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

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

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

Inverting voltage amplifier – switched response Input and response:

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

Dynamic System – example Heating a frying pan

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

Dynamic System Example – continued

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

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

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

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

Example 1 Sketch 5u 0 (t-3)

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

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

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

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

Effect of varying 

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