Unit 13:
Capacitors in DC Circuits
Recall that resistance is opposition
to the flow of current. Capacitance, which we'll study in this
unit, is an entirely different electrical property. It's the ability
to store a charge. Capacitors are components that are manufactured
to have this ability.
When a DC circuit has both resistance and capacitance, the currents
and voltages will usually change with time. In this unit we'll study
how to predict the values of these changing DC currents and voltages
at different times. We'll also see how to figure out how long
it will take a capacitor in a DC circuit to charge up.
Unit 12 Review
 This unit will build on material that you studied in Unit
12. So let's begin by taking this selftest to review what you
learned in that unit.

Capacitance & Capacitors
 Capacitance is a measure of a component's ability to store
charge.
 A capacitor is a device specially designed to have a certain
amount of capacitance.
 This ability to store charge means that capacitors can be dangerous.
Some common electronic devices, such as televisions, contain large
capacitors that can hold a deadly charge long after the device
has been turned off and unplugged. Just
as you should always assume that a firearm is loaded, you should
always assume that a capacitor is charged.
Capacitor Application: A Camera Flash
 A simple, everyday use of capacitors is in the flash unit for a
camera. You need a large charge in a very short time to light up the
camera's flash bulb. The camera's battery cannot provide such a large
charge in such a short time. So the charge from the battery is gradually
stored in a capacitor, and when the capacitor is fully charged, the
camera lets you know that it's ready to take a flash picture.
Schematic Symbol, and Appearance
 Here's the schematic symbol for a standard capacitor:
Often, one of the lines in this symbol is drawn slightly curved, so that
people won't confuse it with the symbol for a voltage source.
 While most resistors look more or less the same, capacitors come
in many different types of package. Here are a few examples of what
they may look like.
ParallelPlate Capacitor
 Most capacitors are parallelplate capacitors, which means that
they consist of two parallel pieces of conducting material separated
by an insulator.
 The insulator between the plates is called the dielectric.
Charging a Capacitor
 When a capacitor is connected across a voltage source, charge flows
between the source and the capacitor's plates until the voltage
across the capacitor is equal to the source voltage.
 In this process, the plate connected to the voltage source's negative
terminal becomes negatively charged, and the other plate becomes positively
charged.


Unit of Capacitance
 Capacitance is abbreviated C.
 The unit of capacitance is the farad, abbreviated F.
 Typical capacitors found in electronic equipment are in the microfarad
(μF) or picofarad (pF) range. Recall that micro means 10^{6} and
that pico means 10^{12}.
 You'll also remember that nano means 10^{9}. But
for some reason, the nanofarad has traditionally not been used, even
in cases where that might make the most sense.
 For example, if a capacitance is equal to 1×10^{9} F
(or 0.000000001 farads), you might think that you'd write that
as 1 nF. But in fact, most people would write this as either
1000 pF or 0.001 μF. This is strange and confusing,
but you just have to get used to it.
 In recent years, however, it's becoming more common to see nanofarads
(nF) used.
 For instance, the capacitance meters that the EET department
bought 10 or 15 years ago displayed all capacitance values in
either μF or pF. But the capacitance meters that we've bought
in the past 5 years display capacitance values in μF, nF,
or pF.

Charge per Voltage
Energy Stored by a Capacitor
 Recall that when current flows through a resistance,
energy is dissipated as heat.
 But capacitance does not behave like resistance. A capacitance does
not dissipate energy; rather it stores energy,
which can later be used to do something useful (such as light up a
camera flash) or returned to the circuit.
 The energy W stored by a capacitance C is given
by
W = ½ CV^{ 2}
where V is the voltage across the capacitor. Here energy
(W) is in joules (J), capacitance (C) is in farads
(F), and voltage (V) is in volts (V).
Capacitor Ratings
 Commercially available capacitors have several important specifications:
 nominal value and tolerance
 temperature coefficient
 DC working voltage
 leakage resistance
 Read on for discussion of these specifications.
