# 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*s*-domain; - 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 sin5*t* 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 200*t*. 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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