The RC Time Constant
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University of Nevada, Las Vegas *
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Course
180L
Subject
Electrical Engineering
Date
Dec 6, 2023
Type
Pages
5
Uploaded by cervad1
The
RC
Time
Constant
A.
Introduction
and
Objectives
When
a
capacitor
is
connected
to
a
DC
power
supply
or
battery,
charge
builds
up
on
the
capacitor
plates,
and
thus
the
potential
difference
(.e.
voltage)
across
the
plates
increases
until
it
equals
the
voltage
of
the
DC
source.
This
voltage
difference
can
be
easily
measured
in
lab
by
attaching
a
DC
voltmeter
to
a
capacitor
such
that
the
two
probes
are
contacting
opposite
terminals
of
the
capacitor,;
if
a
"negative"
voltage
is
shown
on
the
voltmeter
display,
simply
reverse
the
position
of
the
two
probes.
At
any
time,
the
charge
Q
of
the
capacitor
is
related
to
the
voltage
across
the
capacitor
plates
by
Q
=
CV,
where
C
is
the
capacitance
of
the
capacitor.
During
charging
(or
discharging),
the
rate
of
voltage
rise
(or
fall)
depends
on
1)
the
capacitance,
C
(in
Farads),
in
the
circuit,
and
2)
the
resistance
R
(in
Q)
in
the
circuit.
Both
the
charge
and
discharge
times
of
a
capacitor
are
characterized
by
a
quantity
1,
called
the
"time
constant”,
where
1
=
RC
(in
seconds).
The
voltage
rise
during
charge
from
V=0
to
V,
(i.e.
maximum
V)
and
the
voltage
fall
during
discharge
from
V,
(i.e.
maximum
V)
to
V=0
can
be
easily
observed
using
the
voltmeter
if
the
rise
and
fall
are
slow
enough,
i.e.
if
T
is
large
enough.
Obviously,
considering
the
form
of
the
equation,
either
a
large
R
oralarge
C,
or
a
large
R
and
a
large
C
will
suffice.
In
this
lab
you
will
investigate
the
charge
and
discharge
characteristics
of
capacitors, and
how
1
affects
them.
B.
Equipment
Used
«
One
capacitor:
10,000
uF
(Phys181L)
1000
uF
(Phys152L)
»
Oneresistor:
10,000
Q
(Phys181L)
100,000
Q
(Phys152L)
DC
power
supply
High-resistance
digital
voltmeter
(10
Mega-Q)
Single-pole,
double-throw
switch
Timer
Connecting
wires
(8)
Graph
Paper
~O-
l—
b
-
AAN——S
R
v
a
t
|
vO
Figure
33.1
Circuit
diagram
for
charging
(switch
in
position
"a")
and
discharging
(switch
in
position
"b")
a
capacitor
through
a
resistor.
C.
Theory
When
a
capacitor
is
charged
through
a
resistor
by
a
DC
voltage
source
(the
switch
in
Figure
33.1
would
be
in
position
"a"),
the
charge
in
the
capacitor
and
hence
the
voltage
across
the
capacitor
increase
through
time.
When
charging,
the
voltage
V
as
a
function
of
time
is:
V=V,
(1-e
"%
=v,(1-e
V)
(33.1)
where
e
=
2.718
is
the
base
of
natural
logarithms,
V,
is
the
voltage
of
the
source
(in
V),
R
is
the
resistance
of
the
charging
circuit
(in
Q),
and
t
is
the
elapsed
time
(in
seconds).
The
curve
of
the
exponential
rise
of
the
voltage
through
time
is
shown
in
Figure
33.2.
To
see
the
result
of
charging
at
a
specific
time
where
t
=
1
=
RC
(i.e.
at
a
time
of
one
time
constant),
simply
substitute
RC
for
tin
equation
33.1.
The
result
indicates
that
the
voltage
across
the
capacitor
will
have
increased
to
a
value
of
(1-1/e)
of
V..
Plug
in
the
value
of
e,
and
you
will
see
that
during
charging
the
exact
value
of
V
at
one
time
constant
is
0.63V,,.
When
a
fully
charged
capacitor
is
discharged
through
a
resistor
(the
switch
in
Figure
33.1
would
be
in
position
"b"),
the
charge
in
the
capacitor
and
hence
the
voltage
across
the
capacitor
decrease
through
time.
When
discharging,
the
voltage
V
as
a
function
of
time
is:
V=V,
e
URC
(33.2)
The
curve
of
the
exponential
decrease
of
the
voltage
through
time
is
shown
in
Figure
33.2.
To
see
the
result
of
discharging
at
a
specific
time
where
t
=1
=
RC
(i.e.
at
a
time
of
one
time
constant),
simply
substitute
RC
for
t
in
equation
33.2.
The
result
indicates
that
the
voltage
across
the
capacitor
will
have
decreased
to
a
value
of
1/e
of
V,
(i.e.
e
'
=
1/e).
