Bài giảng Electric circuit theory - Chapter VII: First-Order Circuits - Nguyễn Công Phương

Electric Circuit Theory First-Order Circuits Nguyễn Công Phương First-Order Circuits - sites.google.com/site/ncpdhbkhn 2 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 First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 3 Introduction to Transient Analysis (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 4 +– 10 V 5 Ω 0.1 H +– t = 010 V 5 Ω 0.1 Hi t i 0 +– 10 V 5 Ω 0.1 mF +– t = 012 V 6 Ω v 0.1 mF + – t v 0 Steady-state Transient-state Transient-stateSteady-state Steady-state Steady-state Any change in an electrical circuit, which brings about a change in energy distribution, will result in a transient-state. 2 A 12 V Introduction to Transient Analysis (2) First-Order Circuits - sites.google.com/site/ncpdhbkhn 5 Inductors in DC circuits Capacitors in DC circuits vL(t) t0 Short-circuit Short-circuitNot short-circuit iC(t) t0 Open-circuit Open-circuitNot open-circuit Old steady-state New steady-stateTransient -state Old steady-state New steady-stateTransient -state Introduction to Transient Analysis (3) First-Order Circuits - sites.google.com/site/ncpdhbkhn 6 First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 7 Initial Conditions (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 8 10 V 5 Ω 0.1 Hi t = 0 +– t i 0 Steady-state/Initial condition 1 2 A 10 V 5 Ω 0.1 Hi t = 0 +– +– 20 V A Steady-state/Initial condition 2 4 0– Prior to switching 0+ After switching Initial Conditions (2) • 1st switching rule/law: the current (magnetic flux) in an inductor just after switching is equal to the current (flux) in the same inductor just prior to switching iL(0+) = iL(0–) λ(0+) = λ(0–) • 2nd switching rule/law: the voltage (electric charge) in a capacitor just after switching is equal to the voltage (electric charge) in the same capacitor just prior to switching vC(0+) = vC(0–) q(0+) = q(0–) First-Order Circuits - sites.google.com/site/ncpdhbkhn 9 Initial Conditions (3) First-Order Circuits - sites.google.com/site/ncpdhbkhn 10 Ex. 1 The switch has been at A for a long time, and it moves to B at t = 0; find I0? (0 ) (0 )i i+ −= (0 ) 0 Ai − = (0 ) 0Ai +→ = 0 0AI→ = Ex. 2 The switch has been at A for a long time, and it moves to B at t = 0; find I0? (0 ) (0 )i i+ −= 20(0 ) 4 A 5 i − = = (0 ) 4 Ai +→ = 0 4AI→ = 10 V 5 Ω 0.1 Hi t = 0 +– A B 10 V 5 Ω 0.1 Hi t = 0 +– +– 20 V A B Initial Conditions (4) First-Order Circuits - sites.google.com/site/ncpdhbkhn 11 Ex. 3 The switch has been at A for a long time, and it moves to B at t = 0; find V0? (0 ) (0 )v v+ −= (0 ) 0 Vv − = (0 ) 0 Vv +→ = 0 0VV→ = Ex. 4 The switch has been at A for a long time, and it moves to B at t = 0; find V0? (0 ) (0 )v v+ −= (0 ) 20 Vv − = (0 ) 20Vv +→ = 0 20VV→ = 10 V 5 Ω 0.1 mF t = 0 +– V0 + – A B 10 V 5 Ω t = 0 +– +– 20 V 0.1 mF V0 + – A B First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 12 The Source-free RC Circuit (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 13 00(0) tv v V== = 0dv vC dt R → + = 0R Ci i+ = 0dv v dt RC → + = 1dv dt v RC → = − ln lntv A RC → = − + ln v t A RC → = − ( ) t RCv t Ae − → = 00(0) tv v V== = 0 0 ( ) t RC t v t V e V e τ − − → = = tτ v V0 0.368V0 0 R t = 0 +– E C R Cv iR iC+ – The Source-free RC Circuit (2) First-Order Circuits - sites.google.com/site/ncpdhbkhn 14 Ex. 