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Good, Better, Best

Good, Better, Best. Elder Oaks (October 2007) :

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Good, Better, Best

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  1. Good, Better, Best Elder Oaks (October 2007): As we consider various choices, we should remember that it is not enough that something is good. Other choices are better, and still others are best. Even though a particular choice is more costly, its far greater value may make it the best choice of all. I have never known of a man who looked back on his working life and said, “I just didn't spend enough time with my job.” Discussion #15 – 1st Order Transient Response

  2. Lecture 15 – Transient Response of 1st Order Circuits DC Steady-State Transient Response Discussion #15 – 1st Order Transient Response

  3. Rs R1 R2 vs vs + – + – L1 L2 1st Order Circuits Electric circuit 1st order system: any circuit containing a single energy storing element (either a capacitor or an inductor) and any number of sources and resistors R1 R2 C 1st order 2nd order Discussion #15 – 1st Order Transient Response

  4. Capacitor/Inductor Voltages/Currents Review of capacitor/inductor currents and voltages • Exponential growth/decay NB: note the duality between inductors and capacitors Capacitor current iC(t) Inductor voltage vL(t) NB: both can change instantaneously Capacitor voltage vC(t) Inductor current iL(t) NB: neither can change instantaneously Discussion #15 – 1st Order Transient Response

  5. vs R1 + – t = 0 R2 C Transient Response Transient response of a circuit consists of 3 parts: • Steady-state response prior to the switching on/off of a DC source • Transient response – the circuit adjusts to the DC source • Steady-state response following the transient response DC Source Energy element Switch Discussion #15 – 1st Order Transient Response

  6. 1. DC Steady State 1st and 3rd Step in Transient Response Discussion #15 – 1st Order Transient Response

  7. DC Steady-State DC steady-state: the stable voltages and currents in a circuit connected to a DC source Capacitors act like open circuits at DC steady-state Inductors act like short circuits at DC steady-state Discussion #15 – 1st Order Transient Response

  8. R1 R1 t → ∞ t = 0 vs vs + – + – R2 R2 R3 R3 C C DC Steady-State Initial condition x(0): DC steady state before a switch is first activated • x(0–): right before the switch is closed • x(0+): right after the switch is closed Final condition x(∞): DC steady state a long time after a switch is activated Final condition Initial condition Discussion #15 – 1st Order Transient Response

  9. R1 t = 0 vs + – R2 R3 C DC Steady-State • Example 1: determine the final condition capacitor voltage • vs= 12V, R1 = 100Ω, R2 = 75Ω, R3 = 250Ω, C = 1uF Discussion #15 – 1st Order Transient Response

  10. R1 t → 0+ vs + – R2 R3 C DC Steady-State • Example 1: determine the final condition capacitor voltage • vs= 12V, R1 = 100Ω, R2 = 75Ω, R3 = 250Ω, C = 1uF • Close the switch and find initial conditions to the capacitor NB: Initially (t = 0+) current across the capacitor changes instantly but voltage cannot change instantly, thus it acts as a short circuit Discussion #15 – 1st Order Transient Response

  11. R1 t → ∞ vs + – R2 R3 C DC Steady-State • Example 1: determine the final condition capacitor voltage • vs= 12V, R1 = 100Ω, R2 = 75Ω, R3 = 250Ω, C = 1uF Close the switch and apply finial conditions to the capacitor NB: since we have an open circuit no current flows through R2 Discussion #15 – 1st Order Transient Response

  12. DC Steady-State Remember – capacitor voltages and inductor currents cannot change instantaneously • Capacitor voltages and inductor currents don’t change right before closing and right after closing a switch Discussion #15 – 1st Order Transient Response

  13. t = 0 is iL L R DC Steady-State • Example 2: find the initial and final current conditions at the inductor is = 10mA Discussion #15 – 1st Order Transient Response

  14. t = 0 is is iL L R iL DC Steady-State • Example 2: find the initial and final current conditions at the inductor is = 10mA • Initial conditions – assume the current across the inductor is in steady-state. NB: in DC steady state inductors act like short circuits, thus no current flows through R Discussion #15 – 1st Order Transient Response

  15. is iL DC Steady-State • Example 2: find the initial and final current conditions at the inductor is = 10mA • Initial conditions – assume the current across the inductor is in steady-state. Discussion #15 – 1st Order Transient Response

  16. t = 0 is iL iL L R L R DC Steady-State • Example 2: find the initial and final current conditions at the inductor is = 10mA • Initial conditions – assume the current across the inductor is in steady-state. • Throw the switch NB: inductor current cannot change instantaneously Discussion #15 – 1st Order Transient Response

  17. t = 0 is – R + iL iL L R L DC Steady-State • Example 2: find the initial and final current conditions at the inductor is = 10mA • Initial conditions – assume the current across the inductor is in steady-state. • Throw the switch • Find initial conditions again (non-steady state) NB: polarity of R NB: inductor current cannot change instantaneously Discussion #15 – 1st Order Transient Response

  18. is DC Steady-State • Example 2: find the initial and final current conditions at the inductor is = 10mA • Initial conditions – assume the current across the inductor is in steady-state. • Throw the switch • Find initial conditions again (non-steady state) • Final conditions (steady-state) t = 0 iL L R NB: since there is no source attached to the inductor, its current is drained by the resistor R Discussion #15 – 1st Order Transient Response

  19. 2. Adjusting to Switch 2nd Step in Transient Response Discussion #15 – 1st Order Transient Response

