networks of capacitors in a d.c. circuit

International Journal of Electrical Engineering Education 46/4

Networks of capacitors in a d.c. circuitVelimir Labinac,1 Marko Jusup,2 Tarzan Legovic2 and Ljubomir Spiric1

1Department of Physics, University of Rijeka, Rijeka, Croatia2Rudjer Boskovic Institute, Zagreb, CroatiaE-mail: [email protected]

Abstract Suppose that an arbitrary electric circuit with a network of capacitors is connected to a d.c. e.m.f. device. Herein we address several related questions. What is the equivalent capacitance of the network? What is the voltage and charge on each capacitor? How long will it take to charge the network and will all capacitors be charged at the same time? Finally, the most challenging among these questions: how, in fact, are capacitors being charged? We answer these questions using theory and experiment adapted to the undergraduate level.

Keywords capacitance measurement; capacitors; d.c. circuit; Kirchhoff rules; network; quasi-static approximation

Popular introductory physics textbooks do not cover complex networks of capacitors beyond the usual connection in series and parallel. Moreover, standard analysis of a simple RC circuit assumes that one can apply Kirchhoff’s loop rule without an explanation of its validity.1–3 Such treatment (i) ignores the transient effects which occur upon switching on the circuit and (ii) does not lead to understanding of general networks of capacitors. Our article is an effort to clearly and concisely address these issues in a manner appropriate to the undergraduate level.

We propose a simple experiment which is an extension of the standard experiment with an RC circuit from the basic physics laboratory.4 The experiment will serve as a motivation for discussion on d.c. circuits with network of capacitors and associated effects. In essence, the students are given a chance to re-examine this topic and do some substantial and novel work.

Let us consider the electric circuit in Fig. 1(a). The circuit is comprised of a d.c. electromotive force (e.m.f.) device maintaining a constant terminal voltage Vs, resis-tor with resistance R through which the network of capacitors is charged and fi ve capacitors with different capacitance Ci, i = 1, 2, . . . , 5. This is the simplest, but still a nontrivial RC circuit. It is simple enough for easy calculation of the equivalent capacitance and thus for the demonstration of underlying analytical technique appli-cable to more complex networks of capacitors. On the other hand, it is complex enough that common rules for capacitors connected in series and parallel are un -usable. Tricks5 and symmetries in the circuit6 will not be helpful because Ci have, in general, different values.

Such a system, whose physical principles are well known, will result in surpris-ingly complex solution.

Quasi-static approximation

Continuous time-independent currents in circuits with resistors can be calculated using Kirchhoff’s rules. Are we allowed to apply Kirchhoff’s rules in circuits

334 V. Labinac et al.


containing capacitors, where the currents are time-dependent and have a discontinu-ity at the capacitor’s plates? The answer to this question is provided by the quasi-static approximation.7,8 If the following condition is met:

τ >> L/c, (1)

where τ represents the characteristic time interval in which currents and voltages are changing within the circuit, c stands for the speed of light and L is a characteristic size of the circuit, then the Kirchhoff’s rules are valid, roughly, in any time instant t > 3L/c.9 Characteristic time τ of the considered circuit (see Fig. 1(a)) has the same order of magnitude as the circuit’s time constant, τ ∼ RCeqv ∼ 10 s, where the values of the resistance R and equivalent capacitance Ceqv are given.10 Size of the circuit is L ∼ 10 cm, so that L/c ∼ 10−9 s and the condition (1) is fulfi lled.

Let us recapitulate the physical interpretation of condition (1). Assume that steady current is fl owing through an electric circuit which is comprised of the resistor and e.m.f. device, for example, a battery. In such conditions, the charge in the battery, on the battery’s terminals and stationary surface charge density on the circuit wires generate electric fi eld that exerts force on the electrons and ensures the current fl ow (for a nice discussion on the role of surface charge on circuit wires, see Heald, Jackson and Preyer).11–13 Let us suppose that we change the surface charge density at some point within the circuit. This creates an electric fi eld disturbance which will redistribute the charges on the wires and create a new electric fi eld. According to Ohm’s law, the current in the circuit must change as well. Electric fi eld disturbance caused by the initial change in surface charge density will spread through the circuit with the velocity comparable to the speed of light and in time L/c will reach all parts of the circuit. Thus we have described a transient state which occurs in a very short time interval: the circuit approaches a new equilibrium state in a few L/c units of time and during that time the transient currents fl ow through the wires creating a new surface charge density and new electric fi eld.14 After establishing the new equilibrium state, the current through the circuit becomes steady again.15

Charging of a capacitor can be viewed as a discrete sequence of processes, where each process is carried out as described in the previous paragraph. A small change

e

e

g

h

h

fC1 C4

C2 C5

CeqvC3

R RVS

(a) (b)

VS

Fig. 1 Schemes of (a) considered electrical circuit with network of capacitors and (b) its corresponding equivalent circuit.

