9/14/29 1 EEEB113 CICUIT ANALYSIS I Chaper 7 Firs-Order Circuis Maerials from Fundamenals of Elecric Circuis 4e, Alexander Sadiku, McGraw-Hill Companies, Inc. 2 Firs-Order Circuis -Chaper 7 7.2 The Source-Free C Circui 7.3 The Source-Free L Circui 7.4 Uni-sep Funcion 7.5 Sep esponse of an C Circui 7.6 Sep esponse of an L Circui 1
9/14/29 7.1 The Source-Free C Circui (1) 3 A firs-order circui is characerized by a firsorder differenial equaion. By KCL i i C v dv C d Ohms law Capacior law Apply Kirchhoff s laws o purely resisive circui resuls in algebraic equaions. Apply he laws o C and L circuis produces differenial equaions. 7.1 The Source-Free C Circui (2) 4 The naural response of a circui refers o he behavior (in erms of volages and currens) of he circui iself, wih no exernal sources of exciaion. Time consan C Decays more slowly Decays faser The ime consan of a circui is he ime required for he response o decay by a facor of 1/e or 36.8% of is iniial value. v decays faser for small and slower for large. 2
9/14/29 7.1 The Source-Free C Circui (3) 5 The key o working wih a source-free C circui is finding: v( ) V e / where C 1. The iniial volage v() = V across he capacior. 2. The ime consan = C. 7.1 The Source-Free C Circui (4) 6 P.P.7.1 efer o he circui below. Le v C () = 45 V. Deermine v C, v x, and i o for. Answer: v C = 45e.25 V ; v x = 15e.25 ; i o = 3.75e.25 A 3
9/14/29 7 7.1 The Source-Free C Circui (5) Soln. P.P.7.1 Volage divider rule KVL 8 7.1 The Source-Free C Circui (6) P.P.7.2 If swich in circui below is opened a =, find v() for. Answer: V() = 8e 2 V, w c () =5.33 J 4
9/14/29 9 7.1 The Source-Free C Circui (7) Soln. P.P.7.2 1 7.1 The Source-Free C Circui (8) con. Soln. P.P.7.2 5
9/14/29 7.2 The Source-Free L Circui (1) 11 A firs-order L circui consiss of a inducor L (or is equivalen) and a resisor (or is equivalen) By KVL v L v di L d i Inducors law Ohms law di i L d i( ) I e / L 7.2 The Source-Free L Circui (2) 12 A general form represening a L i( ) I e / where L The ime consan of a circui is he ime required for he response o decay by a facor of 1/e or 36.8% of is iniial value. i() decays faser for small and slower for large. The general form is very similar o a C source-free circui. 6
9/14/29 7.2 The Source-Free L Circui (3) 13 Comparison beween a L and C circui A L source-free circui A L source-free circui i( ) i( ) I e I e / / where where L L A C source-free circui A C source-free circui v( ) V e v( ) V e / / where where C C 7.2 The Source-Free L Circui (4) 14 The key o working wih a source-free L circui is finding: i( ) I e / where L 1. The iniial volage i() = I hrough he inducor. 2. The ime consan = L/. 7
9/14/29 7.2 The Source-Free L Circui (5) 15 P.P.7.3 Find i and v x in he circui. Le i() = 5 A. Answer: i() = 5e 53 A 7.2 The Source-Free L Circui (6) Soln. P.P.7.3 16 8
9/14/29 17 7.2 The Source-Free L Circui (7) con. Soln. P.P.7.3 7.2 The Source-Free L Circui (8) con. Soln. P.P.7.3 18 9
9/14/29 7.2 The Source-Free L Circui (9) con. Soln. P.P.7.3 19 7.2 The Source-Free L Circui (1) 2 P.P.7.4 For he circui, find i() for >. Answer: i() = 2e 2 A 1
9/14/29 21 7.2 The Source-Free L Circui (11) Soln. P.P.7.4 22 7.2 The Source-Free L Circui (12) con. Soln. P.P.7.4 11
9/14/29 7.3 Uni-Sep Funcion (1) 23 The uni sep funcion u() is for negaive values of and 1 for posiive values of. u( ), 1, u( o ), 1, o o u( o ), 1, o o 7.3 Uni-Sep Funcion (2) 24 epresen an abrup change for: 1. volage source. 2. for curren source: 12
9/14/29 7.4 The Sep-esponse of a C Circui (1) 25 The sep response of a circui is is behavior when he exciaion is he sep funcion, which may be a volage or a curren source. Iniial condiion: v(-) = v(+) = V Applying KCL, dv c d v Vsu( ) or dv d v V C s u() Where u() is he uni-sep funcion 7.4 The Sep-esponse of a C Circui (2) 26 Inegraing boh sides and considering he iniial condiions, he soluion of he equaion is: V v( ) / V ( V V ) e s s Final value a -> Iniial value a = Source-free esponse Complee esponse = Naural response + Forced esponse (sored energy) (independen source) = V e /τ + V s (1 e /τ ) 13
9/14/29 7.4 The Sep-esponse of a C Circui (3) 27 Three seps o find ou he sep response of an C circui: 1. The iniial capacior volage v(). 2. The final capacior volage v( ) DC volage across C. 3. The ime consan. v ( ) v ( ) [ v ( ) v ( )] e / Noe: The above mehod is a shor-cu mehod. You may also deermine he soluion by seing up he circui formula direcly using KCL, KVL, ohms law, capacior and inducor VI laws. 7.5 The Sep-response of a L Circui (1) 28 The sep response of a circui is is behavior when he exciaion is he sep funcion, which may be a volage or a curren source. Iniial curren i(-) = i(+) = I o Final inducor curren i( ) = Vs/ Time consan = L/ i( ) Vs ( I o Vs ) e 14
9/14/29 7.5 The Sep-esponse of a L Circui (2) 29 Three seps o find ou he sep response of an L circui: 1. The iniial inducor curren i() a = +. 2. The final inducor curren i( ). 3. The ime consan. i ( ) i ( ) [ i ( ) i ( )] e / Noe: The above mehod is a shor-cu mehod. You may also deermine he soluion by seing up he circui formula direcly using KCL, KVL, ohms law, capacior and inducor VI laws. 15