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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 with V_C(0) = 1.0 V. Solve for the current i(t) in the circuit.

Answer

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 sin5t V, R = 4 W and L = 2 H.

Answer

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 i₁ and i₂, and the charge on C at time t greater than zero.

Answer

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.

Answer

Alternative answer using Scientific Notebook. (.tex file)

 

EXAMPLE 5

The system is quiescent. Find the loop current i₂(t).

Answer

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 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.

Answer

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 vR(t):

Note: v(t) = 0 V for all t < 0 s implies v(0^-) = 0 V.

Answer

Alternative answer using Scientific Notebook. (.tex file)

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Chapter Contents

• Laplace Transforms
• 1a. The Unit Step Function - Definition
• 1b. The Unit Step Function - Products
• 2. Laplace Transform Definition
• Table of Laplace Transformations
• 3. Properties of Laplace Transform
• 4. Transform of Unit Step Functions
• 5. Transform of Periodic Functions
• 6. Transforms of Integrals
• 7. Inverse of the Laplace Transform
• 8. Using Inverse Laplace to Solve DEs
• 9. Integro-Differential Equations and Systems of DEs
• 10. Applications of Laplace Transform

Following are the original SNB files (.tex or .rap) used in making this chapter. For more information, go to SNB info.

SNB files

1a. The Unit Step Function - Definition

1b. The Unit Step Function - Products

2. Laplace Transform Definition

3. Properties of Laplace Transform

4. Transform of Unit Step Functions

5. Transform of Periodic Functions

6. Transforms of Integrals

7. Inverse of the Laplace Transform

8. Using Inverse Laplace to Solve DEs

9. Integro-Differential Equations and Systems of DEs

10. Applications of Laplace Transform

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