# Capacitors. Chapter How capacitors work Inside a capacitor

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1 Chapter 6 Capacitors In every device we have studied so far sources, resistors, diodes and transistors the relationship between voltage and current depends only on the present, independent of the past. In this chapter, we will study a device that stores electrical energy, so its behavior depends on the past. This characteristic enables a wealth of new applications, including memory, frequency-dependent response and power management. Understanding capacitors will also allow us to refine our model of a MOS transistor, to explain why transistors (and hence computers) can t operate infinitely quickly. In order to describe this behavior, we ll need to start thinking of voltage and current as functions of time, which we might denote v(t) and i(t). It is common to omit the (t) part, so that v and i are implicitly understood to be functions of time. The voltage v across and current i through a capacitor with capacitance C are related by the equation i = C dv, C i v where dv is the rate of change of voltage with respect to time. Note that v and i are labeled according to the passive sign convention, with current going into the terminal that is labeled positive. 6.1 How capacitors work Inside a capacitor Capacitors store energy in an electric field between charged parallel plates separated by a dielectric, seen in Figure Current can t pass through the dielectric, because it s an electrical insulator. When there is a voltage between the two plates, positive charge builds up on one plate, and negative charge builds up on the other, creating an electric field in the dielectric. The amount of charge Q in coulombs that is stored in the capacitor is related to the voltage across the capacitor v by the equation Q = Cv, (6.1)

2 116 CHAPTER 6. CAPACITORS Figure 6.1: Capacitors store energy between two charged plates non-polarized capacitor polarized capacitor Figure 6.2: Symbols for capacitors Geometry of capacitor where C is the capacitance of the capacitor. The larger the capacitor, the more charge would be stored for the same voltage. The capacitance C of a capacitor is governed by the area of these plates, the distance between them, and the material used for the dielectric, according to the equation C = εa d (6.2) Polarized capacitors where ε is the dielectric constant, a property of the material, A is the area of the plates, and d is the distance between them. The larger the area, and the closer together the plates are from each other, the higher the capacitance. Capacitors are made of many different materials; you ll encounter several different types in your labs in this class. In most capacitors, there s no positive or negative terminal, just like with resistors. However, some capacitors are made in a way that means they can only accept voltages in one direction. These capacitors are called polarized capacitors. If a capacitor is polarized, we often use a special symbol for it, as shown in Figure 6.2. In particular, in a type of capacitor called electrolytic capacitors (Figure 6.3), the dielectric is a very thin oxide layer that would be depleted if a negative voltage is applied, causing the capacitor to start conducting as if it were a short circuit; the resulting huge amount of current then boils the electrolyte, pressure builds up, and the capacitor explodes. In other words, double check the polarity before connecting these! What does a capacitor do? Derivation of i = C dv

3 6.1. HOW CAPACITORS WORK 117 Figure 6.3: Polarized Electrolytic Capacitors Circuit designers rarely think of capacitors in terms of Q = Cv. The more convenient form is to relate voltage to current, as we have in other parts of this course. Recall that current is the flow of charge per unit time, that is, i = dq. Then, i = dq = d dv (Cv) = C. The equation i = C dv is how we normally think of capacitors in electrical engineering, that is, current is proportional to the change in voltage with respect to time. It s worth pausing for a moment to consider what this means, and what it does not mean. The above equation tells us nothing about the voltage across the capacitor itself only the rate of change of voltage. If (and only if) positive current is flowing into the positive terminal, then the capacitor s voltage is increasing. If (and only if) positive current is flowing into the negative terminal, then the capacitor s voltage is decreasing. If (and only if) there is no current through the capacitor, the capacitor s voltage is constant. It s worth pointing out two corollaries of the equation i = C dv. First, if the circuit has reached a state where everything is steady, then (by assumption) dv = 0, which implies that i = 0, independent of C. We often phrase this as the maxim, at steady state, capacitors look like an open circuit. We ll come back to this in Section 6.4. Another corollary of the equation i = C dv is that the voltage across a capacitor can t change instantaneously. A sudden change would require dv =, implying that i =, which isn t physically possible. This fact that even if something else suddenly changes, the voltage across a capacitor cannot will become useful in our study of transient response shortly. More generally, since rapid changes in voltage require large currents, one good way to think about capacitors is that they tend to resist changes in voltage, or that they ll try to hold the voltage for a little bit. The larger the capacitance, the more they tend to do so. Note that, in capacitors, this observation only applies to voltage. There s nothing preventing the current through a capacitor from changing abruptly. (6.3) Capacitor in steady state looks like open circuit No instantaneous voltage change

