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Capacitance
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What is a capacitor?
A capacitor is an electronic component that
stores electric charge (and electric energy)
A simple version of a capacitor is the
parallel-plate capacitor; it consists of two
conducting plates separated by an insulating
material
– The insulating material is called a dielectric in
this context.
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Parallel-plate capacitor
1. Battery (not
shown) pushes
charges out
I
2. Positive charges
accumulates on
first plate;
dielectric does not
let them through
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conducting plates
+
+
+
+
+
+
+
-
dielectric
3. Neutral atoms on
other plate are made
up positive and
negative charges
I
4. Negative charges are
attracted and stay behind,
while positive charges are
repelled and move out giving
rise to the current on other
side
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We can’t go on like this
1. There are
positive charges
coming from the
battery
I
conducting plates
2. There are
positive charges on
the first plate. Like
charges repel.
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+
+
+
+
+
+
+
-
dielectric
3. When the push from the
battery is equal to the push
back from the plate, the
capacitor stops charging
I
4. The bigger the push from
the battery (i.e. voltage), the
more charge goes onto the
plate. The voltage and
charge are proportional.
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Q = VC
When we talk about the charge on a
capacitor, we mean the charge on one of the
plates.
The charge is proportional to the voltage: Q
 V.
The proportionality constant C is called the
capacitance.
Q
Solving for V, the equation becomes V =
C
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Farad
The standard unit of capacitance is the farad
(F).
A farad is quite large, usually you see
–
–
–
–
millifarad
microfarad
nanofarad
picofarad
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mF
F
nF
pF
(1 mF = 10-3 F)
(1  F = 10-6 F)
(1 nF = 10-9 F)
(1 pF = 10-12 F)
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Capacitance Q=VC
The capacitance is a measure of how easy it is to
put charge on the plates, it is
– directly proportional to the surface area of the plates,
bigger plates can hold more charge
– inversely proportional to the distance between the
plates, the interaction between the positive and negative
charges is greater when they are closer
– dependent on the material (dielectric) separating the
plates, having a good insulator between them is like
their being further apart
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Capacitive keyboards
Capacitive keyboards detect which key was pressed by
looking at changes in capacitance.
Under each key are two oppositely charged plates ( a
capacitor).
Pressing a key moves its upper plate closer to its lower
plate, changing its capacitance, and hence changes the
amount of charge the plates can hold for a given voltage.
The keyboard circuitry detects this change and sends the
appropriate information (interrupt request and ASCII code)
to the CPU.
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DRAM
Capacitors can be put on integrated circuit (IC)
chips.
Together with transistors, they are used in
dynamic random access memory (D-RAM).
The charge or lack thereof of the capacitor
corresponds to a stored bit. Since these capacitors
are small, their capacitance is pretty low.
DRAM has to be recharged (refreshed) thousands
of times per second or it loses its data.
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Power Supply
Large capacitors are used in the power supplies of
computers and peripherals.
The capacitors (along with diodes) are used in
rectifying: turning the alternating current (AC)
into smoothed out direct current (DC).
Capacitors do the smoothing part.
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Monitors
Monitors (CRT’s) are another place that
large capacitors are found.
Since capacitors store charge, monitors and
power supplies can be dangerous even when
the power is off.
– If you don’t know what you’re doing, don’t
even open them up.
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Equivalent Capacitance
When a combination of capacitors can be
replaced by a single capacitor, which has
the same effect as the combination, the
capacitance of the single capacitor is called
the equivalent capacitance.
Having the same effect means that same
voltage results in the same amount of
charge being stored.
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Capacitors in parallel
Recall that things in parallel have the same voltage
V
C1
C2
And the charge is split between the capacitors.
The charge has a choice, some will go onto one
capacitor, the rest on the other.
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Capacitors in parallel
Individual capacitors obey basic equation
Q1 = C1 V and
Q2 = C2 V
Equivalent capacitor obeys basic equation
Qtotal = Ceq V
The total charge is the sum of the individual charges
Qtotal = Q1 + Q2
Solve basic equations for Q’s and substitute
Ceq V = C1 V + C2 V
Divide by common factor V
Ceq = C1 + C2
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Capacitors in series
V
C1
C2
Capacitors in series have the same charge
– When the capacitors are uncharged, the region including the
lower plate of C1 and the upper plate of C2 is electrically
neutral. This region is isolated and so this remains true; the
negative charge on the lower plate of C1 when added to the
positive charge on the upper plate of C2 would give zero
The voltage is split between them
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Capacitors in series
Individual capacitors obey basic equation
V1 = Q1 / C1 and V2 = Q2 / C2
Equivalent capacitor obeys basic equation
Vtotal = Qtotal / Ceq
The total voltage is the sum of the individual voltages
Vtotal = V1 + V2
Substitute basic equations into voltage equation
Qtotal / Ceq = Q1 / C1 + Q2 / C2
All of the charges are the same (Qtotal = Q1 = Q2 = Q),
divide out the common factor
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1 / Ceq = 1 / C1 + 1 / C2
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Example
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Example (Cont.)
