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 = E0 sin ωt,
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
In a RCL series circuit, R = 10 Ω, 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 6 (Using Scientific Notebook)
Here is an example using Scientific Notebook showing the effect of a forced response. (You need SNB to see the result. See SNB information).
a. Natural Response
In an RLC circuit we have L = 1 H , R = 10 Ω and C = 0.0025 F and at t = 0, the current is 0 and i'(0) = 0.1 A/s.
Solve for i.
Solution:
We need to solve:
[In Scientific Notebook, go to Compute menu, Solve ODE..., Exact]
Get the SNB file.
b. Forced Response
Let us now take the RLC circuit in Example 6a but have a non-constant EMF
E = -0.08 cos 2.5t
Using the result (from Section 8)
with L = 1, R = 10 and 1/C = 400, we differentiate throughout with respect to t to give the following 2nd order DE, with initial conditions shown:
See the SNB file (.TEX file) for the (long, long) result...
[Go here for SNB information.]
Now for the graphs that we obtain using our SNB solution:
We will also use the Laplace Transform in a later section to solve this type of DE.
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SNB files