Matthew's Asgn

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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)=cosw*t

V(s)=s/(s2+w2)

I(0)=i

The Laplace transform for an inductor:

{f(t)} = Ls+Li

The Laplace transform for a resistor:

{f(t)} = R

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

0=s(s2+w2)+RI(s)+LsI(s)Li

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


s/(s2+w2)=(R+Ls)I(s)+Li


I(s)=s(s2+w2)(R+Ls)Li(R+Ls)

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

s(s2+w2)(R+Ls)=A(s+jw)+A*(sjw)+B(Ls+R)

Failed to parse (unknown function "\dfra"): {\displaystyle \ \dfrac{s/L}{(s^2+w^2)(R/L+s)} = \dfra{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)} \,\!}

S/L=A((sjw)(s+R/L))+A*((s+jw)(s+R/L))+B(s2+w2)=A(s2jws+R/LsjwR/L)+a*(s2+jws+R/L+jwr/L)+B(s2+w2)