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First-Order Circuits (6.1-6.2)

First-Order Circuits (6.1-6.2). Dr. Holbert November 15, 2001. 1st Order Circuits. Any circuit with a single energy storage element , an arbitrary number of sources , and an arbitrary number of resistors is a circuit of order 1 .

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First-Order Circuits (6.1-6.2)

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  1. First-Order Circuits (6.1-6.2) Dr. Holbert November 15, 2001 ECE201 Lect-22

  2. 1st Order Circuits • Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 1. • Any voltage or current in such a circuit is the solution to a 1st order differential equation. ECE201 Lect-22

  3. Important Concepts • The differential equation • Forced and natural solutions • The time constant • Transient and steady-state waveforms ECE201 Lect-22

  4. vr(t) + – R + + – vc(t) vs(t) C – A First-Order RC Circuit • One capacitor and one resistor • The source and resistor may be equivalent to a circuit with many resistors and sources. ECE201 Lect-22

  5. Applications Modeled bya 1st Order RC Circuit • Computer RAM • A dynamic RAM stores ones as charge on a capacitor. • The charge leaks out through transistors modeled by large resistances. • The charge must be periodically refreshed. ECE201 Lect-22

  6. vr(t) + – R + + – vc(t) vs(t) C – The Differential Equation(s) KVL around the loop: vr(t) + vc(t) = vs(t) ECE201 Lect-22

  7. Differential Equation(s) ECE201 Lect-22

  8. What is the differential equation for vc(t)? ECE201 Lect-22

  9. + R L v(t) is(t) – A First-Order RL Circuit • One inductor and one resistor • The source and resistor may be equivalent to a circuit with many resistors and sources. ECE201 Lect-22

  10. Applications Modeled by a 1st Order LC Circuit • The windings in an electric motor or generator. ECE201 Lect-22

  11. + R L v(t) is(t) – The Differential Equation(s) KCL at the top node: ECE201 Lect-22

  12. The Differential Equation ECE201 Lect-22

  13. 1st Order Differential Equation Voltages and currents in a 1st order circuit satisfy a differential equation of the form ECE201 Lect-22

  14. Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE201 Lect-22

  15. The Particular Solution • The particular solutionvp(t) is usually a weighted sum of f(t) and its first derivative. • That is, the particular solution looks like the forcing function • If f(t) is constant, then vp(t) is constant. • If f(t) is sinusoidal, then vp(t) is sinusoidal. ECE201 Lect-22

  16. The Complementary Solution The complementary solution has the following form: Initial conditions determine the value of K. ECE201 Lect-22

  17. Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE201 Lect-22

  18. The Time Constant () • The complementary solution for any 1st order circuit is • For an RC circuit, t = RC • For an RL circuit, t = L/R ECE201 Lect-22

  19. What Does vc(t) Look Like? t = 10-4 ECE201 Lect-22

  20. Interpretation of t • The time constant, t, is the amount of time necessary for an exponential to decay to 36.7% of its initial value. • -1/t is the initial slope of an exponential with an initial value of 1. ECE201 Lect-22

  21. Implications of the Time Constant • Should the time constant be large or small: • Computer RAM • A sample-and-hold circuit • An electrical motor • A camera flash unit ECE201 Lect-22

  22. Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE201 Lect-22

  23. Transient Waveforms • The transient portion of the waveform is a decaying exponential: ECE201 Lect-22

  24. Steady-State Response • The steady-state response depends on the source(s) in the circuit. • Constant sources give DC (constant) steady-state responses. • Sinusoidal sources give AC (sinusoidal) steady-state responses. ECE201 Lect-22

  25. LC Characteristics ECE201 Lect-22

  26. Class Examples • Learning Extension E6.1 • Learning Extension E6.2 ECE201 Lect-22

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