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Step natural

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Step natural

  1. 1. Natural and step response Ahsan Khawaja
  2. 2. Cont. inductor V L di dt • Where V= voltage between inductor in (volt) L= inductor in (henery) = rate of change of current flow in amper di = rate of change of time in second dt
  3. 3. dv Cont. capacitor • Mathematical relation . i C dv dt • Where i= current that in capacitor in (ampere) C= capacitance in (farad) d v = rated of change of voltage. d t = rate of change of time in second
  4. 4. First-Ordersources, resistors and an Circuits A circuit that contains only • inductor is called an RL circuit. • A circuit that contains only sources, resistors and a capacitor is called an RC circuit. • RL and RC circuits are called first-order circuits because their voltages and currents are described by first-order differential equations. R i i L vs – + – + vs R C
  5. 5. 6 Different First-Order Circuits There are six different STC circuits. These are listed below. • An inductor and a resistance (called RL Natural Response). • A capacitor and a resistance (called RC Natural Response). • An inductor and a Thévenin equivalent (called RL Step Response). • An inductor and a Norton equivalent (also called RL Step Response). • A capacitor and a Thévenin equivalent (called RC Step Response). • A capacitor and a Norton equivalent (also called RC Step Response). RX LX CX RX RX + vS LX - iS RX iS RX LX RX + vS CX - These are the simple, first-order cases. Many circuits can be reduced to one of these six cases. They all have solutions which are in similar forms. CX
  6. 6. 6 Different First-Order Circuits These are the simplest cases, so we handle them first. There are six different STC circuits. These are listed below. • An inductor and a resistance (called RL Natural Response). RX CX RX LX • A capacitor and a resistance (called RC Natural Response). RX • An inductor and a Thévenin equivalent + vS iS RX LX LX (called RL Step Response). • An inductor and a Norton equivalent (also called RL Step Response). • A capacitor and a Thévenin equivalent RX + vS iS RX CX (called RC Step Response). • A capacitor and a Norton equivalent (also called RC Step Response). These are the simple, first-order cases. CX They all have solutions which are in similar forms.
  7. 7. The Natural Response of a Circuit • The currents and voltages that arise when energy stored in an inductor or capacitor is suddenly released into a resistive circuit. • These “signals” are determined by the circuit itself, not by external sources!
  8. 8. Step Response • The sudden application of a DC voltage or current source is referred to as a “step”. • The step response consists of the voltages and currents that arise when energy is being absorbed by an inductor or capacitor.
  9. 9. Circuits for Natural Response • Energy is “stored” in an inductor (a) as an initial current. • Energy is “stored” in a capacitor (b) as an initial voltage.
  10. 10. General Configurations for RL • If the independent sources are equal to zero, the circuits simplify to
  11. 11. Natural Response of an RL Circuit • Consider the circuit shown. • Assume that the switch has been closed “for a long time”, and is “opened” at t=0.
  12. 12. What does “for a long time” Mean? • All of the currents and voltages have reached a constant (dc) value. • What is the voltage across the inductor just before the switch is opened?
  13. 13. Just before t = 0 • The voltage across the inductor is equal to zero. • There is no current in either resistor. • The current in the inductor is equal to IS.
  14. 14. Just after t = 0 • The current source and its parallel resistor R0 are disconnected from the rest of the circuit, and the inductor begins to release energy.
  15. 15. The Source-Free RL Circuit • A first-order RL circuit consists of a inductor L (or its equivalent) and a resistor (or its equivalent) By KVL vL L vR di 0 iR 0 dt Inductors law Ohms law di R i L dt i (t ) I0 e Rt/L
  16. 16. The Source-Free RC Circuit • A first-order circuit is characterized by a first-order differential equation. By KCL iR iC Ohms law • • 0 v C R dv dt Capacitor law Apply Kirchhoff’s laws to purely resistive circuit results in algebraic equations. Apply the laws to RC and RL circuits produces differential equations. 0
