User:Egm4313.s12.team8.lucas/R1.4

=R1.4=

Problem Statement
Derive (i) Equation 3 ($$\displaystyle LI+RI'+\frac{1}{C}I=V'$$) and (ii) Equation 4 ($$\displaystyle LQ+RQ'+\frac{1}{C}Q=V$$) using Equation 2 ($$\displaystyle V=LC\frac{\mathrm{d^2v_{c}} }{\mathrm{d} t^2}+RC\frac{\mathrm{d} v_{c}}{\mathrm{d} t}+v_c$$) on page 2-2 in the lecture notes.

Part i
Equation 2:

Equation 1:

Equation 1 can be rewritten as Eqn 1-b:

Differentiating Eqn 1-b yields:

Differentiating Eqn 1-c gives:

Then we differentiate Equation 2 to get:

Then we subsitute Eqn. 1-b, Eqn. 1-c, and Eqn. 1-d into Equation 2-b to get:

By cancelling like terms we derive Equation 3:

Which can be rewritten as:

Part ii
To derive Equation 4, we again start with Equation 2:

Then we will use another version of Equation 1 on p. 2-2 in the lecture notes.

Then rearranging the variables, we get:

Differentiating this yields:

And differentiating again yields:

Substituting these derivatives into Equation 2 gives:

Cancelling like terms gives Equation 4:

Authorship:

Solved and typed by: Brian Lucas Egm4313.s12.team8.lucas 20:14, 30 January 2012 (UTC)

Checked by: