HISTORY OF TECHNOLOGY

The graphical methods of Sumpner,Drysdale and Marchant: solving the

Kelvin equationC.F. Amor, C.Eng. F.I.E.E., M.I.E.R.E.

Indexing terms: History, Instrumentation and measuring science, Measurement and measuring, Electromagnetics

Abstract: The paper commences with the equation derived in 1853 by Kelvin, for circuits containing induc-tance, resistance and capacitance. From this, the history of a graphical analysis is traced from Sumpner in 1888,to Drysdale who in 1910 published details of his planimeter, a manual device to solve LR and CR circuits. Littlewas added to the method, until Marchant extended it, in 1960, to cover all three circuit parameters, L, R and C,by one graphical solution. The methods are useful in explaining circuit operation and examples of use are given.

1 Introduction

Lord Kelvin had mathematically solved the transient con-dition in a circuit containing combinations of inductance,capacitance and resistance in 1853, enabling instantaneousvalues to be calculated. However, to obtain the currentand voltage waveforms, a number of calculations had to bemade and the points plotted on a graph (the Duddelloscillograph was not invented until 1897).

So several engineers devised graphical methods, deriv-ing the various waveforms and providing an insight intocircuit behaviour of some educational value.

One of the first of these appears to have been Sumpnerin 1888, for L R circuits, but his powerful method seems tohave had little impact until Drysdale promoted it in 1910,inventing a mechanism to facilitate curve drawing. Themethod was not extended to the solution of the generalLCR circuit transient until 1960, by Marchant.

These techniques and the order of their proposal aredescribed in greater detail in the remainder of the paper.

2 The Kelvin equation

The father of circuit theory will be generally acknowledgedto be Lord Kelvin, for, in 1853, as Professor WilliamThomson, he described the transient electric current in aseries circuit comprising inductance, capacitance andresistance [1].

In the 18th Kelvin lecture, in 1927, Professor W.E. Mar-chant stated that Thomson had given a clear mathematicalexplanation by solving the differential equation for the dis-charge of a 'condenser' (nowadays a capacitor), showingthe oscillatory character of a discharge, its frequency andrate of decay of oscillations [2, 3].

The equation proposed and solved by Lord Kelvin wasexpressed as

d2q k dq 1+-7 + (1)

where k = galvanic resistanceA = electrodynamic capacity of the discharger

(explained by Marchant as now called selfinduction)

Paper 4559A(S5,S7), first received 8th August 1985 and in revised form 6th May1986The author is with the Design Council, 28 Haymarket, London SW1Y 4SU, UnitedKingdom

IEE PROCEEDINGS, Vol. 133, Pt. A, No. 6, SEPTEMBER 1986

C = electrical capacity of the principle conductor(electrical capacity of the 'condenser')

q = quantity of electricity at time t

and potential energy = - q2/C

(The now familiar form of eqn. 1 is

dt2 L dt CL

(2)

(3)

where R = resistance and L — inductance.)

Solutions were shown to be in the now accustomedexponential and damped/oscillatory form. Kelvin solvedthe equations for oscillatory and nonoscillatory conditionsand, additionally, with only capacitance and resistancepresent. ('If electrical inertia is insensible'), i.e. no induc-tance.)

Kelvin had seen multiple lightning flashes 'in Europe'and suggested that this evidenced an oscillatory phenome-non, similar to that described by the equation. He referredto the value of C for a Leyden phial and used the symbol yfor the quantity of electricity discharged per second. (It isalso interesting to note that Kelvin referred to experimentsby Weber, in which a wet cord was used to form a resistor,varying the cord length to adjust resistance.)

All our subsequent calculations in LCR, CR and LRcircuits are based on the original Kelvin equation of 1853.However, the analytic solution of the equation is only fea-sible with constant inductance, resistance and capacitance,which is not always so in practice. Graphical methodswere therefore evolved to cope with variables.

