how to model connection wires in a circuit ajp13

8/10/2019 How to Model Connection Wires in a Circuit AJP13

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How to model connection wires in a circuit: From physical vector fields to circuit scalar

quantities

Guy A. E. Vandenbosch

Citation: American Journal of Physics 81, 676 (2013); doi: 10.1119/1.4812592

View online: http://dx.doi.org/10.1119/1.4812592

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Published by the American Association of Physics Teachers

This article is copyrighted as indicated in the abstract. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to I

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How to model connection wires in a circuit: From physical vector fieldsto circuit scalar quantities

Guy A. E. Vandenboscha)

Department of Electrical Engineering, Katholieke Universiteit Leuven,Kasteelpark Arenberg 10, 3001 Leuven, Belgium

(Received 8 January 2013; accepted 14 June 2013)

Starting from the basic equations of electromagnetism, Maxwells equations, the concepts

of inductive coupling in a loop and capacitive coupling between two pieces of wire areformally explained. Inductive coupling is linked to Faradays law and capacitive coupling to the

Ampere-Maxwell law. Capacitive coupling is also inherently linked to the phenomenon of surface

charges, which has been recently studied thoroughly in the literature, especially in static

situations. It is shown that, when applied to the connecting wires in a circuit at higher frequencies,

simple circuit theory must be significantly modified in order to take into account the effects of the

two types of coupling between the wires. VC 2013 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4812592]

I. INTRODUCTION

Inductive and capacitive coupling are among the most

basic concepts used in electronic circuit theory, taught incircuit courses all over the world. In most textbooks, theseconcepts are defined starting from the topology of the corre-sponding lumped components, the inductor and capacitor,respectively.1,2 Moreover, in many curricula, this basic cir-cuit course is scheduled in parallel with a course on electro-magnetic fields, without really explaining the fundamentallinks between the two. Although this approach is education-ally sensible, unfortunately it does not give the student anunderstanding of the deeper links between couplings in cir-cuits and Maxwells fundamental laws. As long as traditionallow frequency topologies are involved, this shortcomingdoes not pose any practical difficulty. However, at higherfrequencies things become more complicated.

Although the average engineer or physicist knows how tosolve an electronic circuit, many of them get into troublewhen they must consider such a circuit at higher frequencies.The reason is that they are so used to applying circuit theorythat most of them forget that this theory yields only an ap-proximate solution, and the higher the frequency, the worsethe approximation. In principle, the circuit theory solution isonly rigorously correct at zero frequency. Circuit theorydoes not give an answer to questions such as: Does the shapeof a circuit loop affect the current through the loop? What isthe physical voltage drop over an inductor? Why does thecurrent follow the path of a conductor?

Because circuit theory is so easy to understand and towork with, mainstream electronic engineering tries to take

into account the (small) deviations from circuit theory bytranslating them into additional lumped elements in thecircuit, but it is essential to do this in the right way. In thispaper, the link between inductive and capacitive coupling,Maxwells laws, and the modeling of connection wires isanalyzed thoroughly. The line of reasoning for capacitivecoupling is closely related to the lines of reasoning alreadyfound in the literature.36 Here, this reasoning is extended tothe frequency domain. Limitations of circuit theory are alsodescribed in the literature, but only concerning inductivecoupling.7 To the best of the authors knowledge, a generaldiscussion on these limitations, as described here, has notbeen published previously.

II. THE BASIC EQUATIONS

All electromagnetic phenomena are described by

Maxwells laws. In time-harmonic situations (i.e., in thefrequency domain), these equations are

r E jxB; (1)

r B jxelE lJind lJs

jxe rlE lJs: (2)

Here e is the permittivity, l the permeability, and r theconductivity of the medium considered. Meanwhile,J Jind Jsis the electric volume current density, the sumof an imposed source current Js and an Ohmic currentdensity Jind rE induced by the electric field in the con-ductor. The fields are thus generated by the source

currentJs.

