First Order Circuits II: mathematical tools we use are a model - PowerPoint PPT Presentation

3/4/20 Contemplating Mathematical Models • One thing that we must remember is that all the First Order Circuits II: mathematical tools we use are a model intended to describe the observed characteristics of the real Step Response to world. • Mother Nature doesn't know anything about any Complete Response equations • The real world does what it does and we as engineers develop models to understand and predict what will happen in any given situation. EGR 220, Chapter 7 part 2 March 5, 2020 - Steve Umans 2 1 2 Steady-State Behavior: Behave as… Overview Short Circuit or Open Circuit? • Previous Class: Natural response as found in source-free circuits • Time dependent functions v(t) & i(t) behavior in first order circuits (circuits with a single storage element) • Today: Response to a dc ‘step’ input = Forced response • Input is a switch or unit step, u(t) function 3 5 3 5 1

3/4/20 When the Circuit is in DC Steady-State: When the Circuit is in DC Steady-State: Which is Inductor V & I Which is Capacitor V & I 6 7 6 7 * Continuity Relationship * Derive the Natural Response Expression i ( t ) = I 0 e − tR / L • Show that • Stored energy cannot change instantaneously à it is “continuous” • Capacitor: i c = ---- • v c ( 0 – ) = v c ( 0 + ) ≡ V 0 • Capacitor current? • Inductor: v L = ---- • i L ( 0 – ) = i L ( 0 + ) ≡ I 0 • Inductor voltage? 8 9 8 9 2

3/4/20 Solve RL Circuit à with u (- t ) Recap: Natural Response u(-t) • What is the role of u (-t)? • Form of solution? • Find i (t): time • Time periods of interest? • Find I o for t < 0 • Use i L ( 0 – ) = i L ( 0 + ) ≡ I 0 • Which values do you calculate using information from which time periods? • Find τ for t > 0 • R = R Th at L 1) • 5Ω resistor? 2) 10 11 10 11 RL Circuit Natural Response • Write i (t) expression • Write v R (t) expression • Discuss polarity of current flow, and v R (t) and v L (t) 12 13 12 13 3

3/4/20 Com Complete Response of an RL Circuit The Complete Response • Find i (t) for all time t > 0 i ( t ) = i n ( t ) + i f ( t ) • Complete response = Step response = u(-t) • Natural response (stored energy) + 2 u ( -t ) • Forced response (independent source) time • The superposition of the response to stored energy & to a power source • But first – to learn the new concept, find only the forced response 14 15 14 15 RL Circuit: Forced Response RL Circuit: Forced Response • Determine i L (t) and v (t) for all time. • Assume that the current through the inductor is zero for t<0 (for the forced response, assume no stored energy) . 1. What is i L (t =0) ? 2. What is v (t =0) ? L i s u (t) R i s u (t) R L KVL: 3. What is i L (t>0) ? KCL: 17 18 17 18 4

3/4/20 RL Circuit: Forced Response RL Circuit: Forced Response • Determine i L (t) and v (t) for all time. v = L di L • Assume that the current through the inductor is zero for t<0 (for the forced KVL: dt = i R R response, assume no stored energy) . L i s u (t) R KCL: i L + i R = i S so i R = − i L − i S ( ) 1. What is i L (t =0) ? 2. What is v (t =0) ? L Substitute KCL into KVL and rearrange i s u (t) R 1) 3) KVL: 3. What is i L (t>0) ? 2) 4) KCL: 21 19 19 21 RL Circuit: Forced Response RL Circuit: Forced Response Substitute KCL into KVL and rearrange R - t - = - L i ( t ) i i e L i s u (t) R L i s u (t) R L S s Rearrange to get our desired expressions 22 24 22 24 5

3/4/20 RL Circuit: Forced Response Graph of Forced Response? • What do each of these terms represent? • What is the graph of this response? • Note change in notation: i s to i ∞ # & − R Lt i L ( t ) = i ∞ 1 − e % ( $ ' { v c ( t ) = V ∞ 1 − e − t / RC ( ) } 26 27 26 27 Graph of Complete Response? The Complete Response i ( t ) = i n ( t ) + i f ( t ) • Complete response = Step response = • Natural response (stored energy) + • Forced response (independent source) • The superposition of the response to stored energy & to a power source 28 29 28 29 6

3/4/20 1) Initial Conditions Complete Response of an RL Circuit • Find the initial conditions 2 u ( -t ) Find i (t) for t > 0 • t = 0 – leads to t = 0 + 1) Write the form of the solution • “Continuity relationship” for L and C • At t = 0 – we know i L (0 – ), so therefore... 2) Identify what you need to calculate, and for which time periods 2 u ( -t ) 30 31 30 31 2) Time Constant 3) Form the Natural Response • Find τ = L/R v n ( t ) = V 0 e − t / τ or • This often means finding R eq from the storage element 2 u ( -t ) i n ( t ) = I 0 e − t / τ 2 u ( -t ) 32 33 32 33 7

3/4/20 4) Final Condition, I ∞ 5) Form the Forced Response i f (t) = i L ( t ) = I ∞ 1 − e − tR / L ( ) • Find the value of current at time ( ) i L ( t ) = I ∞ 1 − e − tR / L t = ∞ (again in DC steady-state) 2 u ( -t ) 2 u ( -t ) 34 35 34 35 Text Formulas for Step Response 6) Total, Complete Response − t τ • RC circuit v ( t ) = v ( ∞ ) + [ v (0) − v ( ∞ )] e i ( t ) = i f ( t ) + i n ( t ) • RL circuit − t τ i ( t ) = i ( ∞ ) + [ i (0) − i ( ∞ )] e v ( t ) = v f ( t ) + v n ( t ) • Be careful not to use these equations without understanding how to develop them – you may be asked to explain each term 36 37 36 37 8

3/4/20 Recap: Unit Step Function Recap: Unit Step Function u ( a ) a a u ( t ) = 0, t < 0 ⎧ ⎧ u ( t ) = 0, t < 0 ( a ) ⎨ ⎨ a 1, t > 0 1, t > 0 ⎩ ⎩ t ( 0 – ) t ( 0 ) t ( 0 + ) ⎧ ⎧ u ( − t ) = ___ t < 0 u ( − t ) = ___ t < 0 ⎨ ⎨ ___ t > 0 ___ t > 0 ⎩ ⎩ v in (t) a time 0 t ( 0 – ) t ( 0 ) t ( 0 + ) 38 39 38 39 Summary • Complete response = step response = total response = the sum of • Natural response + Questions? • Forced response • Practice the analysis method, step by step • Know what each term means in the i(t) and v(t) step response expressions 40 40 41 9