1 R1 = 6 Ω; R2 = 12 Ω; vC(0) = 10 V; C = 0.01F; find vC ? C R1 R2vC + – C R12 vC + – 12 6 12 4 6 12 R ×= = Ω + 12 4 0.01 0.04sR Cτ = = × = 250.04(0) 10 10 V t t t C Cv v e e e τ − − − = = = The Source-free RC Circuit (3) First-Order Circuits - sites.google.com/site/ncpdhbkhn 15 Ex. 2 E = 24 V; R1 = 8 Ω; R2 = 12 Ω; C = 0.01F; the switch has been closed for a long time, and it is opened at t = 0; find vC for t ≥ 0? C R1 R2 vC + – t = 0 + – E R1 R2 V0 + – + – E t < 0 R1 R2 V0 = 24 V + – t > 0 0 24VV E= = 1 2( ) (8 12) 0.01 0.2sR R Cτ = + = + × = 50.2 0 24 24 V t t t Cv V e e eτ − − − = = = First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 16 The Source-free RL Circuit (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 17 00(0) ti i I== = 0R Lv v+ = 0diRi L dt → + = di R dt i L → = − ln lnRi t A L → = − + ln i R t A L → = − ( ) R t Li t Ae − → = 00(0) ti i I== = 0 0 ( ) R t L t i t I e I e τ − − → = = tτ i I0 0.368I0 0 R t = 0 +– E L R L i vR vL + – + – The Source-free RL Circuit (2) First-Order Circuits - sites.google.com/site/ncpdhbkhn 18 Ex. E = 24 V; R1 = 5 Ω; R2 = 4 Ω; R3 = 12 Ω; L = 0.01H; the switch has been closed for a long time, and it is opened at t = 0; find iL for t ≥ 0? + – t = 0 E R1 R2 R3 L iL + – E R1 R2 R3 I0 t < 0 4 125 8 4 12eq R ×= + = Ω + 1 24 3A 8 i→ = = 2 0 1 2 3 43 0.75A 4 12 RI i R R → = = = + + R2 R3 L I0 = 0.75 A t > 0 23 0.01 0.000625s 4 12 L R τ = = = + 1600 0( ) 0.75 A t t Li t I e eτ − − = = First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 19 Step Response of an RC Circuit (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 20 E2 R t = 0 +– +– E1 C v + – i 0 1(0 ) (0 )V v v E+ −= = = 2Ri v E+ = 2 dvRC v E dt → + = 2 Edv v dt RC RC → + = 2v Edv dt RC − → = − 2 dv dt v E RC → = − − 0 ( ) 2 0 ln( ) t v t V t v E RC → − = − 2 0 2 ln v E t V E RC − → = − − 2 0 2 t RCv E e V E − − → = − 2 0 2 2 1 2( ) ( ) ( ) , 0 t t RCv t E V E e E E E e tτ − − → = + − = + − > Step Response of an RC Circuit (2) First-Order Circuits - sites.google.com/site/ncpdhbkhn 21 E2 R t = 0 +– +– E1 C v + – i 2 0 2 2 1 2 ( ) ( ) ( ) , 0 t RC t v t E V E e E E E e tτ − − = + − = + − > t v E2 0 V0 = E1 t v E2 0 t v E2 0 V0 = E1 Forced response/steady-state response Natural response/transient-state response First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 22 Step Response of an RL Circuit (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 23 1 0 (0 ) (0 ) EI i i R + − = = = 2 diRi L E dt + = 2 E Ridi dt L − → = 2 di dt E Ri L → = − 2 di R dtE Li R → = − 0 2 0 ln t t I E Ri t R L   → − = −    2 2 0 ln E i RR tE LI R − → = − − 2 2 0 R t L E i R eE I R − − → = − 2 2 2 1 2 0( ) , 0 tR t LE E E E Ei t I e e t R R R R R τ −−    → = + − = + − >        E2 R L i t = 0 +– +– E1 Step Response of an RL Circuit (2) First-Order Circuits - sites.google.com/site/ncpdhbkhn 24 2 2 0 2 1 2 ( ) , 0 R t L t E Ei t I e R R E E E e t R R R τ − −   = + −      = + − >    Forced response/steady-state response Natural response/transient-state response E2 R L i t = 0 +– +– E1 t i 0 1 0 EI R = 2E R t i 0 2E R t v 0 1 0 EI R = 2E R First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 25 The Classical Method (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 26 E2 R L i t = 0 +– +– E12 1 2( ) R t LE E Ei t e R R R −  = + −    t i 0 1 0 EI R = 2E R E2 R L i t = 0 +– +– E1 E2 R