Bài giảng Electric circuit theory - Chapter XIV: The Laplace Transform - Nguyễn Công Phương

Electric Circuit Theory The Laplace Transform Nguyễn Công Phương Contents I. Basic Elements Of Electrical Circuits II. Basic Laws III. Electrical Circuit Analysis IV. Circuit Theorems V. Active Circuits VI. Capacitor And Inductor VII. First Order Circuits VIII.Second Order Circuits IX. Sinusoidal Steady State Analysis X. AC Power Analysis XI. Three-phase Circuits XII. Magnetically Coupled Circuits XIII.Frequency Response XIV.The Laplace Transform XV. Two-port Networks The Laplace Transform - sites.google.com/site/ncpdhbkhn 2 F(s) = 0 (algebraic) I(s), V(s), Laplace Transform Inverse Transform The Laplace Transform The Laplace Transform - sites.google.com/site/ncpdhbkhn 3 f(t) = 0 (integrodifferential) i(t), v(t), Circuit The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 4 Definition The Laplace Transform - sites.google.com/site/ncpdhbkhn 5 t ( )f t 0[ ] 0( ) ( ) ( ) stF s L f t f t e dt ∞ − = = ∫ s jσ ω= + 0 ( ) tf t e dtσ∞ − < ∞∫ [ ] 1 1 1 1( ) ( ) ( ) 2 j st j f t L F s F s e dsj σ σpi + ∞ − − ∞ = = ∫ The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 6 Two Important Singularity Functions (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 7 t ( )u t 0 1 t ( )u t a− 0 1 a 0 0( ) 1 0 t u t t < =  > 0( ) 1 t a u t a t a < − =  > Two Important Singularity Functions (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 8 t ( )u t 0 1 Ex. 1 Determine the Laplace transform for the waveform? 0 ( ) ( ) stF s u t e dt∞ −= ∫ 0 1 ste dt ∞ − = ∫ 0 1 st e s ∞ − = − 1 s = Two Important Singularity Functions (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 9 Ex. 2 Determine the Laplace transform for the waveform? 0 ( ) ( ) stF s u t a e dt∞ −= −∫ 0 0 1 a st a dt e dt ∞ − = +∫ ∫ 1 st a e s ∞ − = − as e s − = t ( )u t a− 0 1 a Two Important Singularity Functions (4) The Laplace Transform - sites.google.com/site/ncpdhbkhn 10 Ex. 3 Determine the Laplace transform for the waveform? 0 ( ) [ ( ) ( )] stF s u t u t a e dt∞ −= − −∫ 0 1( ) stu t e dt s ∞ − =∫ 0 ( ) st st e u t a e dt s − ∞ − − =∫ 1 1( ) as ase eF s s s s − − − → = − = t ( )u t a− − 1− 0 a t ( )u t 0 1 t0 1 a Two Important Singularity Functions (5) The Laplace Transform - sites.google.com/site/ncpdhbkhn 11 t ( )tδ 0 t ( )t aδ − 0 a ( ) 0 0 ( ) 1 0 t t t dt ε ε δ δ ε − = ≠ = >∫ ( ) 0 ( ) 1 0a a t a t a t a dt ε ε δ δ ε+ − − = ≠ − = >∫ 2 1 1 2 1 2 ( )( ) ( ) 0 , t t f a t a tf t t a dt a t a t δ < <− =   ∫ Two Important Singularity Functions (6) The Laplace Transform - sites.google.com/site/ncpdhbkhn 12 Ex. 4 Determine the Laplace transform of an impulse function? 0 ( ) ( ) stF s t a e dtδ∞ −= −∫ 2 1 1 2 1 2 ( )( ) ( ) 0 , t t f a t a tf t t a dt a t a t δ < <− =   ∫ ( ) asF s e−→ = The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 13 Transform Pairs (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 14 Ex. 1 Find the Laplace transform of f(t) = t? 0 ( ) stF s te dt∞ −= ∫ 1Let & &st st stu t dv e dt du dt v e dt e s − − − = = → = = = −∫ 20 0 0 1( ) 0 st st stt e eF s e dt s s s s ∞∞ − − ∞ −→ = − + = − =∫ Transform Pairs (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 15 Ex. 2 Find the Laplace transform of f(t) =cosωt? 0 ( ) cos stF s te dtω∞ −= ∫ 0 2 j t j t ste e e dt ω ω− ∞ − + = ∫ ( ) ( ) 0 2 s j t s j te e dt ω ω− − − + ∞ + = ∫ 1 1 1 2 s j s jω ω   = +  − +  2 2 s s ω = + f(t) F(s) Transform Pairs (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 16 ( )tδ 1 ( )u t 1 s at e − 1 s a+ t 2 1 s atte− 2 1 ( )s a+ sin at 2 2 a s a+ cos at 2 2 s s a+ The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 17 Properties of the Transform (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 18 Property f(t) F(s) 1. Magnitude scaling 2. Addition/subtraction 3. Time scaling 4. Time shifting 5. Frequency shifting 6. Differentiation 7. Multiplication by t 8. Division by t 9. Integration 10. Convolution ( )Af t ( )AF s 1 2( ) ( )f t f t± 1 2( ) ( )F s F s± ( )f at 1 sF a a       ( ) ( ), 0f t a u t a a− − ≥ ( )ase F s− ( )ate f t− ( )F s a+ ( ) ( ), 0f t u t a a− ≥ [ ( )]ase L f t a− + ( ) /n nd f t dt 1 2 1 1( ) (0) (0) ... (0)n n n o ns F s s f s f s f− − −− − − ( )nt f t ( 1) ( ) /n n nd F s ds− ( ) /f t t ( ) s F dλ λ∞∫ 0 ( )t f dλ λ∫ ( ) /F s s 1 2 1 20 ( ) * ( ) ( ) ( )tf t f t f f t dλ λ λ= −∫ 1 2( ) ( )F s F s Properties of the Transform (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 19 Ex. 1 Find the Laplace transform of 10( ) 5 cos20 ?tf t e t−= + − 1 2 1 2( ) ( ) ( ) ( )f t f t F s F s± → ± 10( ) [5] [ ] [cos20 ]tF s L L e L t−→ = + − ( ) ( )Af t AF s→ [5] 5 [1]L L→ = 1[1]L s = 5[5]L s → = 10 1[ ] 10 tL e s − = + 2 2 2[cos20 ] 20 400 s sL t s s = = + + 3 2 2 5 1 5 2400 4000( ) 10 400 ( 10)( 400) s s sF s s s s s s s + + → = + − = + + + + Properties of the Transform (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 20 Ex. 2 Find the Laplace transform of the waveform? t0 5 1 2 3 t0 5 1 2 3 t0 5 1 2 3 ( ) 5 ( 1) 5 ( 2)f t u t u t= − − − 2 25( ) 5 5 ( ) s s s se eF s e e s s s − − − −→ = − = − t0 1 1 2 3 Properties of the Transform (4) The Laplace Transform - sites.google.com/site/ncpdhbkhn 21 Ex. 3 Find the Laplace transform of the waveform? t0 5 1 2 3 t0 5 1 2 3 ( ) ( 5 10)[ ( 1) ( 2)]f t t u t u t= − + − − − 5 ( 1) 10 ( 1) 5 ( 2) 10 ( 2) tu t u t tu t u t = − − + − + + − − − 5 ( 1) 5( 1 1) ( 1)tu t t u t− − = − − + − 5( 1) ( 1) 5 ( 1)t u t u t= − − − − − 5 ( 2) 5( 2 2) ( 2)tu t t u t− = − + − 5( 2) ( 2) 10 ( 2)t u t u t= − − + − ( ) 5( 1) ( 1) 5 ( 1) 10 ( 1) 5( 2) ( 2) 10 ( 2) 10 ( 2) f t t u t u t u t t u t u t u t → = − − − − − + + − + − − + − − − Properties of the Transform (5) The Laplace Transform - sites.google.com/site/ncpdhbkhn 22 Ex. 3 Find the Laplace transform of the waveform? t0 1 1 2 3 t0 5 1 2 3 t0 5 1 2 3 ( ) ( 5 10)[ ( 1) ( 2)]f t t u t u t= − + − − − 5( 1) ( 1) 5 ( 1) 10 ( 1) 5( 2) ( 2) 10 ( 2) 10 ( 2) t u t u t u t t u t u t u t = − − − − − + + − + − − + − − − 5( 1) ( 1) 5 ( 1) 5( 2) ( 2) t u t u t t u t = − − − + − + + − − 2 2 2 2 5( ) 5 5 5 (1 ) s s s s s e eF s e s s s e s e s − − − − − → = − + + = − − − Properties of the Transform (6) The Laplace Transform - sites.google.com/site/ncpdhbkhn 23 Ex. 4 Find the Laplace transform of the waveform? t0 5 1 2 3 The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 24 The Laplace Transform The Laplace Transform - sites.google.com/site/ncpdhbkhn 25 F(s) = 0 (algebraic) I(s), V(s), Laplace Transform Inverse Transform f(t) = 0 (integrodifferential) i(t), v(t), Circuit Inverse Transform (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 26 1 1 1 0 1 1 1 0 ( ) ...( ) ( ) ... m m m m n n n n P s a s a s a s aF s Q s b s b s b s b − − − − + + + + = = + + + + 1 2 1 2 Simple poles : ( ) ... n n K K KF s s p s p s p = + + + + + + 1 1 1 11 12 1 2 1 1 ( )Multiple poles : ( ) ( )( ) ... ...( ) ( ) ( ) n n n P sF s Q s s p K K K s p s p s p = + = + + + + + + + 1 1 * 1 1 ( )Complex- conjugate poles : ( ) ( )( )( ) ... P sF s Q s s j s j K K s j s j α β α β α β α β = + − + + = + + + − + + Inverse Transform (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 27 1 2 1 2 ( )Simple poles : ( ) ...( ) n n P s K K KF s Q s s p s p s p= = + + ++ + + ( )( ) 0 ... 0 0 ... 0( ) i i i s p P s s p KQ s =− + = + + + + + + 1 ip ti i i KL K e s p −−   = +  1 2 1 2( ) ... np tp t p t nf t K e K e K e−− −= + + + Inverse Transform (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 28 Ex. 1 Find the inverse Laplace transform of 225 300 640( ) ( 4)( 8) s sF s s s s + + = + + 1 2 3 ( )( ) ; ( ) 0 ... 0 0 ... 0 4 8 ( ) i i i s p K K K P sF s s p K s s s Q s = = + + + = + + + + + + + + 2 2 1 0 0 0 25 300 640 25 300 640 640( ) 20( 4)( 8) ( 4)( 8) 4 8s s s s s s sK sF s s s s s s s= = = + + + + = = = = = + + + + × 2 2 2 4 4 4 2 25 300 640 25 300 640( 4) ( ) ( 4) ( 4)( 8) ( 8) 25( 4) 300( 4) 640 10( 4)( 4 8) s s s s s s sK s F s s s s s s s=− =− =− + + + + = + = + = = + + + − + − + = = − − + Inverse Transform (4) The Laplace Transform - sites.google.com/site/ncpdhbkhn 29 Ex. 1 Find the inverse Laplace transform of 225 300 640( ) ( 4)( 8) s sF s s s s + + = + + 1 220; 10K K= = 2 2 3 8 8 8 2 25 300 640 25 300 640( 8) ( ) ( 8) ( 4)( 8) ( 4) 25( 8) 300( 8) 640 5( 8)( 8 4) s s s s s s sK s F s s s s s s s=− =− =− + + + + = + = + = = + + + − + − + = = − − − + 20 10 5( ) 4 8 F s s s s → = + − + + 4 8( ) 20 10 5t tf t e e− −→ = + − 1 2 3 ( )( ) ; ( ) 0 ... 0 0 ... 0 4 8 ( ) i i i s p K K K P sF s s p K s s s Q s = = + + + = + + + + + + + + Inverse Transform (5) The Laplace Transform - sites.google.com/site/ncpdhbkhn 30 Ex. 1 Find the inverse Laplace transform of 225 300 640( ) ( 4)( 8) s sF s s s s + + = + + 1 2 3( ) 4 8 K K KF s s s s = + + + + 2 1 25 300 640s sK s + + = 2 0 0 25 300 640 640 20( 4)( 8) 4 8( 4)( 8) s s s s s ss s = = + + = = = + + ×+ + 2 2 25 300 640 ( 4) s sK s s + + = + 2 44 25 300 640 10( 8)( 8) ss s s s ss =− =− + + = = ++ 2 3 25 300 640 ( 4) ( 8) s sK s s s + + = + + 2 88 25 300 640 5( 4) ss s s s s =− =− + + = = − + 20 10 5( ) 4 8 F s s s s → = + − + + 4 8( ) 20 10 5t tf t e e− −→ = + − Inverse Transform (6) The Laplace Transform - sites.google.com/site/ncpdhbkhn 31 Ex. 2 Find the inverse Laplace transform of 100( 6)( ) ( 1)( 3) sF s s s + = + + Inverse Transform (7) The Laplace Transform - sites.google.com/site/ncpdhbkhn 32 * 1 1 1 1 ( )Complex- conjugate poles : ( ) ...