Nominal Value & Tolerance
 Capacitors are available in a wide range of nominal values, from
1 picofarad to several farads.
 A specific capacitor's actual value is subject to the manufacturer's tolerance
specification. Typical capacitor tolerances range from ±5%
to ±20%.

Capacitor Identification Game
 In lab you'll learn to read the codes on capacitors. You'll need
to become an expert at reading these capacitor codes. To work on this
skill, play the Capacitor
Identification Game. This game has a Study mode that reviews the
relevant theory, a Practice mode that lets you practice with no time
pressure, and a Challenge mode that tests your skill while the clock
is running.
Temperature Coefficient
 Ideally, a capacitor's capacitance would be the same at all temperatures.
But in reality, capacitance changes as the capacitor gets warmer or
cooler. In many cases, you can ignore this change in capacitance,
but if you need a very precise capacitance value, or if you're dealing
with very large temperature swings, you may not be able to ignore
it.
 A capacitor's temperature coefficient tells you
how the capacitance changes with temperature.
 A positive temperature coefficient means
that as the capacitor's temperature increases, its capacitance
also increases. (And vice versaas the capacitor's temperature
decreases, its capacitance also decreases.)
 A negative temperature coefficient means
that as the capacitor's temperature increases, its capacitance
decreases. (And vice versaas the capacitor's temperature decreases,
its capacitance increases.)
 In addition to telling which way the capacitance changes (increase
or decrease), the temperature coefficient also quantifies the amount
of change. Usually it does this by saying how many parts per million
(ppm) the capacitance changes for each °C change in the temperature.
 For example, a temperature coefficient of 200 ppm/°C
means that for each °C change in the temperature, the
capacitance changes by 0.02%. (That's because 200 ÷ 1,000,000
= 0.0002, which is the same as 0.02%.)
DC Working Voltage
 The DC working voltage (also called the breakdown
voltage) is the maximum voltage at which a capacitor is
designed to operate continuously.
 Usually, the higher the capacitance value, the lower the DC working
voltage.
 Typical values of DC working voltage are a few volts for very large
capacitors to several thousand volts for small capacitors.
Leakage Resistance
 An ideal capacitor would have infinite resistance, with absolutely
no current flowing between the plates.
 In reality, a capacitor's resistance is finite, resulting in a small leakage
current between the plates.
 Typical values of leakage resistance are 1 MΩ to
100,000 MΩ or more. This is large enough that, from a practical
standpoint, we can often pretend that the resistance is infinite.
Capacitor Types
 Capacitors are often classified by the materials used for the dielectric
(the insulator between the capacitor's plates).
 Some types are air, paper, plastic film, mica, ceramic, electrolyte,
and tantalum.
 Often you can tell a capacitor's type by the appearance of the package.
For example, ceramic capacitors typically look like this:
Here's a typical plasticfilm capacitor:
Here's how electrolytic capacitors usually look:
Electrolytic Capacitors
 Of the different types of capacitors just mentioned, one in particular
deserves special discussion: electrolytic capacitors, which are available
in very large values, up to 100,000 μF and above.
 Unlike most capacitors, they are polarized: one side must remain
positive with respect to the other. Therefore you
must insert them in the proper direction. Inserting them backwards
can result in injury to you or in damage to equipment.
 In this photo of an electrolytic cap, notice that it has little
arrows with negative signs pointing to one end:
The lead that the arrows are pointing to is the negative lead.
 Also, the schematic symbol for an electrolytic cap has a positive
sign to tell you which way to hook up the capacitor:
Variable Capacitors
 Variable capacitors are also available. These contain a
knob or screw that lets you adjust the capacitor's capacitance.
 The schematic symbol has an arrow to show that the component's value
can be adjusted:
Stray Capacitance
 Stray capacitance exists between any two conductors that
are separated by an insulator, such as two wires separated by air.
This means that a circuit may contain some capacitance even if there's
no capacitor in the circuit.
 Stray capacitance is usually small (a few pF), and you can usually
ignore it, but it can have undesirable effects in highfrequency AC
circuits.