3 Sumpner's graphical construction

The first graphical method, following from Kelvin's equa-tion, appears to be that described by Mr. (later Dr.)William Edward Sumpner in 1888, in considering thevariation of the coefficients of induction (in iron-coredinductors) and the effect on current waveshape [4].

Sumpner stated that the assumptions of a sinusoidalapplied voltage and a constant coefficient of induction aremore convenient than true. The coefficient of self induction(of a ferromagnetic device) can be deduced from the mag-netisation curve, but cannot be expressed exactly as amathematical function, so that graphical methods are pref-erable to analytical.

387

For a series circuit comprising inductance and resist-ance, then

L % + iR = Edt

(4)

where i = current at time t (Sumpner used c rather than i)E = applied voltage (impressed EMF)

E LLet 70 = - and x = -

from which

(H _ 7 0 -

Jt=~i~(5)

(Again, Sumpner's Co has been changed to 70, to avoidconfusion with capacitance. 70 is the value of currentobtained if there were no self inductance. Sumpner calledT, the time ratio, but this is usually now known as the timeconstant, and he expressed L in secohms (Qs).)

The method is explained by Sumpner's original illustra-tion, reproduced as Fig. 1. The curve of x with i is plotted

time ratio r time tFig. 1 Reproduction from 'Philosophical Magazine and Journal ofScience', 5th series, Volume 25, June 1888, LVII, W.E. Sumpner (plate III,Fig. 3)

in the negative direction, for convenience, to the left of thevertical current axis, from a knowledge of the relationshipbetween L and i.

From initial current and time at F t project horizontallyto Tx on L/R curve and vertically to Q, on 70 curve.Project from Q± horizontally to Rx on current axis. FromPj draw a line parallel to R^, to P 2 , a short time inter-val from Pl (i.e. dt). P2 is the next point on the currenttime curve. This process is repeated to form P3, and so onuntil the required curve is complete.

Sumpner makes a number of comments. First, if appliedvoltage E and resistance R are constant, then 70 is a hori-zontal line cutting the current axis at a point 7?.* Linesjoining R to any point T represent the rate of increase ofcurrent at that point. If L is constant, this will be largeinitially, but continually diminishing with time. If,however, L varies, increasing at first and then diminishing,the current will increase rapidly, at first, then slowly, then

* Note that Sumpner used R for both resistance and to indicate a point on thediagram, corresponding to E/R.

more rapidly, to a maximum again slowly. This is mostmarked if the iron is magnetised beyond saturation andProfessor Silvanus Thompson had observed this in practi-cal experiments.

He also commented that if E is alternating, L will bevariable and have 2 values for a given current, eitherincreasing or decreasing (although the difference may besmall, in some instances).

0.24 0.20 0.16 0.12 0.08time ratio, s

Fig. 2 See Fig. 1, but originally Plate 111, Fig. 4

Fig. 2 shows an example on an unloaded Kapp andSnell transformer. Circuit resistance was 1Q and the differ-ence in L for increasing or decreasing current was small,and so disregarded. E was taken as a sinewave, with aperiod of 0.16 s (the largest curve 70). The current curvesIu 72, 73 and 74 represent the first halfwave, but with dif-ferent initial values. The third is that which will eventuallybe the continuous magnetising current, the iron operatingbelow saturation. ('It has been found best not to allow theiron of transformers to be magnetized beyond saturationpoint', per Gisbert Kapp.)

It is shown that Sumpner's graphical method coveredboth variable applied voltage and varying time constant,in inductive circuits. It is surprising that it has not receivedwider recognition, but possibly this is because it has appar-ently rather lengthy, with the introduction of the variabletime constant. Sumpner does not appear to have extendedthe method to other than LR problems.

4 Drysdale's graphical construction

4.1 Drysdale 's methodIn 1910, Dr Charles Vickery Drysdale, published a modifi-cation of Sumpner's method, and extended it to a numberof first-order linear differential equations. Drysdale con-sidered combinations of friction, inertia and elasticity, orof resistance, inductance, or capacity [5].