( )( )( ) P s K KF s Q s s j s j s j s jα β α β α β α β= = + ++ − + + + − + + 1 1 ( )( ) ( ) s j P s s j K KQ s α β α β =− + + − = = θ * 1 1K K= θ− 1( ) KF s→ = 1K s j θ α β ++ − 1 1 ... ... j jK e K e s j s j s j θ θθ α β α β α β − − + = + + + + + − + + ( ) ( ) ( ) ( ) 1 1 1( ) ... ...j j t j j t t j t j tf t K e e K e e K e e eθ α β θ α β α β θ β θ− − − − + − + − + → = + + = + +  cos sinje jφ φ φ= + [ ]1( ) cos( ) sin( ) cos( ) sin( ) ...tf t K e t j t t j tα β θ β θ β θ β θ−→ = + + + + − − + − − + 12 cos( ) ...tK e tα β θ−= + + Inverse Transform (8) The Laplace Transform - sites.google.com/site/ncpdhbkhn 33 Ex. 3 Find the inverse Laplace transform of 2 2 4 76( ) ( 2)( 6 25) s sF s s s s + = + + + 1 1 ( )( ) ; ( ) 2 cos( ) ...( ) t s j P sK s j f t K e tQ s α α β α β β θ− =− + = + − = + + 2 2 3 2 2 22 4 76 4 76( 2) 8( 2)( 6 25) 6 25 ss s s s sK s s s s s s =−=−  + + = + = = − + + + + +  2 1 2 3 4 4 76( 3 4) 6 8 10( 2)( 6 25) s j s sK s j j s s s =− + + = + − = − = + + + o53.1− 3 o 2 3 o 2( ) 2 10 cos(4 53.1 ) 8 20 cos(4 53.1 ) 8t t t tf t e t e e t e− − − −→ = × − − = − − 1 2 3( ) 3 4 3 4 2 K K KF s s j s j s= + ++ − + + + Inverse Transform (9) The Laplace Transform - sites.google.com/site/ncpdhbkhn 34 Ex. 3 Find the inverse Laplace transform of 2 2 4 76( ) ( 2)( 6 25) s sF s s s s + = + + + 2 3 4 76 ( 2) s sK s + = + 2 22 22 4 76 8 6 25( 6 25) ss s s s ss s =− =− + = = − + ++ + 2 1 4 76 ( 2) ( 3 4) s sK s s j + = + + − 3 4 6 8 10( 3 4) s j j s j =− + = − = + + o53.1− 3 o 2 3 o 2( ) 2 10 cos(4 53.1 ) 8 20 cos(4 53.1 ) 8t t t tf t e t e e t e− − − −→ = × − − = − − 1 2 3( ) 3 4 3 4 2 K K KF s s j s j s= + ++ − + + + Inverse Transform (10) The Laplace Transform - sites.google.com/site/ncpdhbkhn 35 Ex. 4 Find the inverse Laplace transform of 2 5( 2)( ) ( 4 5) sF s s s s + = + + Inverse Transform (11) The Laplace Transform - sites.google.com/site/ncpdhbkhn 36 1 11 12 1 2 1 1 1 1 ( )Multiple poles : ( ) ... ...( )( ) ( ) ( ) ( ) n n n P s K K KF s Q s s p s p s p s p= = + + + ++ + + + 1 1 1( ) ( )n ns ps p F s K=−+ = 1 2 1 1 22 [( ) ( )] (2!)n n s p d s p F s K ds − =− + = 1 1 1 1[( ) ( )]n n s p d s p F s K ds − =− + = 1 1 1 1 [( ) ( )]( )! n j n j n j s p dK s p F s n j ds − − =− = + − Inverse Transform (12) The Laplace Transform - sites.google.com/site/ncpdhbkhn 37 Ex. 5 Find the inverse Laplace transform of 2 2 10 34 27( ) ( 3) s sF s s s + + = + 2 2 2 2 12 23 3 3 10 34 27 10 34 27( 3) ( ) ( 3) 5( 3)s s s s s s sK s F s s s s s=− =− =− + + + + = + = + = = − + 2 2 11 3 3 2 2 3 10 34 27[( 3) ( )] (20 34) (10 34 27) 7 s s s d d s sK s F s ds ds s s s s s s =− =− =−  + + = + = =    + − + + = = 2 2 20 0 10 34 27( ) 3( 3)s s s sK sF s s s s= = + + = = = + 1 11 12 2 1 12 1( ) ; [( ) ( )] 3 ( 3) ( )! n j n j n j s p K K K dF s K s p F s s s s n j ds − − =− = + + = + + + − Inverse Transform (13) The Laplace Transform - sites.google.com/site/ncpdhbkhn 38 Ex. 5 Find the inverse Laplace transform of 2 2 10 34 27( ) ( 3) s sF s s s + + = + 2 7 5 3( ) 3 ( 3)F s s s s→ = − ++ + 3 3( ) 7 5 3t tf t e te− −→ = − + 11 12 27; 5; 3K K K= = − = 1 11 12 2 1 12 1( ) ; [( ) ( )] 3 ( 3) ( )! n j n j n j s p K K K dF s K s p F s s s s n j ds − − =− = + + = + + + − Inverse Transform (14) The Laplace Transform - sites.google.com/site/ncpdhbkhn 39 Ex. 5 Find the inverse Laplace transform of 2 2 10 34 27( ) ( 3) s sF s s s + + = + 11 12 2 2( ) 3 ( 3) K K KF s s s s = + + + + 2 2 10 34 27s sK s + + = 2 22 0 0 10 34 27 3( 3)( 3) s s s s ss = = + + = = ++ 2 12 2 10 34 27 ( 3) s sK s s + + = + 2 33 10 34 27 5 s s s s s =− =− + + = = − 2 11 2 10 34 27 ( 3) d s sK ds s + + = + 2 33 2 2 3 10 34 27 (20 34) (10 34 27) 7 s s s d s s ds ss s s s s s =− =− =−      + +    =           + − + + = = 2 7 5 3 3 ( 3)s s s= − ++ + 3 3( ) 7 5 3t tf t e te− −→ = − + Inverse Transform (15) The Laplace Transform - sites.google.com/site/ncpdhbkhn 40 Ex. 6 Find the inverse Laplace transform of 2 5( 3)( ) ( 1)( 2) sF s s s + = + + The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 41 Initial-Value & Final-Value Theorems (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 42 0 Initial value theorem : lim ( ) lim ( ) t s f t sF s → →∞ − = 0 Final value theorem : lim ( ) lim ( ) t s f t sF s →∞ → − = Initial-Value & Final-Value Theorems (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 43 