if +– R L in 2 1 2( ) R t LE E Ei t e R R R −  = + −    1 2 R t LE E e R R −  −    2E R The Classical Method (2) First-Order Circuits - sites.google.com/site/ncpdhbkhn 27 E2 R L i t = 0 +– +– E1 E2 R if +– R L in 2 f Ei R → = 2 0fRi E− = ' 0n nRi Li+ = st ni Ae= 0st stRAe LAse→ + = ( ) 0stR Ls Ae→ + = 0R Ls→ + = R s L → = − R t L n i Ae − → = 2 1 2( ) R t LE E Ei t e R R R −  = + −    1 0 (0) EI i R = = 2 R t LEi Ae R − = + 02 2 1(0) R LE E Ei Ae A R R R − = + = + = 1 2E EA R R = − R L i(0) +– E1 The Classical Method (3) First-Order Circuits - sites.google.com/site/ncpdhbkhn 28 2fv E= ( )' 0n nv R Cv+ = st n v Ae= 0st stAe RCAse→ + = (1 ) 0stRCs Ae→ + = 1 0RCs→ + = 1 s RC → = − 1 t RC n v Ae − → = 1 2 1 2( ) ( ) t RCv t E E E e − = + − 0 1(0)V v E= = 1 2 t RCv E Ae − = + 1 0 2 2 1(0) RCv E Ae E A E − = + = + = 1 2A E E= − E2 R Ct = 0 +– +– E1 v + – E2 R+– vf + – R C vn + – R C +– E1 v(0) + – The Classical Method (4) First-Order Circuits - sites.google.com/site/ncpdhbkhn 29 E2 R L i t = 0 +– +– E1 E2 R Ct = 0 +– +– E1 v + – 0 12. /I E R= 1. f nv v v= + 23. /fi E R= 4. R t L n i Ae − = 1. f ni i i= + 2 05. EI A A R = + → 2 1 26. R t LE E Ei e R R R −  = + −    0 12. V E= 23. fv E= 1 4. t RC n v Ae − = 0 25. V E A A= + → 1 2 1 26. ( ) t RCv E E E e − = + − 1. Write the general form 2. Find the initial condition 3. Find the forced response 4. Deactivate source(s), find the natural response (with the unknow integration constant) 5. Find the integration constant 6. Write the complete response The Classical Method (5) First-Order Circuits - sites.google.com/site/ncpdhbkhn 30 Ex. 1 The switch has been at A for a long time, and it moves to B at t = 0; find v for t ≥ 0? f nv v v= + 0 12 VV = 36 Vfv = 3 200100 0.05 10 tt tRC nv Ae Ae Ae − − − −× × = = = 0 0 36 36 12 24V Ae A A= + = + = → = − 20036 24 Vtv e−→ = − + – + – + –12 V 36 V 50 Ω 100 Ω 0.05 mF A B v t = 0 1. Write the general form 2. Find the initial condition 3. Find the forced response 4. Deactivate source(s), find the natural response (with the unknow integration constant) 5. Find the integration constant 6. Write the complete response The Classical Method (6) First-Order Circuits - sites.google.com/site/ncpdhbkhn 31 + – + – + –12 V 36 V 50 Ω 100 Ω 0.05 mF A B v t = 020036 24 Vtv e−= − t v 0 36V 12V 0 12 VV = 200tnv Ae − = 36 Vfv = + – 100 Ω 0.05 mF B vn + – + – 36 V 100 Ω B vf + – + –12 V 50 Ω A V0 The Classical Method (7) First-Order Circuits - sites.google.com/site/ncpdhbkhn 32 Ex. 2 The switch has been at A for a long time, and it moves to B at t = 0; find v for t ≥ 0? f nv v v= + 0 12 10 2 V 50 10 V −= = − + 80 36 16V 100 80f v = = + 380 100 0.05 10 45080 100 t t tRC nv Ae Ae Ae − − − × × × −+ = = = 0 0 16 16 2 18V Ae A A= + = + = − → = − 45016 18 Vtv e−→ = − + – + – + –12 V 36 V 50 Ω 100 Ω 0.05 mF A B v10 Ω 80 Ω t = 0 1. Write the general form 2. Find the initial condition 3. Find the forced response 4. Deactivate source(s), find the natural response (with the unknow integration constant) 5. Find the integration constant 6. Write the complete response The Classical Method (8) First-Order Circuits - sites.google.com/site/ncpdhbkhn 33 Ex. 3 The switch has been at A for a long time, and it moves to B at t = 0; find i for t ≥ 0? 