lt4 - Boise State University Department of Electrical and...

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Department of Electrical and Computer Engineering ECE 225 – Circuit Analysis and Design The Laplace Transform IV Lecture Objectives : 1. To discuss the unit impulse (or Dirac delta) function and its applications. 2. To review important inverse Laplace transforms. 0 ε ε ε 1 - ε 2 ε ε u (t) (t) ε 0.5 1 0 2 δ u(t) (t) δ 0 (.)dt 0 1 2 ε - 2 ε 0 0 (.)dt δ ² ( t ) = 1 ² u ( t + ² 2 ) - 1 ² u ( t - ² 2 ) δ ( t ) = lim ² 0 δ ² ( t ) ² ( s ) = 1 ² ˆ e ²s/ 2 s - e - ²s/ 2 s ! = ˆ e ²s/ 2 - e - ²s/ 2 ²s ! L{ δ ( t ) } = lim ² 0 ² ( s ) = lim ² 0 ˆ e ²s/ 2 - e - ²s/ 2 ²s ! = lim ² 0 ˆ ( s/ 2) e ²s/ 2 - ( - s/ 2) e - ²s/ 2 s ! = 1 L{ u ( t ) } = L{ δ ( t ) } s = 1 s Property 1 : R -∞ δ ( t ) dt = R 0+ 0 - δ ( t ) dt = 1 = R τ + τ - δ ( t - τ ) dt = 1 Property 2 : R -∞ δ ( t ) f ( t ) dt = R 0+ 0 - δ ( t ) f (0) dt = f (0) = R -∞ δ ( t - τ ) f ( t ) dt = f ( τ ) Property 3 : L{ δ ( t ) } = R 0 - δ ( t ) e - st dt
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lt4 - Boise State University Department of Electrical and...

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