ELEG 205 Fall 2017 Leure #13 Mark Miroznik, Ph.D. Professor The Universiy of Delaware Tel: (302831-4221 Email: mirozni@ee.udel.edu
Chaper 8: RL and RC Ciruis 1. Soure-free RL iruis (naural response 2. Soure-free RC iruis (naural response 3. Driven RL iruis (fored response 4. Driven RC iruis (fored response
Summary: RL and RC Naural and Fored Response Naural Response Fored Response Time onsan RL i L i (0 L e τ i L ( τ i (0 i ( e i ( L L L τ L R eq RC v v (0 e τ v ( τ v (0 v ( e v ( τ R eq C
100 Ω 100 Ω 800 Ω 0 0.1 A 0 100 Ω i ( 1 µf 10 V Find he urren in he apaior for all ime. Is his a naural response or fored response problem?
100 Ω 100 Ω 800 Ω 0 0.1 A 0 100 Ω i ( 1 µf 10 V v Fored Response Find apaior volage firs ( τ v (0 v ( e v ( τ R eq C final value iniial value ime onsan
100 Ω 100 Ω 800 Ω 0.1 A 0 100 Ω i ( 1 µf 0 10 V v ( τ v (0 v ( e v ( τ R eq C Problem #1: Solve for he iniial ondiion (0 -. Nohing is hanging a his ime so his is a seady-sae problem. This means you an replae he apaior by an open irui.
100 Ω 100 Ω 800 Ω 0.1 A 100 Ω v ( 0-10 V Problem #1: Solve for he iniial ondiion (0 -. Nohing is hanging a his ime so his is a seady-sae problem. This means you an replae he apaior by an open irui. v ( 0 0.1 100 10 V v ( 0 v (0 10 V
100 Ω 100 Ω 800 Ω 0.1 A 100 Ω v ( 0-10 V Problem #2: Solve for he final ondiion (infiniy. Nohing is hanging a his ime so his is a seady-sae problem. This means you an replae he apaior by an open irui. 100 v ( 10 1. 11V 100 800
100 Ω 100 Ω 800 Ω 0.1 A 100 Ω i ( 1 µf 10 V Problem #3: Find he ime onsan. Do his a 0. To find he equivalen resisane kill all he soures. C 1 µf τ R C eq R eq 100 800 100 800 88.9 Ω τ 88.9 10 6 se
100 Ω 100 Ω 800 Ω 0.1 A 0 100 Ω i ( 1 µf 0 10 V Now pu i all ogeher. v ( 1. 11V v ( v (0 v e τ v ( ( τ 88.9 10 6 v ( 0 10V
100 Ω 100 Ω 800 Ω 0.1 A 0 100 Ω i ( 1 µf 0 10 V Now pu i all ogeher. v 1.11 ( 10 1.11 e 88.9 10 6 v 1.11 8.89 e 11,249 How do we find I ( now?
100 Ω 100 Ω 800 Ω 0.1 A 0 100 Ω i ( 1 µf 0 10 V v 1.11 8.89 e 11,249 i C dv d 10 6 ( 11,249 11,249 11,249 8.89 e 0.1 e > 0 i 0 < 0
100 Ω 100 Ω 800 Ω 0.1 A 0 100 Ω i ( 1 µf 0 10 V i 0 < 0 i 0.1 e 11,249 > 0
0 500 Ω 10 V 5 V 500 Ω 2 H i( Find he urren in he induor for all ime. Is his a naural response or fored response problem?
0 500 Ω 10 V 5 V 500 Ω 2 H i( Find he urren in he induor for all ime. fored response i L L ( τ i (0 i ( e i ( L L τ L R eq
0 500 Ω 10 V 5 V 500 Ω 2 H i( Firs find iniial ondiion a 0-500 Ω 10 V 5 V 500 Ω 0 ( i
Firs find iniial ondiion a 0-500 Ω 10 V 5 V 500 Ω i ( 0 KVL : 10 5 500i i (0 5 1000 5mA (0 500i (0 i ( 0 i (0 5mA 0
Nex find iniial ondiion a infiniy 500 Ω 10 V 5 V 500 Ω i ( 10 i ( 20 ma 500
0 500 Ω 10 V 5 V 500 Ω 2 H i( Las find he ime onsan 500 Ω 10 V 5 V 500 Ω 2 H
Las find he ime onsan 500 Ω 10 V 5 V 500 Ω 2 H L τ Req 2 500 1 250 τ 1 250 se
0 500 Ω 10 V 5 V 500 Ω 2 H i L L i( ( τ i (0 i ( e i ( L L i L 20 ( 250 5 20 e ma i L 20 15e 250 ma
20 kω 3 kω 5 µf 12 V 5 kω 1 kω v( 0-10 kω Find he volage v( for all ime. Is his a naural response or fored response problem?
