Second Order Circuits: Difference between revisions
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<math>iR + L \frac{di}{dt} + V_C = 0</math> | <math>iR + L \frac{di}{dt} + V_C = 0</math> | ||
Next, we can use <math>i = C \frac{dV_C}{dt}</math> to get | Next, we can use <math>i = C \frac{dV_C}{dt}</math> and substitution to get | ||
<math>RC \frac{dV_C}{dt} + L \frac{d}{dt} \frac{C V_C} {dt} V_C = 0</math> | <math>RC \frac{dV_C}{dt} + L \frac{d}{dt} \frac{C V_C} {dt} V_C = 0</math> | ||
Changing the order and moving the constants, | |||
<math>LC \frac{d^2 V}{dt^2} + RC \frac{dV_C}{dt} + V_C = 0</math> | <math>LC \frac{d^2 V}{dt^2} + RC \frac{dV_C}{dt} + V_C = 0</math> | ||
Moving constants away from the first term to get a ''second-order differential equation,'' | |||
<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> | ||
Revision as of 03:46, 1 March 2024
Second order circuits are circuits that have two energy storage
elements, resultingin second-order differential equations. The circuits have to be cast in terms of irreducible elements (i.e. combine L's and C's into one when possible).
Consider an un-forced RLC circuit. We want to find .
First, we can use KVL and KCL
Next, we can use and substitution to get
Changing the order and moving the constants,
Moving constants away from the first term to get a second-order differential equation,