III. POTENTIAL AND VOLTAGE

The term potential can easily be introduced by analyz-ing Eq. (1) at zero frequency. At zero frequency, the right-hand side is zero, so

r E 0: (3)

This means that the line integral ofE over any closed loopthe circulation ofEis zero. A vector field, whose circula-tion is zero can be derived from a scalar function; thus, wecan write

E rU; (4)

where U is called the electric potential. The differencebetween the value ofU at an arbitrary point (x,y,z) in spaceand the value at some reference point is defined as the volt-age at (x,y,z).

In principle, the solution of a problem obtained by work-ing with potentials and voltages is rigorous only in staticcases, where the frequency is zero. In practice, the solutionremains very accurate as long as the dimensions of the struc-ture under consideration are much smaller than a radiationwavelength.

676 Am. J. Phys.81(9), September 2013 http://aapt.org/ajp VC 2013 American Association of Physics Teachers 676


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IV. INDUCTIVE COUPLING

Inductors are in fact used as a means to take into accountthe right-hand side of Eq.(1). The idea is to keep on workingwith potentials and voltages but to introduce a correctionrepresenting this term. We start with the integral form ofFaradays law and rewrite it as

c

E dC jx

S

B dS 0; (5)

whereS is an arbitrary surface in space and C is its boundarycurve. This equation says that the integral over a closed loopdoes not equal zero but is related to the changing magneticflux through the loop. For an electronic circuit this meansthat the voltage applied by the source is not completely com-pensated by the voltage drops over the lumped elementsin the circuit. In fact, this is a violation of Kirchhoffs volt-age law.

The usual way to deal with this is to introduce additionallumped elements, inductances, which account for thisclosed-loop integral. The integral is thus represented by addi-tional voltage drops over additional lumped elements (see

Fig.1), and Eq.(5)translates into

Vs RIjx/tot 0; with /tot /inc /self; (6)

with/inc the incident flux due to other circuits, and /self theflux generated through the loop of the circuit by the currentflowing through the circuit itself. (Note that B in Eq. (5) isthe total magnetic field, which is due both to the current inthe loop itself and to any other nearby currents.)

Using the well-known relation between current and flux,Eq.(6)becomes

Vs jxLincIm R jxLselfI; (7)

withLincthe mutual (incident) inductance between the othercircuit with current Im (for simplicity only one other circuitloop is assumed, but the generalization is straightforward),

and Lself the self inductance of the loop under consideration.In textbooks, indeed, the concept of inductance in manycases is explained starting from the lumped componentinductor, where the winding of the wire invokes a strongconcentration of the inductive effect at the inductor. Fromthese textbooks, it is hard to grasp the general nature and thedeep physical meaning of the inductive effect.

It has to be emphasized that the voltage drop over aninductance is by no means a physical voltage drop, i.e., a

voltage drop to be calculated as the integral of the electricfield inside this lumped component. This explains an appa-rent paradox in circuit theory: how one can see a voltagedrop of jxLIover a component that is ideally made of per-fectly conducting material, with electric field zero inside?

It also has to be emphasized that Eq.(5) does not provideany information on where exactly in the loop an inductanceshould be placed. In practice, the position of this additionallumped element in the circuit always has to be chosen care-fully so as not to make fundamental errors. In the case of asingle loop, the positioning of the inductance is arbitrary(see Fig.1).

Because the flux through a loop for a given current Iincreases with increasing loop area, loop sizes are the main

determining factor when considering inductive coupling.

V. CAPACITIVE COUPLING

In circuit theory (at low frequencies), currents flowing in acircuit are independent of the shape of the circuit and of thesurroundings of the circuit. They depend only on the lumpedcomponents and on their connection topology. In this sec-tion, the proof of this very logical intuitive characteristic of acircuit will be formally linked to Maxwells laws by analyz-ing the charge and current distribution in a circuit.

