Physics
Capacitors in Series and Parallel
When capacitors are connected in series, their total capacitance decreases, and the voltage across each capacitor is the same. In contrast, when capacitors are connected in parallel, their total capacitance increases, and the voltage across each capacitor is also the same. These configurations are important in understanding the behavior of capacitors in electronic circuits.
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12 Key excerpts on "Capacitors in Series and Parallel"
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Capacitors in Parallel and in Series When there is a combination of capacitors in a circuit, we can sometimes replace that combination with an equivalent capacitor —that is, a single capacitor that has the same capacitance as the actual combination of capacitors. With such a replacement, we can simplify the circuit, affording easier solutions for unknown quantities of the circuit. Here we discuss two basic combinations of capacitors that allow such a replacement. Capacitors in Parallel Figure 25.3.1a shows an electric circuit in which three capacitors are connected in parallel to battery B. This description has little to do with how the capacitor plates are drawn. Rather, “in parallel” means that the capacitors are directly wired together at one plate and directly wired together at the other plate, and that the same potential difference V is applied across the two groups of wired-together plates. Thus, each capacitor has the same potential difference V, which produces charge on the capaci- tor. (In Fig. 25.3.1a, the applied potential V is maintained by the battery.) In general: When a potential difference V is applied across several capacitors connected in parallel, that potential difference V is applied across each capacitor. The total charge q stored on the capacitors is the sum of the charges stored on all the capacitors. Capacitors connected in parallel can be replaced with an equivalent capaci- tor that has the same total charge q and the same potential difference V as the actual capacitors. When we analyze a circuit of capacitors in parallel, we can simplify it with this mental replacement: (You might remember this result with the nonsense word “par-V,” which is close to “party,” to mean “capacitors in parallel have the same V.”) Figure 25.3.1b shows the equivalent capacitor (with equivalent capacitance C eq ) that has replaced the three capacitors (with actual capacitances C 1 , C 2 , and C 3 ) of Fig.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Multiple connections of capacitors behave as a single equivalent capacitor. The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. Capacitors can be arranged in two simple and common types of connections, known as series and parallel, for which we can easily calculate the total capacitance. These two basic combinations, series and parallel, can also be used as part of more complex connections. The Series Combination of Capacitors Figure 8.11 illustrates a series combination of three capacitors, arranged in a row within the circuit. As for any capacitor, the capacitance of the combination is related to the charge and voltage by using Equation 8.1. When this series combination is connected to a battery with voltage V, each of the capacitors acquires an identical charge Q. To explain, first note that the charge on the plate connected to the positive terminal of the battery is +Q and the charge on the plate connected to the negative terminal is −Q . Charges are then induced on the other plates so that the sum of the charges on all plates, and the sum of charges on any pair of capacitor plates, is zero. However, the potential drop V 1 = Q/C 1 on one Chapter 8 | Capacitance 355 capacitor may be different from the potential drop V 2 = Q/C 2 on another capacitor, because, generally, the capacitors may have different capacitances. The series combination of two or three capacitors resembles a single capacitor with a smaller capacitance. Generally, any number of capacitors connected in series is equivalent to one capacitor whose capacitance (called the equivalent capacitance) is smaller than the smallest of the capacitances in the series combination. Charge on this equivalent capacitor is the same as the charge on any capacitor in a series combination: That is, all capacitors of a series combination have the same charge. This occurs due to the conservation of charge in the circuit.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
626 CHAPTER 25 CAPACITANCE When we analyze a circuit of capacitors in parallel, we can simplify it with this mental replacement: Capacitors in Parallel and in Series When there is a combination of capacitors in a circuit, we can sometimes replace that combination with an equivalent capacitor— that is, a single capacitor that has the same capacitance as the actual combination of capacitors. With such a replacement, we can simplify the circuit, affording easier solutions for unknown quantities of the circuit. Here we discuss two basic combinations of capacitors that allow such a replacement. Capacitors in Parallel Figure 25-8a shows an electric circuit in which three capacitors are connected in par- allel to battery B. This description has little to do with how the capacitor plates are drawn. Rather, “in parallel” means that the capacitors are directly wired together at one plate and directly wired together at the other plate, and that the same potential difference V is applied across the two groups of wired-together plates. Thus, each capacitor has the same potential difference V, which produces charge on the capacitor. (In Fig. 25-8a, the applied potential V is maintained by the battery.) In general: Figure 25-8 (a) Three capacitors connected in parallel to battery B. The battery maintains potential difference V across its terminals and thus across each capacitor. (b) The equivalent capacitor, with capacitance C eq , replaces the parallel combination. V +q 3 V – + Terminal C 3 B (a) –q +q V + – (b) C eq B Terminal –q 3 +q 2 –q 2 C 2 V +q 1 –q 1 C 1 V Parallel capacitors and their equivalent have the same V (“par-V”). (You might remember this result with the nonsense word “par-V,” which is close to “party,” to mean “capacitors in parallel have the same V.”) Figure 25-8b shows the equivalent capacitor (with equivalent capacitance C eq ) that has replaced the three capacitors (with actual capacitances C 1 , C 2 , and C 3 ) of Fig.
