When you finish this lesson, you should be able to:
The problems in this unit are rather special. They all have sinusoidal forcing functions. They all involve only the steady state solution. This may seem rather restrictive, but a vast number of problems in electricity fall into this category. Almost all problems in AC circuits do. The power of the phasor method lies in the fact that when a sinusoidal forcing function is used in a linear system, the results (voltage and current) are always sinusoidal, and of the same frequency as the forcing function. Thus, if a 120 V, 60 Hz voltage is used to drive a system of resistors, inductors and capacitors, the steady-state (what happens after waiting a long time) current in the system will be a 60 Hz sinusoid. Its amplitude and phase angle are the only unknowns that must be calculated.
Click on this link: http://micro.magnet.fsu.edu/electromag/java/generator/ac.html to see how an AC generator works.
Using the phasor method to get a complete solution to a problem requires 3 steps:
This method may at first seem long and tedious, but it is far easier than solving the original time domain problem using differential equations.
Try this site for a good explanation of phasors: http://people.clarkson.edu/~svoboda/eta/phasors/Phasor10.html (created by Dr. James A. Svoboda of Clarkson University).
Below is an example of a network and its transform.
Continuing with the same example, I can be calculated as,
I = 85/27°/(-j88.9 + 110 + j135) = 85/27°/(110 + j46.1) = 85/27°/119.3/22.7° = 0.712/4.3° A
An inverse transform can then be done on I to get:
i(t) = 0.712cos(450t + 4.3°) A
The complex version of resistance is called impedance. It is a combination of resistance and reactance. Resistance is the real part of the impedance, and reactance is the magnitude of the imaginary part. It's expressed this way:
Z = R + jX
R = resistance, the real part of impedance
X = reactance, the imaginary part, without the j.
The reactance for an inductor:
XL = ωL
The units of XL are ohms: radians/second × henries is ohms.
The reactance for a capacitor:
XC = -1/ωC
Note the minus sign. X is real, not imaginary. You have to add the j to make it imaginary. The units of XC are ohms: radians/second × farads is inverse ohms. Capacitors always have negative reactance and negative impedance.
To convert reactance into equivalent impedance, just add the j:
ZL = jXL = jωL
ZC = jXC = -j/ωC
Ohm's Law has a complex version:
V = I Z
It's the same a Ohm's Law for resistors, except that all the numbers are complex. The analysis for series/parallel combinations of impedances is the same as it is for resistances.
At this time you should complete Tutorial 11 on phasors.