In any circuit, at least two media are of concern: a con-ducting medium 1, forming the circuit wires, and a non-conducting medium 2, surrounding the wires. The net currentand charge everywhere inside the volume of medium 2 arezero (because it is non-conducting), and the current in me-dium 1 is induced. The charge in medium 1 can be analyzedby taking the divergence of Eq.(2), where the volume sourcedensity of electric current is set zero (there is no source cur-rent inside the wires):

r r B r jxe rlE: (8)

It is well known in vector calculus that the divergence of acurl is always zero, so we have simply

r jxe rlE 0: (9)

Equation(9)is valid in both medium 1 and 2. Inside medium1 this yields r E1 0, because the permittivity, conductiv-ity, and permeability are constants. Multiplying again by theconductivity, we can write

r r1E1 r J1 0; (10)

which means that the current flowing into a small volumeinside medium 1 equals the current flowing out of thisvolume. In other words, the volume charge distribution inmedium 1 (the conductor) equals zero. The only possiblecharge is thus a surface charge distribution at the interfacebetween medium 1 and medium 2. Integration of Eq. (10)

Fig. 1. Principle of inductive coupling. Top: physical circuit with incident

flux generated by other circuits; bottom: circuit model including model for

the inductive effects of connection wires.

677 Am. J. Phys., Vol. 81, No. 9, September 2013 Guy A. E. Vandenbosch 677


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over an elementary volume at the interface, as shown inFig.2, yields

n jxe1 r1E1 jxe2E2 0: (11)

If medium 2 is a dielectric with zero conductivity, n is theunit vector normal to the interface from medium 2 to me-dium 1. This can be written as

jxe1 r1

r1 r1E1n

jxe1 r1

r1 J1n jxe2E2n: (12)

From Eq. (12), several properties can be easily derived, asdescribed in the following subsections.

A. Independence of a dc current on the circuit shape andsurroundings

At zero frequency, Eq.(12)yields

J1n 0; (13)

which means that no current flows to the interface. The sur-face charge distribution at the interface is thus constant in

time, but not necessarily zero. Applying Eq. (10)in combi-nation with Eq. (13) to a conducting wire segment of me-dium 1, we see that the entire current flowing into thissegment flows out of this segment at the other side. Thismeans that the current is constant along the length of thewire. In other words, the charge has no effect on the ampli-tude of the current in medium 1. One can wonder, what isthe purpose of the charge distribution at the surface? Thisquestion is best answered by considering the simplest possi-ble circuit, a voltage source feeding a resistor (Fig. 3, top).Here the (surface) charge distribution at the boundary of thewire has two functions:

Inside the wireto point the electric field in a direction

parallel to the wire and with the proper magnitude, whichresults in a current flowing in the proper direction insidethe wire in such a way that the current is conserved;

Outside the wireto generate the proper voltage dropbetween the top wire and the bottom wire connectingsource and resistor. This voltage drop is VRVsRIR.The ratio between this voltage and the total charge (inte-grated surface charge distribution) on the top wire (Q)and bottom wire (Q) gives rise to the definition of theconcept capacitance, as it is explained in classicaltextbooks.

These roles of the surface charge have already been stud-ied thoroughly, mainly in the electrostatic case. Interestedreaders can find more information in the literature.

35

This surface charge distribution can be considered as con-

sisting of two parts: The average over the circumference of the cross section of

the wire, which by definition does not vary (spatially) overthis circumference. Its integral over the length of the topand bottom wire yields Q and Q for the top and bottomwire, respectively. It is this charge that is mainly responsi-ble for producing the correct voltage drop between thetwo wires;

The deviation with respect to this average, which doesshow a spatial variation over the circumference of thewires cross section. For simple cross-sectional shapes,this part is essentially a dipole-type charge separation. Theintegral of this part over the circumference is by definitionzero, so that it generates a negligible field outside the

wire. It just helps to point the electric field in the correctdirection inside the wire, with the correct magnitude.

The surface charge that has been built up at the interfacethus generates the force keeping the current within the wire,pushing it in the right direction. Because this surface chargedistribution adapts itself automatically to the shape of thecircuit and the surroundings, as soon as dc equilibrium isreached the current flowing in the circuit is independent ofthe circuit shape and the surroundings, and always equals

IRVs/R. Note that the distribution of the charge over thesurface may be quite complex.4,5 However, from a (quasi-)static circuit point of view, only the total charge on the topand bottom wires is important, and this can be calculated

fromQCVs.

