 # 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.

##### 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.   • ##### Parallel-Plate Capacitor
• Most capacitors are parallel-plate 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.

##### 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
• As a mathematical quantity, capacitance is defined to be the ratio of the charge stored by a capacitor to the voltage across it:

C = Q ÷ V

where capacitance (C) is in farads (F), charge (Q) is in coulombs (C), and voltage (V) is in volts (V).
• We’ll use this basic formula frequently in analyzing capacitor circuits.
• Using algebra, we can rearrange that equation to yield two other equally useful equations:

Q = C × V       (Charge equals capacitance times voltage.)

and

V = Q ÷ C       (Voltage equals charge divided by capacitance.)

• ##### 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.
##### 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 versa–as 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 versa–as 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 plastic-film 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 high-frequency AC circuits.

##### Shortcut Rules for Capacitors in Series
• In Unit 8 you learned two shortcut rules that you can use for special cases of parallel resistors.
• The first shortcut rule said that if you have just two resistors in parallel, you can find their total resistance using the product-over-sum rule: RT = (R1 × R2) ÷ (R1 + R2). Similarly, if you have just two capacitors in series, you can find their total capacitance using the use the product-over-sum rule:

CT = (C1 × C2) ÷ (C1 + C2)

• The second shortcut rule applied to several parallel resistors that all have the resistance. This rule said that if you have n parallel resistors, each with resistance R, the total resistance is given by RT = R ÷ n. Similarly, if you have n series capacitors, each with capacitance C, the total capacitance is given by:

CT = C ÷ n

• Of course, the reciprocal formula given earlier applies to all cases of capacitors in series, so it will give the same answer that the shortcut rules give for these special cases.
• ##### Charge on Capacitors in Series
• Now let’s think about connecting series capacitors to a voltage source, as in the picture below: • Series-connected 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, Q1 = Q2 = Q3 = …
• We call this charge QT, which stands for total charge.  It’s given by:

QT = CT × VT

where VT 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.
• ##### Charge and Voltage on Capacitors in Parallel
• Now let’s think about connecting parallel capacitors to a voltage source, as in the picture below: • Parallel-connected capacitors have the same voltage.
• So in a parallel circuit, all of the capacitors will have the same voltage. In symbols, V1 = V2 = V3 = … = VT, where VT 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,

Q1 = V1 × C1

Q2 = V2 × C2

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.
• ##### 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
• The currents and voltages in a DC RC circuit depend on whether the capacitors are fully discharged, partially charged, or fully charged.
• Here’s an important rule of thumb that you must memorize:

A fully discharged capacitor is equivalent to a short circuit.

• So to find currents and voltages in a DC RC circuit whose capacitors are fully discharged, replace all capacitors with short circuits (in other words, with wires). Then you’ll be left with a circuit containing just a power supply and resistors, which you can analyze using the skills you learned earlier in this course.
• Here’s another important rule of thumb:

A fully charged capacitor is equivalent to an open circuit.

• So to find currents and voltages in a DC RC circuit whose capacitors are fully charged, replace all capacitors with open circuits. Then you’ll be left with a circuit containing just a power supply and resistors, which you can analyze using the skills you learned earlier in this course.
• • 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:
1. the initial period, when the switch is first closed.
2. the transient period, while the capacitors are being charged or discharged.
3. the steady-state 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).
• • When the switch in a DC RC circuit has been closed for a long time, currents and voltages have reached their steady-state 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 steady-state 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 steady-state 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 in-between 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 steady-state values. We call this the transient period of a DC RC circuit.
• Transient means temporary, or short-lived. 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 steady-state 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 steady-state 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 peak-to-peak values. (Remember, the instantaneous voltages and currents in an AC circuit change with time, but the peak and peak-to-peak 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, vR, and vC. 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
• In the series DC RC circuit shown, after the switch is closed at time t = 0, the current is given by the equation:

i = (VS ÷ R) e−t ÷ RC

where e is a constant equal to approximately 2.7183.
• You don’t need to remember the value of e, because scientific calculators have a special button to give you this value.
• ##### 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 VS, 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 VS, 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
• We’ve been considering the equation

i = (VS ÷ R) e−t ÷ RC

The quantity RC in this equation is called the time constant of the series RC network.  It is represented by the Greek letter τ, and its units are seconds:

