Second Order Circuits: Difference between revisions

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* '''Underdamped''', where there are complex solutions
* '''Underdamped''', where there are complex solutions
* '''Critically damped''', where the solutions are not distinct.
* '''Critically damped''', where the solutions are not distinct.
For an overdamped response, we have
<math>
v = A_1 e^{s_1 t} + A_2 e^{s_2 t}
</math>
The A's can be solved by substituting in v(0) and dv(0)/dt




[[Category:Electrical Engineering]]
[[Category:Electrical Engineering]]

Revision as of 07:59, 8 March 2024

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.

There are primarily two types of second order circuits:

  • Parallel RLC circuits
  • Series RLC circuits

This page will analyze them and derive some useful equations.

Series RLC Circuits

Natural Response

An unforced series RLC circuit

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,

Parallel RLC Circuits

Natural Response

A parallel unforced RLC circuit

By KCL,

By differentiating once with respect to and rearranging some constants,

we get a homogeneous second-order differential equation, which has a standard solution that I will not go into detail. Briefly, it is solved by assuming since derivatives of must take the same form to cancel out to zero.

By applying the standard solution, we have

Characteristic Equation

The above simplifies to

This is the characteristic equation of the differential equation, as the root of the quadratic determines properties of

where

and

It can be pretty easily proven that the sum of the two roots is also a solution

Forms

Depending on the root, there are three forms:

  • Overdamped, where there are real, distinct solutions
  • Underdamped, where there are complex solutions
  • Critically damped, where the solutions are not distinct.

For an overdamped response, we have

The A's can be solved by substituting in v(0) and dv(0)/dt