Resource Lesson
RC Time Constants
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Capacitors are used in DC circuits to provide "bursts of energy." Typical examples would be a capacitor used to jump start a motor or a capacitor used to charge a camera's flash or a capacitor used to provide a large voltage across the chest in an emergency defibrillator when a person suffers from cardiac arrest.
Charging Capacitors
When the switch is closed, charges immediately start flowing onto the plates of the capacitor. As the charge on the capacitor's plates increases, this
transient current
decreases; until finally, the current ceases to flow and the capacitor is fully charged. In the diagram shown above, the right plate of the capacitor would be positively charged and its left plate negatively charged since the plates are arbitrarily assigned as + and - according to their proximity to the nearest battery terminal.
Graphs of
current vs time
and
charge vs time
are shown below. Mathematically, both of these graphs are exponential functions - current is an example of exponential decay, while charge is an example of exponential growth.
Charging Capacitor Graphs
current vs time
charge vs time
At t = 0 seconds, when the switch is initially closed, the capacitor does not have carry any charge and Kirchoff's loop rule would result in the equation:
where I
_{max}
represents the initial, maximum current flowing off the battery onto the plates of the capacitor. However, this current is not steady. As time passes, more and more charges accumulate on the capacitor. This in turn increases the voltage across the capacitor, Q = CV. When the voltage of the capacitor equals the voltage of the battery, charges will cease to flow. The current decreases with time.
We will now derive the equations for the transient charge on the capacitor and the transient current in the circuit. In this derivation,
i
or
i (t)
represents the transient current in the circuit as the capacitor charges at any time,
t
,
q
or
q (t)
represents the amount of charge on the capacitor at any given time,
t
,
R
is the resistance of the resistor, and
C
is the capacitance of the capacitor.
Applying Kirchoff's Loop Rule we have
Separating variables and integrating both sides yields
In order to solve for
q(t)
we must first extricate it from within its natural log expression:
. To do this, we will raise both sides of our equation to a power of
e
since
.
In the equation above the final total charge present on the plates is
. This allows us to rewrite our final equation for the build up of charge on the capacitor's plates as
In these equations, the product of RC must have the units of time, since the exponent in the function f(x) = e
^{x}
must be dimensionless. Let's investigate this relationship.
t
RC
sec
ohms (farads)
(volts/amps) (coulombs/volts)
coulombs/amps
coulombs/(coulombs/sec)
sec
This product is called the
RC time constant
and it allows us the ability to determine when certain percentages of change have occurred. To calculate the equation for the transient current, we will use the fact that
and differentiate the equation we just derived for q(t).
Let's use this information to work an example of a charging capacitor.
Refer to the following information for the next eleven questions.
A 9-V battery, a resistor of 1000 ohms, and a capacitor of 10 µF are connected in series with a switch. When the switch is closed, the capacitor begins charging.
What is the initial current in the circuit?
What is this circuit's RC time constant?
What will be the final charge on the capacitor?
How much charge will have been placed on the capacitor just as one time constant is reached?
What is the value of the transient current just as one time constant is reached?
How much time passed before the capacitor became 75% charged?
How much time passed before the capacitor became 99% charged?
What is the final current in the circuit when the capacitor is fully charged?
What is the
total energy stored in the fully-charged capacitor
?
What was the total work done
by the battery
while charging the capacitor?
Why are these values not equal?
Discharging Circuits
When the switch is closed in the circuit shown below, charges immediately start flowing off of the plates of the capacitor. As the charge on the capacitor's plates decreases, the current decreases; until finally, the current ceases to flow and the capacitor is fully discharged.
Both of the graphs of
current vs time
and
charge vs time
will now be decay functions since the current flowing through the resistor will fall off according to the flow of charge leaving the capacitor's plates.
Discharging Capacitor Graphs
current vs time
charge vs time
We will now derive the equation for the transient charge on the capacitor. In this derivation,
i
or
i (t)
represents the transient current in the circuit as the capacitor discharges at any time,
t
,
q
or
q (t)
represents the amount of charge remaining on the capacitor at any given time,
t
,
R
is the resistance of the resistor, and
C
is the capacitance of the capacitor.
Once again, by applying Kirchoff's Loop Rule, separating variables and integrating, we have
To calculate the equation for the transient current we must differentiate the equation we just derived for charge and use the fact that the charge on the capacitor is decreasing,
.
Let's practice once again, but with discharging capacitors.
Refer to the following information for the next five questions.
A 10 µF capacitor is charged to 9 V and subsequently connected through a switch to a resistor. When the switch is closed, the capacitor begins discharging.
What was the initial charge on the capacitor?
If the initial current flowing through the resistor is measured to be 3 µA, what is the resistance of the resistor?
What is this circuit's RC constant?
How large will the current be after one RC time constant has passed?
How much charge will be still be present on the capacitor's plates at that instant?
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Electric Power
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Ohm's Law
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Shock!
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Kirchhoff's Laws: DC Circuits with Capacitors
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Kirchhoff's Laws: Sample Circuit
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Resistance, Wattage, and Brightness
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35A: Series and Parallel
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Advanced Capacitors
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Basic Capacitors
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Basic DC Circuits
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Multiple-Battery Circuits
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Textbook Set #6: Circuits with Multiple Batteries
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