Demonstration - Faculty Pages
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Transcript Demonstration - Faculty Pages
In-Class Problems
1. Sketch the following functions:
a) x(t) = 3sin(40pt) for 0≤ t ≤ 0.2 sec
b) z(t) = 10e-4t for 0≤ t ≤0.5 sec
d
4t
y
(
t
)
10
e
2. What is
?
dt
3. What
d
is f (t ) dt 3 sin( 40pt )
d
4. What is g (t ) 6t ?
dt
?
Capacitor-Resistor Circuits
J1
2
1
Key = Space
V1
12 V
R1
1k
4
C1
1uF
0
• In-class activity:
• In pairs build this circuit in Multisim.
• Look at the voltage across the capacitor on the
oscilloscope.
• Describe what you see when the switch moves
between positions (let the switch stay in each
position until the capacitor voltage stops changing).
Discharge of a Capacitor
Through a Resistor
• In the following circuit, when the switch
moves from the battery to the wire
connected to ground, the voltage across
the capacitor is vc(t) = Vse-t/RC
J1
2
1
Key = Space
Vs
R1
1k
4
V1
12 V
vc
0
C1
1uF
-
Class Activity
• What are the units of RC in vc(t) = Vse-t/RC ?
dv
c
-t/RC
i
t
C
• Since vc(t) = Vse
and
, what
dt
is i(t) in terms of Vs, R and C?
• What do vc(t) and i(t) look like on a graph?
Step Response: RC Time Constants
• Now, what happens when the switch moves the
other way?
J1
2
1
Key = Space
R1
1k
4
V
V1
12 Vs
C1
1uF
0
The response of the capacitor voltage will be to
charge up to the supply voltage.
Vc Response to Constant Voltage Vs
• The voltage across
the capacitor will rise
and asymptotically
approach Vs
How can we describe this mathematically?
Analysis of RC Circuits
J1
2
1
Key = Space
Vs
R1
1k
4
V1
12 V
vc
C1
1uF
0
Kirchhoff’s voltage loop law
VS vC vR
Ohm’s law across resistor
vR iR
Substituting for VR gives
Vs vc iR
Analysis of RC Circuits
J1
2
1
Key = Space
Vs
R1
1k
4
V1
12 V
vc
C1
1uF
0
From previous page
VS vC iR
dvC
Substitute C
in for i
dt
dvC
VS vC RC
dt
Analysis of RC Circuits
dvC
• The equation VS vC RC
is called a differential
dt
equation.
t
• The solution is of the form: vC VS 1 e
where RC is defined as the time constant
= the circuit time constant, in seconds if and only if
C = the total (connected) capacitance Farads
R = the total (connected) resistance Ohms
Team Activity
t
• Substitute vC VS 1 e
dvC
into the equation VS vC
dt
to show that LHS = RHS
Team Activity
VS 100V
vC
1 e
VS
t
Show that when t is 5 times the time constant, ,
the capacitor voltage is 99.33% of the peak voltage.
Team Activity – Discharge Process
J1
2
R1
1
Key = Space
1k
V1
12 V
4
C1
1uF
0
Kirchhoff’s voltage loop law ?
Ohm’s law across resistor?
Substituting for VR gives?
Team Activity
• From previous activity the equation
• Substitute
vC VS e
t
into the above equation
and show that LHS =
RHS
dvC
vC
dt
Rectangular Wave
• If you repeatedly switch between the battery and
the short you are effectively applying a
rectangular time pulse to the RC circuit.
Rectangular Wave Response
• The voltage across the capacitor will behave as
below in response to such a wave: