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

V OUT

   

R f R in

 

V in

• 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

Heat Dissipation, q out Body with: mass

m

, specific heat

c P

, temperature

T B

Heat Input, q in Ambient Temperature, T 0

Dynamic System Example – continued

• The rate at which the temperature can respond is dictated

by the body’s mass and material properties

mc p dT B dt



q in



q out

• The heat out of the mass is governed by the difference in

temperature between the body and the surroundings:

q out



R ( T B



T

0

)

• The mass is storing heat as temperature

q in (t) t=0

Dynamic System Example – continued

t T B (t)

Final Temperature

t=0

Initial Temperature

t

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:

u

0

( t )

 0  1

, , t t

  0 0 • Circuit to generate the

signal:

Scaled and shifted step functions

• Scaling • Multiply by a constant

K



u

0

( t )

    0

K , , t t

  0 0 • Shifting • Moving in time

u

0

( t



a )

   0 1

, , t t



a



a

•

Sketch 5u

0 (t-3)

Example 1

Example 2

•

Represent v(t) in the circuit below in terms of step functions t = 3 sec t = 1 sec

• •

Example 3

f ( t )



cos( t



t

 0

),

0

, otherwise

2 

function defined over -





.

Exponential Functions

• An exponential

function is defined by

f ( t )



Ae



t

 • •  

is the time constant > 0

Exponential Functions – continued

• Our exponential

functions will generally be limited to t≥0:

f ( t )



Ae



t



, t



0 0.368A

•

or:

f ( t )



Ae



t

 

u

0

( t )

• 

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