1. 1. 9/14/2009 1 EEEB113 CIRCUIT ANALYSIS I Chapter 7 First-Order Circuits 1 Materials from Fundamentals of Electric Circuits 4e, Alexander Sadiku, McGraw-Hill Companies, Inc. First-Order Circuits -Chapter 7 2 7.2 The Source-Free RC Circuit 7.3 The Source-Free RL Circuit 7.4 Unit-step Function 7.5 Step Response of an RC Circuit 7.6 Step Response of an RL Circuit
2. 2. 9/14/2009 2 7.1 The Source-Free RC Circuit (1) A first-order circuit is characterized by a first- order differential equation. 3 Apply Kirchhoffs laws to purely resistive circuit results in algebraic equations. Apply the laws to RC and RL circuits produces differential equations. Ohms law Capacitor law 0 dt dv C R v 0CR ii By KCL 7.1 The Source-Free RC Circuit (2) The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation. 4 The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value. v decays faster for small and slower for large . CRTime constant Decays more slowly Decays faster
3. 3. 9/14/2009 3 7.1 The Source-Free RC Circuit (3) The key to working with a source-free RC circuit is finding: 5 1. The initial voltage v(0) = V0 across the capacitor. 2. The time constant = RC. / 0)( t eVtv CRwhere 7.1 The Source-Free RC Circuit (4) P.P.7.1 Refer to the circuit below. Let vC(0) = 45 V. Determine vC, vx, and io for t 0. 6 Answer: vC = 45e0.25t V ; vx = 15e0.25t ; io = 3.75e0.25t A
4. 4. 9/14/2009 4 7.1 The Source-Free RC Circuit (5) 7 Soln. P.P.7.1 Voltage divider rule KVL 7.1 The Source-Free RC Circuit (6) P.P.7.2 If switch in circuit below is opened at t = 0, find v(t) for t 0. 8 Answer: V(t) = 8e2t V, wc(0) =5.33 J
5. 5. 9/14/2009 5 7.1 The Source-Free RC Circuit (7) Soln. P.P.7.2 9 7.1 The Source-Free RC Circuit (8) cont. Soln. P.P.7.2 10
6. 6. 9/14/2009 6 7.2 The Source-Free RL Circuit (1) A first-order RL circuit consists of a inductor L (or its equivalent) and a resistor (or its equivalent) 11 0RL vvBy KVL 0iR dt di L Inductors law Ohms law dt L R i di LtR eIti / 0)( 7.2 The Source-Free RL Circuit (2) 12 The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value. i(t) decays faster for small and slower for large . The general form is very similar to a RC source-free circuit. / 0)( t eIti R L A general form representing a RL where
7. 7. 9/14/2009 7 7.2 The Source-Free RL Circuit (3) 13 / 0)( t eIti R L A RL source-free circuit where / 0)( t eVtv RC A RC source-free circuit where Comparison between a RL and RC circuit / 0)( t eIti R L A RL source-free circuit where / 0)( t eVtv RC A RC source-free circuit where 7.2 The Source-Free RL Circuit (4) The key to working with a source-free RL circuit is finding: 14 1. The initial voltage i(0) = I0 through the inductor. 2. The time constant = L/R. / 0)( t eIti R Lwhere
8. 8. 9/14/2009 8 7.2 The Source-Free RL Circuit (5) P.P.7.3 Find i and vx in the circuit. Let i(0) = 5 A. 15 Answer: i(t) = 5e53t A 7.2 The Source-Free RL Circuit (6) Soln. P.P.7.3 16
9. 9. 9/14/2009 9 7.2 The Source-Free RL Circuit (7) cont. Soln. P.P.7.3 17 7.2 The Source-Free RL Circuit (8) cont. Soln. P.P.7.3 18
10. 10. 9/14/2009 10 7.2 The Source-Free RL Circuit (9) cont. Soln. P.P.7.3 19 7.2 The Source-Free RL Circuit (10) P.P.7.4 For the circuit, find i(t) for t > 0. 20 Answer: i(t) = 2e2t A
11. 11. 9/14/2009 11 7.2 The Source-Free RL Circuit (11) Soln. P.P.7.4 21 7.2 The Source-Free RL Circuit (12) cont. Soln. P.P.7.4 22
12. 12. 9/14/2009 12 7.3 Unit-Step Function (1) The unit step function u(t) is 0 for negative values of t and 1 for positive values of t. 23 0,1 0,0 )( t t tu o o o tt tt ttu ,1 ,0 )( o o o tt tt ttu ,1 ,0 )( 7.3 Unit-Step Function (2) 1. voltage source. 2. for current source: 24 Represent an abrupt change for:
13. 13. 9/14/2009 13 7.4 The Step-Response of a RC Circuit (1) The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source. 25 Initial condition: v(0-) = v(0+) = V0 Applying KCL, or Where u(t) is the unit-step function 0 )( R tuVv dt dv c s )(tu RC Vv dt dv s 7.4 The Step-Response of a RC Circuit (2) Integrating both sides and considering the initial conditions, the solution of the equation is: 26 0)( 0 )( / 0 0 teVVV tV tv t ss Final value at t -> Initial value at t = 0 Source-free Response Complete Response = Natural response + Forced Response (stored energy) (independent source) = V0et/ + Vs(1et/)
14. 14. 9/14/2009 14 7.4 The Step-Response of a RC Circuit (3) Three steps to find out the step response of an RC circuit: 27 1. The initial capacitor voltage v(0). 2. The final capacitor voltage v( ) DC voltage across C. 3. The time constant . / )]()0([)()( t evvvtv Note: The above method is a short-cut method. You may also determine the solution by setting up the circuit formula directly using KCL, KVL , ohms law, capacitor and inductor VI laws. 7.5 The Step-response of a RL Circuit (1) The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source. 28 Initial current i(0-) = i(0+) = Io Final inductor current i() = Vs/R Time constant = L/R t s o s e) R V I( R V )t(i
15. 15. 9/14/2009 15 7.5 The Step-Response of a RL Circuit (2) Three steps to find out the step response of an RL circuit: 29 1. The initial inductor current i(0) at t = 0+. 2. The final inductor current i( ). 3. The time constant . Note: The above method is a short-cut method. You may also determine the solution by setting up the circuit formula directly using KCL, KVL , ohms law, capacitor and inductor VI laws. / )]()0([)()( t eiiiti