eBook - ePubMeasurement, Instrumentation, and Sensors Handbook
Electromagnetic, Optical, Radiation, Chemical, and Biomedical Measurement
- John G. Webster, Halit Eren, John G. Webster, Halit Eren(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
Figure 28.11c .28.5.1 Series and Parallel Connections
The formulae for series and parallel connection of capacitors can be obtained from the impedances of such connections as shown in Figure 28.12a and b. For the series connection, the impedances are added such that1=s C1+sC 11+ ⋯ +sC 21sC n(28.20) where C 1 , C 2 , …, C n are the capacitance of the capacitors connected in series.Thus, the equivalent value of the capacitances in series connection can be expressed by
The final capacitance value will always be smaller than the smallest value. In a similar way, the equivalent capacitance of parallel connections can be expressed asC ={}1+C 11+ ⋯ +C 21C n- 1(28.21) C =C 1+C 2+ ⋯ +C n(28.22) and the final value of C is always larger than the largest capacitance in the circuit.FIGURE 28.12Series and parallel connection of capacitors: (a) series connection and (b) parallel connection. In the series connection, the final capacitance value will always be smaller than the smallest value of the capacitor in the circuit, whereas in parallel connection, the final value is greater than the largest capacitance.28.5.2 Distributed Capacitance Circuits
Capacitance is inherent whenever an electric potential exists between two conducting bodies, albeit in electric circuits, communication equipment, electronic circuit boards, computer devices, and so on. For example in electric circuits, its effect will be most noticeable in the coils and in the transmission lines. In the case of coils, there are small capacitances between adjacent turns, between turns that are not adjacent, between terminal leads, between turns and ground, etc. Each of the various capacitances associated with the coil stores a quantity of electrostatic energy that is determined by the capacitance involved and the fraction of the total coil voltage that appears across it. The total effect is that the numerous small coil capacitances can be replaced by a single capacitor of appropriate size shunted across the coil terminals. This equivalent capacitance is called either the distributed capacitance or the self-capacitance
- Jerry D. Wilson, Anthony J. Buffa, Bo Lou(Authors)
- 2022(Publication Date)
- CRC Press(Publisher)
FOLLOW-UP EXERCISE. In this Example, suppose instead that the spacing between the plates were increased by 1.00 mm from the normal value of 3.00 mm. Would charge flow onto or away from the capacitor? How much charge would this entail? 16.5 Capacitors in Series and in Parallel Capacitors can be connected in two basic ways: series or parallel. In series, the capacitors are connected head to tail (▼ Figure 16.23a). When connected in parallel, all the leads on one side of the capacitors have a common connection. (Think of all the “tails” connected together and all the “heads” connected together; Figure 16.23b.) 16.5.1 Capacitors in Series When capacitors are in series, the charge Q must be the same on all the plates. To see why this must be true, examine ▶ Figure 16.24. Note that only plates A and F are connected to = + + C 1 2 3 1 + – C 2 C 3 +Q 1 – Q 1 +Q 2 – Q 2 +Q 3 – Q 3 +Q – Q C s Q = Q 1 = Q 2 = Q 3 (Q's equal) C 1 +Q 1 – Q 1 C 2 C 3 +Q 2 – Q 2 +Q 3 Q 1 Q 2 Q 3 – Q 3 C p +Q – Q Q = Q 1 + Q 2 + Q 3 (Q's not necessarily equal) C p = C 1 + C 2 + C 3 + – + – + – (a) Capacitors in series (b) Capacitors in parallel (c) Capacitors in parallel 1 C s 1 C 1 1 C 2 1 C 3 ΔV ΔV ΔV ΔV ΔV ΔV ΔV ΔV ΔV ΔV ΔV = ΔV 1 + ΔV 2 + ΔV 3 ΔV ΔV Q total Q total = Q 1 + Q 2 + Q 3 + ▲ FIGURE 16.23 Capacitors in series and in parallel (a) All capacitors connected in series have the charge, and the sum of the voltage drops is equal to the voltage of the battery. The total series capacitance is equivalent to one capacitor of capacitance C s . (b) When capacitors are in parallel, the voltage drops across the capacitors are the same as the battery voltage. The total charge is the sum of the charges on the individual capacitors. The total parallel capacitance is equivalent to one capacitor of capacitance C p . (c) To visualize a parallel arrangement, think of the charge on each pair of plates. 433 Electric Potential, Energy, and Capacitance the battery.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
