Laplace transforms:Series RLC circuit - Revision history

Nathan.reeves: /* State Equations */

2009-12-15T04:31:55Z

State Equations

← Older revision		Revision as of 21:31, 14 December 2009
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	<math> i=\begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} \int{idt} \\ i \end{bmatrix} +0 </math>		<math> i=\begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} \int{idt} \\ i \end{bmatrix} +0 </math>

			Next, we solve the system using the matrix exponential method.

	MATLAB tells us that <math>e^{At}</math> is		MATLAB tells us that <math>e^{At}</math> is

Nathan.reeves: /* State Equations */

2009-12-04T01:14:31Z

State Equations

← Older revision		Revision as of 18:14, 3 December 2009
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	<math>\begin{bmatrix} i \\ \dfrac{di}{dt} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ 1000000 & 10 \end{bmatrix} \begin{bmatrix} \int{idt} \\ i \end{bmatrix} + \begin{bmatrix} 0 \\ 100cos(20t) \end{bmatrix}</math>		<math>\begin{bmatrix} i \\ \dfrac{di}{dt} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ 1000000 & 10 \end{bmatrix} \begin{bmatrix} \int{idt} \\ i \end{bmatrix} + \begin{bmatrix} 0 \\ 100cos(20t) \end{bmatrix}</math>

	~~The initial conditions mean that~~		<math> i=\begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} \int{idt} \\ i \end{bmatrix} +0 </math>

	<math> ~~\int{idt}~~=\begin{bmatrix} 1 ~~& 0~~ \end{bmatrix} \begin{bmatrix} \int{idt} \\ i \end{bmatrix} +0 </math>

	MATLAB tells us that <math>e^{At}</math> is		MATLAB tells us that <math>e^{At}</math> is
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	<math>		<math>
	e^{At}x(0) + \int_{0}^{3} e^{A(t-\tau)}Bu(\tau) \, d\tau		x(t)=e^{At}x(0) + \int_{0}^{t} e^{A(t-\tau)}Bu(\tau) \, d\tau
			</math>

			Since x(0)=0,

			<math>
			x(t)=\int_{0}^{t} e^{A(t-\tau)}Bu(\tau) \, d\tau
	</math>		</math>

			This gives the same solution as we got above.
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	Written by Nathan Reeves ~ Checked by		Written by Nathan Reeves ~ Checked by

Nathan.reeves: /* State Equations */

2009-12-04T01:06:07Z

State Equations

@@ Line 124: / Line 124: @@
 <math> \int{idt}=\begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} \int{idt} \\ i \end{bmatrix} +0 </math>
 ----
 Written by Nathan Reeves ~ Checked by

Nathan.reeves: /* Laplace Transform Example: Series RLC Circuit */

2009-11-02T06:02:18Z

Laplace Transform Example: Series RLC Circuit

@@ Line 93: / Line 93: @@
 Looking at the top part of the Bode plot, we see that the graph is indeed going down at roughly 40db/decade at 1000.
 ----
 Written by Nathan Reeves ~ Checked by

Nathan.reeves: /* Bode Plot */

2009-11-02T05:24:32Z

Bode Plot

← Older revision		Revision as of 22:24, 1 November 2009
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	To get a Bode plot, we use the transfer function:		To get a Bode plot, we use the transfer function:

	<math>H(s)=\dfrac{~~s^2~~}{~~(s^2+400)(~~0.01s^2+s+10000~~)}=\dfrac{s^2}{0.01s^4+s^3+10004s^2+400s+4000000~~}</math>		<math>H(s)=\dfrac{1}{0.01s^2+s+10000}</math>

	We then use a program such as Octave or MATLAB to obtain the Bode plot, which looks like this:		We then use a program such as Octave or MATLAB to obtain the Bode plot, which looks like this:
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	The break points of our function are determined by the transfer function		The break points of our function are determined by the transfer function

	<math>H(s)=\dfrac{~~s^2~~}{~~(s^2+400)(~~0.01s^2+s+10000)}=\dfrac{~~100s^2~~}{~~(s^2+400)~~(s^2+100s+1000000)}</math>		<math>H(s)=\dfrac{1}{0.01s^2+s+10000}=\dfrac{100}{(s^2+100s+1000000)}</math>

	The break points are:		The break points are:

	~~<math>0 \upuparrows</math>(40db/decade up)~~
	~~<math>20 \downdownarrows</math>(40db/decade down)~~
	<math>1000 \downdownarrows</math>(40db/decade down)		<math>1000 \downdownarrows</math>(40db/decade down)

	Looking at the top part of the Bode plot, we see that the graph is indeed ~~going up from 0 to 20, and then is flat from 20 to 1000 since the break point at 20 counteracted the break point at 0. Finally, the graph is~~ going down ~~after 1000, and~~ at ~~about~~ 40db/decade.		Looking at the top part of the Bode plot, we see that the graph is indeed going down at roughly 40db/decade at 1000.

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	Written by Nathan Reeves ~ Checked by		Written by Nathan Reeves ~ Checked by

Nathan.reeves: /* Laplace Transform Example: Series RLC Circuit */

2009-10-29T05:34:04Z

Laplace Transform Example: Series RLC Circuit

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 <br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
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 Written by Nathan Reeves ~ Checked by

Nathan.reeves at 06:31, 26 October 2009

2009-10-26T06:31:02Z

← Older revision		Revision as of 23:31, 25 October 2009
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	We then use a program such as Octave or MATLAB to obtain the Bode plot, which looks like this:		We then use a program such as Octave or MATLAB to obtain the Bode plot, which looks like this:

			[[Image:ReevesBodePlot.jpg\|400px\|thumb\|left\|Bode Plot]]

			<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>

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	Written by Nathan Reeves ~ Checked by		Written by Nathan Reeves ~ Checked by

Nathan.reeves at 06:26, 26 October 2009

2009-10-26T06:26:46Z

← Older revision		Revision as of 23:26, 25 October 2009
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	This time, when we actually evaluate ''i''(∞) from the equation for <math>i(t)</math>, we find it to be undefined. So here, the Final Value Theorem tells us something that is not necessarily true (in fact, because we have oscillating functions, we know that ''i''(∞) will not be zero).		This time, when we actually evaluate ''i''(∞) from the equation for <math>i(t)</math>, we find it to be undefined. So here, the Final Value Theorem tells us something that is not necessarily true (in fact, because we have oscillating functions, we know that ''i''(∞) will not be zero).

			===Bode Plot===
			To get a Bode plot, we use the transfer function:

			<math>H(s)=\dfrac{s^2}{(s^2+400)(0.01s^2+s+10000)}=\dfrac{s^2}{0.01s^4+s^3+10004s^2+400s+4000000}</math>

			We then use a program such as Octave or MATLAB to obtain the Bode plot, which looks like this:


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	Written by Nathan Reeves ~ Checked by		Written by Nathan Reeves ~ Checked by

Nathan.reeves at 22:12, 22 October 2009

2009-10-22T22:12:12Z

@@ Line 46: / Line 46: @@
 which is our answer.
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 Written by Nathan Reeves ~ Checked by

Nathan.reeves at 04:14, 20 October 2009

2009-10-20T04:14:47Z

← Older revision		Revision as of 21:14, 19 October 2009
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	Written by Nathan Reeves		Written by Nathan Reeves ~ Checked by
	Checked by