Plug
in
the
value
of
e,
and
you
will
see
that
during
discharge
the
exact
value
of
V
at
one
time
constant
is
0.37V,,.
In
order
to
analyze
voltage
versus
time,
it
is
useful
to
put
Equations
33.1
and
33.2
in
the
form
of
a
straight
line.
From
Equation
33.1
we
obtain:
(Vo-V)=Vye
™
and
taking
the
natural
logarithm
of
both
sides
of
the
equation
gives
the
charge
equation:
In(Vo-V)=-tY(RC)
+InV,
{33.3)
Then
from
Equation
33.2,
taking
the
natural
logarithm,
we
obtain
this
discharge
equation:
InV
=-Y(RC)+/InV,
(33.4)
Both
of
these
equations
have
the
form
of
the
equation
of
a
straight
line,
y
=
mx
+
b
;
you
should
be
able
to
identify
the
variables
and
constants.
Both
have
negative
slopes
of
magnitude
1/RC.
Hence,
the
time
constant
can
be
found
from
the
slopes
of
the
charging
graph
of
/n
(V,
-
V)
versus
t
and/or
the
discharge
graph
of
In
V
versus
t
(i.e.
RC
is
a
system
constant
whether
that
system
is
charging
or
discharging).
V,
==
-
e
O
e
1
!
V
P
T
°
(
—'5')
a
!
3
i
S
v
|
o
:
Dls‘""’a
:
’9/};9
|
|
|
t=RC
Time,
t
Figure
33.2
This
graph
illustrates
voltage
versus
time
for
a
charging
and
discharging
capacitor.
The
“steepness”
of
the
curves
depends
on
the
time
constant
RC.
D.
Experimental
Procedure,
Calculations
and
Graphs
1)
Build
the
circuit
shown
in
Figure
33.1;
place
the
switch
in
the
open
position.
Make
sure
that
the
negative
end
of
the
capacitor
is
connected
to
the
negative
terminal
of
the
power
supply.
Lastly,
connect
a
"shorting
wire"
to
one
terminal
of
the
capacitor
so
that
you
can,
when
needed.
temporarily
complete
the
connection
across
the
capacitor
to
rapidly
and
completely
discharge
it.
Record
the
capacitance
C
and
the
resistance
R
in
the
blanks
above
the
right
side
of
the
Data
Table.
In
this
circuit,
to
the
degree
that
the
voltmeter
resistance
is
very
large
(10
Mega-Q)
compared
to
a
very
small
R,
charging
and
discharging
will
occur
through
R
(and
not
through
the
voltmeter)
2)
Set
the
maximum
system
voltage
(V,)
before
you
collect
data
by
adjusting
the
power
supply
output.
To
do
this,
make
sure
the
circuit
switch
is
in
the
open
position,
then
set
the
voltmeter
range
to
20
V,
disconnect
the
voltmeter
from
the
terminals
of
the
capacitor
and
connect
it
to
the
DC
power
supply
outputs,
then
turn
the
power
supply
dial
clockwise
until
the
voltmeter
displays
5.0
V.
For
the
duration
of
this
lab
DO
NOT
change
the
position
of
the
power
supply
dial.
Record
this
value
of
V,
in
the
two
labeled
cells
in
the
Data
Table.
Re-connect
the
voltmeter
across
the
capacitor
terminals;
it
will
now
display
the
potential
across
the
capacitor.
3)
Test
the
circuit
by
connecting
the
shorting
wire
across
the
capacitor
to
completely
discharge
it
(the
voltmeter
will
then
display
zero),
then
disconnect
one
end
of
the
shorting
wire.
Now,
with
no
charge
in
the
capacitor,
close
the
switch
to
position
a
and
note
the
voltage
rise
of
the
capacitor
through
time
on
the
voltmeter.
If
you
wait
two
minutes
or
so
the
capacitor
will
eventually
charge
up
to,
or
very
close
to,
V_.
Next,
move
the
switch
to
position
b
and
note
the
voltage
decrease
as
the
capacitor
discharges
through
time.
The
data
collected
in
this
lab
will
be
voltage
readings
through
time
as
the
capacitor
charges,
and
then
as
it
discharges.
4)
To
collect
the
charging
data,
start
with
the
switch
in
the
open
position
and
the
capacitor
fully
discharged
(use
the
shorting
wire);
the
voltmeter
should
display
zero.
Now,
simultaneously
close
the
switch
to
position
a
and
start
the
timer.
Record
the
time
and
voltage
at
approximately
0.5
V
intervals
until
the
capacitor
is
fully
charged
(to
Vy);
exactly
what
the
intervals
are
is
not
important,
as
the
general
idea
is
just
to
spread
out
the
data
somewhat
evenly.
Due
to
the
logarithmic
nature
of
capacitor
discharge,
these
intervals
will
be
very
short
at
first,
then
get
longer
as
the
capacitor
approaches
full
charge.
