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Solution Method for Solving General Second Order Circuits

The concepts we have learned to use in the solving of RLC series and parallel circuits can be applied to any second order circuit which has one or more independent sources with constant values. Given any second order circuit, the step response is represented by:

$$x(t) = step \; response $$

...where x(t) may be either a voltage or current.

Four steps to determine x(t):

1) Determine initial conditions and final values:

First, determine the numerical values for x(0), x'(0), as well as the final value for x where:

$$x(\infty)= final \; value \; of \; x $$

2) Determine transient response:

Next, we turn off any independent sources and determine the transient response for t>0 using Kirchoff's Voltage Law (KVL) and Kirchoff's Current Law (KCL).

$$x_t(t) = transient \; response $$

In the process of doing this, we will obtain a second order differential equation which we use to get a "characteristic equation" along with its roots. It may be interesting to note that the turning off of the independent sources is equivalent to solving for the "homogeneous case if we were to use the Annihilator Method

Depending on whether the response is over/critically or underdamped, we obtain an expression for x(t) with two unknown constants.

3) Determine the steady state response for t>0:

Recall that: $$x(\infty) = x_{ss}(t) = steady \; state \; response $$

4) Obtain the Complete Response:

We can now express the total/complete response as: $$x(t) = x_{ss}(t) + x_t(t) $$ The values for any constants associated with the transient response are determined via initial conditions x(0) and x'(0).

Recalling the definitions for alpha and omega for series and parallel RLC circuits:

For series circuits: $$\alpha = \frac{R}{2L} $$ $$\omega_0 = \frac{1}{\sqrt{LC}} $$ For parallel circuits: $$\alpha = \frac{1}{2RC} $$ $$\omega_0 = \frac{1}{\sqrt{LC}} $$

If Overdamped:

then: $$\alpha \gt \omega_0$$ ...and solutions will be of the form: $$x(t) = x_{ss}(t) + Ae^{s_1t} + Be^{s_2t} $$ ...where s1 and s2 are the roots of the "characteristic equation"

If Critically Damped:

then: $$\alpha = \omega_0 $$ ...and solutions will be of the form: $$x(t) = x_{ss}(t) + Ae^{-\alpha t} + Bte^{-\alpha t} $$

If Underdamped:

then: $$\alpha \lt \omega_0 $$ ...and solutions will be of the form: $$x(t) = x_{ss}(t) + Ae^{-\alpha t}cos(\omega_d t) + Be^{-\alpha t}sin(\omega_d t) $$ ...where: $$\omega_d = \sqrt{\omega_0^2 - \alpha^2} $$

Lets look at an example

Continue on to General 2nd order circuits (example 1)