Something must take the brunt of the
voltage when the battery is first connected,
that’s why the 0.2-k resistor is there.
The 2.5-F and 1.5-F are in parallel, so
they can be replaced with one 4.0-F
capacitor.
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Example (Cont.)
The 3.5-F and 4.0-F capacitors are in
series, so they can be replaced with one
1.87-F capacitor.
The charge on it Q=CV would be 9.33 C
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Example (Cont.)
Going back to the 3.5-F and 4.0-F
capacitors are in series, capacitors in series
have the same charge, so Q3.5 = 9.33 C
Then V3.5 = Q3.5 / C3.5 or V3.5 = 2.67 V
And V4.0 = Q4.0 / C4.0 or V4.0 = 2.33 V
(Note that V3.5 + V4.0 = 5)
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Example (Cont.)
The 4.0-F capacitor was really a 1.5-F
and a 2.5-F in parallel
Things in parallel have the same voltage, so
V1.5 = V2.5 = 2.33 V
Since Q1.5 = V1.5 C1.5, Q1.5 = 3.50 C
Similarly, Q2.5 = 5.83 C
(Note Q1.5 + Q2.5 = 9.33 C)
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Testing in Electronics
Workbench
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Two Cautions
When the switches are closed, there is an easy
path from one side of the capacitor to another, this
makes sure that the capacitors are discharged.
Even when the switches are open there is a path
from one side to another for the upper capacitors,
it is through the voltmeter. The voltmeter must be
made extremely ideal (very high resistance) to get
agreement with theory.
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RC circuits: Charging
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What happens
When you connect an uncharged capacitor and a
resistor in series to a battery, the voltage drop is
initially all across the resistor.
– Because the voltage drop across a capacitor is
proportional to the charge on it and there is not charge
on it at the beginning.
But charge starts to build up on the capacitor, so
some voltage is dropped across the capacitor now.
– Capacitors have a gap and while current gets all the
way around the circuit, individual charges are
trapped on one side of the capacitor.
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What happens (Cont.)
Since some voltage is dropped across the capacitor, less
voltage (than before) is dropped across resistor. With less
voltage being dropped across the resistor, the current drops
off.
– V = IR (smaller V  smaller I)
With less current, the rate at which charge goes onto the
capacitor decreases.
The charge continues to build up, but the rate of the build
up continues to decrease.
In mathematical language, the charge as a function of time
Q(t) increases but its slope decreases.
Theory says the charge obeys Q(t) = C V (1 - e- t /  )
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Time Constant
Q(t) = C V (1 - e- t /  )
 in that equation is known as the “time constant” and
is given by  = RC
Note that since R = V / I , resistance has units volt/ampere
Since C = Q / V, capacitance has units coulomb/volt
RC = (V / I)  (Q / V) = Q / I
Then RC has units coulomb/ampere but an ampere is
coulomb/second
RC = Q / (Q / T) = T
So RC has units of second
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Charging Capacitor
Charge on capacitor
RC Circuit (Charging)
95%
63%
Time
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1
3
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Voltage is similar
If the charge on the capacitor varies
according to the expression
Q(t) = C V (1 - e- t /  )
then since the voltage across a capacitor is
V=Q/C the voltage is
V(t) = V (1 - e- t /  )
the voltage approaches its “saturation” value,
which in a simple RC circuit in the battery’s
voltage.
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Time Constant
The time constant ( = RC) is the time required
for a certain percentage (63%) of the saturation
charge (the charge after a very long time) to be put
on the capacitor.
If the resistance is large, the currents are small,
even from the start and it requires more time to
charge up the capacitor (  R)
If the capacitance is large, the capacitor can take a
lot of charge and again it requires more time to
charge up the capacitor (  C)
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RC circuits: Discharging
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What happens
This time we imagine that there are already
charges on the capacitor, but there is no battery.
With no battery to "push" the charges around, the
opposite charges on the two capacitor plates
would prefer to be together.
They must pass through the resistor before they
can reunite.
With all those like charges on one plate, there is a
strong incentive for charges to leave the plate.
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What happens (Cont.)
However, as charges leave the plate, the voltage
across the capacitor decreases (V = Q C) and the
incentive for charges to leave the capacitor
decreases, thus the rate at which charges leave
decreases as well.
In mathematical language, this time the charge as
a function of time Q(t) decreases and its slope
decreases.
Theory says the charge obeys Q(t) = Q0 e- t /  .
Same time constant as before.
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Discharging Capacitor
Charge on the capacitor
RC Circuit (Discharging)
37%
5%
Time
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1
3
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