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Understanding First Order Transient Response in Electrical Circuits

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This discussion focuses on the transient response of first-order circuits and highlights the importance of making informed choices in circuit design. We examine the steady-state and transient responses of circuits containing capacitors and inductors, discussing their behaviors during switching events. Key concepts include initial and final conditions of voltages and currents, along with practical examples to illustrate these principles. Understanding these foundational concepts is crucial for anyone studying electrical engineering or circuit analysis.

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Understanding First Order Transient Response in Electrical Circuits

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  1. Schedule… Discussion #15 – 1st Order Transient Response

  2. 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

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

  4. 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

  5. Capacitor/Inductor Voltages/Currents Review of capacitor/inductor currents and voltages • exponential growth/decay 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

  6. 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

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

  8. 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

  9. 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

  10. R1 t = 0 vs + – R2 R3 C DC Steady-State • Example1: determine the final condition capacitor voltage vs= 12V, R1 = 100Ω, R2 = 75Ω, R3 = 250Ω, C = 1uF Discussion #15 – 1st Order Transient Response

  11. R1 t → 0+ vs + – R2 R3 C DC Steady-State • Example1: 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

  12. R1 t → ∞ vs + – R2 R3 C DC Steady-State • Example1: 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

  13. 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

  14. t = 0 is iL L R DC Steady-State • Example2: find the initial and final current conditions at the inductor is = 10mA Discussion #15 – 1st Order Transient Response

  15. t = 0 is is iL L R iL DC Steady-State • Example2: 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

  16. is iL DC Steady-State • Example2: 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

  17. t = 0 is iL iL L R L R DC Steady-State • Example2: 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

  18. t = 0 is – R + iL iL L R L DC Steady-State • Example2: 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

  19. is DC Steady-State • Example2: 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

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

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

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

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

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

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

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

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

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

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

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

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

  32. 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

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

  34. General Solution of 1st Order Circuits Forced response (particular 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

  35. 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

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

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

  38. 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

  39. 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

  40. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example3: find vc(t) for all t vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF Discussion #15 – 1st Order Transient Response

  41. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example3: 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

  42. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example3: 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

  43. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example3: 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 Discussion #15 – 1st Order Transient Response

  44. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example3: 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

  45. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example3: 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

  46. R t = 0 + vC(t) – C vs i(t) + – Transient Response • Example3: 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

  47. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example4: find vc(t) for all t va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF Discussion #15 – 1st Order Transient Response

  48. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example4: 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 will be equal to vT (the open circuited Thévenin equivalent) Discussion #15 – 1st Order Transient Response

  49. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example4: 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

  50. R1 R3 t = 0 + vC(t) – + – R2 C vb va + – Transient Response • Example4: 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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