Capacitors in Series
Shortcut Rules for Capacitors in Series
Charge on Capacitors in Series
 Now let's think about connecting series capacitors to a voltage
source, as in the picture below:
 Seriesconnected capacitors have the same charge as
each other, regardless of their individual capacitance values.
 So in a series circuit, all of the capacitors will have the same
charge. In symbols, Q_{1} = Q_{2} = Q_{3} = ...
 We call this charge Q_{T}, which stands for total
charge. It's given by:
Q_{T} = C_{T} × V_{T}
where V_{T} is the source voltage.
 Note: this equation is basically a rearranged version of the equation
that we saw earlier:
C = Q ÷ V
The only difference is that and we've added "T" subscripts
to show that we're talking about total capacitance, total charge,
and total voltage.
Voltage on Capacitors in Series
 Once you know the charge on each capacitor in a series circuit,
find the voltage drops by using the equation:
V = Q ÷ C
for each capacitor. In other words,
V_{1} = Q_{1} ÷ C_{1}
V_{2} = Q_{2} ÷ C_{2}
and so on.
 Again, this is just a rearranged version of our basic formula. Right?
Capacitors in Parallel
Charge and Voltage on Capacitors in Parallel
 Now let's think about connecting parallel capacitors to a voltage
source, as in the picture below:
 Parallelconnected capacitors have the same voltage.
 So in a parallel circuit, all of the capacitors will have the same
voltage. In symbols, V_{1} = V_{2} = V_{3} = ... = V_{T},
where V_{T} is the source voltage.
 To find the charge on each capacitor in a parallel circuit
, use
Q = V × C
for each capacitor. In other words,
Q_{1} = V_{1} × C_{1}
Q_{2} = V_{2} × C_{2}
and so on.
Once again, this is not a new equation, just a rearranged version
of our basic equation for capacitors.
 For capacitors in parallel, the total charge delivered by the source
equals the sum of the charges on the individual capacitors.
SeriesParallel Capacitors
 For seriesparallel capacitor circuits, the strategy is very similar
to the strategy that you've learned for seriesparallel
resistor circuits:
 Combine series capacitors and parallel capacitors to obtain
progressively simpler equivalent circuits, until you've combined
all of the capacitors into a single total capacitance, C_{T}.

 Then work backwards, using our basic formula C = Q ÷ V (along
with rearranged versions of that formula) and remembering that seriesconnected
capacitors have the same charge and parallelconnected
capacitors have the same voltage.
Don't Connect Capacitors Directly Across a Voltage Source
Above we've been looking at circuits such as the one pictured
above, in which a capacitor (or combination of capacitors) is connected
directly across a voltage source. In fact, though, you should not connect
a capacitor (or combination of capacitors) directly across a voltage source,
since the resulting surge of current could damage the capacitor or the
voltage source. So the circuits that we've looked at so far are ones
that you should not build on a breadboard.
 Instead, you should always have a resistance in series with the
capacitor(s), to limit the amount of current that flows.
 If you should never build circuits like these, then why did we bother
looking at them? Because the rules that you learned there
do apply to capacitors in more complicated circuits. For example,
it's true in any circuit that:
 Total capacitance of series capacitors is given by the reciprocal
formula.
 Total capacitance of parallel capacitors is equal to the sum
of the capacitances.
 Series capacitors have the same charge.
 Parallel capacitors have the same voltage.
 If you know a capacitor's capacitance and its voltage, then
you can find its charge using the equation Q = C × V.
So even though the circuits that you studied above are not circuits
that you should actually build, the techniques you learned by studying
these circuits do apply to more realistic circuits.
Series RC Network
 A resistor and capacitor connected in series are called a series
RC network.
 Series RC networks have many practical uses. They are often used
in timing circuits to control events that must happen repeatedly at
a fixed time interval.
 One example is a circuit that causes an LED to blink on and
off once every second. There are several ways to design a circuit
to do this, but one of the most common ways uses a series RC circuit. By adjusting the value of the resistor or the capacitor,
the designer can cause the LED to blink faster or slower.