B. Dependence of an ac current on the circuit shape andsurroundings

If the frequency does not equal zero, Eq. (12) can bewritten as

J1n jxe2r1

jxe1 r1E2n; (14)

which means there is a current to the interface proportionalto the field just external and normal to the interface. TheFig. 2. Relation between the two normal components of the electric field.

Fig. 3. Principle of capacitive coupling. The charges deposited at the surfa-

ces of the top and bottom conductors (top figure) are modeled by a capacitor

(bottom figure).



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surface charge distribution is thus not constant in time. Thisaffects the current distribution in medium 1. Considering awire segment of medium 1, the current flowing into this seg-ment partially flows out of this segment at the other side, andpartially flows to the surface of the segment. In what follows,we will only consider cases where jxe1 r1 (for metals,this is the case up to frequencies in the visible range),yielding

J1n jxe

2E

2n: (15)

The portion of the current flowing to the surface is deter-mined by the electric field just outside and normal to the seg-ment surface. Integrated over the circumference of the crosssections and lengths of the top and bottom wire in Fig.3, thisportion of the current also causes a change of the surfacecharges Q and Q on these pieces of wire. In turn, thechange of these charges produces a change of the voltagedrop between the two pieces of wire. Assuming that the twopieces of wire are still at potentials that are constant overtheir volume, but vary now in time in the frequency domain,this gives rise to the following analysis of the circuit. Therelation between current through R and voltage over R is

IR

(t)VR

(t)/R. Because the potential on the wires is constantover their volume, this is also the voltage drop between thewires, i.e., VR(t)Vs(t). This voltage is linked to a totalchargeQ and Q, which takes care of the proper genera-tion of electric field distribution between top and bottomwires, with CVs(t)Q(t). Differentiating with respect totime yields

CdVst

dt

dQt

dt ICt: (16)

This last current is the current given in Eq. (15) inte-grated over the wire surfaces, flowing to or from thesesurfaces within the top and bottom wire, respectively. In

fact, this is a violation of Kirchhoffs current law. Theusual way to deal with this is to introduce an additionallumped element, a capacitor, that accounts for this changein current (see Fig. 3). In textbooks, the concept of capaci-tance in many cases is explained starting from the lumpedcomponent capacitor, where the small distance betweenthe capacitor surfaces results in a very strong concentrationof the capacitive effect at the capacitor. From these text-books it is hard to grasp the general nature and the deepphysical meaning of the capacitive effect in circuits. Thecurrent in Eq.(16)is thus identical to the current flowing toa capacitor C. This C is the capacitance between the twopieces of wire.

It is evident from this line of reasoning that any pair

of two conductors shows this capacitive phenomenon,not only lumped capacitors. The only difference is themagnitude of the capacitive couplingin lumped capaci-tors it is very large; between pieces of wire it is normallysmall, as long as we are dealing with sufficiently low fre-quencies. The fundamental cause of capacitive couplingis thus the necessary change in Q for a Vs changing intime, since this Q is responsible for the correct voltagedrop (integrated electric field) between the top andbottom wires.

It is well known that the value of C depends stronglyon the exact geometry of the top and bottom wires in thisspecific simple circuit.

VI. INDUCTIVE1 CAPACITIVE COUPLING

In the previous sections, we have studied inductive andcapacitive coupling due to the connection wires in a circuit.Inductive coupling is linked to Faradays law, capacitivecoupling to the Ampere-Maxwell law. The couplings are dis-tributed effects and are not localized. The separationbetween the two types of coupling was possible because wemade some approximations in comparison with reality.While deriving the mechanism of inductive coupling, we

assumed that the current flowing through the loop is constantalong its length (Fig. 1). While deriving the mechanism ofcapacitive coupling, we assumed that the voltage dropbetween the two pieces of wire is constant, whatever theintegration path between the two wires (Fig. 3). Theseassumptions are only rigorously true at zero frequency (forstatic situations). In the time-harmonic regime, theseassumptions degrade with increasing frequency. Actually, itis evident that in practice the two forms of coupling alwaysoccur simultaneously. This complicates things considerably.

