Second Order Circuits: Difference between revisions
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'''Second order circuits''' are circuits that have two energy storage | '''Second order circuits''' are circuits that have two energy storage elements, resulting in second-order differential equations. | ||
elements, | |||
primarily two types: | There are primarily two types: | ||
* Parallel RLC circuits | * Parallel RLC circuits | ||
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== Unforced == | == Unforced == | ||
[[File:Unforced RLC Circuit.png|thumb|An unforced RLC circuit]] | [[File:Unforced RLC Circuit.png|thumb|An unforced series RLC circuit]] | ||
Consider an un-forced RLC circuit. We want to find <math>V_C</math>. | Consider an un-forced RLC circuit. We want to find <math>V_C</math>. | ||
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<math>\frac{d^2V_C}{dt^2} + \frac{R}{L} \frac{dV_C}{dt} + \frac{1}{LC} V_C = 0</math> | <math>\frac{d^2V_C}{dt^2} + \frac{R}{L} \frac{dV_C}{dt} + \frac{1}{LC} V_C = 0</math> | ||
= Parallel RLC Circuits = | |||
== Unforced == | |||
[[File:Parallel Unforced RLC Circuit.png|thumb|A parallel unforced RLC circuit]] | |||
[[Category:Electrical Engineering]] | [[Category:Electrical Engineering]] | ||
Revision as of 06:54, 8 March 2024
Second order circuits are circuits that have two energy storage elements, resulting in second-order differential equations.
There are primarily two types:
- Parallel RLC circuits
- Series RLC circuits
Series RLC Circuits
Unforced
Consider an un-forced RLC circuit. We want to find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_C} .
First, we can use KVL and KCL
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_R + V_L + V_C = 0}
Next, we can use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = C \frac{dV_C}{dt}} and substitution to get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RC \frac{dV_C}{dt} + L \frac{d}{dt} \frac{C V_C} {dt} V_C = 0}
Changing the order and moving the constants,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle LC \frac{d^2 V}{dt^2} + RC \frac{dV_C}{dt} + V_C = 0}
Moving constants away from the first term to get a second-order differential equation,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2V_C}{dt^2} + \frac{R}{L} \frac{dV_C}{dt} + \frac{1}{LC} V_C = 0}
Parallel RLC Circuits
Unforced