  17. 17. Natural Response of an RL Circuit • Consider the following circuit, for which the switch is closed for t < 0, and then opened at t = 0: t=0 Io Ro i L + R v – Notation: 0– is used to denote the time just prior to switching 0+ is used to denote the time immediately after switching • The current flowing in the inductor at t = 0– is Io
  18. 18. Solving for the Current (t 0) • For t > 0, the circuit reduces to i Io Ro L + R v – • Applying KVL to the LR circuit: • Solution: i (t ) i(0)e ( R / L ) t = I e-(R/L)t 0
  19. 19. Solving for the Voltage (t > 0) i (t ) ( R / L )t Ioe + Io Ro L R v – • Note that the voltage changes abruptly: v( 0 ) 0 for t 0, v(t) v( 0 ) iR I0R I o Re ( R /L )t
  20. 20. Time Constant • In the example, we found that i (t ) I oe ( R / L )t and v (t ) I o Re L • Define the time constant ( R / L )t (sec) R – At t = , the current has reduced to 1/e (~0.37) of its initial value. – At t = 5 , the current has reduced to less than 1% of its initial value.
  21. 21. The Source-Free RL Circuit Comparison between a RL and RC circuit A RL source-free circuit i(t ) I0 e t/ where A RC source-free circuit L R v (t ) V0 e t/ where RC
  22. 22. The Complete Solution R i (t ) Ie 0 L t ,t 0
  23. 23. The voltage drop across the resistor v iR R v I Re L 0 t ,t v(0 ) 0 v(0 ) I R 0 0 .
  24. 24. The Power Dissipated in the Resistor p vi v 2 iR 2 R p 2 I Re 0 2 R L t ,t 0
  25. 25. The Energy Delivered to the Resistor t w t pdx w 2 0 0 2 2 I R (1 e 0 L t ,w R L I Re 0 1 R 2 2 1 2 LI 2 0 x dx R L t ),t 0.
  26. 26. The Source-Free RL Circuit A general form representing a RL i(t ) where I0 e t/ L R • • • 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.
  27. 27. The Source-Free RC Circuit • 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. Time constant RC Decays more slowly Decays faster • • 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 .
  28. 28. Natural Response Summary RL Circuit RC Circuit i + L C R v R – • Inductor current cannot change instantaneously i(0 ) i (t ) • i(0 ) i(0)e • time constant Capacitor voltage cannot change instantaneously v (0 ) t/ v (t ) L R • v (0 ) v (0)e time constant t/ RC
  29. 29. General Solution for Natural and Step Responses of RL and RC Circuits ( t t0 ) x (t ) xf [ x (t 0 ) x f ]e Final Value Time Constant Initial Value Determine the initial and final values of the variable of interest and the time constant of the circuit. Substitute into the given expression.
  30. 30. Example b R1 400kOhm V1 90 V a R3 20 Ohm J1 Key = Space + vC(t) R2 60 Ohm V2 40 V C 0.5uF - • What is the initial value of vC? • What is the final value of vC? • What is the time constant when the switch is in position b? • What is the expression for vC(t) when t>=0?
  31. 31. Initial Value of vC b R1 400kOhm a R3 20 Ohm J1 + V1 90 V Key = Space V60 + vC(0) C 0.5uF R2 60 Ohm V2 40 V - - • The capacitor looks like an open circuit, so the voltage @ C is the same as the voltage @ 60Ω. v C (0 ) 60 4 0V 20 3 0V 60
  32. 32. Final Value of vC b R1 400kOhm V1 90 V a R3 20 Ohm J1 Key = Space + vC(∞) R2 60 Ohm V2 40 V C 0.5uF - • After the switch is in position b for a long time, the capacitor will look like an open circuit again, and the voltage @ C is +90 Volts.
  33. 33. The time constant of the circuit when the switch is in position b R1 400kOhm V1 90 V b a R3 20 Ohm J1 Key = Space R2 60 Ohm C 0.5uF • The time constant τ = RC = (400kΩ)(0.5μF) • τ = 0.2 s V2 40 V
  34. 34. The expression for vC(t) for t>=0 t vC (t ) vC ( ) [ v C (0) v C ( )]e t vC (t ) 90 [ 30 vC (t ) 90 120 e 90]e 5t V 0.2
  35. 35. The expression for i(t) for t>=0 b R1 a 400kOhm R3 20 Ohm J1 Key = Space V1 90 V i(t) 30V R2 60 Ohm V2 40 V C 0.5uF + • Initial value of i is (90 - - 30)V/400kΩ = 300μA • Final value of i is 0 – the capacitor charges to +90 V and acts as an open circuit • The time constant is still τ = 0.2 s
  36. 36. The expression for i(t) (continued) t i (t ) i( ) [ i (0 ) i ( )]e t i (t ) i (t ) 0 [300 10 300 e 5t A 6 0] e 0.2
  37. 37. How long after the switch is in position b does the capacitor voltage equal 0? vC (t ) 120e e 5t 90 5t 120e 5t 0 90 90 120 5t ln 90 0 .2 8 7 6 8 120 t 0 .0 5 7 5 4 s 5 7 .5 4 m s
  38. 38. Plot vC(t)
  39. 39. Plot i(t)

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