1 / 28

Lecture 20 (parts A & B)

Lecture 20 (parts A & B). First order circuit step response Nonzero initial conditions and multiple sources Steady-state response and DC gain Bias points and nominal operating conditions Introduction to second order systems Related educational materials: Chapters 7.5, 8.1.

reiff
Download Presentation

Lecture 20 (parts A & B)

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 20(parts A & B) First order circuit step response Nonzero initial conditions and multiple sources Steady-state response and DC gain Bias points and nominal operating conditions Introduction to second order systems Related educational materials: Chapters 7.5, 8.1

  2. First order system step response • Block diagram: • So far, we have considered only circuits which are initially relaxed  y(0) = 0 • We now consider circuits with non-zero initial conditions

  3. Example 1 • The switch moves from A to B at time t=0 • Find v(t), t>0

  4. Sketch input function on previous slide

  5. Example 1 – initial condition

  6. Example 1 – Differential equation for t>0

  7. Example 1 – Check , steady-state response

  8. Example 1 – circuit response • Differential equation: • Initial, final conditions: , • Form of solution:

  9. Example 1 – sketch input, output

  10. Alternate representation of example 1 • The circuit of example 1 can be written as: • Now determine the response using superposition

  11. Annotate previous slide to show input function

  12. Example 1 – superposition approachResponse to (constant) 2V source

  13. Example 1 – superposition approach (cont’d)Response to 3V step input • Input-output equation:

  14. Example 1 – superposition approach (cont’d)Response to 3V step input • Governing equation: • Form of solution: • Initial condition: • Final condition:

  15. Example 1 – superposition approach (cont’d)Overall response

  16. Note on overall approach • Both the input and output can be decomposed into a constant value and a time-varying value • It is sometimes convenient to analyze these components independently • For example, the DC gain of the system applies to both the constant input and the time varying input

  17. Graphical interpretation • The system DC gain =

  18. Why is this approach useful? • Decomposing the input and output into constant and time-varying components can simplify analysis and interpretation of results • The constant part of the input and output is the bias point or nominal operating point • The system dynamic response is often characterized by the time-varying part of the input-output relationship • A nonlinear system, for example, can be approximated as a linear system with a bias point

  19. Introduction to second order systems • Second order systems are governed by second order differential equations • Input-output relation contains a second order derivative term, but no derivatives higher than second order • The physical system has two independent energy storage elements • The natural response of a second order system can oscillate with time (but doesn’t necessarily have to) • The response can overshoot its final value

  20. Introduction to second order systems – continued • The oscillations in the natural response are due to energy being traded between the energy storage elements • Increasing energy dissipation reduces the amplitude of the oscillations (the system is said to be more highly damped) • If energy dissipation is above a critical value, the response will no longer oscillate • In general, increasing the energy dissipation will also cause the system to respond to changes more “slowly”

  21. On previous slide, talk about damping and energy dissipation • Example: suspension system in car

  22. Example: Series RLC circuit • Write the differential equation governing iL(t)

  23. Series RLC circuit – continued

  24. Example: Parallel RLC circuit • Write the differential equation governing vC(t)

  25. Parallel RLC circuit – continued

More Related