25.3.5 Identify that capacitors in series have the same charge, which is the same value that their equivalent capacitor has. 25.3.6 Calculate the equivalent of series capacitors. 25.3.7 Identify that the potential applied to capacitors in series is equal to the sum of the potentials across the individual capacitors. 25.3.8 For a circuit with a battery and some capacitors in parallel and some in series, simplify the circuit in steps by finding equivalent capacitors, until the charge and potential on the final equivalent capacitor can be determined, and then reverse the steps to find the charge and potential on the individual capacitors. 25.3.9 For a circuit with a battery, an open switch, and one or more uncharged capacitors, determine the amount of charge that moves through a point in the circuit when the switch is closed. 25.3.10 When a charged capacitor is connected in parallel to one or more uncharged capacitors, determine the charge and potential difference on each capacitor when equilibrium is reached. Capacitors in Parallel and in Series When there is a combination of capacitors in a circuit, we can sometimes replace that combination with an equivalent capacitor —that is, a single capacitor that has the same capacitance as the actual combination of capacitors. With such a replacement, we can simplify the circuit, affording easier solutions for unknown quantities of the circuit. Here we discuss two basic combinations of capacitors that allow such a replacement. Capacitors in Parallel Figure 25.3.1a shows an electric circuit in which three capacitors are connected in parallel to battery B. This description has little to do with how the capacitor plates are drawn. Rather, “in parallel” means that the capacitors are directly wired together at one plate and directly wired together at the other plate, and that the same poten- tial difference V is applied across the two groups of wired-together plates.
eBook - ePubCollege Physics Essentials, Eighth Edition
Electricity and Magnetism, Optics, Modern Physics (Volume Two)
- Jerry D. Wilson, Anthony J. Buffa, Bo Lou(Authors)
- 2019(Publication Date)
- CRC Press(Publisher)
Figure 16.23b .)16.5.1 Capacitors in Series▲ Figure 16.23 Capacitors in series and in parallel (a) All capacitors connected in series have the charge, and the sum of the voltage drops is equal to the voltage of the battery. The total series capacitance is equivalent to one capacitor of capacitance C s . (b) When capacitors are in parallel, the voltage drops across the capacitors are the same as the battery voltage. The total charge is the sum of the charges on the individual capacitors. The total parallel capacitance is equivalent to one capacitor of capacitance C p . (c) To visualize a parallel arrangement, think of the charge on each pair of plates.When capacitors are in series, the charge Q must be the same on all the plates. To see why this must be true, examine ▶ Figure 16.24 . Note that only plates A and F are connected to the battery. Because the plates labeled B and C are isolated, the total (net) charge on them must be zero. Thus if the battery puts a charge of + Q on plate A, then − Q is induced on B at the expense of plate C, which must acquire a charge of + Q . This charge in turn induces − Q on D, and so on down the line.▲ Figure 16.24 Charges on capacitors in series Plates B and C together had zero net charge to start. When the battery placed + Q on plate A, charge − Q was induced on B; thus, C must have acquired + Q for the BC combination to remain neutral. Continuing this way through the string, we see that all the charges must be the same in magnitude.When all the capacitor voltage drops are added (see Figure 16.24a ), the total must be the same as the voltage of the battery. Thus, in series Δ V = Δ V 1 + Δ V 2 + Δ V 3 + ⋯ . The equivalent series capacitance ( C s ) is defined as the value of a single capacitor that could replace the series combination and store the same charge and energy at the same voltage. Because the combination of capacitors stores a charge of Q at a voltage of V , it follows that C s = Q /Δ V , or Δ V = Q /C s . However, the individual voltages are related to the individual charges by Δ V 1 = Q /C 1 , Δ V 2 = Q /C 2 , Δ V 3 = Q /C 3