2 5( 1)(0) lim ( ) lim 0 2 2s s sf sF s s s→∞ →∞ + = = = + + 20 0 5( 1)( ) lim ( ) lim 2.5 2 2s s sf sF s s s→ → + ∞ = = = + + Ex. Find the initial and final values of 2 5( 1)( ) ( 2 2) sF s s s s + = + + The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 44 The Laplace Transform - sites.google.com/site/ncpdhbkhn 45 Ex. Find the current i(t)? t = 0 + – 200Ω 100 mH 1V ( )i t1L R di v v e L Ri dt + = → + = 0n n diL Ri dt + = tni Ke α−→ = 0t tLK e RKeα αα − −→ − + = 0L Rα→ − + = 3 200 2000 100 10 R L α − → = = = × 2000t ni Ke −→ = 1 0.005A 200f ei R = = = 20000.005 tf ni i i Ke − = + = + 2000 0(0) 0.005 0.005 0 0.005i Ke K K− ×= + = + = → = − 2000( ) 0.005(1 )Ati t e−→ = − Method 1 Laplace Circuit Solutions (1) Laplace Circuit Solutions (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 46 F(s) = 0 (algebraic) I(s), V(s), Laplace Transform Inverse Transform f(t) = 0 (integrodifferential) i(t), v(t), Circuit Laplace Circuit Solutions (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 47 Ex. Find the current i(t)? t = 0 + – 200Ω 100 mH 1V ( )i t Method 2 0.1 200 1L R di v v e i dt + = → + = 0.1 200 [1] 0.1 [200 ]di diL i L L L i dt dt     + = = +       1[1]L s = [200 ] 200 ( )L i I s= 1 2 1 1( ) ( ) (0) (0) ... (0) n n n n o n n d f t s F s s f s f s f dt − − −→ − − − 0.1 0.1[ ( ) (0)] 0.1 ( )diL sI s i sI s dt   → = − =   10.1 ( ) 200 ( )sI s I s s → + = Laplace Circuit Solutions (4) The Laplace Transform - sites.google.com/site/ncpdhbkhn 48 Ex. Find the current i(t)? t = 0 + – 200Ω 100 mH 1V ( )i t Method 2 0.1 200 1L R di v v e i dt + = → + = 10.1 ( ) 200 ( )sI s I s s → + = 1 21 10( ) (0.1 200) ( 2000) 2000 K KI s s s s s s s → = = = + + + + 1 0 10 0.005 2000 s K s = = = + 0.005 0.005( ) 2000 I s s s → = − + 2 2000 10 0.005 s K s =− = = − 2000( ) 0.005(1 ) Ati t e−→ = − The Laplace Transform - sites.google.com/site/ncpdhbkhn 49 F(s) = 0 (algebraic) I(s), V(s), Laplace Transform Inverse Transform f(t) = 0 (integrodifferential) i(t), v(t), Circuit Circuit in s-domain Circuit Element Models The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 50 Circuit Element Models (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 51 v Ri= ( ) ( )Af t AF s→ ( ) ( )V s RI s→ = R( )v t ( )i t + − R( )V s ( )I s + − Circuit Element Models (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 52 di v L dt = ( ) [ ( ) (0)]df tA A sF s f dt → − ( ) [ ( ) (0)] ( ) (0) V s L sI s i sLI s Li → = − = − L ( )v t ( )i t + − (0)i sL ( )V s ( )I s + − +– (0)Li Circuit Element Models (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 53 0 1 ( ) (0)tv i x dx v C = +∫ 0 ( )( )t F sf d s λ λ →∫ 1 (0)( ) ( ) vV s I s sC s → = + (0)(0) vv s → C ( )v t ( )i t + − 1 sC( )V s ( )I s + − – + (0)v s Circuit Element Models (4) The Laplace Transform - sites.google.com/site/ncpdhbkhn 54 1 2 1 1 ( ) ( )( ) di t di tv t L M dt dt = + 1 1 1 1 1 2 2( ) ( ) (0) ( ) (0)V s sL I s L i sMI s Mi→ = − + − 1( )i t 1L 2L + – M 2 ( )i t 2 ( )v t + – 1( )v t 2 1 2 2 ( ) ( )( ) di t di tv t L M dt dt = + 2 2 2 2 2 1 1( ) ( ) (0) ( ) (0)V s sL I s L i sMI s Mi→ = − + − 1( )I s 1sL 2sL + – sM 2 ( )I s 2 ( )V s + – 1( )V s + –+– 1 1 2(0) (0)L i Mi+ 2 2 1(0) (0)L i Mi+ 1(0)i 2 (0)i Circuit Element Models (5) The Laplace Transform - sites.google.com/site/ncpdhbkhn 55 R( )v t ( )i t + − R( )V s ( )I s + − L ( )v t ( )i t + − (0)i sL ( )V s ( )I s + − +– (0)Li Circuit Element Models (6) The Laplace Transform - sites.google.com/site/ncpdhbkhn 56 C ( )v t ( )i t + − 1 sC( )V s ( )I s + − – + (0)v s 1( )i t 1L 2L + – M 2 ( )i t 2 ( )v t + – 1( )v t 1( )I s 1sL 2sL + – sM 2 ( )I s 2 ( )V s + – 1( )V s + –+– 1 1 2(0) (0)L i Mi+ 2 2 1(0) (0)L i Mi+ Circuit Element Models (7) The Laplace Transform - sites.google.com/site/ncpdhbkhn 57 Ex. 1 Transfer the circuit into Laplace domain? t = 0 + – 200Ω 100 mH 1V ( )i t + – 200 0.1s 1 s ( )I s + – 0.1 (0)i + – 1 s + – 1V 200 200 0.1s + – 0.1 (0)i 0.1H (0)i Circuit Element Models (8) The Laplace Transform - sites.google.com/site/ncpdhbkhn 58 Ex. 2 Transfer the circuit into Laplace domain? t = 0 + – 4A 0.01H 1mF 24V 6Ω 4Ω i + – v 2Ω The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 59 Analysis Techniques (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 60 I(s), V(s), Inverse Transform i(t), v(t), Circuit Circuit in s-domain Circuit Element Models DC circuit analysis techniques (KVL, KCL, nodal analysis, mesh analysis, source transformation, superposition, Thevenin/Norton equivalent, ) 1 2KVL/KCL : ( ) ( ) ... ( ) 0nx t x t x t+ + + = 1 2KVL/KCL : ( ) ( ) ... ( ) 0nX s X s X s+ + + = Analysis Techniques (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 61 Ex. 1 Find the current i(t)? t = 0 + – 200Ω 100 mH 1V ( )i t + – 200 0.1s 1 s ( )I s + – 0.1 (0)i + – 1 s + – 1V 200 200 0.1s + – 0.1 (0)i 0.1H (0)i Analysis Techniques (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 62 Ex. 1 Find the current i(t)? t = 0 + – 200Ω 100 mH 1V ( )i t + – 200 0.1s 1 s ( )I s + – 0.1 (0)i (0) 0i = 1200 ( ) 0.1 ( ) 0.1 (0) 200 ( ) 0.1 ( )I s sI s i I s sI s s + − = = + 1 21 10( ) (0.1 200) ( 2000) 2000 K KI s s s s s s s → = = = + + + + 1 0 10 0.005 2000 s K s = = = + 0.005 0.005( ) 2000 I s s s → = − + 2 2000 10 0.005 s K s =− = = − 2000( ) 0.005(1 ) Ati t e−→ = − The Laplace Transform - sites.google.com/site/ncpdhbkhn 63 Analysis Techniques (4) Inverse Transform Circuit Element Models t = 0 + – 200Ω 100 mH 1V ( )i t + – 200 0.1s 1 s ( )I s + – 0.1 (0) 0i = 1200 ( ) 0.1 ( )I s sI s s + = 10( ) ( 2000)I s s s= + 2000( ) 0.005(1 )Ati t e−= − 1. Solve for initial capacitor voltages & inductor currents 2. Draw an s-domain circuit 3. Use one of DC circuit analysis techniques to solve for voltages or/and currents in s-domain 4. Find the inverse Laplace transform to convert them back to the time domain (0) 0i = Ex. 1 Find the current i(t)? The Laplace Transform - sites.google.com/site/ncpdhbkhn 64 Analysis Techniques (5) 4 6 4 6 1101 525 10( ) // ( ) 1 210 25 10 sV s R J s sC s s − −   × = = ×  +  + × 1. Solve for initial capacitor voltages & inductor currents 2. Draw an s-domain circuit 3. Use one of DC circuit analysis techniques to solve for voltages or/and currents in s-domain 4. Find the inverse Laplace transform to convert them back to the time domain (0) 0v = Ex. 2 Find the voltage v(t)? t = 0 10kΩ 25 Ate− 25 Fµ – + v 10k 5 A 2s + 6 1 25 10 s−× – + ( )V s



4 1 24 10 ( 2)( 4) 2 4 K K s s s s × = = + + + + + 4 4 1 2 4 10 2 10 4 s K s =− × = = × + 4 4 2 4 4 10 2 10 2 s K s =− × = = − × + 4 2 4( ) 2 10 ( ) Vt tv t e e− −→ = × − The Laplace Transform - sites.google.com/site/ncpdhbkhn 65 Analysis Techniques (6)Ex. 2 Find the voltage v(t)? t = 0 10kΩ 25 Ate− 25 Fµ – + v 4 2 4( ) 2 10 ( ) Vt tv t e e− −= × − -500 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 The Laplace Transform - sites.google.com/site/ncpdhbkhn 66 Analysis Techniques (7) 122 ( ) 2 4 sI s s + = + 1. Solve for initial capacitor voltages & inductor currents 2. Draw an s-domain circuit 3. Use one of DC circuit analysis techniques to solve for voltages or/and currents in s-domain 4. Find the inverse Laplace transform to convert them back to the time domain 8(0) 1A 8 i = = Ex. 3 Find the current i(t)?



1 26 ( 2) 2 s K K s s s s + = = + + + 1 0 6 3 2 s sK s = + = = + 2 2 6 2 s sK s =− + = = − 2( ) 3 2 Ati t e−→ = − L ( )v t ( )i t + − (0)i sL ( )V s ( )I s + − +– (0)Li + – 4Ω 2 H8V ( )i t 0t = 8Ω + – 12 V + – 4 2s 2 ( )I s + – 12 s The Laplace Transform - sites.google.com/site/ncpdhbkhn 67 Analysis Techniques (8)Ex. 3 Find the current i(t)? 2( ) 3 2 Ati t e−= − + – 4Ω 2 H8V ( )i t 0t = 8Ω + – 12 V -2000 -1000 0 1000 2000 3000 4000 5000 0 0.5 1 1.5 2 2.5 3 + – 8V 0i 8Ω 4Ω 2 H ( )ni t 4Ω ( )fi t + – 12 V The Laplace Transform - sites.google.com/site/ncpdhbkhn 68 Analysis Techniques (9) 12 8 1 8( ) 12 4 2 s sV s s s s − = × + + 1. Solve for initial capacitor voltages & inductor currents 2. Draw an s-domain circuit 3. Use one of DC circuit analysis techniques to solve for voltages or/and currents in s-domain 4. Find the inverse Laplace transform to convert them back to the time domain (0) 8 Vv = Ex. 4 Find the voltage v(t)?



1 28 1.5 ( 0.125) 0.125 s K K s s s s + = = + + + 1 0 8 1.5 12 0.125 s sK s = + = = + 2 0.125 0.5 4 s K s =− = = − 0.125( ) 12 4 Vtv t e−→ = − + – 4Ω 2 F8V ( )v t 0t = 8Ω + – 12 V + – – + 4 1 2s 8 s ( )V s + – 12 s + – C ( )v t ( )i t + − 1 sC ( )V s ( )I s + − –+ (0)v s The Laplace Transform - sites.google.com/site/ncpdhbkhn 69 Analysis Techniques (10)Ex. 4 Find the voltage v(t)? 