He suggested that mass divided by the friction constant,could be termed the time constant of this combination xalthough this term had previously only been employed foran electric circuit having resistance and inductance. Drys-dale also pointed out that the function attains 63.4% of itsfinal value in a time equal to the time constant, a fact now

388 IEE PROCEEDINGS, Vol. 133, Pt. A, No. 6, SEPTEMBER 1986

probably better known than the graphical constructionitself.

Drysdale first demonstrated the graphical constructionfor a curve of velocity and time, with a potential finalvelocity (of a body) being the propulsive force divided bythe friction constant (which it would have attained imme-diately without inertia). The resulting curve is at everyinstant directed to a point, a distance T in front of it, on ahorizontal line at the potential final velocity. (If there is nodriving force, this final velocity will be on the zero axis.)

It was suggested that curves obtained in this manner(and by step-by-step calculation) are not exact owing tothe length of time interval dt. If, however, half of this inter-val is added to the time constant (i.e. T + dt/2), the slope ofthe curve at mid interval is obtained and 'the results arevery close to the theoretical curve'.

Next Drysdale considered the rise of current in aninductive coil (having inductance of L henry and resistanceR ohm), e.g. a field magnet coil. This is, of course, thesimplest version addressed by Sumpner, with an applied(constant) DC supply. He set the ultimate or maximumcurrent at E/R on a graph with current as the vertical andtime as the horizontal axis. Then a line drawn from zero toa length equal to the time constant T on a horizontal lineat E/R represents the initial slope (which would be main-tained if only inductance was present). A point on thisinitial line is marked off at the first time interval dt fromwhich another line is drawn to x + dt on the ultimatecurrent line, and so on from T + 2dt, etc. Thus the com-plete curve is constructed.

Drysdale then showed the curve for falling current,when the coil is short-circuited. Construction is similarwith the ultimate current being, of course, zero (see Fig. 3).

2dt)

Fig. 3 Reproduction from IEE SQJ, June I960, Vol. 30, (120), p. 156(originally Fig. 2), C.F. Amor

The curve for the charge and discharge of a capacitorthrough a resistor was also demonstrated by Drysdale,plotting charge in coulomb vertically, against time hori-zontally. Again, an ultimate charge horizontal line is set atCV coulomb (farad x volt) and the time constant x = CRseconds. (Note that Drysdale used the symbol K for capac-ity and C for current (coulomb per second) in his originalpublication of 1910.)

42 The planimeterThe next step for Drysdale was to design an apparatus toconstruct these curves more readily. He pointed out that,in the general case, the rate of change of the variable y is(ypk — y)/x corresponding to the slope of the curve, whereypk is its ultimate value. Thus the curve of y with time risesor falls, 'always directed to a point on the line representing

the final y, a time T ahead'. 'It is quite simple to make aninstrument which will do this automatically and trace outthe curve'. This instrument Drysdale called a planimeter,although this title is usually now associated with a differ-ent, unconnected, instrument (for area measurement).

The basic mechanism as shown in his book [5] is repro-duced here as Fig. 4.

Fig. 4 Reproduction from book 'The foundations of alternate currenttheory', by C.V. Drysdale, published by Edward Arnold 1910, p. 62

A small metal 'Tee-square', with a slot in the verticalarm, carries a free sliding block Bx on which a swivelling'crutch' C is mounted. Crutch C carries a sharp edgedwheel W which rolls on the paper and marks out the finalcurve. A rod R is attached to the crutch in line with theedge of the wheel, the other end of the rod passing freelythrough a swivelling block B2. Block B2 can be clamped ina groove in the horizontal arm of the Tee-square, which isgraduated, so that the distance between the centre lines ofB2 and the vertical arm can be set to represent the timeconstant t.

Operation of the planimeter was to draw it across asheet of paper, parallel to the time axis, so that the pointP, the junction of the arm centre lines, follows the line ofypk. The wheel then traces the final curve of y; because thewheel is a distance (ypk — y), at any instant, from P andhaving a sharp edge, it can only roll in the direction of therod. This points to block B2, a fixed distance ahead, equalto the time constant x, SO that the wheel traces dyjdt at anyinstant.

