Second Order Circuits - Revision history

Rice: /* Series RLC Circuits Natural Response */

2024-03-08T22:26:25Z

Series RLC Circuits Natural Response

← Older revision		Revision as of 22:26, 8 March 2024
Line 204:		Line 204:

	<math>		<math>
	v(t) = B_1 e^{-\alpha t} \cos \omega_d t + B_2 e^{-\alpha t} \sin \omega_d t		i(t) = B_1 e^{-\alpha t} \cos \omega_d t + B_2 e^{-\alpha t} \sin \omega_d t
	</math>		</math>

2024-03-08T18:26:30Z

← Older revision		Revision as of 18:26, 8 March 2024
Line 7:		Line 7:
	* Parallel RLC circuits		* Parallel RLC circuits
	* Series RLC circuits		* Series RLC circuits

			each with two responses:
			* Natural response (no power source)
			* Forced response (has power source)

	This page will analyze them and derive some useful equations.		This page will analyze them and derive some useful equations.

2024-03-08T18:25:23Z

← Older revision		Revision as of 18:25, 8 March 2024
Line 1:		Line 1:
	'''Second order circuits''' are circuits that have two energy storage elements, resulting in second-order differential equations.		'''Second order circuits''' are circuits that have two energy storage elements, resulting in second-order differential equations.

	One application of second order circuits is in timing computers. As we will see, an RLC circuit can generate a sinusoidal wave.		One application of second order circuits is in timing computers. As we will see, an underdamped RLC circuit can generate a sinusoidal wave.

	There are primarily two types of second order circuits:		There are primarily two types of second order circuits:

2024-03-08T11:21:14Z

Characteristic Equation

← Older revision		Revision as of 11:21, 8 March 2024
Line 78:		Line 78:

	== Forms ==		== Forms ==
			[[File:Equations for Analyzing the Step Response of Parallel RLC Circuits.png\|alt=Equations for Analyzing the Step Response of Parallel RLC Circuits\|thumb\|Equations for Analyzing the Step Response of Parallel RLC Circuits]]
	Depending on the root, there are three forms:		Depending on the root, there are three forms:
	* '''Overdamped''', where there are real, distinct solutions		* '''Overdamped''', where there are real, distinct solutions

2024-03-08T10:56:46Z

← Older revision		Revision as of 10:56, 8 March 2024
Line 181:		Line 181:
	The form equations are identical to parallel, only with different		The form equations are identical to parallel, only with different
	<math>\alpha</math>:		<math>\alpha</math>:
			[[File:Equations for Analyzing the Step Response of Series RLC Circuits.png\|thumb\|Equations for Analyzing the Step Response of Series RLC Circuits]]

	'''Overdamped:'''		'''Overdamped:'''

2024-03-08T10:55:17Z

← Older revision		Revision as of 10:55, 8 March 2024
Line 10:		Line 10:
	This page will analyze them and derive some useful equations.		This page will analyze them and derive some useful equations.

	= Parallel RLC Circuits =		= Parallel RLC Circuits Natural Response =

	~~== Natural Response ==~~
	[[File:Parallel Unforced RLC Circuit.png\|thumb\|A parallel unforced RLC circuit]]		[[File:Parallel Unforced RLC Circuit.png\|thumb\|A parallel unforced RLC circuit]]

Line 42:		Line 41:
	</math>		</math>

	=== Characteristic Equation ===		== Characteristic Equation ==

	The above simplifies to		The above simplifies to
Line 150:		Line 149:
	and the circuit is damped.		and the circuit is damped.

	= Series RLC Circuits =		= Series RLC Circuits Natural Response =

			[[File:Unforced RLC Circuit.png\|thumb\|An unforced series RLC circuit]]

	Calculations are very similar to that of parallel circuits, so I'll		Calculations are very similar to that of parallel circuits, so I'll
	speed things up.		speed things up.

	~~== Natural Response ==~~
	~~[[File:Unforced RLC Circuit.png\|thumb\|An unforced series RLC circuit]]~~

	The second order differential from KVL is		The second order differential from KVL is
Line 186:		Line 185:

	<math>		<math>
	i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}		i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}
	</math>		</math>

2024-03-08T10:22:16Z

Series RLC Circuits

2024-03-08T10:21:43Z

@@ Line 179: / Line 179: @@
 with neper frequency <math>\alpha = \frac{R}{2L} rad/s</math> and
 resonant radian frequency <math>\omega_0 = \frac{1}{\sqrt{LC}} rad / s</math>
 [[Category:Electrical Engineering]]

2024-03-08T10:14:34Z

@@ Line 9: / Line 9: @@
 This page will analyze them and derive some useful equations.
 = Parallel RLC Circuits =
@@ Line 173: / Line 149: @@
 Notice that when <math>R \neq 0</math>, there is <math>\alpha > 0</math>
 and the circuit is damped.
 [[Category:Electrical Engineering]]

2024-03-08T08:41:28Z

@@ Line 158: / Line 158: @@
 -\alpha B_1 + \omega_d B_2 =  \frac{1}{C} \left( \frac{-V_0}{R} - I_0 \right)
 </math>
 [[Category:Electrical Engineering]]