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laplace transform for engineering students
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- 3. Applications of LaplaceTransforms 1.Laplace transform is used in electronic engineering to solve differential equations occurring in the analysis of electronic circuits. 2.Laplace transform are used in solving digital signal processing problem 3.Laplace transform are used to get the true form of radioactive decay
- 4. Objective: Define Laplacetransform of periodic function Solve the problems based on Laplace transform of periodic function
- 5. Definition: A function𝑓ሺ 𝑡ሻis said to be a periodic function of period 𝑇> 0 if 𝑓ሺ 𝑡+ 𝑛𝑇 ሻ= 𝑓ሺ 𝑡ሻwhere 𝑛 = 1,2,3….. Example: 𝑠𝑖𝑛𝑡,𝑐𝑜𝑠𝑡 are periodic functions of period 2𝜋since 𝑠𝑖𝑛ሺ 𝑡+ 2𝑛𝜋 ሻ= 𝑠𝑖𝑛𝑡,𝑐𝑜𝑠ሺ 𝑡+ 2𝑛𝜋 ሻ= 𝑐 𝑜𝑠𝑡 Result: If 𝑓ሺ 𝑡ሻis a periodic function of period 𝑇 , then 𝐿 ሾ 𝑓ሺ 𝑡ሻ ሿ= 1 1− 𝑒−𝑠𝑇 න𝑒−𝑠𝑡 𝑇 0 𝑓ሺ 𝑡ሻ 𝑑𝑡 LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
- 6. 𝐿 ሾ 𝑓ሺ 𝑡ሻ ሿ= 𝐸 1− 𝑒−𝜋 𝑠 𝜔 Τ ቈ 𝑒−𝑠𝑡 𝑠2+ 𝜔2 ሺ −𝑠𝑠𝑖𝑛𝜔𝑡− 𝜔𝑐𝑜𝑠𝜔𝑡ሻ 0 𝜋 𝜔 Τ Problems 1. Find the Laplace transform of the full wave rectifier 𝑓ሺ 𝑡ሻ= 𝐸𝑠𝑖𝑛𝜔𝑡, 0 < 𝑡 < 𝜋 𝜔 Τ Soln: Here period 𝑇= 𝜋 𝜔 Τ and 𝑓ሺ 𝑡ሻ= 𝐸𝑠𝑖𝑛𝜔𝑡 𝐿 ሾ 𝑓ሺ 𝑡ሻ ሿ= 1 1− 𝑒−𝜋 𝑠 𝜔 Τ න 𝑒−𝑠𝑡 𝜋 𝜔 Τ 0 𝐸𝑠𝑖𝑛𝜔𝑡𝑑𝑡 We Know 𝐿 ሾ 𝑓ሺ 𝑡ሻ ሿ= 1 1−𝑒−𝑠𝑇 𝑒−𝑠𝑡 𝑇 0 𝑓ሺ 𝑡ሻ 𝑑𝑡 𝐿 ሾ 𝑓ሺ 𝑡ሻሿ= 𝐸 1− 𝑒−𝜋 𝑠 𝜔 Τ න 𝑒−𝑠𝑡 𝜋 𝜔 Τ 0 𝑠𝑖𝑛𝜔𝑡𝑑𝑡 We Know 𝑒𝑎𝑥 𝑠𝑖𝑛𝑏𝑥𝑑𝑥= 𝑒𝑎𝑥 𝑎2+𝑏2 ሺ 𝑎𝑠𝑖𝑛𝑏𝑥− 𝑏𝑐 𝑜𝑠𝑏𝑥ሻ
- 7. = 𝐸 𝜔൫ 1+ 𝑒−𝜋 𝑠 𝜔 Τ ൯ ሺ 𝑠2+ 𝜔2ሻ ሺ 1− 𝑒−𝜋 𝑠 𝜔 Τ ሻ Multiply both the numerator and denominator by 𝑒𝜋 𝑠 2𝜔 Τ in RHS, we get 𝐿 ሾ 𝑓ሺ 𝑡ሻሿ= 𝐸 𝜔 ሺ 𝑠2+𝜔2ሻ . ൫ 𝑒𝜋 𝑠 2𝜔 Τ +𝑒−𝜋 𝑠 2𝜔 Τ ൯ ൫ 𝑒𝜋 𝑠 2𝜔 Τ −𝑒−𝜋 𝑠 2𝜔 Τ ൯ = 𝐸 ሺ 1− 𝑒−𝜋 𝑠 𝜔 Τ ሻ ሺ 𝑠2 + 𝜔2ሻ ൣ 𝑒−𝜋 𝑠 𝜔 Τ .𝜔− ሺ −𝜔ሻ ൧
- 8. 𝐿 ሾ 𝑓ሺ 𝑡ሻሿ= 𝐸 𝜔 ሺ 𝑠2 + 𝜔2ሻ . 2𝑐𝑜𝑠ℎሺ 𝜋 𝑠2𝜔 Τ ሻ 2𝑠𝑖𝑛ℎሺ 𝜋 𝑠 2𝜔 Τ ሻ 𝐿 ሾ 𝑓ሺ 𝑡ሻ ሿ= 𝐸 𝜔 ሺ 𝑠2 + 𝜔2ሻ 𝑐 𝑜𝑡ℎሺ 𝜋 𝑠 2𝜔 Τ ሻ We know 2𝑐 𝑜𝑠ℎ𝑥= ሺ 𝑒𝑥 + 𝑒−𝑥ሻand 2𝑠𝑖𝑛ℎ𝑥= ሺ 𝑒𝑥 − 𝑒−𝑥ሻ
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