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
25.14 For a circuit with a battery, an open switch, and one or more uncharged capacitors, determine the amount of charge that moves through a point in the circuit when the switch is closed. 25.15 When a charged capacitor is connected in parallel to one or more uncharged capacitors, determine the charge and potential difference on each capacitor when equilib- rium is reached. Key Idea ● The equivalent capacitances C eq of combinations of individual capacitors connected in parallel and in series can be found from C eq = ∑ n j = 1 C j (n capacitors in parallel) and 1 C eq = ∑ n j =1 1 C j (n capacitors in series). Equivalent capacitances can be used to calculate the capac- itances of more complicated series – parallel combinations. 25-3 CAPACITORS IN PARALLEL AND IN SERIES 724 CHAPTER 25 CAPACITANCE When we analyze a circuit of capacitors in parallel, we can simplify it with this mental replacement: Capacitors in Parallel and in Series When there is a combination of capacitors in a circuit, we can sometimes replace that combination with an equivalent capacitor— that is, a single capacitor that has the same capacitance as the actual combination of capacitors. With such a replacement, we can simplify the circuit, affording easier solutions for unknown quantities of the circuit. Here we discuss two basic combinations of capacitors that allow such a replacement. Capacitors in Parallel Figure 25-8a shows an electric circuit in which three capacitors are connected in par- allel to battery B. This description has little to do with how the capacitor plates are drawn. Rather, “in parallel” means that the capacitors are directly wired together at one plate and directly wired together at the other plate, and that the same potential difference V is applied across the two groups of wired-together plates. Thus, each capacitor has the same potential difference V, which produces charge on the capacitor.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
KEY IDEA • The equivalent capacitances C eq of combinations of individual capacitors connected in parallel and in series can be found from C eq = ∑ n j=1 C j and 1 C eq = ∑ n j=1 C j . Pdf_Folio:555 CHAPTER 25 Capacitance 555 Equivalent capacitances can be used to calculate the capacitances of more complicated series–parallel combinations. When there is a combination of capacitors in a circuit, we can sometimes replace that combination with an equivalent capacitor — that is, a single capacitor that has the same capacitance as the actual combination of capacitors. With such a replacement, we can simplify the circuit, affording easier solutions for unknown quantities of the circuit. Here we discuss two basic combinations of capacitors that allow such a replacement. Capacitors in parallel Figure 25.8a shows an electric circuit in which three capacitors are connected in parallel to battery B. This description of being in parallel has little to do with how the capacitor plates are drawn. Yes, in the figure all the plates are neatly drawn parallel, but that is not the test for parallel capacitors. Rather, ‘in parallel’ means that the capacitors are directly wired together at one plate and directly wired together at the other plate, and that the same potential difference V is applied across the groups of wired‐together plates. Thus, each capacitor has the same potential difference V, which produces charge on the capacitor. (In the figure, the applied potential V is maintained by the battery.) In general, this is as follows. When a potential difference V is applied across several capacitors connected in parallel, that potential difference V is applied across each capacitor. The total charge q stored on the capacitors is the sum of the charges stored on all the capacitors. When we analyse a circuit of capacitors in parallel, we can simplify it with this mental replacement.
