Download Lecture 1 - Digilent Inc.

Lecture 15
•Review:
•Energy storage and dynamic systems
•Basic time-varying signals
•Capacitors
•Related educational materials:
–Chapter 6.3
Dynamic Systems
• We now consider circuits containing energy storage
elements
• The circuits are dynamic systems
• They are governed by differential equations
• We need to be concerned with the input and output
of the system as functions of time
• The system output depends upon the state of the system
at previous times
Basic Time-Varying Signals
• Step functions
0 , t  0
u0 ( t )  
1, t  0
• Exponential functions
f ( t )  Ae
t

, t 0
f(t)
A
0.368A
0

t
Example: Sliding mass with friction
• Do forced, natural response; input and output
response plots
• Time constant and effect of mass on time
constant
• Notes:
– Mention transient, steady-state
– Natural vs. forced response
– Homogeneous vs. particular solution
Energy storage elements – capacitors
• Capacitors store energy as an electric field
• In general, constructed of two conductive elements
separated by a non-conductive material
Capacitors
• Circuit symbol:
• Voltage-charge relation:
q( t )  Cv( t )
• Recall:
i( t ) 
dq( t )
dt
• So:
• C is the capacitance
• Units are Farads (F)
i( t ) 
d
Cv( t )  C dv( t )
dt
dt
Capacitor voltage-current relations
• Differential form:
• Integral form:
• Annotate previous slide to show initial
voltage, define times on integral, sketchy
derivation of integration of differential form to
get integral form.
Important notes about capacitors
1. If voltage is constant, no current
flows through the capacitor
• If nothing in the circuit is changing
with time, capacitors act as open
circuits
2. Sudden changes in voltage
require infinite current
• The voltage across a capacitor must
be a continuous function of time
Capacitor Power and Energy
• Power:
• Energy:
t
1 2
 Cv ( t )
2

Example
v(t), V
• The voltage applied to the capacitor by the source is as
shown. Plot the power absorbed by the capacitor and the
energy stored in the capacitor as functions of time.
10
5
0
0
1
2
3
4
Time, ms
Example – continued
Example – continued
Series combinations of capacitors
+ v1(t) -
+ v2(t) +
vN(t)
-
Series combinations of capacitors
• A series combination of capacitors can be
represented as a single equivalent capacitance
Þ
Parallel combinations of capacitors
i1(t)
i2(t)
iN(t)
Parallel combinations of capacitors
• A parallel combination of capacitors can be
represented as a single equivalent capacitance
Þ
Example
• Determine the equivalent capacitance, Ceq