Drysdale arranged the planimeter on a parallelogramlinkage so that it always moved across a drawing board,parallel to itself. The tracing point P was mounted at theupper end of the vertical arm, so that the final curve of ywas drawn with its x (time) axis displaced by that distance,below the curve of ypk.

Illustrations in Drysdale's book show examples ofcurves drawn by the planimeter for growth and decay ofcurrent in an inductive-resistive circuit for various timeconstant values, with a rectangular pulse of direct voltageapplied. Solutions were also shown for applied alternatingvoltages, both sinusoidal and nonsinusoidal and withvarious switch-on points. In the latter cases, he pointedout that the graphs clearly demonstrated how the differentinitial current transients, with various switch-on pointsunite to form the steady-state wave. In addition, the

IEE PROCEEDINGS, Vol. 133, Pt. A, No. 6, SEPTEMBER 1986

smoothing effect of the inductance on an irregular wavewas apparent.

In 1910, the planimeter was obtainable from G. CussonsLtd, scientific instrument makers of Manchester.

This can be seen to be similar in form to eqn. 5 and theconstruction proceeds in similar manner, with a numericalintegration at each step to evaluate i'.

Note that, from eqn. 8,

5 After Drysdale

A number of textbooks of electrical engineering sciencesubsequently described graphical methods, similar toeither Sumpner or Drysdale's, but generally not takenfurther than the simple growth or decay of current in anLR (or voltage in a CR) circuit.

For example, Marchant described a fixed-time-constantversion of Sumpner's method for an LR circuit [6].

Wall described a version of Drysdale's construction fora fixed-time-constant LR circuit and notes that the methodis particularly useful where the inductance is a function ofcurrent [7].

Hughes also describes fixed-time-constant examples ofDrysdale's construction for both growth and decay ofcurrent in inductive-resistive and voltage in capacitive-resistive circuits [8]. (Wall further proposed a constructionfor heating and cooling curves, where z is a fixed (thermal)time constant, similar to Sumpner's method [9].)

Other authors have used the general exponential solu-tion to determine points on a curve, where x is fixed andthe final level (e.g. direct voltage) is constant. Benson andHarrison show that the exponential function attains 50%of its final value, relative to a considered point, in a timeequal to 0.693 z and so on. Thus an approximate curvemay be drawn rapidly. While theoretically infinite time isrequired to achieve 100%, 99% is attained at time = 4.6 z[10].

However, all these examples were restricted to a DCsupply with fixed time constant, ignoring the more usefulvariable supply and variable-time-constant possibilities.

6 Graphical solution of the Kelvin equation

6.1 No external supplyA graphical solution of the complete Kelvin equation (eqn.3) was proposed by Professor E.W. Marchant, not until1960 [3], 50 years after Drysdale's book.

Marchant considered the series LRC circuit, with noexternal supply, but the capacitor initially charged to Qo( = CV0) at voltage Vo. When the circuit is closed, on itself,the charge causes current to flow, grow and eventually fallto zero, possibly with some oscillations.

He rearranged the Kelvin equation (eqn. 3) to

dt CR(10)

Lf + iR + = 0dt c

while at time t, the charge on the capacitor

Jo

Marchant proposed that

From eqns. 6, 7 and 8, then

di (»' - 0dt

(6)

(7)

(8)

(9)

and 5t is chosen to have a convenient value, e.g. dt = CR.Marchant evaluated i at the mid point of dt.