eBook - PDF- David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Capacitors Connected in Parallel Figure 30-7a shows two capacitors connected in parallel. There are three properties that characterize a parallel con- nection of circuit elements. (1) In traveling from a to b, we can take any of several (two, in this case) parallel paths, each of which goes through only one of the parallel ele- ments. (2) When a battery of potential difference V is con- nected across the combination (that is, one terminal of the battery is connected to point a in Fig. 30-7a and the other terminal to point b), the same potential difference V ap- pears across each element of the parallel connection. The wires and capacitor plates are conductors and therefore equipotentials under electrostatic conditions. The potential at a appears on the wires connected to a and on the two left-hand capacitor plates; similarly, the potential at b ap- pears on all the wires connected to b and on the two right- hand capacitor plates. (3) The total charge that is delivered by the battery to the combination is shared among the ele- ments; some charge “pumped” by the battery ends up on C 1 and some on C 2 . With these principles in mind, we can now find the equivalent capacitance C eq that gives the same total capaci- tance between points a and b, as indicated in Fig. 30-7b. We assume a battery of potential difference V to be con- nected between points a and b. For each capacitor, we can write (using Eq. 30-1) (30 -13) In writing these equations, we have used the same value of the potential difference across the capacitors, in accordance with the second characteristic of a parallel connection stated previously. The battery extracts charge q from one side of the circuit and moves it to the other side. This charge is shared among the two elements according to the q 1 C 1 V and q 2 C 2 V. 30-4 Capacitors in Series and Parallel 683 Figure 30-7. (a) Two capacitors in parallel. (b) The equiva- lent capacitance that can replace the parallel combination.
eBook - ePubElectronics
A First Course
- Owen Bishop(Author)
- 2010(Publication Date)
- Routledge(Publisher)
decreasing . Ohm’s Law still applies, so the current through R1 is decreasing. This means that the rate of charge of C1 is decreasing and the voltage across it rises more slowly.The voltage rises more and more slowly until C1 is charged to 6 V. Then there is NO voltage difference across R1 and therefore NO current flows through it. The graph levels out. The capacitor is fully charged.A curve shaped like the graph above is called an exponential curve .The reverse happens when the capacitor is discharged. At first, there is a voltage difference of 6 V across R1, so 600 μA flows out of the capacitor, through R1 to the 0 V line. The voltage becomes less as the capacitor discharges. The voltage drops more and more slowly. When it reaches zero, the capacitor is uncharged.Figure 17.4CAPACITORS IN PARALLEL
Wiring two or more capacitors in parallel is the equivalent of adding together the areas of their plates. For this reason, the effective capacitance is the sum of the individual capacitances. Figure 17.5Example
In the diagram above, the effective capacitance is: The effective series capacitance is always greater than the greatest of the individual capacitances.Self Test
1 If the supply voltage in the circuit in column 2, p. 59 was increased to 10 V and R1 was increased to 18 kΩ, what would be the current into C1 when the switch was first set to B? What would be the final current into C1?2 If C1 was decreased to 470 μF, will the time to charge the capacitor be longer? Or the same? Or shorter?Design Tip — Capacitor Values
Capacitors are made in a range of values that is similar to the range of preferred resistor values (p. 43). Capacitors have wider tolerances than resistors so there is no point in having 24 basic values in the range. Instead, capacitors values are the E12 series:
eBook - ePubEnergy Storage Devices for Electronic Systems
Rechargeable Batteries and Supercapacitors
- Nihal Kularatna(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
However, most high power applications require considerably higher voltages. For example, a car battery pack situation may mean either 12 or 42 V, which means that we need anything from about 5 to 20 elements in series to meet the nominal voltage value. In order to operate at higher application voltages super capacitors are connected in series and parallel configurations to form a module, to meet the DC voltage, energy capability and the total ESR value, which determines the maximum power capability of the bank. Supercapacitors coming out of mass scale production lines are not exactly identical and hence the series-connected supercapacitor stacks develop unequal voltage distributions (Aiguo et al., 2009a, b ; Chunhe et al., 2009 ; Gai Xiao et al., 2008 ; Rengui et al., 2008). Situation could become worse, due to device capacitance is also varying with different DC bias voltages. Voltage distribution in a series stack of super capacitors is initially a function of capacitance. After the stack has been held at voltage for a period of time, voltage distribution becomes a function of internal parallel resistance (leakage current). For example, consider a stack of 20 capacitors initially charged to 50 V. If the cells have identical capacitance, the voltage should divide evenly so that each capacitor charges up to 2.5 V. If the cells have any variation in capacitance, individual cell voltages will vary based on capacitance. The cells with greater capacitance will be charged to lower voltages, and the cells with smaller capacitance will be charged to higher voltages
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