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Review of Phasors

Module by: Bill Wilson. E-mail the author

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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 passé. 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 Figure 1.

Figure 1
Sinusoidal Excitation of a Loaded Transmission Line
Sinusoidal Excitation of a Loaded Transmission Line (7.01.png)
The usual set-up includes a source, with a sinusoidal output, a source impedance Z g Z g a transmission line with impedance Z 0 Z 0 , LL meters long, and a load of impedance Z L Z L at the end.

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

Vt= V g cosωt+θ V t V g ω t θ (1)
Which when plotted lookes like Figure 2.
Figure 2
Excitation Waveform
Excitation Waveform (7.02.png)
The oscillating waveform has a period TT and its angular frequency ωω is given as
ω=2πT=2πf ω 2 T 2 f (2)
The angle, θθ, which specifies how much the wave is leading a cosine function with zero off-set is given by
θ=2πτT θ 2 τ T (3)
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 Euler's Identity

V g ⅇⅈωt+θ= V g cosωt+θ+ⅈsinωt+θ V g ω t θ V g ω t θ ω t θ (4)
If we take a real part of V g ⅇⅈωt+θ V g ω t θ we will extract the voltage waveform we desire. We will re-write this function as
V g ⅇⅈωt+θ= V g ⅇⅈθⅇⅈωt V g ω t θ V g θ ω t (5)
and then define V ∼ g V ∼ g as the phasor voltage where
V ∼ g = V g ⅇⅈθ V ∼ g V g θ (6)
Note that V ∼ g V ∼ g is a complex quantity, with both a magnitude | V g | V g and a phase angle θθ. In order to retrieve a real voltage signal from a phasor, we have to multiply the phasor by ⅇⅈωt 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 ωt ω t Figure 3.
Figure 3
Phasor Representation
Phasor Representation (7.03.png)
This method of visualization will sometimes help make results seem a little easier to understand, or at least check for reasonableness.

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