05-10-2016

Lecture: Introduction to phase shifting, phasor representation in frequency domain

I. Introduction to Phase shifting:
For today lecture (no lab today), we will learn about how to modify the orginal graph using the idea of pre-cal, which is known as vertical shift and horizontal shift, but for engineering, horizontal shift would be phase shift.

We have to parametrized the two graphs. The graph y(t) is really close to the orignal which only had the vertical shift;therefore we chose to do it first; whereas the green one also had the horizontal shift

II. Phasor representation:

The idea is genius, it is really hard to add the trig identities together. We assumed that voltage and current had the same frequency, therefore, we could use the phasor with polar and imaginary part to simplify our calculation

Given the original time domain function, we could convert it to the polar coordination to multiply and division operation, or to imaginary representation to do the add or subtracting equations

In trig, to find the phase angle, simply take the acrtan of y.x

We practice some operations between polar and imaginary part of phasor representation

For adding, imaginary part will be a better choice

Some more practice for phasors, recall that to to convert from sin to cos , we subtract the angle by 90.

Here given the time domain voltages, we have to combine them into one new voltage. We performed many steps of phasors to get the answer

We find the current with the given impedance of the inductor

III. Conclusion:
Today we spent most of our time to understand the phasor representation of the trig in the frequency domain. For phasors,  we have to use polar to divide or multiply, and imaginary to subtract and add. The capacitor and inductor act like a complex resistors in the AC circuits.