Matthew's Asgn - Revision history

Btheredude at 01:40, 2 November 2010

2010-11-02T01:40:04Z

← Older revision		Revision as of 18:40, 1 November 2010
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	<math> B = \dfrac{-R}{R^2+w^2L^2} \,\!</math>		<math> B = \dfrac{-R}{R^2+w^2L^2} \,\!</math>

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

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

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

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

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

			<math> i(t) = \mathcal{L}^{-1} \{I(s)\} \,\!</math>

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

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

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

Btheredude at 01:03, 2 November 2010

2010-11-02T01:03:45Z

← Older revision		Revision as of 18:03, 1 November 2010
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	<math>\ \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)} \,\!</math>		<math>\ \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)} \,\!</math>


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

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

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

			After a lot of messy math and work:

			<math> A = \dfrac{1/2}{R-jwL} \,\!</math>

			<math> A* = \dfrac{1/2}{R+jwL} \,\!</math>

	<math> = ~~A(s^2~~-~~jws+~~R~~/Ls-jwR/L) + a*(s^2+jws+~~R~~/L+jwr/L) + B(s~~^2+w^2) \,\!</math>		<math> B = \dfrac{-R}{R^2+w^2L^2} \,\!</math>

Btheredude at 00:56, 2 November 2010

2010-11-02T00:56:27Z

← Older revision		Revision as of 17:56, 1 November 2010
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	<math>\ \dfrac{s}{(s^2+w^2)(R+Ls)} = \dfrac{A}{(s+jw)} + \dfrac{A*}{(s-jw)} + \dfrac{B}{(Ls+R)} \,\!</math>		<math>\ \dfrac{s}{(s^2+w^2)(R+Ls)} = \dfrac{A}{(s+jw)} + \dfrac{A*}{(s-jw)} + \dfrac{B}{(Ls+R)} \,\!</math>

	<math>\ \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)} \,\!</math>		<math>\ \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)} \,\!</math>

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

	= A(s^2-jws+R/Ls-jwR/L) + a*(s^2+jws+R/L+jwr/L) + B(s^2+w^2) \,\!</math>
			<math> = A(s^2-jws+R/Ls-jwR/L) + a*(s^2+jws+R/L+jwr/L) + B(s^2+w^2) \,\!</math>

Btheredude at 00:51, 2 November 2010

2010-11-02T00:51:15Z

← Older revision		Revision as of 17:51, 1 November 2010
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	<math>\ \dfrac{s}{(s^2+w^2)(R+Ls)} = \dfrac{A}{(s+jw)} + \dfrac{A*}{(s-jw)} + \dfrac{B}{(Ls+R)} \,\!</math>		<math>\ \dfrac{s}{(s^2+w^2)(R+Ls)} = \dfrac{A}{(s+jw)} + \dfrac{A*}{(s-jw)} + \dfrac{B}{(Ls+R)} \,\!</math>

	<math>\ (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)) \,\!</math>		<math>\ \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)} \,\!</math>

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

Btheredude at 00:32, 2 November 2010

2010-11-02T00:32:48Z

← Older revision		Revision as of 17:32, 1 November 2010
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	Therefore the Resulting Equation for the system after applying the Laplace Transform:		Therefore the Resulting Equation for the system after applying the Laplace Transform:

	<math>\ 0 = -s/(s^2+w^2) + RI(s) + LsI(s) - Li \,\!</math>		<math>\ 0 = \dfrac{-s}{(s^2+w^2)} + RI(s) + LsI(s) - Li \,\!</math>

	A series of algebraic manipulations follows to come up with I(s):		A series of algebraic manipulations follows to come up with I(s):
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	We can then use partial fraction manipulation to expand the expression:		We can then use partial fraction manipulation to expand the expression:

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

	<math>\ (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)) \,\!</math>		<math>\ (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)) \,\!</math>

Btheredude at 00:31, 2 November 2010

2010-11-02T00:31:17Z

← Older revision		Revision as of 17:31, 1 November 2010
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	<math>\ I(s) = s/((s^2+w^2)(R+Ls)) - Li/(R+Ls) \,\!</math>		<math>\ I(s) = \dfrac{s}{(s^2+w^2)(R+Ls)} - \dfrac{Li}{(R+Ls)} \,\!</math>

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

			<math>\ s/((s^2+w^2)(R+Ls)) = A/(s+jw) + A*/(s-jw) + B/(Ls+R) \,\!</math>

			<math>\ (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)) \,\!</math>

			<math>\ 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) \,\!</math>

Btheredude at 00:09, 2 November 2010

2010-11-02T00:09:30Z

← Older revision		Revision as of 17:09, 1 November 2010
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	[[File:RLcircuit.jpg]]		[[File:RLcircuit.jpg]]

	The Laplace transform for an inductor		The Laplace transform for an inductor:

	<math>\displaystyle\mathcal{L} \left\{f(t)\right\}</math> = <math>\ Ls + Li \,\!</math>		<math>\displaystyle\mathcal{L} \left\{f(t)\right\}</math> = <math>\ Ls + Li \,\!</math>

	The Laplace transform for a resistor ~~is just the resistor itself~~		The Laplace transform for a resistor:

	<math>\displaystyle\mathcal{L} \left\{f(t)\right\}</math> = <math>\ R\,\!</math>		<math>\displaystyle\mathcal{L} \left\{f(t)\right\}</math> = <math>\ R\,\!</math>
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	<math>\ 0 = -s/(s^2+w^2) + RI(s) + LsI(s) - Li \,\!</math>		<math>\ 0 = -s/(s^2+w^2) + RI(s) + LsI(s) - Li \,\!</math>

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


			<math>\ s/(s^2+w^2) = (R+Ls)I(s) + Li \,\!</math>


			<math>\ I(s) = s/((s^2+w^2)(R+Ls)) - Li/(R+Ls) \,\!</math>

Btheredude at 00:06, 2 November 2010

2010-11-02T00:06:07Z

← Older revision		Revision as of 17:06, 1 November 2010
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	''I(0)=i''		''I(0)=i''

			[[File:RLcircuit.jpg]]

			The Laplace transform for an inductor

			<math>\displaystyle\mathcal{L} \left\{f(t)\right\}</math> = <math>\ Ls + Li \,\!</math>

			The Laplace transform for a resistor is just the resistor itself

			<math>\displaystyle\mathcal{L} \left\{f(t)\right\}</math> = <math>\ R\,\!</math>

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

			<math>\ 0 = -s/(s^2+w^2) + RI(s) + LsI(s) - Li \,\!</math>

Btheredude at 23:44, 1 November 2010

2010-11-01T23:44:56Z

← Older revision		Revision as of 16:44, 1 November 2010
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	Given:		'''''Given:'''''

	V(t)=<math>\cos w*t \,\!</math>		V(t)=<math>\cos w*t \,\!</math>

Btheredude at 23:44, 1 November 2010

2010-11-01T23:44:08Z

← Older revision		Revision as of 16:44, 1 November 2010
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	I decided that i would do the analysis of a RL circuit with the variables instead of given values.		I decided that i would do the analysis of a RL circuit with the variables instead of given values.


	Given:		Given:

			V(t)=<math>\cos w*t \,\!</math>

			V(s)=<math>\ s/(s^2+w^2) \,\!</math>

			''I(0)=i''