The construction is clear from Fig. 5 (reproduced from

0 6t 26tt for i —»

(t+r) for i '-r r+St r+28t

Fig. 5 Reproduction from IEE J. September I960, Vol. 6, (69), letters, p.537 (originally Fig. 1), E.W. Marchant

Marchant's Fig. 1). The actual current (I) curve commencesat t = 0, i = 0 with initial slope Vo/L, to meet / 0 ( = Vo/R)line, at T = L/R seconds, exactly as with Drysdale. Sec-ondly, another curve commences from /0 at T to representi', proportional to capacitor voltage. Marchant demon-strated, first, an oscillatory example (R less than 2L). Whencurrent reaches a peak, i' = i, so di/dt = 0, and thenbecomes negative. By changing resistance only, (to Rgreater than 2L), this forms a nonoscillatory case, theinitial slope (Vo/L) is identical, but the increased resistancecauses i' to decrease more slowly, so that now i neverbecomes negative.

Marchant considered the advantage of his graphicalmethod is that it shows why the circuit changes from oscil-latory to nonoscillatory with increasing resistance.

He stated that Dr. J. Brown of University College,London, had suggested the introduction of i' in themethod. As an afterthought, Marchant pointed out thatthe method can accommodate variable inductance and,hence, variable time constant, which cannot be determinedmathematically. (This variation, of course, effectively incor-porated Sumpner's original method into the LRC con-struction [11].)

6.2 Kelvin equation with applied voltageIt was an obvious extension of Marchant's method toinclude an applied direct or alternating voltage E [12].

Thus the Kelvin equation becomes

4<dt

(11)

Then Eo = E ± Vo and /0 = Eo/R. The sign of Qo dependson polarity of its initial charge, relative to the initialapplied voltage.

If a steady direct voltage E is applied, the method isexactly as described by Marchant, but

I0 = (E±V0)/R (12)

390

(Derivation of capacitor voltage requires careful consider-ation.)

If the applied voltage is variable, let i = 0 (x-axis) be afixed baseline. Suppose E = Em sin cot, then the /0 refer-ence will vary sinusoidally, Fig. 6.

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Construction is easier if a (second) curve 70 is used., dis-placed by time x from the applied wave E, i.e. above the

2.0

'(=L/R)h

0 • 0.5 1.0 1.5 2 X 1 2 ^ 5 \ 3 5t, ms

Fig. 6 Reproduction from January 1961, IEE J. Vol. 7, (73), letters, p.23 (originally Fig. 2), C.F. Amor

Although the first-order method has been referred to asDrysdale's for brevity and to comply with usage, it is sug-gested that this should be correctly described as Sumpner-Drysdale's method.

Now that the use of computers is common, it is unlikelythat the graphical methods will be used on 'real' problems,but the methods may still be useful in understanding cir-cuits.

9 Acknowledgments

The author wishes to acknowledge the assistance of thestaff of the IEE Library and to thank Messrs. Taylor &Francis Ltd, and Edward Arnold Ltd, the present pub-lishers of References 4 and 5, for permission to reproduceFigs. 1,- 2 and 4 and the IEE for the remainder. Also tothank Professor Dance for the loan of his Unpublishedmaterial (some years ago) and special thanks to P.G.Davis, who introduced this student to the method.

eventual i' curve. (This is also helpful in Drysdale's con-struction.)

It is interesting to note the importance of the ratios ofL/R and CR. A graph may be converted to other time-scales, if these ratios are constant.

With Drysdale's construction, dt should be between \and JQ x for reasonable accuracy (and ease ofconstruction). It may be convenient to express time con-stants as fractions of 2rt on a sinusoidal supply.

7 Practical applications

The first application, by Sumpner, was to determine mag-netising currents in inductors and transformers.

Chin and Moyer used an approximate application ofthe construction (probably derived without knowledge ofDrysdale) to investigate controlled rectifier circuits, in1944, in industry [13].

The author also applied Drysdale's construction to anumber of rectifier circuits, including battery charging andcontrolled rectifiers in industrial applications [14]. Inaddition the method was applied to the intermittentthermal rating of components, e.g. transformers, on diffi-cult duty cycles.

Marchant's method, for the LRC circuit, is ratherlengthy and is unlikely to have been applied in 'real' prob-lems. However, it has been used to provide a better under-standing of the circuit.