4 118 CHAPTER 6. CAPACITORS Capacitors are linear Energy stored in capacitor The relationship between i and v for a capacitor is still linear: if you double v (for all time), i doubles with it. It s a bit hard to imagine, given that i and v aren t really directly related, but recall that constant multiplicative coefficients survive differentiation, which is a property of linear systems. 2 Because capacitors are linear, a lot of circuit analysis techniques we studied with resistors still work with capacitors, most notably superposition. If you think of charge as a fluid, then you can think of a capacitor like a large tank. 3 The height of the water in the tank represents the voltage on the capacitor. While we can instantaneously turn the flow of water off and on (changing the current), we cannot instantaneously change the water level of this tank. Similarly, we cannot instantaneously change the voltage across the capacitor. Its value changes as a result of the integration of the current being added to the tank/capacitor, and the value of the capacitance is related to the area of the tank. Because capacitors are energy storage devices, we might also want to know how much energy is stored in a capacitor. Recall that power is the rate of change of energy with respect to time. Therefore, we can solve for the power P, substitute values from the equations above, and integrate over time to get energy: that is, E = P = iv = C dv V v = Cv dv = CV 2, E = 1 2 CV 2, (6.4) where V is the voltage across the capacitor now. Example 6.1 In the (contrived) circuit below, at t = 0 ms, v a = 10 V. The switch begins on the lower throw, and moves to the upper throw at t = 0 ms. When does the capacitor s voltage to reach 0 V? v a 500 µa 2 µf Solution 6.1: By passive sign convention, the current through the capacitor is negative, thus the capacitor 2 We don t study linearity formally in this course, but you will in EE 102A. 3 You need to be careful with this analogy, since there is plus and minus charge, and there is not negative water. When water flows into a tank to fill it up, it only flows into the top of the tank. With a capacitor there are really two tanks. Charge flows into the top tank, which starts filling it up, but and equal amount of charge also leaves the bottom tank, filling the bottom tank with negative charge. A better fluid analogy would be to model a capacitor as two tanks. The first one is right side up and empty, and the second is upside down and full. When liquid flows into the top tank, to start to fill it up, and equal amount of liquid flows out of the bottom tank, and is replaced by bubbles.

5 6.2. CAPACITORS IN SERIES AND PARALLEL 119 is discharging: 50 µa 250 nf V a Current is given by The rate of change of voltage is given by dv = i C i = C dv = 500 µa 2 µf = 250 V/s Thus, the amount of time for the voltage across the capacitor to reach 0 V is 10 V = 40 ms 250 V/s 6.2 Capacitors in Series and Parallel We would like to create equations to simplify multiple capacitors in a circuit. To do this it would be nice to see if there was an equivalent resistance for a capacitor. We can t really find one, since the current depends on V not V, but we can say that V = i T C. This is interesting in two ways. First is says that the effective resistance is related to 1/C, so circuits where R adds will need to combine 1/C. Second it shows how the resistance depends on the T. If T is small, the effective resistance will be small, but if the time is large, the resistance will also be large Capacitors in Series Like with resistors, when we have multiple capacitors in a circuit, we often want to replace them with one equivalent capacitor. We ll start by analyzing capacitors in series: C 1 i 1 C 2 i 2

6 120 CHAPTER 6. CAPACITORS dv Using our equation for the current through a capacitor, we have i 1 = C 1 dv 1 and i 2 = C 2 2 where V 1 and V 2 are the voltages across C 1 and C 2 respectively. We want to find one capacitor with capacitance C eq such that the voltage across it is V 1 V 2 = V tot and the current through it is i = i 1 = i 2. We know from KCL that i 1 = i 2. With this information, we now rearrange the equations to get: Capacitors in series dv 1 i 1 = C 1 i 1 = dv 1 C 1 ( 1 i 1 ) = dv tot C 1 C 2 ( 1 i = 1 C 1 1 C 2 ) dv 2 i 2 = C 2 i 2 = dv 2 C 2 This looks a lot like our equation for current through a capacitor. Thus, we have the following relation for capacitors in series: dv tot 1 C eq = 1 C 1 1 C 2 1 C n. (6.5) This makes sense intuitively, as both capacitors will be experiencing the same current, and the voltage across both will increase with respect to this current. Thus, the total voltage across both capacitors will increase at a greater rate than either of the voltages accross individual capacitors. We also notice that this relation looks like the relation for resistors in parallel Capacitors in Parallel We also want to be able to replace capacitors in parallel. Let s analyze the following: i i 2 C 1 i 1 C 2 By KVL, we know that the voltage across C 1 must equal the voltage across C 2. Call this voltage V. Also, by KCL, we know that i = i 1 i 2. We substitute the currents through each capacitor: i = i 1 i 2 = C 1 dv C 2 dv = (C 1 C 2 ) dv

