Download 4: Capacitors

• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
Lecture 4
Capacitors
•  The capacitor
–  Hydraulic analogy
–  I/V relation
–  Package, symbol
–  Standard vs. electrolytic types
•  RC circuits
–  Differential equation
–  RC time constant
–  Impact on a circuit
Robert R. McLeod, University of Colorado
38
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
The hydraulic capacitor
and the I/V relation
V
+
I
-
Q
The reverse pressure caused by the stretched membrane is proportional to the
quantity of water that has been pushed into the capacitor. Using electrical units,
Voltage[V =
J
C
] ∝ Charge[C]
The proportionality constant depends on the size of the tank and flexibility of the
membrane. Let’s give it a name and a unit
C [farad] = Q[C] V [V ]
or
Q = CV
To get to an I/V relationship, remember how charge relates to current:
t1
Q(t1 ) = ∫ I dt + Q(t0 )
t0
Which gives us the I/V relationship for the capacitor
Q(t ) 1
V (t ) =
= ∫ I dt + V (t0 )
C
C t0
t
Taking the derivative of both sides (using the fundamental theorem of calculus),
I (t ) = C
Robert R. McLeod, University of Colorado
dV
dt
39
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
The Capacitor
Packages
http://en.wikipedia.org/wiki/Capacitor
Physics
http://
www.maxwellrosspierson.co
m/2009/03/17/how-to-pickaudio-capacitors/
Symbol
Note: Looks a bit like a battery symbol. In some
ways, a capacitor is a substandard battery.
Robert R. McLeod, University of Colorado
40
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
The Electrolytic Capacitor
Packages
http://www.beavisaudio.com/techpages/Caps/
Physics
Symbol
http://www.electronics-tutorials.ws/capacitor/cap_2.html
Most common
Note: The critical requirement is polarity. Use it
backwards and it’s dead.
Robert R. McLeod, University of Colorado
http://en.wikipedia.org/wiki/Electrolytic_capacitor
41
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
The RC circuit
Let the switch be closed for a long
time so that the capacitor is fully
charged and there is no current
flowing through the capacitor. At
t=0, open the switch. The
capacitor will now discharge
through the resistor.
I R = − IC
IC
IR
-
Use the I/V relationship for both components
Re-arrange
− t ( RC )
RC time constant
τ = RC [s]
Solve the differential equation
1.0
• 
• 
• 
• 
0.8
V (t )
V0
V (t )
The current through the resistor came from the cap
V
dV
= −C
R
dt
dV
1
=−
V
dt
RC
V =V 0 e
+
0.6
Switch opens at t=0
Capacitor discharges through R
Voltage falls like exponential
Time constant is RC
e −1 = 0.368
0.4
0.2
t
0.5
Robert R. McLeod, University of Colorado
1.0
1.5
2.0
2.5
3.0
τ
42
• Lecture 44 Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
RC in a circuit
1
2 V peak-to-peak
Square wave
1 KHZ
2
τ = RC = 0.1 [ms]
1
0.1ms
2
Robert R. McLeod, University of Colorado
43
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
Quiz 4.1
Q: A constant (“DC”) voltage is
placed across a capacitor on the left
and a time-varying (“AC”) voltage is
placed across the capacitor on the
right. Current will flow in which
cases?
A: No in the DC and no in the AC
B: Yes in the DC and no in the AC
C: No in the DC and yes in the AC
D: Yes in the DC and yes in the AC
E: It depends on the AC frequency.
Robert R. McLeod, University of Colorado
44
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
Quiz 4.2
Q: At t=0, a switch is closed and the
uncharged capacitor C begins to
charge. How much charge is stored
in the capacitor at t=1 second?
RC = 1ms
A: 0 µC
One second is 1000x the RC time
B: 1 µC
constant, so the capacitor will be fully
charged, no current will be flowing
C: 10 µC
and thus the voltage across the
capacitor is 10V. By the definition of
D: 100 µC
capacitance, Q = C V.
E: I have no idea how to find this.
Robert R. McLeod, University of Colorado
45
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
Quiz 4.3
Q: A function generator is hooked to
the series RC circuit above. The peak
voltage across the capacitor depends
on
A: The peak voltage of the source
B: The frequency of the source
C: The order of R and C in series
D: A and B but not C •  The system is linear, so
the capacitor voltage is
proportional to the source
E: A and C but not B
voltage.
• 
• 
Robert R. McLeod, University of Colorado
The voltage across the
capacitor will drop when
the frequency is > 1/RC
The order makes no
difference
46
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
Bonus Quiz 4.1
Q: The voltage across a capacitor of
capacitance C=2 F is
V(t) = 1 + cos(500 t).
The charge on the capacitor is:
A:
B:
C:
D:
E:
Q(t)= -1000 sin(500 t)
Q(t)=2t + (1/250) sin(500 t)
Q(t)=1/2 + (1/2) cos(500 t)
Q(t)=2/(1 + cos(500 t))
Q(t)=2 + 2 cos(500 t)
Q = CV (t )
Robert R. McLeod, University of Colorado
47
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
Bonus Quiz 4.2
5 KW
1 mF
Q: As the frequency of the function
generator is increased from zero, the peak
voltage on the capacitor stays constant,
then drops. The frequency where this
begins to occur is about
A: 5 mHz
RC = 5 ms
B: 200 Hz
1
f
=
= 200 Hz
C: 500 Hz
T
D: 2 KHz
E: 200 MHz
Robert R. McLeod, University of Colorado
48
• Lecture 4: Capacitors
ECEN 1400 Introduction to Analog and Digital Electronics
Bonus Quiz 4.3
V(t ) = V0e
−
t
RC
t
−
1
V0 = V0e RC
4
⎛1⎞
t = − RC ln⎜ ⎟
⎝4⎠
1 ⎛1⎞
Q: A function generator is A : t = −
ln⎜ ⎟
RC ⎝ 4 ⎠
hooked to the series RC
circuit above. The function
1 ⎛3⎞
B:t = −
ln⎜ ⎟
generator is set to a square
RC ⎝ 4 ⎠
wave which alternates
⎛1⎞
between V0 and 0V (high/ C : t = − RC ln⎜ ⎟
⎝4⎠
low) with a period >> RC.
If a transition from V0 to
⎛3⎞
D : t = − RC ln⎜ ⎟
0V occurs at t=0, when
⎝4⎠
does the voltage across the
1
−
capacitor drop to V0/4?
E:t = e 4
Robert R. McLeod, University of Colorado
49