a rate ω t .

Phasor representation Phasor representation

This method of visualization will sometimes help make results seem a little easier to understand, or at least check for reasonableness.