7 6.2. CAPACITORS IN SERIES AND PARALLEL 121 Thus, we have the following relation for capacitors in parallel: C eq = C 1 C 2... C n (6.6) Going back to our water tank analogy, the summation makes sense. Putting two capacitors in parallel is like putting two water tanks in parallel. We also notice that this relation is similar to resistors in series. In this section, we have seen that the equivalent capacitance equations are the same as the equivalent resistance ones, but the series and parallel behaviors are flipped. Capacitors in parallel Example 6.2 You wish to replace the following network of capacitors with a single capacitor. What should its value be? 300 nf 600 nf 350 nf 150 nf Solution 6.2: We first find the equivalent capacitance of the 300 nf and 600 nf in series The circuit can now be simplified to C s = C 1C nf 600 nf = = 200 nf C 1 C nf 600 nf 200 nf 350 nf 150 nf We now find the equivalent capacitance of the 200 nf and 150 nf in parallel, C p = C 1 C 2 = 200 nf 150 nf = 350 nf The circuit can now be simplified to 350 nf 350 nf

8 122 CHAPTER 6. CAPACITORS The equivalent capacitance of these two 350 nf capacitors in parallel is given by C s = C 1C nf 350 nf = = 175 nf C 1 C nf 350 nf We replace the initial circuit with the a single 175 nf 175 nf 6.3 Capacitors in real circuits Capacitors, like many electrical components, come in a variety of shapes and sizes. Larger capacitance and higher voltage compliance generally comes with larger size. Electrolytic capacitors have the largest capacitors per unit volume, but they have limited voltage and as polarized, as described before Capacitors to Control Supply Voltage In our circuits, we often want a constant, DC voltage source. While this is perfectly fine on paper, in practice, we know that sources are not ideal. As you draw more current, especially if the change is sudden, voltage will drop. We can mitigate these effects by connecting a capacitor between Vdd and Gnd. We know capacitors resist changes in voltage, so the capacitor will work to keep Vdd constant. Essentially, the capacitor acts as a energy reserve for the circuit, supplying energy when the demands of the circuit exceed the battery s (or other voltage source s) capabilities Capacitors in MOS Transistors In a MOS transistor, there is a capacitor between the gate and the source. From the equation of a capacitor i = C dv we see that if i = 0, then dv = 0. As a result, if there is no current, the voltage across the capacitor (from the gate to the source) remains constant. For example, if we simply disconnect the gate terminal from Vdd, without driving it to another voltage, it will remain at Vdd. Thus, we need a current to cause a change in voltage and change the gate voltage of the MOS transistor Real Wires All real wires also have capacitance. As we saw above, this means wires require some charge to change their voltage. Remember that voltage is defined as potential energy per charge, so this observation makes sense in the context of our definition. This means that any time we want to change the voltage of a wire, we need charge to flow into it. The amount of charge we need to

9 6.4. CAPACITORS IN STEADY STATE 123 produce a given change of voltage in a certain amount of time is governed by the equation: i = C dv. We re starting to see a theme. Capacitance governs how fast we can change voltages and the energy we need to do so. This result governs the speed and power consumption of modern electronics, including your computers! 6.4 Capacitors in Steady State If a circuit has reached a state where all currents and voltages have settled into a constant, longterm state, we say that the circuit is in steady state. That is, in steady state, di dv = 0 and = 0 for every current and voltage in the circuit. As far as capacitors go, this means that i = C dv = C 0 = 0, independent of C. If the current through a device is identically zero, then it looks the same as an open circuit. This gives rise to a maxim on capacitors in steady state: We illustrate this with an example. In steady state, capacitors behave like open circuits. Capacitors in steady state Example 6.3 The circuit below has reached steady state. Find v 1 and v kω 10 kω 10 V v 1 C 1 20 kω v 2 C 2 Solution 6.3: Since the circuit is in steady state, all capacitors look like open circuits. Redraw the circuit accordingly: 20 kω 10 kω 10 V v 1 20 kω v 2