Networks of capacitors in a d.c. circuit 335


in charge δQ on a capacitor’s plates (plates are conductors!) causes transient events with the duration of the order of L/c. Afterwards a new equilibrium state is achieved and a new steady-state current fl ows through the circuit. The whole process starts from the beginning when the charge on the plates changes again. In this picture, the time between two equilibrium states is of the order of L/c and according to condition (1) the system continuously remains close to equilibrium during the charging of a capacitor. That is, the quasi-static approximation assumes that the charging of capacitors is very slow in comparison to transient processes in the circuit (except at the moment of switching on, when the current rises sharply). Equations of electro-statics and magnetostatics apply to such conditions and we can derive Kirchhoff’s rules for every time instant t > 3L/c.16

Here we have described the charging of capacitors through a series of quasi-static states with steady current and very short transient events (this discussion is analo-gous to Sherwood and Chabay).18 In case that condition (1) is not fulfi lled, the system is in a transient state for a considerable period of time during the charging of a capacitor. Therefore, we cannot apply Kirchhoff’s rules. In order to calculate the electric fi eld and current one must use the full set of Maxwell’s equations with terms that contain time derivatives.9

Equivalent capacitance

After switching the circuit shown in Fig. 1(a), capacitors (labelled by their capaci-tances Ci) acquire positive charges qi = qi(t), i = 1, 2, . . . , 5 (or more precisely, there is a defi ciency of negative charge) on one of their plates at some time instant t. Cor-responding voltages across capacitors are Vi = Vi(t), i = 1, 2, . . . , 5. We assume that capacitors are uncharged at t = 0. In Fig. 1(a) arrows point in the direction of posi-tive currents, which are chosen almost arbitrarily.19

We apply Kirchhoff’s loop rule to the loops efge and fhgf respectively, which yields

− − + =q

C

q

C

q

C1

1

3

3

2

2

0, (2a)

− + + =q

C

q

C

q

C4

4

5

5

3

3

0, (2b)

where

Vq

Cii

i

i

= =, , , ,1 2 5… (3)

for charges qi and voltages Vi has been used. Charges qi and currents charging capaci-tors Ii = Ii(t) are related as Ii = dqi/dt, which leads to

q t I t ti i

t( ) = ′( ) ′∫ d

0, (4)



because capacitors are uncharged at t = 0. Kirchhoff’s junction rule provides addi-tional equation for currents at junction point e, which upon integration, performed as in eqn (4), results in

q q q= +1 2, (5)

where q = q(t) is the total charge passed through the circuit in time period t. Simi-larly, applying Kirchhoff’s junction rule to points f and g in the circuit respectively, we obtain the following equations

q q q q q q1 3 4 2 3 5= + + =, . (6a), (6b)

The solution to the system of eqns (2a), (2b), (5), (6a) and (6b) for charges qi can be written as

q f q ii i= =, , , , ,1 2 5… (7)

where functions fi = fi(C1, C2, . . . , C5), i = 1, 2, . . . , 5 depend only on capacitances Ci in the circuit. The complete expressions for functions fi are given in the Appendix. Note that internal currents between different capacitors, as for example C1, C3, C4 are due to shifting of charge from the plate of one, to plates of other capacitors. The total charge (positive + negative) on the plates enclosed by the dashed line in Fig. 2 equals zero at every instant t > 3L/c.20

Voltage across equivalent capacitor q/Ceqv equals the potential difference between points h and e. We choose the path hfe so that the potential difference is V1 + V4. Taking into account eqns (3) and (7) the voltage can be written as:

q

CV V

q

C

q

C

f q

C

f q

Ceqv

= + = + = +4 14

4

1

1

4

4

1

1

. (8)

Finally, after dividing eqn (8) by q, the reciprocal of equivalent capacitance emerges:

1 1

1

4

4C

f

C

f

Ceqv

= + . (9)

With the knowledge of f1 = f1(C1, C2, . . . , C5) and f4 = f4(C1, C2, . . . , C5) the equi-valent capacitance Ceqv = Ceqv(C1, C2, . . . , C5) of the circuit in the Fig. 1 can be calculated, resulting in expression

C1 C4

C2 C5C3

R VS

Fig. 2 The total charge on the plates within the dashed rectangle equals zero.



