1 / 21

First Order Circuit Response Review

This lecture reviews the natural, forced, and step responses of first-order circuits, providing examples and related educational materials.

albertor
Download Presentation

First Order Circuit Response Review

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 18 Review: First order circuit natural response Forced response of first order circuits Step response of first order circuits Examples Related educational materials: Chapter 7.4, 7.5

  2. Natural response of first order circuits – review • Circuit being analyzed has a single equivalent energy storage element • Circuit being analyzed is “source free” • Any sources are isolated from the circuit during the time when circuit response is determined • Circuit response is due to initial energy storage • Circuit response decays to zero as t

  3. First order circuit forced response – overview • Now consider the response of circuits with sources • Notes: • We will typically write our equations in terms of currents through inductors and voltages across capacitors • The above circuits are very general; consider them to be the Thévenin equivalent of a more complex circuit

  4. RC circuit forced response

  5. RL circuit forced response

  6. First order circuit forced response – summary • Forced RC circuit response: • Forced RL circuit response:

  7. General first order systems • Block diagram: • Governing differential equation:

  8. Active first order system – example • Determine the differential equation relating Vin(t) and Vout(t) for the circuit below

  9. Active first order system – example • Determine the differential equation relating Vin(t) and Vout(t) for the circuit below

  10. Step Response – introduction • Our previous results are valid for any forcing function, u(t) • In this course, we will be mostly concerned with a couple of specific forcing functions: • Step inputs • Sinusoidal inputs • We will defer our discussion of sinusoidal inputs until later

  11. Applying step input • Block diagram: • Governing equation: • Example circuit:

  12. First order system step response • Solution is of the form: • yh(t) is homogeneous solution • Due to the system’s response to initial conditions • yh(t)0 as t • yp(t) is the particular solution • Due to the particular forcing function, u(t), applied to the system • y(t) yp(t) as t

  13. First order system – homogeneous solution • Assume form of solution: • Substitute into homogeneous D.E. and solve for s : • Homogeneous solution:

  14. First order system – particular solution • Recall that the particular solution must: • Satisfy the original differential equation as t • Have the same form as the forcing function • As t:

  15. First order system particular solution -- continued • As t, the original differential equation becomes: • The particular solution is then

  16. First order system step response • Superimpose the homogeneous and particular solutions: • Substituting our previous results: • K1 and K2 are determined from initial conditions and steady-state response;  is a property of the circuit

  17. Example 1 • The switch in the circuit below has been open for a long time. Find vc(t), t>0

  18. Example 1 – continued • Circuit for t>0:

  19. Example 1 – continued again • Apply initial and final conditions to determine K1 and K2 Governing equation: Form of solution:

  20. Example 1 – checking results

More Related