10 124 CHAPTER 6. CAPACITORS Now this is just a resistor network, so using the voltage divider rule, ( ) 20 kω 10 kω v 1 = 10 V = 6 V, 20 kω 20 kω 10 kω ( ) 20 kω v 2 = 10 V = 4 V. 20 kω 20 kω 10 kω Example 6.4 The circuit below has reached steady state. (a) What is the voltage v x? (b) What is the voltage across the capacitor? (Be sure to specify the direction.) 33 nf v x 6 V 680 Ω 120 kω Solution 6.4: In the steady state, capacitors look like open circuits so we can simplify the circuit to v C v x 6 V 680 Ω 120 kω (a) No current flows through the 120 kω resistors, so v x is effectively connected to ground, v x = 0 V. (b) The voltage across the capacitor is V C = 6 V v x = 6 V 0 V = 6 V. 6.5 Transient Response of RC Circuits If a circuit has only one capacitor (or capacitors that can be reduced using series/parallel rules to one capacitor), we say it is a first-order circuit. In many applications, these circuits respond to a

11 6.5. TRANSIENT RESPONSE OF RC CIRCUITS 125 sudden change in an input: for example, a switch opening or closing, or a digital input switching from low to high. Just after the change, the capacitor takes some time to charge or discharge, and eventually settles on its new steady state. We call the response of a circuit immediately after a sudden change the transient response, in contrast to the steady state. Recall that in a capacitor, i = C dv. If a capacitor experienced a sudden change in voltage even if small sudden change that would be tantamount to dv =, which in turn requires i =. This isn t physically possible you can have large currents, but not infinite ones. This leads to another maxim about capacitors: The voltage across a capacitor can t change instantaneously. Note that this constraint only relates to the voltage across a capacitor. The current through a capacitor is free to change as suddenly as it likes, and voltages across other devices can in principle change instantaneously. But the voltage across a capacitor is an important means of relating voltage and current in a circuit just before a sudden change, to that just after a sudden change. To distinguish between just before and just after, we often use and superscripts next to the time in question. So, for example, v(0 ) means the voltage v(t) just before t = 0, and v(0 ) means the voltage v(t) just after t = 0. No instantaneous voltage change A First Example To get us started, consider the circuit in Figure 6.4, whose behavior we ll derive from first principles. The voltage source provides v in (t) = 0 for t < 0, and v in (t) = 10 V for t 0. We wish to find v out (t). R 10 v (V) v in (t) vin (t) C v out 5 t (a) circuit diagram (b) graph showing v in (t) Figure 6.4: Basic RC charging circuit First, a few observations, using the steady state analysis we discussed in Section 6.4. Just before the step in v in from 0 V to 10 V at t = 0, the input has (by assumption) been constant for a very long time, so the circuit should have reached steady state. Therefore, the capacitor looks like an open circuit, as shown in Figure 6.5(a). Since there s no current through the resistor, v out (0 ) = v in (0 ) = 0 V. Just after the step, v out (0 ) must be the same, v out (0 ) = v out (0 ) = 0 V, because it so happens in this case that v out is the voltage across a capacitor, which can t change instantaneously. Long after the step, if we wait long enough the circuit will reach steady state, then v out ( ) = 10 V, as shown in Figure 6.5(b).

12 126 CHAPTER 6. CAPACITORS R R 0 V (a) t = 0 (just before t = 0) v out 10 V (b) t = (final steady state) v out Figure 6.5: Circuit of Figure 6.4 in steady state What happens in between? Using Kirchoff s current law applied at the top-right node, we could write Solving this differential equation yields 1 RC 10 V v out R = C dv out. 1 RC = 1 dv out 10 V v out 1 = dv out 10 V v out t RC = ln (10 V v out) c e t RC c = 10 V v out 10 V Ae t RC = vout, (where A = e c ) and applying the initial condition v(0 ) = 0 that we found above, So the solution to the differential equation is Ae 0 RC = 10 V 0 = A = 10 V v out (t) = 10 10e t RC. 10 v v in (t) v out (t) 5 τ = RC 2τ 3τ 4τ t