20 kω 3 kω v ( 5 µf 12 V 5 kω 1 kω v( - 0-10 kω v naural response τ v (0 e τ R eq C
20 kω 3 kω - v ( 5 µf 12 V 0 10 kω 5 kω 1 kω - v( Find iniial ondiion a 0-20 kω 3 kω v ( 0 v (0 12V v ( 0 12 V 5 kω 1 kω - 10 kω
20 kω 3 kω v ( 5 µf 12 V 5 kω 1 kω v( - 0-10 kω Find ime onsan 20 kω 3 kω v ( 5 µf 12 V 5 kω 1 kω v( - - 10 kω
20 kω 3 kω - v ( 5 µf 5 kω 1 kω - v( 10 kω τ R eq C C 5 µf R eq 5,000 1,000 5,000 1,000 3,000 10,000 20,000 33,667 Ω τ 0.1683se v τ v (0 e 12 e.1683 0 12 e 5.94
20 kω 3 kω - v ( 5 µf 5 kω 1 kω - v( v i v 12 e dv C d 5.94 5 10 6 10 kω 12 5.94e 5.94 0.3564 e 5.94 i ( i v i KVL ouer loop: 20000 ( 3000 ( ( 10000 0 12 e 5.94 0.3564 e 5.94 ma 33,000 v( 0 ma v( 0.2388 e 5.94 V
Chaper 10: Seady-Sae Sinusoidal Analysis V Aos( ω φ
Chaper 10: Sinusoidal Seady-Sae Review of Sinusoids V Aos( ω φ ime, se ω A radians angular frequeny, se ampliudeor magniude φ phaseangle, radians
Chaper 10: Sinusoidal Seady-Sae ω 2π f V Aos( ω φ V Aos(2π f φ ime, se ω angular frequeny, f frequeny, Hz radians se ( yles / se A ampliudeor magniude φ phase angle, radians
V π φ 5os(2, A 5, f 1Hz, 0 5 1 yle or 2π radians over 1 seond Tperiod
V 2.5os(2π, A 2.5, f 1Hz, φ 0 2.5 1 yle or 2π radians over 1 seond
V 5os(6π, A 5, f 3Hz, φ 0 5 3 yles or 6π radians over 1 seond
π π V π 5os(2, A 5, f 1Hz, φ 2 2 5 1 yle or 2π radians over 1 seond
π φ V 5os(2, A 5, f 1Hz
Oher ways o wrie sinusoids V Aos( ω φ Reall: e jx os( x j sin( x e j ( ω φ j( ω φ e V Aos( ω φ A 2
Oher ways o wrie sinusoids V Aos( ω φ Reall: e jx os( x j sin( x V Aos( ω φ Re [ ( ] A e j ω φ Re [ ] A e jφ e jω Re real par
Before we learn how o solve irui problems using sinusoids Wha is so speial abou sinusoids?
Wha is so speial abou sinusoids? Wha do I ge if I add wo sinusoids of he same frequeny? Aos( ω φ B os( ω φ A B?
Wha is so speial abou sinusoids? Wha do I ge if I add wo sinusoids of he same frequeny? Aos( ω φ B os( ω φ C os( ω A B φ C C φ C [ ( ( ] 2 Aos φ B os φ [ Asin( φ Bsin( φ ] an 1 A Aos Asin A ( φ ( ( ( A B os φa φa Bsin φa A A 2 The resul is a sinusoid of he same frequeny, ω!
Wha is so speial abou sinusoids? Wha do I ge if I subra wo sinusoids of he same frequeny? Aos( ω φ B os( ω φ C os( ω A B φ C C φ C [ ( ( ] 2 Aos φ B os φ [ Asin( φ Bsin( φ ] an 1 A Aos Asin A ( φ ( ( ( A B os φa φa Bsin φa A A 2 The resul is a sinusoid of he same frequeny, ω!
Wha is so speial abou sinusoids? Wha do I ge if I ake he derivaive of a sinusoid? d d ( Aos( ω φ A ω sin( ω φ A ω os( ω φ π 2 The resul is a sinusoid of he same frequeny, ω!
Wha is so speial abou sinusoids? Wha do I ge if I ake he inegral of a sinusoid? ( Aos( ω φ d A ω sin( ω φ A ω os( ω φ π 2 The resul is a sinusoid of he same frequeny, ω!
Wha is so speial abou sinusoids? REASON #1 A os( ω φa Linear Sysem C os( ω φc For any linear sysem if I inpu a sinusoid of frequeny, ω, he oupu is also a sinusoid of he same frequeny.
Wha is so speial abou sinusoids? REASON #2
Wha is so speial abou sinusoids? REASON #2 Any non-sinusoid signal an be onsrued by adding a bunh of sinusoids of differen frequenies. Called Fourier series and Fourier ransforms.
Elerial Symbols Library for Drawing Shemais in PowerPoin For bes resuls: Turn snap-o-grid-on Do no sale he symbols library. Draw enire shemai firs, hen sale o desired size. Afer roaion, press up/down and sideways arrow keys o realign he omponen wih he grid. Aives Passives Soures Disrees