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for SMALL Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York EE 203 Circuit Analysis 2 Lecture 26 Chapter 13.8 Impulse Function in Circuit Analysis Kwang W. Oh, Ph.D., Assistant Professor SMALL (nanobioSensors and MicroActuators Learning Lab) Department of Electrical Engineering University at Buffalo, The State University of New York 215E Bonner Hall, SUNY-Buffalo, Buffalo, NY 14260-1920 Tel: (716) 645-3115 Ext. 1149, Fax: (716) 645-3656 E-mail: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 1/12 Impulse Function by a Switching Operation Capacitor Circuit (1) C1 is charged to an initial voltage of V0 at the time the switch is closed. The initial charge on C2 is zero. Q: find the expression for i(t) as R 0. Vo 1 1 = I( +R+ ) s sC1 sC2 As R gets smaller, the current starts from a larger initial value and then drops off more rapidly idl Vo Q R() Initial Current () R R () Time Constant RCe () EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 2/12 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Impulse Function by a Switching Operation Capacitor Circuit (2) As As R gets smaller, the current starts from a larger initial value and then drops off more rapidly as R zero, i is approaching an impulse function R 0: i VOCe(t) Impulse Function by a Switching Operation Series Inductor Circuit (1) Q: Q: find the time-domain expression for vo after the switch has been opened. Opening the switch forces an instantaneous change in the current of L2, which causes vo to contain an impulsive component. Initial condition at t = 0- When R = 0, a finite amount of charge is transferred transferred to C2 instantaneously. Set R = 0 The current in the 3 H inductor = 10 A The current in the 2 H inductor = 0 A 100 + 30 + 30 =s 10 + 3s + 15 + 2 s 5s + 25 20 + 6 6 s + 20 4 2 =s = =+ s + 5 s( s + 5) s s + 5 I= 100 s i = ( 4 + 2 e 5 t )u ( t ) EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 3/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 4/12 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Impulse Function by a Switching Operation Series Inductor Circuit (2) Vo = (15 + 2 s ) I = (15 + 2 s ) 6s + 20 s( s + 5) 12( s 2 + 5s ) 60s + 90s + 40s + 300 s 2 + 5s 70s + 300 = 12 + s( s + 5) K1 K 60 10 = 12 + + 2 = 12 + + s s+5 s s+5 300 70 s + 300 = 60 K1 = [12 s + ]= s+5 5 s =0 = K1 = [12( s + 5) + 100 + 30 + 30 =s 10 + 3s + 15 + 2 s 5s + 25 20 + 6 6 s + 20 4 2 =s = =+ s + 5 s( s + 5) s s + 5 I= 70s + 300 350 + 300 = 10 ] = s 5 s = 5 vo = 12 (t ) + (60 + 10e 5t )u (t ) EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 100 s i = ( 4 + 2 e 5 t )u ( t ) Lecture 27 | Chapter 13 | 8/8 | 5/12 Impulsive Sources Impulsive Impulsive functions can occur in sources as well as responses A mechanical analogy is striking a bell with an impulsive clapper blow. After the energy has been transferred to the After bell, the natural responses of the bell determines the tone emitted (that the is, frequency of the resulting sound waves) and and the tones duration. Initial condition When the impulse voltage source is applied, the initial energy energy in the inductor is zero Therefore the initial current is zero. There is no voltage drop across R, so the impulsive voltage source appears directly across L. An impulsive voltage at the terminals of an inductor establishes an instantaneous current. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 6/12 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Internally Generated and Externally Applied Impulses Simultaneously (1) 100 + 50 + 30 + 80 20 + 16 =s =s 10 + 3s + 15 + 2 s 5s + 25 s+5 16s + 20 K1 K 4 12 = = + 2= + s( s + 5) s s+5 s s+5 Initial Initial Condition at t = 0- I= i1(0-) = 10 A i2(0-) = 0 A SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York Impulse Function by a Switching Operation 100 s i = ( 4 + 12e 5t )u(t ) Series Inductor Circuit (1) Vo = (15 + 2 s ) I = (15 + 2 s ) Q: find the time-domain expression for vo after the switch has been opened. Opening the switch forces an instantaneous change in the current of L2, which causes vo to contain an impulsive component. Initial condition at t = 0- 16s + 20 s( s + 5) 32( s 2 + 5s ) 160s + 240s + 40s + 300 s 2 + 5s 120s + 300 K K = 32 + 1 + 2 = 32 + s( s + 5) s s+5 60 60 = 32 + + s s+5 vo = 32 (t ) + (60 + 60e 5t )u(t ) = The current in the 3 H inductor = 10 A The current in the 2 H inductor = 0 A 100 100 + 30 s + 30 =s 10 + 3s + 15 + 2 s 5s + 25 20 + 6 6 s + 20 4 2 =s = =+ s + 5 s ( s + 5) s s + 5 I= i = ( 4 + 2 e 5 t )u ( t ) EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Kwang Oh Internally Generated and Externally Applied Impulses Simultaneously (2) Lecture 27 | Chapter 13 | 8/8 | 4/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 7/12 K1 = [32 s + 120s + 300 300 ]= = 60 s+5 5 s =0 K1 = [32( s + 5) + EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 120s + 300 600 + 300 ] = = 60 s 5 s = 5 Lecture 27 | Chapter 13 | 8/8 | 8/12 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Internally Generated and Externally Applied Impulses Simultaneously (3) Summary (1) Initial Initial Condition at t = 0i1(0-) = 10 A i2(0-) = 0 A Switching Off (t 0+) i = ( 4 + 12e 5t )u (t ) it =0+ = 16 A i1 = 10 A 16 A i1 = 6A i2 = 0 A 16 A i2 = 16A it = = 4 A EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 9/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 10/12 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Summary (3) Summary (2) input output, response h(t) (t) impulse source system, circuit, black box impulse response EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 11/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 27 | Chapter 13 | 8/8 | 12/12 ... View Full Document