14 128 CHAPTER 6. CAPACITORS R vin (t) C v out Solution 6.5: Let s first find the steady state voltage, v out ( ). In steady state, capacitors look like open circuits, thus we can simplify the circuit to R vin (t) v out No current is flowing, thus there is no voltage drop across the resistor and v out = v in = 10 V. Since v out (t) is the voltage across a capacitor, it can t change instantaneously, so the voltage just after time t = 0 is equal to the voltage just before t = 0, v(0 ) = v(0 ) = 0. We plug these values for v out (0 ) and v out ( ) into the equation v out (t) = v out ( ) [ v out (0 ) v out ( ) ] e t τ v out (t) = 10 V 10 V e t τ = 10 V 10 V e t RC = 10 V [0 V 10 V] e t τ Charging a capacitor Special Cases: Charging and Discharging a Capacitor In the special case where the capacitor starts fully discharged and charges up to a final voltage V, with no voltage across it (i.e. no charge stored in it), we have v(0 ) = 0 and v( ) = V. Then the transient response equation (6.7) reduces to v(t) = V V e t/τ = V (1 e t/τ ). (6.9) Discharging a capacitor We sometimes call (6.9) the charging equation, reflecting that it starts at zero and charges towards the final steady-state value. The example in Section is an example of this case, where V = 10 V. In the special case where a capacitor starts at an initial voltage V and discharges down to zero, we have v(0 ) = V and v( ) = 0. Here, (6.7) reduces to Unsurprisingly, we sometimes call (6.10) the discharging equation About the Time Constant v(t) = V e t/τ (6.10) The parameter τ (the Greek letter tau) is known as the time constant of the circuit. It has units of seconds (you should verify this for yourself), and it governs the speed of the transient response.

15 6.5. TRANSIENT RESPONSE OF RC CIRCUITS 129 v(0 ) v(t) 63% of [v(0 )v( )] 95% of [v(0 )v( )] v( ) τ 2τ 3τ 4τ 5τ Figure 6.6: Time constant t Circuits with higher τ take longer to get close to the new steady state. Circuits with short τ settle on their new steady state very quickly. More precisely, every time constant τ, the circuit gets 1 e 1 63% of its way closer to its new steady state. This is illustrated in Figure 6.6. Memorizing this fact can help you draw graphs involving exponential decays quickly. After 3τ, the circuit will have gotten 1 e 3 95% of the way, and after 5τ, more than 99%. So, after a few time constants, for practical purposes, the circuit has reached steady state. Thus, the time constant is itself a good rough guide to how long the transient response will take. Of course, mathematically, the steady state is actually an asymptote: it never truly reaches steady state. But, unlike mathematicians, engineers don t sweat over such inconsequential details Examples with Several Capacitors, Resistors or Sources If there are multiple capacitors in the circuit, you might be able to reduce them to a single capacitor using series and parallel rules. If you can, then it s still a first-order circuit. Take, for example, the circuit in Figure 6.7(a). Using the series and parallel rules from Equations (6.5) and (6.6), we can reduce the three capacitors C 1, C 2 and C 3 to a single equivalent capacitor of capacitance C eq = C1C2 C 1C 2 C 3, as shown in Figure 6.7(b). A word of caution: If you can t reduce the capacitors to a single equivalent capacitor (say, because they re not actually in series or parallel), then you probably don t have a first-order circuit, and you can t apply the transient response equation (which only applies to first-order circuits). If there are multiple resistors or sources in the circuit, the workflow to apply the transient response equation (6.7) remains the same. Simply find v(0 ), v( ) and τ = RC using the same methods. We illustrate with two examples. RC circuits with several capacitors RC circuits with several resistors or sources Example 6.6 The switch in the circuit below, having been open for a long time, closes at t = 0. Find an expression for v C (t), the voltage across the capacitor in this circuit.