  20. + R– + C – + – iC vs General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation NB: Review lecture 11 for derivation of these equations Discussion #15 – 1st Order Transient Response

  21. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation Discussion #15 – 1st Order Transient Response

  22. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation NB: Constants Discussion #15 – 1st Order Transient Response

  23. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation NB: similarities Discussion #15 – 1st Order Transient Response

  24. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation Discussion #15 – 1st Order Transient Response

  25. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation Capacitor/inductor voltage/current Discussion #15 – 1st Order Transient Response

  26. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation Forcing function (F – for DC source) Discussion #15 – 1st Order Transient Response

  27. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation Combinations of circuit element parameters (Constants) Discussion #15 – 1st Order Transient Response

  28. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation Forcing Function F Discussion #15 – 1st Order Transient Response

  29. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation DC gain Discussion #15 – 1st Order Transient Response

  30. General Solution of 1st Order Circuits • Expressions for voltage and current of a 1st order circuit are a 1st order differential equation DC gain Time constant Discussion #15 – 1st Order Transient Response

  31. General Solution of 1st Order Circuits The solution to this equation (the complete response) consists of two parts: • Natural response (homogeneous solution) • Forcing function equal to zero • Forced response (particular solution) Discussion #15 – 1st Order Transient Response

  32. General Solution of 1st Order Circuits Natural response (homogeneous or natural solution) • Forcing function equal to zero Has known solution of the form: Discussion #15 – 1st Order Transient Response

  33. General Solution of 1st Order Circuits Forced response (particular or forced solution) F is constant for DC sources, thus derivative is zero NB: This is the DC steady-state solution Discussion #15 – 1st Order Transient Response

  34. General Solution of 1st Order Circuits Complete response (natural + forced) Solve for α by solving x(t) at t = 0 Time constant Final condition Initial condition Discussion #15 – 1st Order Transient Response

  35. General Solution of 1st Order Circuits Complete response (natural + forced) Transient Response Steady-State Response Discussion #15 – 1st Order Transient Response

  36. 3. DC Steady-State + Transient Response Full Transient Response Discussion #15 – 1st Order Transient Response

  37. vs R1 + – t = 0 R2 C Transient Response Transient response of a circuit consists of 3 parts: • Steady-state response prior to the switching on/off of a DC source • Transient response – the circuit adjusts to the DC source • Steady-state response following the transient response DC Source Energy element Switch Discussion #15 – 1st Order Transient Response

  38. Transient Response Solving 1st order transient response: • Solve the DC steady-state circuit: • Initial condition x(0–): before switching (on/off) • Final condition x(∞): After any transients have died out (t → ∞) • Identify x(0+): the circuit initial conditions • Capacitors: vC(0+) = vC(0–) • Inductors: iL(0+) = iL(0–) • Write a differential equation for the circuit at time t = 0+ • Reduce the circuit to its Thévenin or Norton equivalent • The energy storage element (capacitor or inductor) is the load • The differential equation will be either in terms of vC(t) or iL(t) • Reduce this equation to standard form • Solve for the time constant • Capacitive circuits: τ = RTC • Inductive circuits: τ = L/RT • Write the complete response in the form: • x(t) = x(∞) + [x(0) - x(∞)]e-t/τ Discussion #15 – 1st Order Transient Response

  39. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example 3: find vc(t) for all t • vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF Discussion #15 – 1st Order Transient Response

  40. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example 3: find vc(t) for all t vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF • DC steady-state • Initial condition: vC(0) • Final condition: vC(∞) NB: as t → ∞ the capacitor acts like an open circuit thus vC(∞) = vS Discussion #15 – 1st Order Transient Response

  41. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example 3: find vc(t) for all t vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF • Circuit initial conditions: vC(0+) Discussion #15 – 1st Order Transient Response

  42. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example 3: find vc(t) for all t vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF • Write differential equation (already in Thévenin equivalent) at t = 0 Recall: Discussion #15 – 1st Order Transient Response

  43. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example 3: find vc(t) for all t vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF • Write differential equation (already in Thévenin equivalent) at t = 0 NB: solution is of the form: Discussion #15 – 1st Order Transient Response

  44. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example 3: find vc(t) for all t vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF • Find the time constant τ Discussion #15 – 1st Order Transient Response

  45. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example 3: find vc(t) for all t vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF • Write the complete response • x(t) = x(∞) + [x(0) - x(∞)]e-t/τ Discussion #15 – 1st Order Transient Response

  46. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example 4: find vc(t) for all t va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF Discussion #15 – 1st Order Transient Response

  47. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example 4: find vc(t) for all t va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF • DC steady-state • Initial condition: vC(0) • Final condition: vC(∞) For t < 0 the capacitor has been charged by vb thus vC(0–) = vb For t → ∞ vc(∞) is not so easily determined – it is equal to vT (the open circuited Thévenin equivalent) Discussion #15 – 1st Order Transient Response

  48. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example 4: find vc(t) for all t va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF • Circuit initial conditions: vC(0+) Discussion #15 – 1st Order Transient Response

  49. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example 4: find vc(t) for all t va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF • Write differential equation at t = 0 • Find Thévenin equivalent Discussion #15 – 1st Order Transient Response

  50. + vC(t) – R1 R2 ia C R3 ib Transient Response • Example 4: find vc(t) for all t va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF • Write differential equation at t = 0 • Find Thévenin equivalent Discussion #15 – 1st Order Transient Response

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