τ = R×C

• In terms of τ, we can rewrite our equation for the current as

i = (VS ÷ R) e−t ÷ τ

• In this equation, be careful not to confuse t with τ. The variable t represents time; its value is the time at which you want to know the current. The time constant τ is fixed and depends on the size of your circuit’s resistor and capacitor.
• By the way, the name of Greek letter τ is "tau," which rhymes with "cow."
• The time constant τ is an indicator of how long the capacitor takes to charge.  The larger τ is, the longer the charge time.
• ##### 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 steady-state 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 vR and vC
• Let’s continue our analysis of a simple series DC RC circuit, in which we’re assuming that the capacitor starts out being fully discharged. • We’ve seen that the current in this circuit as the capacitor charges is given by the equation:

i = (VS ÷ R) e−t ÷ τ

• In the same circuit, after the switch is closed at time t = 0, the resistor’s voltage drop is given by the equation:

vR = VS e−t ÷ τ

and the capacitor’s voltage drop is given by the equation:

vC = VS(1 – e−t ÷ τ)

• Don’t think of these as three separate equations that you have to remember. Once you’ve got the equation for i, you can easily use Ohm’s Law to derive the expression for vR.
• Remember, Ohm’s Law says that a resistor’s voltage is equal to its current times its resistance. Do you see that if you take the expression above for i and multiply it by R, you’ll get the expression above for vR?
• And once you’ve got the equation for vR, you can easily use Kirchhoff’s Voltage Law (KVL) to derive the expression for vC.
• Remember, KVL says that the sum of the voltage drops around a loop must equal the sum of the voltage rises around that loop. Applying KVL to our simple series DC RC circuit gives us VS = vR + vC. Do you see how this lets you derive the expression above for vC from the expression above for vR?
• ##### 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. • ##### General Exponential Equations
• So far, we’ve been discussing the transient values of i, vR, and vC during the time while a capacitor is being charged up from 0 V. But what if the capacitor is initially partly charged, so that its initial voltage is not 0 V? Or what if we’ve got a charged capacitor that we’re discharging, rather than a discharged capacitor that we’re charging?
• Wouldn’t it be nice to have general equations that we can use in all of these cases? The equations that we wrote above for i, vR, and vC would then be special cases of the general equations.
• We can indeed write down general equations to cover all of these cases. For voltage, the general equation is

v = VF + (Vi − VF) e−t ÷ τ

where Vi is the initial voltage and and VF is the final voltage.
• Make sure that you see how this general equation gives you the equations we got above for vR and vC when you plug in the appropriate values for Vi and VF.
• Hint: In the special case that we discussed above, the capacitor is initially discharged, which means it initially behaves like a short. But in the end the capacitor is fully charged, which means that its final behavior is like an open. Therefore you should be able to see that the capacitor’s initial voltage is 0 V and its final voltage is VS. Using this knowledge, you should be able to use the general equation to derive our earlier equation for vC.
• Similarly, you should be able to see that the resistor’s initial voltage is VS and its final voltage is 0 V. Using this knowledge, you should be able to use the general equation to derive our earlier equation for vR.
• Simply replacing v with i and V with I, we can write down a similar general equation for current:

i = IF + (Ii − IF) e−t ÷ τ

where Ii is the initial current and and IF is the final current.
• Make sure that you see how this general equation gives you the equation we got above for i when you plug in the appropriate values for Ii and IF.
• ##### Series-Parallel Transients
• When a capacitor charges (or discharges) through a series-parallel resistor network, the equations given above for the capacitor’s transient current and voltage still work, as long as you first replace the series-parallel network with its Thevenin equivalent.
• For example, in the series-parallel circuit shown below, which you will analyze in the Self-Test 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 e-Lesson 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 series-parallel
• DC RC circuits
• calculating initial values, steady-state values, and transient values in DC RC circuits
• time constant of a DC RC circuit.
• To finish the e-Lesson, take this self-test to check your understanding of these topics.
• Congratulations! You’ve completed the e-Lesson for this unit.