1 / 20

Introduction to Dynamic Systems: Energy Storage & Time-Varying Signals

Learn about energy storage in dynamic systems, including the role of capacitors and inductors. Explore basic time-varying signals such as step and exponential functions.

sharondah
Download Presentation

Introduction to Dynamic Systems: Energy Storage & Time-Varying Signals

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 14 Introduction to dynamic systems Energy storage Basic time-varying signals Related educational materials: Chapter 6.1, 6.2

  2. Review and Background • Our circuits have not contained any energy storage elements • Resistors dissipate energy • Governing equations are algebraic, the system responds instantaneously to changes

  3. Example: Inverting voltage amplifier • The system output at some time depends only on the input at that time • Example: If the input changes suddenly, the output changes suddenly

  4. Inverting voltage amplifier – switched response • Input and response:

  5. Dynamic Systems • We now consider circuits containing energy storage elements • Capacitors and inductors store energy • The circuits are dynamic systems • They are governed by differential equations • Physically, they are performing integrations • If we apply a time-varying input to the system, the output may not have the same “shape” as the input • The system output depends upon the state of the system at previous times

  6. Dynamic System – example • Heating a frying pan

  7. Dynamic System Example – continued • The rate at which the temperature can respond is dictated by the body’s mass and material properties • The heat out of the mass is governed by the difference in temperature between the body and the surroundings: • The mass is storing heat as temperature

  8. Dynamic System Example – continued

  9. Time-varying signals • We now have to account for changes in the system response with time • Previously, our analyses could be viewed as being independent of time • The system inputs and outputs will become functions of time • Generically referred to a signals • We need to introduce the basic time-varying signals we will be using

  10. Basic Time-Varying Signals • In this class, we will restrict our attention to a few basic types of signals: • Step functions • Exponential functions • Sinusoidal functions • Sinusoidal functions will be used extensively later; we will introduce them at that time

  11. Step Functions • The unit step function is defined as: • Circuit to generate the signal:

  12. Scaled and shifted step functions • Scaling • Multiply by a constant • Shifting • Moving in time

  13. Example 1 • Sketch 5u0(t-3)

  14. Example 2 • Represent v(t) in the circuit below in terms of step functions

  15. Example 3 • Represent the function as a single • function defined over -<t<.

  16. Exponential Functions • An exponential function is defined by •  is the time constant •  > 0

  17. Exponential Functions – continued • Our exponential functions will generally be limited to t≥0: • or: • Note: f(t) decreases by 63.2% every  seconds

  18. Effect of varying 

  19. Exponential Functions – continued • Why are exponential functions important? • They are the form of the solutions to ordinary, linear differential equations with constant coefficients

More Related