DC RC Circuit
 An RC circuit is any circuit containing, in addition to a
power supply, just resistors and capacitors.
 In this course we'll restrict our attention to RC circuits
containing DC voltage sources. We'll refer to such circuits as DC RC circuits.
 Examples: A very simple DC RC circuit just has a resistor,
a capacitor, and a voltage source in series:
 Here's a more complicated DC RC circuit, with several resistors
and capacitors:
Behavior of Capacitors in DC Circuits
Initial, Transient, SteadyState
 In most practical DC RC circuits, the values of current and voltage
change with time as capacitors are charged or discharged. Typically
such circuits contain a switch that is initially open, and you're
interested in finding the values of voltage and current after the
switch has been closed.
 To remind ourselves of this fact, we often include an open switch
in schematic drawings of DC RC circuits, as in the following picture:
 We distinguish three time periods in the analysis of such DC RC circuits:
 the initial period, when the switch is first closed.
 the transient period, while the capacitors are being
charged or discharged.
 the steadystate period, after the capacitors have
been fully charged or fully discharged.
 As we'll see now, we use different rules to figure out voltages
and currents during these three different time periods.
Initial Currents and Voltages
 The currents and voltages in a circuit at the instant when a switch
is first closed are called the initial currents and initial
voltages.
 In most cases, at this initial instant the circuit's capacitors
are either fully discharged or fully charged. Therefore, using the
rules of thumb you learned above, you'll find the circuit's initial
values of voltage and current by replacing the capacitors either with
shorts (if the capacitors are fully discharged) or with opens (if
the capacitors are fully charged).
SteadyState Currents & Voltages
 When the switch in a DC RC circuit has been closed for a long time,
currents and voltages have reached their steadystate values.
 In most cases, in the steady state the circuit's capacitors are
either fully discharged or fully charged. Therefore, using the rules
of thumb you learned above, you'll find the circuit's steadystate
values of voltage and current by replacing the capacitors either with
shorts (if the capacitors have been fully discharged) or by opens
(if the capacitors have been fully charged.)
Transient
 We've just seen how to figure out the initial currents and
voltages in a DC RC circuit and the steadystate currents and
voltages in a DC RC circuit. That covers the first instant when the
switch is first closed, and it also covers times a long time later,
after the capacitors have been fully charged or fully discharged.
But what about the inbetween times, after the switch has been closed
but before the capacitors are fully charged or discharged?
 While a capacitor is being charged (or discharged), currents and
voltages change gradually from their initial values to their steadystate
values. We call this the transient period of a DC
RC circuit.
 Transient means temporary, or shortlived. In DC circuits,
a transient is a voltage or current that changes for a short time.
 As we saw above, it's not too difficult to figure out initial and steadystate currents
and voltages. You just have to replace all capacitors with either
shorts or opens. But it takes more work to figure out transient voltages
and currents. Below we'll write down equations that let us calculate
the values of these currents and voltages during the time while they're
changing from their initial values to their steadystate values.
v and i
 First, let's review some notational conventions. You've seen in
earlier units that we use uppercase italic letters,
such as V and I, for quantities whose values are
constant.
 These constant values might be steady, unchanging values
in a DC circuit.
 Or they might be constant values in an AC circuit, such
as peak values or peaktopeak values. (Remember, the instantaneous
voltages and currents in an AC circuit change with time, but
the peak and peaktopeak values don't change as time passes.)
 On the other hand, we use lowercase italic letters,
such as v and i, to designate quantities whose values
change with time.
 These changing values might be instantaneous values in an
AC circuit.
 Or they might be changing, transient values in a DC circuit.
 So in the material below, we'll be using lowercase letters because
we'll be talking about transient values in a DC circuit. The important
thing to realize is that we use uppercase letters for constant quantities
(whether the circuit is DC or AC), and we use lowercase letters for changing quantities
(whether the circuit is DC or AC).