0.125( ) 12 4 Vtv t e−= − + – 4Ω 2 F8V ( )v t 0t = 8Ω + – 12 V + – -5 0 5 10 x 104 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 + – 8V 0v 8Ω + – 4Ω ( )nv t + – 4Ω ( )fv t + – 12 V + – The Laplace Transform - sites.google.com/site/ncpdhbkhn 70 Analysis Techniques (11)Ex. 5 Write the mesh equations in the s-domain? + – + – 1( )e t 2 ( )e t 1C 1R 2R 2L 3L 3C 3(0)i 2 (0)i –+ – + 1(0)v 3(0)v + – + – 1( )E s 2( )E s 1 1 sC 1R 2R 2sL 3sL 3 1 sC 3 3(0)L i 2 2 (0)L i 1(0)v s 3(0)v s + – + – + – + – L ( )v t ( )i t + − (0)i sL ( )V s ( )I s + − +– (0)Li C ( )v t ( )i t + − 1 sC( )V s ( )I s + − –+ (0)v s The Laplace Transform - sites.google.com/site/ncpdhbkhn 71 Analysis Techniques (12)Ex. 5 Write the mesh equations in the s-domain? + – + – 1( )e t 2 ( )e t 1C 1R 2R 2L 3L 3C 3(0)i 2 (0)i –+ – + 1(0)v 3(0)v ( )AI s ( )BI s 1 1 3 3 1 3 1 3 3 1 : ( ) 1 [ ( ) ( )] (0) (0)( ) (0) A A B A R I s sC sL I s I s sC v vE s L i s s   + +      + + − =    = − + − + – + – 1( )E s 2( )E s 1 1 sC 1R 2R 2sL 3sL 3 1 sC 3 3(0)L i 2 2 (0)L i 1(0)v s 3(0)v s + – + – + – + – ( )2 2 3 3 3 3 3 2 2 2 : ( ) 1 [ ( ) ( )] (0) (0) (0) ( ) B B A B R sL I s sL I s I s sC v L i L i E s s + +   + + − =    = − − − The Laplace Transform - sites.google.com/site/ncpdhbkhn 72 Analysis Techniques (13)Ex. 6 Write the node equations in the s-domain? +– + – 1( )j t 1L 1(0)i 2 (0)i 1R 3R 2L 2C 2(0)v 3(0)v 3C 3( )j t The Laplace Transform - sites.google.com/site/ncpdhbkhn 73 Analysis Techniques (14)Ex. 7 Solve for v(t) ? + – 5 ( ) Au t 15 ( ) Vu t 8Ω 1H 2Ω 1F + – ( )v t 15 ( )5 ( ) : 012 a a V s V ssa s s s − + − = + + – 5 s 15 s 8 s 2 1 s + – ( )V s a 2 3 2 10 35 15( ) 2a s sV s s s s + + → = + + 2 3 2 ( ) 10 35 15( ) 2 21 2 2 12 aV s s s sV s s s s s s + + → = = × + + ++ 11 12 2 2 10( 3) ( 1) 1 ( 1) s K K s s s + = = + + + + (0) 0; (0) 0;L Ci v= = The Laplace Transform - sites.google.com/site/ncpdhbkhn 74 Analysis Techniques (15) Solve for v(t) ? 2 12 2 1 1 10( 3)( 1) 10( 3) 20( 1) s s sK s s s =− =− + = + = + = + ( )211 2 11 10( 3)( 1) 10 3 10( 1) ss d s dK s s ds s ds =− =−  + = + = + = +  + – 5 s 15 s 8 s 2 1 s + – ( )V s a 11 12 2 2 10( 3)( ) ( 1) 1 ( 1) s K KV s s s s + = = + + + + 2 10 20( ) ( ) 10(2 1) V 1 ( 1) tV s v t t e s s −→ = + → = + + + Method 1 Ex. 7 The Laplace Transform - sites.google.com/site/ncpdhbkhn 75 Analysis Techniques (16) Solve for v(t) ? 5 1 15( ) 2 ( )A As I s I s s s s     − + + =      + – 5 s 15 s 8 s 2 1 s + – ( )V s a Method 2 ( )AI s 2 3( ) 5 ( 1)A sI s s + → = + 2 3( ) 10 ( 1) sV s s + → = + Ex. 7 5 s The Laplace Transform - sites.google.com/site/ncpdhbkhn 76 Analysis Techniques (17) Solve for v(t) ? 5 2 12 5 10( 0.5)( ) 1 ( 1)2 ab s s ssV s s ss s  +  +  = × = ++ + + – 5 s 15 s 8 s 2 1 s + – ( )V s a Method 3 5 2 2 10( 0.5) 2 10( ) 1( 1) ( 1)2s s sV s s s s + → = × = + ++ 5 s 8 s 2 1 s + – 5( ) s V s a b Ex. 7 The Laplace Transform - sites.google.com/site/ncpdhbkhn 77 Analysis Techniques (18) Solve for v(t) ? 15 2 15 15( ) 1 ( 1)2 C s sI s ss s = = ++ + + – 5 s 15 s 8 s 2 1 s + – ( )V s a Method 3 15 2 15( ) 2 ( 1)sV s s→ = + + – 15 s 8 s 2 1 s + – 15( ) s V s a Ex. 7 The Laplace Transform - sites.google.com/site/ncpdhbkhn 78 Analysis Techniques (19) Solve for v(t) ? + – 5 s 15 s 8 s 2 1 s + – ( )V s a Method 3 5 2 10( ) ( 1)s sV s s → = + 15 2 15( ) 2 ( 1)sV s s→ = + 5 s 8 s 2 1 s + – 5( ) s V s a b + – 15 s 8 s 2 1 s + – 15( ) s V s a 15 15 2 2 2 ( ) ( ) ( ) 10 30 ( 1) ( 1) 10( 3) ( 1) s s V s V s V s s s s s s → = + = + + + + = + Ex. 7 The Laplace Transform - sites.google.com/site/ncpdhbkhn 79 Analysis Techniques (20) Solve for v(t) ? + – 5 s 15 s 8 s 2 1 s + – ( )V s a Method 4 + –( )E s ( )Z s 2 1 s + – ( )V s ( )Z s s= 2 5 15 3( ) 5 sE s s s s s +  = + =    2 35 3( ) 51 ( 1)2 s ssI s ss s + + = = ++ + 2 2 3 3( ) 2 5 10( 1) ( 1) s sV s s s + + = × = + + Ex. 7 The Laplace Transform - sites.google.com/site/ncpdhbkhn 80 Analysis Techniques (21) Solve for v(t) ? + – 5 s 15 s 8 s 2 1 s + – ( )V s a Method 5 + –( )eqE s ( )eqZ s 2 + – ( )V s 8 s 1 s ( ) eqZ s a + – 5 s 15 s 8 s 1 s + – ( ) eqE s a 1( )eqZ s s s = + 5 15( )eqE s s s s − = 3( ) 5eq sE s s + → = 2 ( )( ) 2 2 ( ) 310 ( 1) eq eq E s V s Z s s s → = + + = + Ex. 7 The Laplace Transform - sites.google.com/site/ncpdhbkhn 81 Analysis Techniques (22)Ex. 8 Solve for i(t) ? 2 ( ) Au t xv + – 0.5 xv 2Ω 2Ω 2Ω 1H 1 F 6 i (0) 0; (0) 0C Lv i= = 2 s ( )xV s + – ( ) 0.5 ( )c xI s V s= 2 2 2 s 6 s ( )I s( )AI s [ ]6( ) 2 ( ) ( ) ( 2) ( ) 0x A c AV s I s I s s I s s   − + + − + + =    [ ]6( ) 2 ( ) 0.5 ( ) ( 2) ( ) 0x A x AV s I s V s s I s s   → − + + − + + =    2 4( ) 2 ( ) 2 ( ) x A AV s I s I s s s   = − = −   4 6 42 ( ) 2 ( ) 0.5 2 ( ) ( 2) ( ) 0 A A A A I s I s I s s s s s I s       → − − + + − − +            + + = 8 12( ) ( )( 2)( 6)A sI s I s s s s + → = = + + The Laplace Transform - sites.google.com/site/ncpdhbkhn 82 Analysis Techniques (23)Ex. 8 Solve for i(t) ? 