As illustrated in Section 5, no doubt the major applica-tion of the methods has been educational. In 1947, Pro-fessor H.E. Dance presented a film strip, describingDrysdale's construction, at a Ministry of Educationsummer school for engineering teachers and at an informalmeeting of the Institution's Education Discussion Circle.

Subsequently, at least one lecturer in electrical engineer-ing introduced the topic for several years (HNC course,Bristol College of Technology).

8 Concluding remarks

The history of the graphical constructions and their placein the history of electrical engineering has been described.

10 References

1 THOMPSON, W.: 'On transient electric currents', The London, Edin-burgh and Dublin Philosophical Magazine and Journal of Science(Philos. Mag.), 1853,5, (34), pp. 393-405

2 MARCH ANT, E.W.: 'High frequency currents', J. IEE, 1927, 65, p.977

3 MARCHANT, E.W.: 'A graphical solution of the Kelvin equation',ibid., 1960, 6, pp. 536-537

4 SUMPNER, WE.: The variation of the coefficients of induction', TheLondon, Edinburgh and Dublin Philosophical Magazine and Journal ofScience, (Philos. Mag.), 1888, 25, (157), pp. 469^t72

5 DRYSDALE, C.V.: 'The foundations of alternate current theory'(Edward Arnold, 1910), pp. 48-63

6 MARCHANT, E.W.: 'An introduction to electrical engineering'(Methuen & Co, 1939), pp. 106-113

7 WALL, T.F.: 'Principles of electrical engineering' (George Newnes,1947), pp. 244-245

8 HUGHES, E.: 'Electrical technology' (Longmans, Green & Co Ltd,1969), pp. 115-121,165-167

9 WALL, T.F.: 'Test papers and solutions in electrical engineering*(George Newnes Ltd, 1947), pp. 12-13

10 BENSON, F.A., and HARRISON, D.: 'Electric-circuit theory'(Edward Arnold Ltd, 1975), pp. 240-246

11 MARCHANT, E.W.: 'A graphical solution of the Kelvin equation', J.IEE, 1960,6, p. 649

12 AMOR, C.F.: 'A graphical solution of the Kelvin equation', ibid.,1961, 7, pp. 22-23

13 CHIN, P.T., and MOYER, E.E.: 'A graphical analysis of the voltageand current wave forms of controlled rectifier circuits', Trans. Amer.Inst. Elect. Engrs., 1944, 63, pp. 501-508

14 AMOR, C.F.: 'Graphical analysis applied to simple rectifier circuits'IEE SQJ, 1960,30, pp. 155-160

11 Appendix: Historical notes

Lord Kelvin, previously Sir William Thomson, was Presi-dent of the Institution in 1874, 1889 and 1907, and anHonorary Fellow.

Dr W.E. Sumpner (1865-1940) was assistant in thePhysical Department, Central Institute, South Kensington,in 1888, when he published his method [4]. He was a Lec-turer at City & Guilds, South Kensington, then Head ofElectrical Engineering, Battersea Polytechnic (1894-5) andfinally, principal of Birmingham Technical College until1930. He worked with Ayrton, Record and Phillips oninstruments, and his transformer test method is well

IEE PROCEEDINGS, Vol. 133, Pt. A, No. 6, SEPTEMBER 1986

known. In 1932 he gave the 23rd Kelvin lecture on the tine instruments and designer of optical and electricalwork of Oliver Heaviside. He was a member of the Institu- devices, including an AC potentiometer. He was a Membertion. of the Institution and clearly stated in his book that his

Dr. C.V. Drysdale, CB, OBE, FRSE (1875-1961) based method was an extension of Sumpner's.his book, published in 1910, [5] on his lectures at the Professor E.W. Marchant was Professor of ElectricalNorthampton Institute. He was Director of Scientific Engineering at Liverpool University. He was a HonoraryResearch, Admiralty, 1929-34. An inventor of many scien- Fellow and President of the Institution in 1932.

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