# 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 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.

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