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Introduction to phasors

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We will not always be dealing with transmission lines excited with a pulse. Although this is a good model fordigital circuitry, it will not always apply. When we go to analog signals (rf, high frequency analog, etc.) we will need more toolsthan are available to us at this point. In the not-too-distant-past, the material we will next consider wasstarting to be considered passé. The rf spectrum was more or less filled up, and the watchword was "digital". Now, in the newage of wireless communication, cell phones, and rf Local Area Networks, demand for engineers who understand ac behavior ontransmission lines and who can design systems which work well with rf signals are very much in demand. Pay heed to what we sayhere, 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

The usual set-up includes a source, with a sinusoidal output, asource impedance Z g a transmission line with impedance Z 0 , L meters long, and a load of impedance Z L at the end.

Let's look at the source first. We can describe the output waveform from the generator as

V t V g ω t θ
Which when plotted lookes like .

Excitation waveform

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 withzero 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 westart multiplying and dividing, the trig turns into a big mess, and gets in the way of our understanding of what is goingon. The way we deal with this, as every good 242 student knows, is to introduce phasors .

Since we know from Euler's Identity

V g ω t θ V g ω t θ ω t θ
If we take a real part of V g ω t θ we will extract the voltage waveform we desire. We will re-write this function as
V g ω t θ V g θ ω t
and then define V ∼ g as the phasor voltage where
V ∼ g V g θ
Note that V ∼ g is a complex quantity, with both a magnitude V g and a phase angle θ . In order to retrieve a real voltage signal from a phasor, we have to multiply the phasor by ω t and then take the real part. Note that this is the same thing as plotting the phasor on the complex plane, and thenobserving the projection of the phasor on the real axis, as the phasor rotates around at a rate ω t .

Phasor representation

This method of visualization will sometimes help make results seem a little easier to understand, or at least check forreasonableness.
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Source:  OpenStax, Introduction to physical electronics. OpenStax CNX. Sep 17, 2007 Download for free at http://cnx.org/content/col10114/1.4
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