C g C C C C C C C C C C C C C C C Ceqv = + + + + + + +( )−3 4 2 3 2 4 3 5 4 5 1 2 1 3 1 5

1 , (10)

where auxiliary function g = g(C1, C2, . . . , C5) is defi ned as

g C C C C C C C C C C C C C C C C C C C C C C C C≡ + + + + + + +1 2 4 1 3 4 2 3 4 1 2 5 1 3 5 2 3 5 1 4 5 2 4 5 . (11)

The described procedure can be generalized to an arbitrarily chosen two-terminal capacitor network in d.c. or a.c. circuit. We are allowed to choose between currents and charges as unknowns to write Kirchhoff’s rules. If capacitors are initially uncharged, it is perhaps more suitable to consider charges as unknowns (as it was done in our example), because the charges on capacitors are proportional to the voltage. Otherwise, in circuits with initially charged capacitors it is more convenient to use currents as unknowns and subsequently calculate charges and voltages according to the relation:

q t q I t ti i i

t( ) − ( ) = ′( ) ′∫0

0d . (12)

In addition, we state two interesting problems which are related to our simple network in Fig. 1(a). The following problem arose from a discussion with students: How should a given set of capacitances be arranged around the network in Fig. 1(a) to reach the minimal or maximal equivalent capacitance? Using experimental values for capacitances which are given10 we have solved this problem in Mathematica 5.0. There are four confi gurations that minimise the equivalent capacitance and also four confi gurations that maximise it. Results are presented in Table 1.

The second problem is the application of computational technique developed in this section to more complicated network. Find the equivalent capacitance between points a and b for networks of capacitors shown in Fig. 3. Directions of positive current are depicted by arrows. Capacitors are initially uncharged and values of capacitances are not equal.

The solutions with source codes (.nb) for either problem are freely available at http://free-ri.t-com.hr/velimirlabinac/docu/papers/paper_01/paper_01.html.

TABLE 1 Four confi gurations that minimise and four that maximise the equivalent capacitance. Capacitances are given in units of μF

Confi g. number

Minimal Maximal

C1 C2 C3 C4 C5 C1 C2 C3 C4 C5

1. 1.95 4.26 6.25 1.10 0.50 1.95 4.26 0.50 1.10 6.252. 0.50 1.10 6.25 4.26 1.95 6.25 1.10 0.50 4.26 1.953. 1.10 0.50 6.25 1.95 4.26 1.10 6.25 0.50 1.95 4.264. 4.26 1.95 6.25 0.50 1.10 4.26 1.95 0.50 6.25 1.10

Ceqv = 1.24 μF Ceqv = 3.26 μF



Charging of a capacitor

Fig. 1(b) shows the equivalent circuit where we replaced capacitor network in Fig. 1(a) with the equivalent capacitor labelled by Ceqv between points e and h. Applica-tion of Kirchhoff’s loop rule to the equivalent circuit yields the following differential equation:

V Rq

t

q

Cs

eqv

d

d− − = 0, (13)

where −Rdq/dt is the potential drop across the resistor with resistance R. The solution of eqn (13) represents the total charge passed through the circuit in time period t, i.e. the instantaneous charge on equivalent capacitor at time t

q t V Ct

RC( ) = − −⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

s eqveqv

1 exp . (14)

The current in the circuit is the time derivative of total charge

I tdq

dt

V

R

t

RC( ) = = −⎛

⎝⎜⎞⎠⎟

s

eqv

exp . (15)

a

a

b

b

C1

C1

C4

C4

C2

C2

C5

C5

C6

C6

C8

C7

C3

C3

Fig. 3 Suggested problems for students. Determine the equivalent capacitance in given electric circuits.



Combining eqns (3), (7) and (14) charges and voltages across capacitors as a func-tion of time can be found. After a suffi ciently long time t >> RCeqv (or more formally, t → ∞) the capacitor network is fully charged, the charge on the equivalent capacitor is q∞ = VsCeqv and the current in the circuit goes to zero. The charges on capacitors in the network become qi = fiq∞ and voltages are Vi = qi/Ci = fiq∞/Ci.

Suppose that fi ≠ 0 which means that currents charging the capacitors are not equal to zero. The rate of charging dqi/dt is in general different for each capacitor. But the percentage of total charge acquired by a single capacitor in a time interval t is

q t

q t

f q t

f q

q t

qi

i

i

i

( )→ ∞( )

=( )

=( )

∞ ∞, (16)

which is the same for every capacitor in any instant t. We conclude that capacitors in the network will be fully charged simultaneously.

