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Response of First-Order Circuits

Response of First-Order Circuits. RL Circuits RC Circuits. 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.

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Response of First-Order Circuits

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  1. Response of First-Order Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I

  2. 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! ECE 201 Circuit Theory I

  3. 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. ECE 201 Circuit Theory I

  4. 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. ECE 201 Circuit Theory I

  5. General Configurations for RL • If the independent sources are equal to zero, the circuits simplify to ECE 201 Circuit Theory I

  6. 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. ECE 201 Circuit Theory I

  7. 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? ECE 201 Circuit Theory I

  8. 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. ECE 201 Circuit Theory I

  9. 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. ECE 201 Circuit Theory I

  10. The expression for the current ECE 201 Circuit Theory I

  11. A first-order ordinary differential equation with constant coefficients. How do we solve it? ECE 201 Circuit Theory I

  12. ECE 201 Circuit Theory I

  13. The current in an inductor cannot change instantaneously • Let the time just before switching be called t(0-). • The time just after switching will be called t(0+). • For the inductor, ECE 201 Circuit Theory I

  14. The Complete Solution ECE 201 Circuit Theory I

  15. The voltage drop across the resistor ECE 201 Circuit Theory I

  16. The Power Dissipated in the Resistor ECE 201 Circuit Theory I

  17. The Energy Delivered to the Resistor ECE 201 Circuit Theory I

  18. Time Constant • The rate at which the current or voltage approaches zero. ECE 201 Circuit Theory I

  19. Rewriting in terms of Time Constant ECE 201 Circuit Theory I

  20. Table 7.1 page 217 of the text ECE 201 Circuit Theory I

  21. Graphical Interpretation of Time Constant • Determine the time constant from the plot of the circuit’s natural response. Straight Line Approximation ECE 201 Circuit Theory I

  22. Graphical Interpretation Tangent at t = 0 intersects the time axis at the time constant ECE 201 Circuit Theory I

  23. Procedure to Determine the Natural Response of an RL Circuit • Find the initial current through the inductor. • Find the time constant,τ, of the circuit (L/R). • Generate i(t) from I0 and τ using ECE 201 Circuit Theory I

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