10. Applications of Laplace Transforms
Circuit Equations
There are two (related) approaches:
 Derive the circuit (differential) equations in the time domain, then transform these ODEs to the sdomain;
 Transform the circuit to the sdomain, then derive the circuit equations in the sdomain (using the concept of "impedance").
We will use the first approach. We will derive the system equations(s) in the tplane, then transform the equations to the splane. We will usually then transform back to the tplane.
Example 1
Consider the circuit when the switch is closed at `t=0`, `V_C(0)=1.0\ "V"`. Solve for the current i(t) in the circuit.
Example 2
Solve for i(t) for the circuit, given that V(t) = 10 sin5t V, R = 4 W and L = 2 H.
Example 3
In the circuit shown below, the capacitor is uncharged at time t = 0. If the switch is then closed, find the currents i_{1} and i_{2}, and the charge on C at time t greater than zero.
Example 4
In the circuit shown, the capacitor has an initial charge of 1 mC and the switch is in position 1 long enough to establish the steady state. The switch is moved from position 1 to 2 at t = 0. Obtain the transient current i(t) for t > 0.
Example 5
The system is quiescent. Find the loop current i_{2}(t).
Example 6
Consider a series RLC circuit where R = 20 W, L = 0.05 H and C = 10^{4} F and is driven by an alternating emf given by E = 100 cos 200t. Given that both the circuit current i and the capacitor charge q are zero at time t = 0, find an expression for i(t) in the region t > 0.
Example 7
A rectangular pulse `v_R(t)` is applied to the RC circuit shown. Find the response, v(t).
Graph of `v_R(t)`:  

Note: v(t) = 0 V for all t < 0 s implies v(0^{}) = 0 V. (We'll use this in the solution. It means we take `v_0,` the voltage right up until the current is turned on, to be zero.)
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