Measurements

A classical laboratory experiment on an RC circuit can be extended to incorporate two types of measurement which can be easily performed by students: indirect measurement of equivalent capacitance by applying eqn (17), given below, and direct measurement of voltages across each capacitor using a peak value instrument. For successful completion of the exercise, adequate equipment is listed in Table 2. The experimental procedure is simple and not particularly time consuming:

1 Connect the network of capacitors according to the given scheme (see Fig. 1(a));

2 For reference, measure the equivalent capacitance directly using an appro-priate multimeter;

3 Connect the e.m.f. device and resistor to the network in order to form an electric circuit;

4 Measure equivalent capacitance indirectly measuring current at point e as a function of time;

5 Measure voltages across each capacitor after time t > 5RCeqv using multi-meter’s peak value function.

Students should be informed that peak value function is necessary because capaci-tors are quickly discharged through the multimeter with standard internal resistance ∼10 MΩ. Additionally, prior to all measurements, capacitors should be discharged by shorting the plates.

TABLE 2 List of equipment for laboratory experiment

Equipment

Stabilized voltage rectifi er, resistance decade box, 5 block capacitors, wires with connectors.UT70A Multimeter (enables measuring capacitance, resistance and voltage), Phywe Multirange meter

with amplifi er 07034.00 (for measuring currents)



In order to check the feasibility of the described exercise we have performed it with readily available equipment listed in Table 2. The specifi cation of each element was checked with a UT70A Multimeter (measured value ± instrument error): Vs = (10.04 ± 0.06) V, R = (4.43 ± 0.09) MΩ, C1 = (1.95 ± 0.05) μF, C2 = (0.50 ± 0.02) μF, C3 = (6.25 ± 0.35) μF, C4 = (1.10 ± 0.03) μF, C5 = (4.26 ± 0.25) μF.

Indirect measurement of the equivalent capacitance was made according to the expression

Ct

R

V

RIeqv

s= ( )⎡⎣⎢

⎤⎦⎥

−

ln ,1

(17)

which follows from eqn (15).The results are displayed in Table 3, Table 4 and Table 5 (value ± standard devia-

tion) together with calculated values. In the calculation of equivalent capacitance, the capacitance of connecting wires was ignored.

TABLE 3 Equivalent capacitance calculated according to eqn (10) and obtained by a direct measurement

Computed Ceqv (μF) Directly measured Ceqv (μF)

1.56 ± 0.03 1.60 ± 0.05

TABLE 4 Results of indirect measurement of equivalent capacitance obtained using eqn (17)

t (s) I (μA)Indirectly measured Ceqv (μA)

5.00 ± 0.05 1.10 ± 0.05 1.56 ± 0.0710.20 ± 0.05 0.60 ± 0.02 1.73 ± 0.1335.00 ± 0.05 ≤ 0.02 1.67 ± 0.35

TABLE 5 Calculated and measured values of voltage across charged capacitors. Calculation was made by

applying eqn Vi = fiVsCeqv/Ci, where functions fi are defi ned in eqns (A1a)–(A1e)

Voltage across the capacitor Calculated (V) Measured (V)

V1 6.14 ± 0.10 5.92 ± 0.04V2 7.37 ± 0.12 7.25 ± 0.05V3 1.23 ± 0.07 1.23 ± 0.01V4 3.90 ± 0.10 3.90 ± 0.03V5 2.67 ± 0.11 2.61 ± 0.02



Conclusion

Popular introductory physics textbooks do not cover d.c. circuits with networks of capacitors beyond simple parallel or serial connection. An application of Kirchhoff’s rules in calculating charge and voltage on individual elements of the network is missing. Moreover, no connection is made between charge and potential on conduct-ing surfaces in electrostatics and current fl ow in simple circuits. As a result, students remain unaware of the conditions in which Kirchhoff’s rules are valid. In practice this shortfall manifests itself as a student’s inability to answer even simple questions. For example, what is the relationship between charging times of different capacitors? We have shown that a combination of theory and experiment can rectify the men-tioned shortfall at the undergraduate level. While the considered problem has a rather complicated solution, it is still attainable using basic algebraic manipulations or, more practically, using any mathematical software package as was done in this paper (example written in Mathematica 5.021 is available at http://free-ri.t-com.hr/velimirlabinac/docu/papers/paper_01/paper_01.html). The experimental exercise is rather inexpensive: it requires only readily available equipment and it is not particu-larly time consuming.