Consider the same circuit, a resistorR fed by a voltage Vs(Fig.1 top or Fig.3 top). The impedance of this circuit, tak-ing into account all couplings, is of course not just the resis-tor R, but is in general a function of frequency that can bewritten as a Taylor series,

Zx R A1xA2x2 A3x

3 ; (17)

where the coefficientsAare uniquely determined by the geom-etry of the circuit, i.e., the actual sizes and shapes of the con-nection wires. The fundamental physical interpretation of theimpedance function is that it incorporates the causal wavebehavior of the electromagnetic field coupled back to the input.It takes time for a signal to reach part of the circuit located far-ther away in space, to interact there with the local geometry ofthe circuit (which depends on the local inductive and capaci-tive coupling there), and to travel back to the input. For sinu-soidal oscillations, the distance relative to the wavelengthchanges with frequency. This means that the phase delay ofany response increases with increasing frequency. Thus thetotal impedance, which is a consequence of a superposition ofresponses, each with its own phase delay in terms of electricaldistance, changes with frequency. In the low-frequency Taylorseries expansion of the impedance, higher order terms becomeimportant with increasing frequency so they must be taken intoaccount. At dc, only the resistor is important.

We have seen that inductive and capacitive coupling canbe modeled by lumped elements L and C. The question nowis how to place these two elements when we know that thetwo types of coupling always occur together. What is thecircuit topology? Is it as in Fig. 4(a) (top), as in Fig. 4(b)(bottom), or something else?

The input impedances of the two circuits in Fig. 4 are (fortop and bottom, respectively)

Za parallel circuit of 1

jxCand R jxL

R jxL

1 jxCR jxL; (18)

Zb parallel circuit of 1

jxCandR; put in series withjxL

R

1 jxCRjxL: (19)



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Expressed as a Taylor series around frequency zero, usingthe expression 1=1 x 1 x x2 , we obtain

Za RjxLR2

C x2

RC2LR2

C ; (20)

Zb R jxL R2C x2RCR2C : (21)

As we can see, the first two terms are identical, implying thatinductive and capacitive effects originating from the connec-tion wires in a circuit can be taken into account in circuittheory to the first order by simply introducing the corre-sponding global inductor in series and capacitor in parallel,no matter where they are located. Global here refers to thetotal loop of the circuit, and the total capacitance betweenthe top and bottom wire. If the global inductance and capaci-tance are correctly introduced, it can be proven that thecorrect coefficient A1 is obtained. The proof is beyond the

scope of this paper.Equations (20) and (21) provide us with a criterion to

judge whether inductive or capacitive coupling is moreimportant. In the case L=R RC, inductive coupling ismore important; in case L=R RC, capacitive coupling ismore important (in this simple circuit). It is seen that thiscriterion depends on the load R, not only on the geometry ofthe circuit, which is reflected in L and C. This illustrates avery well-known fact: in high-impedance circuits, mainlycapacitive coupling has to be taken into accountcouplingas a consequence of charge distributions; in low-impedancecircuits, mainly inductive coupling has to be taken intoaccountcoupling as a consequence of current distributions.

However, as soon as the distributed nature over the con-

nection wires itself starts to be important, this simple correc-tion procedure also degrades. In general, neitherof the twomodels of Fig. 4 provides the correct second-order term.Fundamentally they are both wrong. The reason is that thecorrect treatment would depend on the exact geometry of thecircuit in space. The actual proof, based on a rigorous vectorfield analysis, is very advanced and also beyond the scope ofthis paper. However, a revealing glimpse can be given, firstfrom a mathematical point of view, then from a physicalpoint of view.

How to obtain the second-order term from a mathematicalpoint of view? It can be reconstructed by taking into accountexplicitly the distributed nature of the inductive and

capacitive effects. The simplest step in this regard is to splitthe inductor and/or capacitor in two, as in Fig.5. Note that inthe top figure, the capacitor is not really split up, and in thebottom figure the inductor is not really split up.

