Transcript Lecture 1

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
VOUT
 Rf 
Vin
 
 Rin 
• 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
Body with:
mass m,
specific heat cP,
temperature TB
Heat
Input,
qin
Heat
Dissipation,
qout
Ambient
Temperature,
T0
Dynamic System Example – continued
• The rate at which the temperature can respond is dictated
by the body’s mass and material properties
dT
mc p B  qin  qout
dt
• The heat out of the mass is governed by the difference in
temperature between the body and the surroundings:
qout  R( TB  T0 )
• The mass is storing heat as temperature
Dynamic System Example – continued
TB(t)
qin(t)
Final
Temperature
Initial
Temperature
t=0
t
t=0
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:
0 , t  0
u0 ( t )  
1, t  0
• Circuit to generate the
signal:
Scaled and shifted step functions
• Scaling
• Multiply by a constant
 0,t  0
K  u0 ( t )  
K , t  0
• Shifting
• Moving in time
0 , t  a
u0 ( t  a )  
1, t  a
Example 1
• Sketch 5u0(t-3)
Example 2
• Represent v(t) in the circuit below in terms of step functions
t = 3 sec
t = 1 sec
Example 3
cos( t ), 0  t  2
• Represent the function f ( t )  
as a single
 0 , otherwise
• function defined over -<t<.
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

 u0 ( 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