9. The Forced Response - Second Order Linear DEs

As in first order circuits, the forced response has the form of the driving function. For a constant driving source, it results in a constant forced response.

For non-constant driving functions e.g. when

`E = E_0 sin omega t`,

the complete response of a circuit is the sum of a natural response and a forced response.

Note: Such solutions can also be obtained using the Laplace transformation method (which we meet later) when initial conditions are given.

Constant Forced Response

Example 1

In a RLC series circuit, `R = 10\ Omega`, `C = 0.02\ "F"`, `L = 1\ "H"` and the voltage source is `E = 100\ "V"`. Solve for the current `i(t)` in the circuit given that at time `t = 0`, the current in the circuit is zero and the charge in the capacitor is `0.1\ "C"`.

Natural and Non-Constant Forced Response

Example 2

Here is an example showing the effect of a forced response. We consider the "natural" case first (with constant EMF).

2a. Natural Response

In an RLC circuit we have `L = 1\ "H"`, `R = 10\ Omega` and `C = 0.0025\ "F"` and at `t = 0`, the current is `0` and `i'(0) = 0.1\ "A/s"`.

Solve for `i`.

2b. Forced Response

Let us now take the same RLC circuit we had in Example 2a, but now we have a non-constant EMF (electromotive force) of:

`E = -0.08\ cos\ 2.5t`

Using the following result (from Section 8)

`L(di)/(dt)+Ri+1/Cq=E`

with

`L = 1`, `R = 10` and `1/C = 1/0.0025 = 400`,

we differentiate throughout with respect to t to give the following 2nd order DE, with initial conditions shown:

`(d^2i)/(dt^2)+10(di)/(dt)+400i=0.2\ sin\ 2.5t`

`i(0)=0`

`i'(0)=0.1`

Here is the graph for the Forced Response solution we found just now:

Graph of second roder differential equation forced response

Graph of `i(t)`.

You can see after the initial spike, the current settles down into a sinusoidal pattern.

We will also use the Laplace Transform in a later section to solve this type of DE.