16 130 CHAPTER 6. CAPACITORS R R C 1 VS C 3 VS C eq = C1C2 C 1C 2 C 3 C 2 (a) original circuit (b) equivalent circuit Figure 6.7: RC circuit with several capacitors t = 0 R 1 R 2 VS v C (t) C R 3 Solution 6.6: Before the switch closes, the switch having been open for a long time, the circuit is in steady state. So the circuit reduces to the following: R 2 v C (0 ) R 3 There s no current through the resistors, so v C (0 ) = 0. Since voltage through a capacitor can t change instantaneously, we have v C (0 ) = 0. After the switch closes, the eventual steady state of the circuit is as follows: R 1 R 2 VS v C ( ) R 3

17 6.5. TRANSIENT RESPONSE OF RC CIRCUITS 131 This is just a voltage divider, so ( ) R2 R 3 v C ( ) = V S. R 1 R 2 R 3 Finally, to find the resistance seen by the capacitor during t > 0, note that the switch is closed for t > 0, and short the voltage source: R 1 R 2 seen from here R 3 Seen from where the capacitor was, we see R 1 in parallel with the R 2 R 3 series combination, yielding so the time constant is R eq = R 1 (R 2 R 3 ) = (R 2 R 3 )R 1 (R 2 R 3 ) R 1, τ = R eq C = (R 2 R 3 )R 1 C (R 2 R 3 ) R 1. Putting it all into the transient response equation, we have v(t) = v( ) [v(0 ) v( )]e t/τ ( ) R2 R 3 = V S R 1 R 2 R 3 ) [ 1 exp = V S ( R2 R 3 R 1 R 2 R 3 [ ( R2 R 3 0 V S R 1 R 2 R 3 ( t (R 2 R 3 ) R 1 (R 2 R 3 )R 1 C )] e t/reqc )].

18 132 CHAPTER 6. CAPACITORS Example 6.7 The switch in the circuit below has been open for a long time and closes at t = 0. Find v C (t) for t > kω t=0 18 V v C (t) 22 µf 20 kω 100 µa Solution 6.7: To find v C (0 ), note that v C (0 ) is the voltage across a capacitor, so v C (0 ) = v C (0 ). Then, what is v C (0 )? Just before t = 0, the switch has been open for a long time, so the circuit would have been in steady state with the switch open: 60 kω 18 V v C (0 ) No current can flow through that resistor (there s an open circuit in the way), so the voltage across the resistor is zero, then v C (0 ) = v C (0 ) = 18 V. Next, to find v C ( ), find the steady-state v C when the switch is closed. Label the node at the bottom ground, and then use nodal analysis on the node on the other side of the capacitor: 60 kω v C ( ) 18 V v C ( ) 20 kω 100 µa

19 6.5. TRANSIENT RESPONSE OF RC CIRCUITS 133 v C ( ) 18 V 60 kω v C( ) = 100 µa 20 kω v C ( ) = 6 V. To find the resistance seen by the capacitor, short the voltage source and open the current source: 60 kω capacitor 20 kω The resistance seen by the capacitor is 60 kω 20 kω = 60 kω 20 kω = 15 kω. 60 kω 20 kω The time constant is then τ = 15 kω 22 µf = 330 ms. Putting this all together, Here s a graph of v C (t): v(t) = v( ) [v(0 ) v( )]e t/τ = 6 (18 6)e t/330 ms (V) = 6 12e t/330 ms (V), t > v C (t) (V) 12 6 t (s)

20 134 CHAPTER 6. CAPACITORS 6.6 Summary They can be polarized or non- Capacitors are devices that store energy in electric fields. polarized. Voltage across a capacitor and current through it are related by i = C dv. (Section 6.1.2) The energy stored in a capacitor is 1 2 CV 2. (Section 6.1.2) 1 Capacitors in series can be replaced by an equivalent capacitor C eq = 1 C 1 1 C 2 1 C n. (Section 6.2.1) Capacitors in parallel can be replaced by an equivalent capacitor C eq = C 1 C 2... C n. (Section 6.2.2) In circuits that have reached steady-state at DC, meaning that all voltages and currents have settled at a stable value, capacitors have like open circuits. (Section 6.4) The voltage across a capacitor can t change instantaneously. (Section 6.5) The time constant of a capacitor is τ = RC. Its units are seconds. (Section 6.5) The general equation for voltages in circuits with one capacitor (or one equivalent capacitor) is v(t) = v( ) [v(0 ) v( )]e t/τ, where v(0 ) is the voltage in question just after t = 0, where v( ) is the steady-state voltage after the change at t = 0 happened, and τ is the time constant. (Section 6.5.2) If a capacitor is charging to a known voltage from zero, this reduces to v(t) = V (1 e t/τ ). (Section 6.5.3) If a capacitor is discharging from a known voltage to zero, this reduces to v(t) = V e t/τ. (Section 6.5.3) When you have multiple capacitors in a circuit, you can often combine them into a capacitor with some C eq. (Section 6.5.5)

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