 Here's another convention that we'll use occasionally. To talk about
the value of a changing quantity at a particular time, we write that
time in parentheses.
 Examples: i(0) means the value of a current at 0
seconds, and v(0.8) means the value of a voltage at
0.8 seconds.
Transients While Charging
 In the circuit shown below, assume that the capacitor starts out
being fully discharged. If we close the switch, the capacitor will
gradually charge.
 We wish to be able to calculate i, v_{R},
and v_{C}. By this I mean that we wish to be able
to calculate the circuit's current, the resistor's voltage, and the
capacitor's voltage at any particular time while the capacitor is
charging.
Calculating i
Exponential Decay
 The current given by the equation above is a maximum at t = 0
and then gradually decays (decreases) until it reaches zero when the
capacitor is fully charged.
 Here is a graph of the equation for particular values of V_{S}, R,
and C. The graph shows current on the vertical axis and time
on the horizontal axis.
 For now, don't worry about the numbers on the axes. Just look at
the shape of the curve. The numbers will be different if you change
the values of V_{S}, R, or C, but the
curve will always have this shape.
 Notice that the current decreases as time passes, but it does not
decrease in a straight line. Instead, the current decreases very
quickly at first, and then decreases more slowly.
 In mathematics, a curve with this shape is called an exponential
curve. So, since the current is decreasing (or decaying) along
a curve of this shape, we call this exponential decay.
Time Constant
How Long to Charge?
 Here is a useful rule of thumb:
For most practical purposes, we may assume that all quantities
in a DC RC circuit have reached their steadystate values after
five time constants.
 So, for example, if we're charging a capacitor in a DC RC circuit,
and if that circuit has a time constant of 1 second, then it will
take about 5 seconds to charge up the capacitor.
 Since one time constant is equal to R×C, we can
write this rule of thumb as an equation:
Time to reach steady state ≈ 5×R×C
 Notice that in this equation I used a "squiggly equals sign" ≈ to
indicate that this is an approximation. Actually, after five time
constants the capacitor will be about 99.3% charged, not completely
charged. For most practical purposes, that's close enough.
Calculating v_{R} and v_{C}
More Exponential Curves
 If you plot these equations for the resistor voltage and the capacitor
voltage, you will get exponential curves similar to the curve we saw
above for current.
 In particular, a graph of the resistor's voltage has the
same shape as the graph of current, and so this is another case of
exponential decay. Here is the graph:
 On the other hand, the capacitor voltage starts at 0 V
and gradually increases until it reaches a maximum when the capacitor
is fully charged. The graph is shown below. Notice that this curve
has basically the same shape as the earlier curves, but flipped upside
down. We see again that the values change very quickly at first, and
then gradually approach a final value.

Discharging a Capacitor
 Up to now we've been talking about charging a capacitor. Similar
comments, but in reverse, apply to the case of discharging a
capacitor.

General Exponential Equations
SeriesParallel Transients
 When a capacitor charges (or discharges) through a seriesparallel
resistor network, the equations given above for the capacitor's transient
current and voltage still work, as long as you first replace the seriesparallel
network with its Thevenin equivalent.
 For example, in the seriesparallel circuit shown below, which you
will analyze in the SelfTest questions, we have a capacitor connected
to a network of three resistors and a voltage source. Thevenin's theorem
lets you "collapse" those three resistors and the voltage
source down to a single resistor and voltage source connected in series
with the capacitor. You can then use the equations you've learned
in this unit to analyze that "collapsed" circuit, and the
results you get from this analysis will be correct for the original
circuit as well.
Unit 13 Review
 This eLesson has covered several important topics, including:
 capacitance
 charge and voltage on a capacitor
 capacitor specifications
 types of capacitors
 capacitors in series, in parallel, and in seriesparallel
 DC RC circuits
 calculating initial values, steadystate values, and transient
values in DC RC circuits
 time constant of a DC RC circuit.
 To finish the eLesson, take this selftest to check your understanding
of these topics.

Congratulations! You've completed the eLesson for this unit.