Question

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/video-lectures/6002_l16.pdf

My question is about the complex algebra on page 11 of 18 of the lecture notes.The second equation from the top has the characteristic equation that is multiplied by the complex conjugate which eliminates the j in the denominator. I understand that he intends to then convert that complex conjugate from rectangular to polar then add the exponentials. I don't understand how he got equation 3 from that. HELP!

Answer 1

Eq 3 is general decay of an RC network
use
I R
I = C dvc/dt
vi = RC dvc/dt + vc
solve for vc, with vi DC vc will exponentially increase from 0 to vi with a time constant of RC

Answer 2

I believe you are looking at page 12 which is the Homogenous solution which would be correct for the transients. This whole lecture note is about the sinusoidal steady state and frequency response of an input the homogeneous solution in this case taken as t approaches infinity which equal s zero. Simply, he discards the homogeneous solution. More interested in the particular solution. Exactly how did he do the complex algebra from equations 2 to 3 from the top of page 11.

Answer 3

\[(Vin/1+j \omega R C)*e^j \omega t \rightarrow (Vin * (1-j \omega R C)/1+ \omega^2 R^2 C^2) * e^j \phi *e^j \omega t\]