1. Write the general form 2. Find the initial condition 3. Find the forced response 4. Deactivate source(s), find the natural response (with the unknow integration constant) 5. Find the integration constant 6. Write the complete response f ni i i= + 0 5AI = 120 6A 20f i = = 20 400.5 R tt tL ni Ae Ae Ae −− − = = = 0 0 6 6 5 1I Ae A A= + = + = → = − 406 Ati e−→ = − + – 5 A 120 V 50 Ω 20 Ω 0.5 H A B i t = 0 The Classical Method (9) First-Order Circuits - sites.google.com/site/ncpdhbkhn 34 + – 5 A 120 V 50 Ω 20 Ω 0.5 H A B i t = 0406 Ati e−= − t i 0 6A 5A 0 5AI = 40tni Ae − = 6Afi = 5 A 50 Ω A I0 20 Ω B in0.5 H + – 120 V 20 Ω B if The Classical Method (10) First-Order Circuits - sites.google.com/site/ncpdhbkhn 35 Ex. 4 The switch has been at A for a long time, and it is opened at t = 0; find i for t ≥ 0? 5 A 50 Ω 100 Ω 0.25 H 40 Ω 30 Ω t = 0i f ni i i= + 0 1/ 50 5 1.28A 1/ 50 1/ 40 1/ 30 I = = + + 5 A 50 Ω 100 Ω 0.25 H 30 Ω if 50 Ω 100 Ω 0.25 H 30 Ω in 1/ 50 5 1.88A 1/ 50 1/ 30f i = = + 50 30 3200.25 R tt tL ni Ae Ae Ae + −− − = = = 0 0 1.88 1.28 0.60I Ae A= + = → = − 3201.88 0.6 Ati e−→ = − First-Order Circuits 1. Introduction to Transient Analysis 2. Initial Conditions 3. The Source-free RC Circuit 4. The Source-free RL Circuit 5. Step Response of an RC Circuit 6. Step Response of an RL Circuit 7. The Classical Method 8. First-order Op Amp Circuits First-Order Circuits - sites.google.com/site/ncpdhbkhn 36 First-Order Op Amp Circuits (1) First-Order Circuits - sites.google.com/site/ncpdhbkhn 37 Ex. 1 The switch has been at A for a long time, and it moves to B at t = 0; find vo for t ≥ 0? f nv v v= + (0) 5Vv = 0fv = 6 3 3 01 10 50 10 50 10 n B C Bdv v v v dt − − − × = = × × B nv v= − 0 0.05 n ndv v dt → + = 20t n v Ae−→ = 0(0) 0 5 5v Ae A= + = → = 205 Vtv e−→ = +– + – 5 V vo 50 kΩ 1 µF A B 100 kΩ + – C v – + t = 0 +– von 50 kΩ 1 µF 100 kΩ + – C vn – + B First-Order Op Amp Circuits (2) First-Order Circuits - sites.google.com/site/ncpdhbkhn 38 Ex. 1 The switch has been at A for a long time, and it moves to B at t = 0; find vo for t ≥ 0? 205 Vtv e−= 3 3 100 10 2 50 10o B B v v v × = − = − × Bv v= − 2010 Vtov e −→ = +– vi vo ii io R1 R2 2 1 o i R v v R = − +– + – 5 V vo 50 kΩ 1 µF A B 100 kΩ + – C v – + t = 0 First-Order Op Amp Circuits (3) First-Order Circuits - sites.google.com/site/ncpdhbkhn 39 Ex. 2 Find vo(t)? f nv v v= + (0) 0v = + – vi voii io R1 R2 2 1 1o i R v v R   = +    20 6 4V 10 20Af v = = + 601 12V 30of Af v v   = + =    4 12 8Vf Bf of Af ofv v v v v= − = − = − = − 360 10 0nv i+ × = 3 660 10 10 0n n dv v dt −→ + × × = 16.67t nv Ae −→ = + – + – t = 0 v –+ 30 kΩ 60 kΩ 1 µF A B 10 kΩ 20 kΩ 6 V vo + – 16.67 16.674 ( 8 8 ) 12 8 Vt t o Bv v v e e − − = − = − − + = − First-Order Op Amp Circuits (4) First-Order Circuits - sites.google.com/site/ncpdhbkhn 40 Ex. 2 Find vo(t)? f nv v v= + (0) 0v = 8Vfv = − 16.67t n v Ae−= 0(0) 8 0 8v Ae A= − + = → = 16.678 8 Vtv e−→ = − + + – + – t = 0 v –+ 30 kΩ 60 kΩ 1 µF A B 10 kΩ 20 kΩ 6 V vo + – First-Order Op Amp Circuits (5) First-Order Circuits - sites.google.com/site/ncpdhbkhn 41 Ex. 3 Find v(t)? f nv v v= + 0 0V = 20 20 5 10 V 10 10Bf Af v v= − = − = − +– vi vo ii io R1 R2 2 1 o i R v v R = − B +– 10 kΩ + – t = 0 20 kΩ 40 kΩ 60 kΩ A v + – 1 µF 5 V 60 60 ( 10) 6V 40 60 40 60f Bf v v= = − = − + + 3 3 6 3 3 1 40 10 60 10 1 10 41.6740 10 60 10 t t n v Ae Ae − − × × × × −× + × = = 0 0 (0) 6 0 6V v Ae A= = − + = → = 41.676 6 Vtv e−→ = − + 40 kΩ 60 kΩ vn + – 1 µF