# 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.

**Note: **Throughout this page these problems are also solved using Scientific Notebook. They are TEX files and you need Scientific Notebook or similar, to view them.

Alternative answer using Scientific Notebook. (.tex file)

### Example 2

Solve for *i*(*t*) for the circuit, given that *V*(*t*) = 10 sin5*t* V, *R* = 4 W and *L* = 2 H.

Alternative answer using Scientific Notebook. (.tex file)

### 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.

Alternative answer using Scientific Notebook. (.tex file)

### 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.

Alternative answer using Scientific Notebook. (.tex file)

### Example 5

The system is quiescent. Find the loop current *i*_{2}(*t*).

Alternative answer using Scientific Notebook. (.tex file)

### 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.

Alternative answer using Scientific Notebook. (.tex file)

### 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.)

Alternative answer using Scientific Notebook. (.tex file)

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