The same procedure can be followed to calculate the inputimpedances of these circuits as for the circuits in Fig. 4. Thecalculation is lengthy, but straightforward. The results are

ZaRjxLR2Cx2R2L2RCR

2C2 ; (22)

ZbRjxLR2Cx2R2LC2sR2C2 ; (23)

with

L L2s L2R; (24)

C C2s C2R: (25)

First of all, note that the first-order terms are identical tothose in Eqs. (20) and (21). It can be proven that this is adirect consequence of Eqs. (24)and(25). Since for a givencircuit the second-order behavior is (of course) uniquelydetermined, it is seen that the splitting up ofL and C cannotbe done freely. Equations(17),(22), and(23)are identical to

the second order when

R2L2RC R2C2 R2LC2S R

2C2 A2; (26)

which is equivalent to

L2RC2s C2R L2s L2RC2s

A2=R R2C2=2 (27)

and

L2RC2R L2sC2s: (28)

So the conclusion is that the circuit approximation can bemade accurate up to the second order, basically by splittingup the inductor or the capacitor. Note that from a

Fig. 4. Simultaneous inductive and capacitive coupling. Which is the correct

model: top (a) or bottom (b)?

Fig. 5. Models accurate up to second order, in order to model more accu-

rately the distributed character of inductive and capacitive coupling. Top:

with the inductance split in two; bottom: with the capacitor split in two.



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mathematical point of view, either the inductor or the capaci-tor can be split; it is formally always possible to get the

correct second-order behavior.Now let us ask the interesting question, how is thissplitting linked to the physical geometry of the circuit? Theconnection can be illustrated by considering two distinctgeometries, shown in Fig.6.

For the top circuit in Fig.6, because the two wires comevery close to each other in the middle, the effect of thecapacitive coupling tends to dominate over the inductiveeffect. It is logical to consider Fig.5(a)as the correct physi-cal model for this situation. In many cases, if the two remain-ing loops at the left and right of the capacitor are sufficientlysmall, the inductive effect can even be completely neglected.

For the bottom circuit in Fig.6, the wires are far apart inthe middle, making the capacitive coupling small there. The

main contributions to capacitive coupling are found near thevoltage source and the resistor. Also, the larger distancebetween the wires automatically increases the loop area. It istherefore logical to consider Fig.5(b)as the correct physicalmodel for this situation.

A splitting based on physical insight is intrinsically betterthan a purely mathematical one, because it automaticallydelivers coefficients for higher orders that are closer to thereal valuessomething that is not guaranteed by the purelymathematical approach.

The basic technique of splitting up the inductor and capac-itor, which has been demonstrated here to the second order,can be extended to higher orders for increased accuracy.Again, the proof is advanced and beyond the scope of this

paper.In essence, circuit theory is able to take into account theeffect of the connection wires in a circuit by distributing theinductive and capacitive couplings over several inductorsand capacitors. The higher the order of accuracy required,the more inductors and capacitors are needed.

VII. CONCLUSION

In this paper, the basics of inductive and capacitive cou-pling have been reviewed in the frequency domain. This isdone from a physical point of view, explaining the nature ofthese phenomena, rather than consistently using rigorousmathematical formulations. It is shown that inductive andcapacitive coupling are distributed phenomena, not necessar-ily linked to physical lumped components. If not properlymodeled, these effects cause the two basic laws of circuittheory, Kirchoffs current and voltage laws, to break down athigher frequencies.

a)Electronic mail: [email protected]

1N. P. Cook,Electronics, a Complete Course (Pearson-Prentice Hall, Upper

Saddle River, New Jersey, 2004).2

T. L. Floyd, Principles of Electric Circuits (Pearson-Prentice Hall, Upper

Saddle River, New Jersey, 2007).3

M. A. Heald, Electric fields and charges in elementary circuits, Am. J.

Phys.52(6), 522526 (1984).4

N. W. Preyer, Surface charges and fields of simple circuits,Am. J. Phys.

68(11), 10021006 (2000).5

R. W. Chabay and B. A. Sherwood, Matter & Interactions II: Electric &

Magnetic Interactions(Wiley, New York, 1995).6

B. A. Sherwood and R. W. Chabay, A unified treatment of electrostatics and

circuits .7

A. Zozaya, On the nonradiative and quasistatic conditions and the limita-

tions of circuit theory,Am. J. Phys.75(6), 565569 (2007).

Fig. 6. Two distinct geometries for the basic circuit considered.



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