# 10. Applications of Laplace Transforms

## Circuit Equations

There are two (related) approaches:

1. Derive the circuit (differential) equations in the time domain, then transform these ODEs to the s-domain;
2. Transform the circuit to the s-domain, then derive the circuit equations in the s-domain (using the concept of "impedance").

We will use the first approach. We will derive the system equations(s) in the t-plane, then transform the equations to the s-plane. We will usually then transform back to the t-plane.

### 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 i1 and i2, 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 i2(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).

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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