At
the
points
that
you
want
to
collect
and
record
data
you
should
simultaneously
open
the
switch
(which
temporarily
stops
the
charging)
and
stop
the
timer;
this
will
require
some
deft
control
of
the
switch
so
as
to
open
it
without
continuing
into
the
discharge
position
(b).
DO
NOT
pause
for
more
than
1-2
seconds
before
putting
the
switch
back
in
position
a,
as
the
capacitor
will
slowly
discharge
through
the
voltmeter
whenever
you
pause.
After
the
capacitor
is
fully
charged,
leave
the
switch
in
position
a
for
the
next
part
of
the
lab.
5)
To
collect
the
discharging
data,
the
switch
should
be
in
position
a,
and
the
capacitor
needs
to
be
fully
charged
(from
part
3,
above).
Zero
the
timer,
then
simultaneously
move
the
switch
from
position
a
to
position
b
and
start
the
timer.
Record
the
time
and
voltage
at
approximately
0.5
V
intervals
until
the
capacitor
is
fully
discharged:
again,
exactly
what
the
intervals
are
is
not
important.
Due
to
the
logarithmic
nature
of
capacitor
discharge,
these
intervals
will
be
very
short
at
first,
then
get
longer
as
the
capacitor
approaches
a
completely
discharged
state.
As
when
charging,
at
the
points
that
you
want
to
collect
and
record
data
you
should
simultaneously
open
the
switch
(to
stop
discharging
through
R)
and
stop
the
timer.
DO
NOT
pause
for
more
than
1-2
seconds
before
putting
the
switch
back
in
position
b,
as
the
capacitor
will
continue
to
slowly
discharge
through
the
voltmeter
whenever
you
pause.
Turn
off
all
of
the
equipment
and
disassemble
the
circuit
when
you
are
done.
6)
Calculations
1:
This
step
will
complete
the
data
table
as
you
transform
the
raw
data
into
a
form
that
can
be
used
to
calculate
the
system
RC
time
constant
from
a
linear
regression.
Calculate
the
quantity
(V,
-
V)
for
each
charging
time
interval,
then
for
each
time
interval
determine
In(V,
-
V)
for
charging
and
/nV
for
discharging.
You
will
fill
in
the
blanks
below
the
data
table
later,
according
to
Calculations
2
(below).
7)
Graphs
1
&
2:
These
two
graphs
will
be
used
to
determine
the
system
RC
time
constant.
On
graph
1
(charging
data)
you
will
plot
/n(V,
-
V)
on
the
ordinate
versus
t
on
the
abscissa.
On
graph
2
(discharging
data)
you
will
plot
/nV
on
the
ordinate
versus
t
on
the
abscissa.
Perform
a
linear
regression
on
the
data
to
determine
the
equations
of
the
lines,
then
draw
each
line
(and
the
data
points)
on
the
appropriate
graph.
8)
Calculations
2:
These
calculations
will
summarize
the
results
from
the
data
table,
and
will
be
recorded
in
the
blanks
below
it.
Record
the
slope
of
each
regression
line,
then
calculate
and
record
the
average
of
the
two
slopes.
Refer
to
equations
33.3
(charging)
and
33.4
(discharging)
to
note
that
both
graphs
1
and
2
will
have
negative
slopes
of
magnitude
1/RC;
use
the
average
slope
to
calculate
the
system
experimental
RC
time
constant
according
to
this
relationship.
Finally,
use
percent
error
to
compare
the
experimental
RC
with
the
theoretical
RC
(as
calculated
from
your
known
C
and
R
values).
9)
Graph
3:
This
graph
will
show
the
rate
of
change
of
the
charging
and
discharging
processes
through
time.
Plot
V
on
the
ordinate
versus
t
on
the
abscissa
for
BOTH
your
charging
and
discharging
data,
i.e
on
the
same
graph.
Draw
a
smooth,
labeled
line
though
each
set
of
data
points.
Notice
the
symmetry.
10)
Questions:
Answer
all
three
questions.
They
have
nothing
to
do
with
your
experimental
data
or
results
per
se,
but
with
the
mathematics
of
charging
and
discharging
capacitors.
Some
answers
are
partly
symbolic,
and
all
require
you
to
convert
phrases
in
the
written
question
into
ratios,
coefficients,
etc.
to
get
the
correct
answer.
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D
E
b) After the capacitor is fully charged, rank the potential at points A-E, from lowest to highest.
c) The battery is now disconnected from the capacitor and a piece of dielectric material is
inserted between the plates. Is the potential difference between points A and E,
than, the same as or larger than the potential difference across the battery? Explain your
smaller
reasoning
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a) The voltage across a 10 capacitance is ve = 150 sin( ......) volts and. Frequency equal to 50 Hz. Determine the current through the capacitor i and sketch its waveform.
b) Define the term frequency
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Recommended textbooks for you
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Publisher:Cengage Learning