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:

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

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:

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

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

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

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

${\dfrac {S}{L}}=s^{2}(A+A*+B)+s((A*-A)jws+(A*+A)R/L)+(A*-A)Rjw/L+Bw^{2}\,\!$

After a lot of messy math and work:

$A={\dfrac {1/2}{R-jwL}}\,\!$

$A*={\dfrac {1/2}{R+jwL}}\,\!$

$B={\dfrac {-R}{R^{2}+w^{2}L^{2}}}\,\!$

$I(s)={\dfrac {a+jb}{s+jw}}+{\dfrac {a-jb}{s-jw}}+{\dfrac {B}{s+R/L}}\,\!$

$I(s)={\dfrac {(a+jb)(s-jw)+(a-jb)(s+jw)}{s^{2}+w^{2}}}+{\dfrac {B}{s^{2}+w^{2}}}\,\!$

$I(s)={\dfrac {2as}{s^{2}+w^{2}}}+{\dfrac {2bw}{s^{2}+w^{2}}}+{\dfrac {B}{s+R/L}}\,\!$

$a={\dfrac {1/2R}{R^{2}+w^{2}L^{2}}}\,\!$

$b={\dfrac {1/2wL}{R^{2}+w^{2}L^{2}}}\,\!$

$i(t)={\mathcal {L}}^{-1}\{I(s)\}\,\!$

$i(t)=2a\cos wt+2b\sin wt+BE^{-Rt/L}-ie^{-Rt/L}\,\!$

$i(t)=2{\dfrac {R}{R^{2}+w^{2}L^{2}}}\cos wt+2{\dfrac {wL}{R^{2}+w^{2}L^{2}}}\sin wt+{\dfrac {-R}{R^{2}+w^{2}L^{2}}}e^{-Rt/L}-ie^{-Rt/L}\,\!$

This Solves the Series RL circuit for the current throughout the circuit, given initial conditions or no initial conditions