2 ( ) Au t xv + – 0.5 xv 2Ω 2Ω 2Ω 1H 1 F 6 i 2 s ( )xV s + – ( ) 0.5 ( )c xI s V s= 2 2 2 s 6 s ( )I s 1 2 38 12( ) ( 2)( 6) 2 6 s K K KI s s s s s s s + = = + + + + + + 1 0 8 12 1( 2)( 6) s sK s s = + = = + + 2 2 8 12 0.5( 6) s sK s s =− + = = + 3 6 8 12 1.5( 2) s sK s s =− + = = − + 2 6( ) 1 0.5 1.5 At ti t e e− −→ = + − The Laplace Transform - sites.google.com/site/ncpdhbkhn 83 Analysis Techniques (24) 22 2 152 16.59( ) 2 4 ( 2)( 9) ssI s s s s + ++ = = + + + 8(0) 1A 8 i = = Ex. 9 Find the current i(t)? * 1 2 2 2 3 3 K K K s s j s j= + ++ − + 2 1 2 2 16.5 1.58 9 s sK s =− + = = + 2 2 3 16.5 0.35( 2)( 3) s j sK s s j = + = = + + o146.3− 2 o( ) 1.58 0.70cos(3 146.3 ) Ati t e t−→ = + − + – 4 2s 2 ( )I s + – 2 15 9s + + – 4Ω 2 H8V ( )i t 0t = 8Ω + – 5sin 3 Vt The Laplace Transform - sites.google.com/site/ncpdhbkhn 84 Analysis Techniques (25)Ex. 9 Find the current i(t)? 2 o( ) 1.58 0.70cos(3 146.3 ) Ati t e t−= + − + – 4Ω 2 H8V ( )i t 0t = 8Ω + – 5sin 3 Vt -2000 -1000 0 1000 2000 3000 4000 5000-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 The Laplace Transform - sites.google.com/site/ncpdhbkhn 85 Analysis Techniques (26)Ex. 10 Find the current i(t)? + – 4Ω 2 H 8V ( )i t 0t = 8Ω + – 20sin 3 Vt + – 6j 0I 8Ω 20 V 0 20 2 8 6j= =+I o36,9 A− o 0( ) 2sin(3 36,9 ) Ai t t→ = − o(0) 2sin( 36,9 ) 1.20Ai→ = − = − + – 4 2s 2.40− ( )I s + – 8 s 82.40 1.2 4( ) A 2 4 ( 2) ssI s s s s − + − + = = + + 1 2 2 K K s s = + + 1 0 1.2 4 2; 2 s sK s = − + = = + 2 2 1.2 4 3.2 s sK s =− − + = = − 2( ) 2 3.2 Ati t e−→ = − The Laplace Transform - sites.google.com/site/ncpdhbkhn 86 Analysis Techniques (27)Ex. 10 Find the current i(t)? + – 4Ω 2 H 8V ( )i t 0t = 8Ω + – 20sin 3 Vt 2( ) 2 3.2 Ati t e−= − t (A)i 2 1.2− Analysis Techniques (28) The Laplace Transform - sites.google.com/site/ncpdhbkhn 87 I(s), V(s), Inverse Transform i(t), v(t), Circuit Circuit in s-domain Circuit Element Models DC circuit analysis techniques (KVL, KCL, nodal analysis, mesh analysis, source transformation, superposition, Thevenin/Norton equivalent, ) The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 88 Convolution Integral (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 89 t0 t0 λ0 λ0 2 ( )f t 1( )f t 1( )f λ 1( )f t λ− 2 ( )f t λ− 2( )f λ 1 2 1 2 1 20 0 ( ) ( ) * ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ Convolution Integral (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 90 Ex. 1 t0 1 1 1( )f t 2 3 4 t0 2 2( )f t 1 2 3 4 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ λ0 2 2( )f λ 1 2 3 4 1 1( )f t λ− λ0 2 2 ( )f λ 1 2 3 4 1 1( )f t λ− 1 20 1: 1; 0t f f< < = = 1 2( )* ( ) 0f t f t = Find the convolution of the two signals? Convolution Integral (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 91 Ex. 1 t0 1 1 1( )f t 2 3 4 t0 2 2( )f t 1 2 3 4 λ0 2 2 ( )f λ 1 2 3 4 1 1( )f t λ− 1 21 2 : 1; 2t f f< < = = 1 2 1 2 11 1 ( ) * ( ) ( ) ( ) 1 2 2 2( 1)t t tf t f t f t f d d tλλ λ λ λ λ == − = × = = −∫ ∫ 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ 1 20 1: ( )* ( ) 0t f t f t< < = Find the convolution of the two signals? Convolution Integral (4) The Laplace Transform - sites.google.com/site/ncpdhbkhn 92 Ex. 1 t0 1 1 1( )f t 2 3 4 t0 2 2( )f t 1 2 3 4 λ0 2 2 ( )f λ 1 2 3 4 1 1( )f t λ− 1 22 3: 1; 2t f f< < = = 1 2 1 2 11 1 ( ) * ( ) ( ) ( ) 1 2 2 2t t t tt t f t f t f t f d d λλ λ λ λ λ = − − − = − = × = =∫ ∫ 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ 1 20 1: ( )* ( ) 0t f t f t< < = 1 21 2 : ( ) * ( ) 2( 1)t f t f t t< < = − Find the convolution of the two signals? Convolution Integral (5) The Laplace Transform - sites.google.com/site/ncpdhbkhn 93 Ex. 1 t0 1 1 1( )f t 2 3 4 t0 2 2( )f t 1 2 3 4 λ0 2 2 ( )f λ 1 2 3 4 1 1( )f t λ− 1 23 4 : 1; 2t f f< < = = 3 3 3 1 2 1 2 11 1 ( ) * ( ) ( ) ( ) 1 2 2 8 2 tt t f t f t f t f d d tλλ λ λ λ λ = − − − = − = × = = −∫ ∫ 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ 1 20 1: ( )* ( ) 0t f t f t< < = 1 21 2 : ( ) * ( ) 2( 1)t f t f t t< < = − 1 22 3: ( ) * ( ) 2t f t f t< < = Find the convolution of the two signals? Convolution Integral (6) The Laplace Transform - sites.google.com/site/ncpdhbkhn 94 Ex. 1 t0 1 1 1( )f t 2 3 4 t0 2 2( )f t 1 2 3 4 λ0 2 2 ( )f λ 1 2 3 4 1 1( )f t λ− 1 24 : 1; 0t f f> = = 1 2( )* ( ) 0f t f t = 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ 1 20 1: ( )* ( ) 0t f t f t< < = 1 21 2 : ( ) * ( ) 2( 1)t f t f t t< < = − 1 22 3: ( ) * ( ) 2t f t f t< < = 1 23 4 : ( )* ( ) 8 2t f t f t t< < = − Find the convolution of the two signals? Convolution Integral (7) The Laplace Transform - sites.google.com/site/ncpdhbkhn 95 Ex. 1 t0 1 1 1( )f t 2 3 4 t0 2 2( )f t 1 2 3 4 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ 1 20 1: ( )* ( ) 0t f t f t< < = 1 21 2 : ( ) * ( ) 2( 1)t f t f t t< < = − 1 22 3: ( ) * ( ) 2t f t f t< < = 1 24 : ( )* ( ) 0t f t f t> = 1 23 4 : ( )* ( ) 8 2t f t f t