Acknowledgements

We thank Igor Zutic for discussions. This research was funded by the Croatian Ministry for Science, Education and Sports.

References

1 D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 7th edn (Wiley, New York, 2005).

2 H. D. Young and R. A. Freedman, University Physics, 11th edn (Pearson Education, San Francisco, 2004).

3 R. A. Serway and J. W. Jewett, Physics for Scientists and Engineers, 6th edn (Brooks/Cole, London, 2004).

4 J. D. Wilson, Physics Laboratory Experiments, 5th edn (Houghton Miffl in, Boston, 1998), pp. 395–399.

5 B. Korsunsky, ‘Three Is a Charm . . . Six Is Not’, Phys. Teach., 43(3) (2005), 186. 6 ref. 3, p. 829. 7 M. H. Nayfeh and M. K. Brussel, Electricity and Magnetism (Wiley, New York, 1985), pp. 391–

395. 8 R. K. Wangsness, Electromagnetic fi elds, 2nd edn (Wiley, New York, 1986), pp. 449–451. 9 N. W. Preyer, ‘Transient behavior of simple RC circuits’, Am. J. Phys., 70(12) (2002), 1187–

1193.10 See section Measurements, this paper.11 M. A. Heald, ‘Electric fi elds and charges in elementary circuits’, Am. J. Phys., 52(6) (1984), 522–

526.12 J. D. Jackson, ‘Surface charge on circuit wires and resistors play three rules’, Am. J. Phys., 64(7)

(1995), 855–870.13 N. W. Preyer, ‘Surface charges and fi elds of simple circuits’, Am. J. Phys., 68(11) (2000), 1002–

1006.



14 W. G. V. Rosser, ‘What makes an electric current fl ow’, Am. J. Phys., 31(11) (1963), 884–885.

15 N. W. Preyer, ‘Surface charges and feedback in simple circuits’, (1999), URL http://galaxy.cofc.edu/pubs/tpt99/

16 Maxwell’s equations in quasi-static limit, when applied to electric circuits with coils, must contain the term ∂B/∂t. The full set of equations can be found in Nayfeh.7 Derivation of Kirchhoff’s rules for circuits with resistances was done by Moreau.17

17 W. R. Moreau, ‘Charge distribution on d.c. circuits and Kirchhoff’s laws’, Eur. J. Phys., 10(4) (1989), 286–290.

18 B. A. Sherwood and R. W. Chabay, ‘A unifi ed treatment of electrostatics and circuits’, (1999), URL http://www4.ncsu.edu:8030/_rwchabay/mi/circuit.pdf

19 Direction of positive current is defi ned to be in the direction of fl ow of positive charge from higher to lower potential.

20 ref. 1, p. 663.21 Wolfram Research Inc., http://www.wolfram.com

Appendix

We solved the system of eqns (2a), (2b), (5), (6a) and (6b) for charges on capacitors using the software package Mathematica 5.0. The solutions for charges are qi = fiq, i = 1, 2, . . . , 5 where q = q(t) is the total charge passed through the circuit in time period t and functions fi = fi(C1, C2, . . . , C5), i = 1, 2, . . . , 5 are defi ned as

f g C C C C C C C C C C C Cf g C C C C C C C1

11 2 4 1 3 4 1 3 5 1 4 5

21

2 3 4 1 2 5 2

≡ + + +( )≡ + +

−

−,

CC C C C Cf g C C C C C Cf g C C C C C C

3 5 2 4 5

31

2 3 4 1 3 5

41

1 2 4 1 3 4

+( )≡ − +( )≡ + +

−

−

,,CC C C C C C

f g C C C C C C C C C C C C2 3 4 1 4 5

51

1 2 5 1 3 5 2 3 5 2 4 5

+( )≡ + + +( )−

,.

(A1a), (A1b),

(A1c), (A1d), (A1e)

Equations (A1a)–(A1e) contain another auxiliary function g = g(C1, C2, . . . , C5), defi ned as

g C C C C C C C C C C C C C C C C C C C C C C C C≡ + + + + + + +1 2 4 1 3 4 2 3 4 1 2 5 1 3 5 2 3 5 1 4 5 2 4 5 . (A2)

Substituting functions f1 and f4 from (A1a) and (A1d) into eqn (9) we arrive at the fi nal expression for equivalent capacitance of circuit in Fig. 1(a)

C g C C C C C C C C C C C C C C C Ceqv = + + + + + + +( )−3 4 2 3 2 4 3 5 4 5 1 2 1 3 1 5

1. (A3)

networks of capacitors in a d.c. circuit

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