Matthew's Asgn

I decided that I would attempt to perform a simple analysis of a series RL circuit, which could then be used to do a more complex analysis on a basic transformer. I have always had interest in electronics, and transformers are key to basic electronics.

I decided that i would do the analysis of a RL circuit with the variables instead of given values.

Given:

V(t)=$\cos w*t\,\!$

V(s)=$\ s/(s^{2}+w^{2})\,\!$

I(0)=i

The Laplace transform for an inductor:

$\displaystyle {\mathcal {L}}\left\{f(t)\right\}$ = $\ Ls+Li\,\!$

The Laplace transform for a resistor:

$\displaystyle {\mathcal {L}}\left\{f(t)\right\}$ = $\ R\,\!$

Therefore the Resulting Equation for the system after applying the Laplace Transform:

$\ 0=-s/(s^{2}+w^{2})+RI(s)+LsI(s)-Li\,\!$

A series of algebraic manipulations follows to come up with I(s):

$\ s/(s^{2}+w^{2})=(R+Ls)I(s)+Li\,\!$

$\ I(s)={\dfrac {s}{(s^{2}+w^{2})(R+Ls)}}-{\dfrac {Li}{(R+Ls)}}\,\!$

We can then use partial fraction manipulation to expand the expression:

$\ s/((s^{2}+w^{2})(R+Ls))=A/(s+jw)+A*/(s-jw)+B/(Ls+R)\,\!$

$\ (s/L)/((s^{2}+w^{2})(R/L+s))=(A(s+jw)(s+R/L))/((s+R/L)(s^{2}+w^{2}))+(A*(s+jw)(s+R/L))/((s^{2}+w^{2})(s+R/L))+(B(s^{2}+w^{2}))/((s+R/L)(s^{2}+w^{2}))\,\!$

$\ S/L=A((s-jw)(s+R/L))+A*((s+jw)(s+R/L))+B(s^{2}+w^{2})=A(s^{2}-jws+R/Ls-jwR/L)+a*(s^{2}+jws+R/L+jwr/L)+B(s^{2}+w^{2})\,\!$