t< < = − t0 2 1 2( )* ( )f t f t 1 2 3 4 Find the convolution of the two signals? Convolution Integral (8) The Laplace Transform - sites.google.com/site/ncpdhbkhn 96 Ex. 2 t0 1 1 ( )w t 2 3 4 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ t0 2 2 te− 1 2 3 4 ( ) 00 0 2 : ( ) 2 1 2 2(1 )t tt t tt f t e d e eλλ λ λ λ =− − − − = < < = × = = −∫ 2 2( ) 2 00 2 : ( ) 2 1 2 2( 1)t t tt f t e d e e eλλ λ λ λ =− − − − = > = × = = −∫ t0 2 2e λ− 1 2 3 4 ( )w t t0 2 1 2 3 4 Method 1 Find the convolution of the two signals? Convolution Integral (9) The Laplace Transform - sites.google.com/site/ncpdhbkhn 97 Ex. 2 Find the convolution of the two signals? t0 1 1 ( )w t 2 3 4 1 2 1 2 1 20 0 ( ) ( )* ( ) ( ) ( ) ( ) ( )t tf t f t f t f t f d f f t dλ λ λ λ λ λ= = − = −∫ ∫ t0 2 2 te− 1 2 3 4 00 0 2 : ( ) 1 2 2 2(1 )t t tt f t e d e eλλ λ λ λ =− − − = < < = × = − = −∫ 2 22 2 : ( ) 1 2 2 2( 1)t t t tt t f t e d e e eλλ λ λ λ =− − − = − − > = × = − = −∫ t0 2 2 te− 1 2 3 4 ( )w λ− t0 2 1 2 3 4 Method 2 Convolution Integral (10) The Laplace Transform - sites.google.com/site/ncpdhbkhn 98 Property f(t) F(s) 1. Magnitude scaling 2. Addition/subtraction 3. Time scaling 4. Time shifting 5. Frequency shifting 6. Differentiation 7. Multiplication by t 8. Division by t 9. Integration 10. Convolution ( )Af t ( )AF s 1 2( ) ( )f t f t± 1 2( ) ( )F s F s± ( )f at 1 sF a a       ( ) ( ), 0f t a u t a a− − ≥ ( )ase F s− ( )ate f t− ( )F s a+ ( ) ( ), 0f t u t a a− ≥ [ ( )]ase L f t a− + ( ) /n nd f t dt 1 2 1 1( ) (0) (0) ... (0)n n n o ns F s s f s f s f− − −− − − ( )nt f t ( 1) ( ) /n n nd F s ds− ( ) /f t t ( ) s F dλ λ∞∫ 0 ( )t f dλ λ∫ ( ) /F s s 1 2 1 20 ( ) * ( ) ( ) ( )tf t f t f f t dλ λ λ= −∫ 1 2( ) ( )F s F s Convolution Integral (11) The Laplace Transform - sites.google.com/site/ncpdhbkhn 99 Ex. 3 Find vo(t)? +– + – ( )iv t 1Ω 0.2 F ( )ov t t0 iv 5 te− 1 2 3 41 ( ) 5 5 50.2( ) ( ) 1( ) 1 5 11 0.2 C o i C Z s sV s V s R Z s s s s s = = × = × + + + ++ Method 1: 5( ) ( ) 6.25( ) Vt t o o V s v t e e− −→ = − Method 2: ( ) ( ) ( ) ( ) ( ) * ( ) o i o iV s H s V s v t h t v t= → = 55( ) ( ) 5 5 tH s h t e s − = → = + 5( ) 5 4 5 4 0 0 0 0 5 ( ) ( ) ( ) 5 5 25 6.25 6.25( ) V t t t tt t t o i t t v t h t v d e e d e d e e e λλ λ λ λ λ λ λ λ λ λ =− − − − + − + = − − = − = = = = − ∫ ∫ ∫ The Laplace Transform 1. Definition 2. Two Important Singularity Functions 3. Transform Pairs 4. Properties of the Transform 5. Inverse Transform 6. Initial-Value & Final-Value Theorems 7. Laplace Circuit Solutions 8. Circuit Element Models 9. Analysis Techniques 10. Convolution Integral 11. Transfer Function The Laplace Transform - sites.google.com/site/ncpdhbkhn 100 Transfer Function (1) The Laplace Transform - sites.google.com/site/ncpdhbkhn 101 ( )inI s + – ( )inV s ( )outI s + – ( )outV s( )H s ( )( ) ( ) Out sH s In s = If ( ) ( ) ( ) 1 ( ) ( )in t t In s H s Out sδ= → = → = Transfer Function (2) The Laplace Transform - sites.google.com/site/ncpdhbkhn 102 Linear bandpass filter + – + – ( )iv t ( )ov t Ex. 1 Find the transfer function h(t) of the filter? ( ) 10 ( )iv t u t= 10( ) ( ) ( ) ( )o iV s H s V s H s s = = 1( ) ( ) 10 o H s sV s→ = 1 ( )( ) 10 odv th t dt → = Transfer Function (3) The Laplace Transform - sites.google.com/site/ncpdhbkhn 103 Ex. 2 Find the transfer function H(s)? +– + – ( )iv t 1Ω 0.2 F ( )ov t 1 ( ) 50.2( ) ( ) ( ) ( ) ( ) ( )1( ) 51 0.2 C o i i i i C Z s sV s V s V s V s H s V s R Z s s s = = = = + ++ ( ) 5( ) ( ) 5 o i V sH s V s s → = = + Transfer Function (4) The Laplace Transform - sites.google.com/site/ncpdhbkhn 104 A circuit is stable if : lim ( ) finite x h t →∞ = 1 2 ( )( ) ( )( )...( )n N sH s s p s p s p = + + + 1 2 1 2( ) ( ... ) ( )np tp t p t nh t k e k e k e u t−− −→ = + + + σ jω A circuit is stable when all the poles of its transfer function H(s) lie in the left half of the s-plane Transfer Function (5) The Laplace Transform - sites.google.com/site/ncpdhbkhn 105 Ex. 3 2 1,2 (4 ) (4 ) 4 2 k k p − − ± − − = An active filter has the transfer function 2( ) (4 ) 1 kH s s k s = + − + For what values of k is the filter stable? σ jω A circuit is stable when all the poles of its transfer function H(s) lie in the left half of the s-plane 4 0k→ − > 4k→ < Transfer Function (6) The Laplace Transform - sites.google.com/site/ncpdhbkhn 106 Ex. 4 +– + – ( )iv t 1Ω C ( )ov t 1 ( ) 1( ) ( ) ( ) ( )1( ) 11 C o i i i C Z s sCV s V s V s V s R Z s Cs sC = = = + ++ ( ) 1 5( ) ( ) 1 5 5 o i V sH s V s Cs Cs → = = = + + Given the transfer function 5( ) 5 H s s = + Find C? +– + – ( )iV s 1 1 sC ( )oV s 5 1C→ = 0.2 FC→ =