Mechatronics Engineering MT-144 NETWORK ANALYSIS Mechatronics Engineering (12) 1

Laplace Transform

Definition of Laplace Transform The Laplace Transform is an integral transformation of a function f(t) from the time domain into the complex frequency domain, giving F(s) s: complex frequency Called “The One-sided or unilateral Laplace Transform”. In the two-sided or bilateral LT, the lower limit is -. We do not use this.

Definition of Laplace Transform Example 1 Determine the Laplace transform of each of the following functions shown below:

Definition of Laplace Transform Solution: The Laplace Transform of unit step, u(t) is given by

Definition of Laplace Transform Solution: The Laplace Transform of exponential function, e-atu(t),a>0 is given by

Definition of Laplace Transform Solution: The Laplace Transform of impulse function, δ(t) is given by

Functional Transform

TYPE f(t) F(s) Impulse Step Ramp Exponential Sine Cosine

TYPE f(t) F(s) Damped ramp Damped sine Damped cosine

Properties of Laplace Transform Step Function The symbol for the step function is K u(t). Mathematical definition of the step function:

f(t) = K u(t)

Properties of Laplace Transform Step Function A discontinuity of the step function may occur at some time other than t=0. A step that occurs at t=a is expressed as:

f(t) = K u(t-a)

Ex:

Three linear functions at t=0, t=1, t=3, and t=4 Y=mx+c

Expression of step functions Linear function +2t: on at t=0, off at t=1 Linear function -2t+4: on at t=1, off at t=3 Linear function +2t-8: on at t=3, off at t=4 Step function can be used to turn on and turn off these functions

Step Functions ON OFF

Properties of Laplace Transform Impulse Function The symbol for the impulse function is (t). Mathematical definition of the impulse function:

Properties of Laplace Transform Impulse Function The area under the impulse function is constant and represents the strength of the impulse. The impulse is zero everywhere except at t=0. An impulse that occurs at t = a is denoted K (t-a)

f(t) = K (t)

Properties of Laplace Transform Linearity If F1(s) and F2(s) are, respectively, the Laplace Transforms of f1(t) and f2(t) Example:

Properties of Laplace Transform Scaling If F (s) is the Laplace Transforms of f (t), then Example:

Properties of Laplace Transform Time Shift If F (s) is the Laplace Transforms of f (t), then Example:

Properties of Laplace Transform Frequency Shift If F (s) is the Laplace Transforms of f (t), then Example:

Properties of Laplace Transform Time Differentiation If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its derivative is Example:

Properties of Laplace Transform Time Integration If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its integral is Example:

Properties of Laplace Transform Frequency Differentiation If F(s) is the Laplace Transforms of f (t), then the derivative with respect to s, is Example:

Properties of Laplace Transform Initial and Final Values The initial-value and final-value properties allow us to find the initial value f(0) and f(∞) of f(t) directly from its Laplace transform F(s). Initial-value theorem Final-value theorem

The Inverse Laplace Transform Suppose F(s) has the general form of The finding the inverse Laplace transform of F(s) involves two steps: Decompose F(s) into simple terms using partial fraction expansion. Find the inverse of each term by matching entries in Laplace Transform Table.

The Inverse Laplace Transform Example 1 Find the inverse Laplace transform of Solution:

Partial Fraction Expansion Distinct Real Roots of D(s) s1= 0, s2= -8 s3= -6

1) Distinct Real Roots To find K1: multiply both sides by s and evaluates both sides at s=0 To find K2: multiply both sides by s+8 and evaluates both sides at s=-8 To find K3: multiply both sides by s+6 and evaluates both sides at s=-6

Find K1

Find K2

Find K3

Inverse Laplace of F(s)

2) Distinct Complex Roots S2 = -3+j4 S3 = -3-j4

Partial Fraction Expansion Complex roots appears in conjugate pairs.

Find K1

Find K2 and K2* Coefficients associated with conjugate pairs are themselves conjugates.

Inverse Laplace of F(s)

Inverse Laplace of F(s)

Useful Transform Pairs

The Convolution Integral It is defined as Given two functions, f1(t) and f2(t) with Laplace Transforms F1(s) and F2(s), respectively Example:

Operational Transform

Operational Transforms Indicate how mathematical operations performed on either f(t) or F(s) are converted into the opposite domain. The operations of primary interest are: Multiplying by a constant Addition/subtraction Differentiation Integration Translation in the time domain Translation in the frequency domain Scale changing

OPERATION f(t) F(s) Multiplication by a constant Addition/Subtraction First derivative (time) Second derivative (time)

OPERATION f(t) F(s) n th derivative (time) Time integral Translation in time Translation in frequency

OPERATION f(t) F(s) Scale changing First derivative (s) n th derivative s integral

Translation in time domain If we start with any function: we can represent the same function translated in time by the constant a, as: In frequency domain:

Ex:

Translation in frequency domain Translation in the frequency domain is defined as:

Ex:

Ex:

APPLICATION

Problem Assumed no initial energy is stored in the circuit at the instant when the switch is opened. Find the time domain expression for v(t) when t≥0.

Integrodifferential Equation A single node voltage equation:

s-domain transformation =0

Ex Obtain the Laplace transform for the function below: 1 2 3 t h(t) 4 1 2 3 t h(t) 4 5

Find the expression of f(t): Expression for the ramp function with slope, m =2 and period, T=2: For a periodic ramp function, we can write:

Different time occurred: Expanding: Different time occurred: t=0 and t=1

Equal time shift:

Inverse Laplace using translation in time property:

Time periodicity property: