a solid-state low-voltage tesla coil...

A solid-state low-voltage Tesla coil demonstratorDonald G. Bruns5195HearthstoneLane, ColoradoSprings,Colorado80919

(Received 18 October 1991; accepted 19 March 1992)

A low-voltage demonstration Tesla coil using a solid-state photovoltaic relay to replace theconventional spark gap has been analyzed and then built. This relay incorporates an isolatedLED to illuminate a silicon photovoltaic stack which drives abidirectional FET. Componentvalues for the inductances and capacitances have been determined theoretically from measuredparameters. Computer simulation by integrating the coupled circuit equations shows excellentagreement with oscilloscope traces. Energy transfer between the primary and secondary circuitsis demonstrated, along with continuous secondary oscillations after the primary circuit isinterrupted. This low-voltage design is easier to build and diagnose than high-voltage Tesla coils.

I. INTRODUCfION

Nikola Tesla invented the Tesla coil late in the nine-teenth century, exploring many high-power variations inhis Colorado Springs laboratory. I They were all basicallyair-cored high-frequency transformers, generating veryhigh voltages. Many of his experiments were complicated,using large coils made with heavy copper wires to conductvery high currents. His high-voltage capacitors used hun-dreds of salt water-filled Leyden jars made from the localManitou Springs mineral water bottling plant. Tesla doc-umented his achievements with multiple-exposure photo-graphs which show his small wooden building filled withcurved sparks up to 40 m in length. Using spark length ishow Tesla often diagnosed his experiments.

Tesla's ultimate goal was to generate high enough volt-ages that he could transmit useful electrical power freelythrough the atmosphere. One contemporary accountclaimed he succeeded in sending enough power to energizea bank of light bulbs 40 km away.2 However, he nevercompleted his final and largest experiment on Long Island,New York, which he designed inside a 60-m-high woodentower. Although lack of funding was the primary reasonthe tower was torn down, in the light of today's knowledge,it never would have succeeded in the manner he eilvi-

191 Am. J. Phys. 60 (9), September 1992

sioned. While Tesla was advanced for his time, he didn'thave electronic diagnostic tools to study his experiments.

Even though Tesla's grandiose plans would not haveworked, we remain fascinated with high-voltage Teslacoils. Generating fiery arcs and lighting fluorescent tubes ata distance are always exciting demonstrations. In the last60 years, instructions for building high-voltage Tesla coilsappeared occasionally in popular magazines, journals,newsletters, and books.3-13 A useful instrument in manyphysics laboratories is the hand-held Tesla coil used toexcite gas discharges and find leaks in vacuum systems.There has also been some interest in using very large Teslacoils to test military aircraft with simulated lightning, 14and using smaller coils to generate electron beams. 15Whileany of these Tesla coils' can. be. experimental subjects, de-tailed measurements to compare with theoretical predic-tions requires sophisticated equipment to deal with highvoltages.

A conventional Tesla coil consists of tuned primary andsecondary circuits. An interrupter in the primary circuitstimulates oscillations from the charge stored in a largecapacitor. The primary circuit interrupter design is criticalto maximize power transfer to the secondary circuit. High-power Tesla coils use variations on rotating spark gaps toextinguish the high-voltage spark,16 a technique that hasn't

@ 1992 American Association of Physics Teachers 191

Rc

high voltageoutput terminal

spark gap Cs

Cp Lp

-

Fig. 1. Tbe classical Tesla coil circuit. A high-voltage source charges acapacitor which discharges across a spark gap, stimulating oscillations inthe primary and secondary circuits.

changed since Tesla used it! This paper replaces that "an-cient" element with modern integrated circuit technology.A BOSFET (Bidirectional Output Switch Field EffectTransistor), while not capable of high powers, has the nec-essary properties which make a demonstration Tesla coilpractical. MOSFETs make up the BOSFET, which arecombined with an optically coupled LED driver in a smallintegrated circuit package, manufactured as a photovoltaicrelay.17 It turns on and off quickly, exhibits a low ON-resistance and a high OFF-resistance, and has the capabil-ity of transmitting alternating currents. In these respects, itis better than mechanical relays and easier to work withthan a spark gap. A Tesla coil with this component caneasily be built and diagnosed using a simple oscilloscope. Alow-voltage Tesla coil adequately demonstrates all the im-portant theories of a high-power Tesla coil. The example inthis paper produces peak secondary voltages of only 20 V,far too small for theatrical demonstrations. The nonlinear-ities and complications which arise from evaluating a haz-ardous high-voltage circuit, such as corona leakage, highacoustic and electrical noise levels, and expensive compo-nents, are absent from this design. This inexpensive, low-voltage circuit provides easily measurable and reproducibleresults.

This paper covers classical Tesla coil theory and uses acomputer simulation to explain the operation of a smallTesla coil. Excellent agreement between theory and exper-iment is reached.

II. THEORY

The classical Tesla coil theory is presented here. Thecircuit elements are described, and the differential equa-tions describing the charge are integrated with a shortcomputer program. These results are compared to experi-mental data in a later section.

Figure 1 shows the classical Tesla coil configuration. Ahigh-voltage source, often ac, usually with moderately highimpedance, charges the primary capacitor Cp up to thespark gap breakdown voltage. The heated air provides alow impedance path as long as a high current flows be-tween the electrodes. In the absence of secondary coupling,the primary current oscillates with a fre9uency wp deter-


mined by the primary circuit components, and decays ac-cording to the spark gap resistance. When the capacitor ispractically discharged and the primary circuit energy isdissipated, the spark is extinguished and the capacitor re-charges. Depending on the spark gap electrode type andseparation, this cycle may operate several times during one60-Hz cycle, or it may only occur once.

When the secondary circuit is included, energy transfersto it at a rate determined by the inductive coupling coeffi-cient k. When all the energy is in the secondary circuit, theprocess reverses, and the energy begins transferring back tothe primary circuit at the same rate. In this case, the op-timum design will have the spark gap extinguish just whenall the energy is in the secondary circuit. The key designparameters of a Tesla coil optimize this energy transfer.

The coupled circuit equations18 are

M a2qs a2qp Rp aqp 2L af2+ ar2+:L ai+ wpqp=0 (1)p p

and

M a2qp a2qs Rs aqs 2r; --af2+af2+r;ai+wsqs=O, (2)s s

where M = k ~(LpLs) is the mutual inductance, wp= 1/ ~(LpCp), and Ws = 1/ ~(LsCs)' With the boundaryconditions at t=O that qs=O, qp=CpVo, aq/at=o, andaq/at=o, the equations have closed solutions only for Rsand Rp equal to zero. The secondary voltage Vs=q/Csconsists of two sine waves beating at the difference fre-quency

wp wp

WB= ~l-k- ~l+k

with amplitudes depending onWhen the coupling is such that

(n+ 1)2_n2k=. ..1 1,

for n=0,1,2..., and ws=wp, the two sine waves beat in sucha way that the maxima occur simultaneously, and the peaksecondary voltage reaches a maximum. This voltage is

Vspeak= Vo*~(L/Lp); all the primary circuit energy hastransferred to the secondary circuit.

If the tuning ratio w/ws is not unity and the resistancesare not zero, then more complicated equations result, butthey can be solved numerically. The larger coupling coef-ficients, represented by small values of n, result in the high-est peak secondary voltages because the transfer of poweroccurs before much energy is dissipated in the resistances.In these cases, families of curves are required to present allthe possibilities of various resistances and tuning ratios;Refs. (18 J and (20) give examples.

(3)

(4)

III. COMPUTER SIMULATION

A short computer program solves the differential equa-tions in the general case of R=ftO. While sophisticatedRunge-Kutta integration or other routines21 would work,the solution is smooth enough that simple step-by-step in-tegration with a step size of P/lOO is adequate, where Pisthe oscillation period. The differential equations solutions

Donald G. Bruns 798

are expected to be accurate, since the nonlinear losses dueto corona leakage and spark gap resistance are absent.

To set up the computer simulation, Eqs. (1) and (2) areintegrated once, then the differentials are changed into fi-nite deltas and the integrals into sums. After a little alge-bra, solving for the circuit current increments leads to

Deltalp= (VO-Rp*lp-SumQp/Cp-M

*Deltals/DeltaT) *DeltaT /Lp

and

Deltals= (-Rs*ls-SumQs/Cs-M*Deltalp/DeltaT)

*DeltaT/Ls.

The computer program first defines the component values,the initial primary voltage, and the integration step size.The coupling coefficient k was used to calculate M. Perfecttuning was assumed, so the values of Wf2and w; were equal.

Inside a loop incremented by Delta , the computer cal-culated the values of Deltalp and Deltals, then the valueswere updated by calculating

Ip=lp+Deltalp,

SumQp=SumQp+lp*DeltaT,

Is=ls+Deltals,

SumQs=SumQs+ls*DeltaT, and

Time = Time + DeltaT. (7)

To simulate the BOSFET turn-off, Rp was changed to1000 0 when the time shown in the oscilloscope photoswas reached.

The time-dependent values of Vp=SumQp/Cp- VOand Vs=SumQs/Cs were plotted in real time. Outputchanges resulting from small changes in any of the param-eters were studied, but the results in this paper are from themeasured values.

IV. COMPONENT VALUE DETERMINATION

The inductance values and the coupling coefficient inthis experiment were calculated from as-built coil dimen-sions. The capacitances were determined from frequencymeasurements. The minimum resistances were determinedfrom theoretical considerations, but the effective resis-tances were used as free parameters in the computer sim-ulation. Table I shows the physical specifications.

The primary and secondary inductances have a length/diameter ratio large enough so they can be calculated bythe Nagaoka formula22 for closely spaced air solenoids;

L _ i2?-n2-- - _~'1,,,~,

where n is the number of turns, r the radius in cm, I is thelength in cm, and L is given in microhenries. The induc-tance error is less than 1% at low frequencies. The primarycoil dc resistance was too low to measure accurately with asimple multimeter, so it was calculated using a wire resis-tance table. The secondary coil's calculated and measuredvalues agreed. The relative impedance change at the oper-ating frequency, however, differs significa,ntly for the two


Table I. Tesla coil physical specifications.

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Specification

Coil radius r

Coil length INumber of turns nCalculated coilinductance L

Measured frequency fdc coil resistance

High-frequencyresistanceEffective totalcircuit resistance RCalculated total

capacitance C

Primary circuit

Secondarycircuit

2.45 cm6.2 cm

67 turns of doubled

# 18 wire127 ILH148 kHz

0.1 !l

1.68 cm21.8cm

805 turns of # 30wire

3100 ILH148 kHz

29 !l

0.23 !l 33 !l

(6) 12 !l 45 !l

0.00911LF 373 pF

(8)

coils. Using the formulas and charts from Ref. 23, the skindepth for copper wire at 148 kHz is approximately 175microns. The high-frequency impedance of # 18 copper islarger by approximately 50%, while for #30 wire, thevalue is practically unchanged. The proximity effects of aclose-wound coil causes a bigger correction, adding about14% to the secondary coil impedance and about 75% tothe primary coil. The final results lead to a primary coilimpedance about 2.3 times, and the secondary coil imped-ance about 1.15 times, the dc value. The minimum primarycoil impedance calculates to 0.23 0, and the secondaryimpedance to 33 O.

The actual effective resistance was even larger, deter-mined by fitting the computer simulation to oscilloscopemeasurements. The primary coil measurement is shown inFig. 2. The primary and secondary circuits were pulsedindividually to determine their resonant frequencies. TheBOSFET impedance drives up the effective resistance morethan expected; their dc resistance was directly measured at1.2 0, but the simulation required 12 0 for the primarycoil. The secondary coil value agreed better, requiring 45 0to match the measurements. The resistances used in thesimulations do affect the decay constants, but do notchange the overall curve shapes. The resonant frequencydetermined by the oscilloscope measurements led to accu-rate capacitance values. The primary circuit is not tunable,so its resonant frequency is defined as wp' With the primaryoscillating and the secondary coil placed about 30 cmaway, its variable capacitor could' be tuned through itsresonant peak. At this distance, mutual coupling betweenthe circuits is small enough not to cause any frequencyshifts, but strong enough to provide a measurable second-ary circuit signal.

The primary capacitance Cp is a single capacitor with anominal O.OI-JLF.component marking. A more precisevalue was determined by measuring the free oscillation fre-quency and letting

Cp= l/wi,Lp (9)

since both Lp and wp could be determined with only a fewpercent error. This led to a determination of 0.0091 JLFforCp, only 10% lower than nominal.

In a conventional Tesla coil, the secondary capacitanceCsis the sum of the top electrode capacitance C; in parallel

Donald G. Bruns 799

(a)

4

3

2

o

-1

-2

- 3

-4

(b) I 0 5 10 15 20 25 30 35 40 45 50

Fig. 2. Primary coil effective resistance measurement. (a) The oscillo-scope photograph, (b) a computer simulation. The vertical axis numbersare volts. The bottom axis numbers are microseconds.

with the self-capacitance C;'. The self-capacitance is esti-mated from the chart in Ref. 24; for single layer solenoids,the self-capacitance depends weakly on the length. Formost typical Tesla coil geometries, the self-capacitance (inpicofarads) equals the diameter in cm. Spherical top elec-trodes contribute a capacitance equal to Cr=41T€r,where ris the top electrode radius. For this demonstration Teslacoil, all of these number are too small to produce a low-enough frequency for BOSFET operation, so a large tun-able capacitor was added in parallel. The secondary circuitcapacitance Cs was set by simply tuning that circuit to theprimary's resonant frequency.

The coupling coefficient k is calculated for typical Teslacoil geometries using the formula (Ref. 25) for concentriccoils,

(nanohenries ) ,

(10)

where

(11 )

np and ns are the number of turns per centimeter on thecoil, and

where r=x2+A2, s is the center to center distance, a is theradius of secondary coil, A is the radius of primary coil(A> a), 2m, is the length of secondary coil, 2m2 is thelength of primary coil, and all measurements are in centi-meters. Knowing Ls and Lp determines k.

While this formula appears complicated, it is the onlyformula I found which converges for typical Tesla coilgeometries and k values. Gray's formula (Ref. 26), whilesimpler, does not converge for close coupling. The coeffi-cient k is plotted in Fig. 3 over the range used in thisexperiment. The coupling near s= 10 cm slopes about 0.05per cm, so a significant error occurs for inaccurate mea-surements. Nevertheless, the computer simulation showsexcellent agreement with the predicted value of k for eachexperiment.

V. BOSFET SWITCHING EXPERIMENT

Preliminary experiments were conducted to assist in thedesign and evaluation of this Tesla coil demonstrator andto provide measurements for comparison. with the com-


I 0.5 - -~-

I " ..._.__I 0.4I

I Coupling 0.3ICoefficient

I k 0.2

I

I 0.1

L"

I

o ~ 5 10 ~-1~ :0

Center Separation (em). J

Fig. 3. Calculated coupling coefficient k, using the Tesla coil parameters

from Table I and the equations from Ref. 25.

puter simulation. The BOSFET's drive circuit design was

Donald G. Bruns 800

Fig. 4. Tesla coil primary driver schematic. Two dual BOSFET units are

used in parallel to replace the spark gap with a low-resistance interrupter.

explored for the fastest switching times. The Tesla coiloperating frequency was designed around this switchingtime.

A switching circuit based on the PVR1300 solid-staterelay with internal BOSFET replaced the spark gap in thisdemonstration. An isolated LED inside the PVR1300 illu-minates a silicon photovoltaic stack surrounded by a re-flective cavity to increase efficiency. The voltage generatedby the stack drives internal circuits connected to theBOSFET gate. Since this circuitry is not tied to an externalreference voltage, the device can be made to pass alternat-ing currents. When turned on, the BOSFET acts as anearly pure resistance with practically no offset voltage.When turned off, the device offers over 100-MO resistance.The turn on and off are essentially limited only by the LEDdrive circuits, so 100 kHz operation is possible. No othersolid-state device has this combination of properties oversuch a wide frequency range.

The Tesla coil master timing was designed around thesimple integrated circuit pulser shown in Fig. 4. While theBOSFETs are OFF, the primary capacitor charges throughthe 2700-0 resistor Re. After the O.Ol-ILFcapacitor Cp ischarged, the NE555 pulser triggers the BOSFETs and theprimary coil current starts flowing. For the first few mi-croseconds, the BOSFETs have a high and varying resis-tance, so the circuit simulation starts after this initial pe-riod. The Tesla coil was designed with a low enoughfrequency that this initial time is short compared to oneoscillation period. After that, the BOSFET was treated asa constant resistance.

Since the lowest possible ON resistance was important,two dual BOSFET packages were paralleled. The nominalON-switching time is 18 ILs with the maximum recom-mended 25-mA LED drive current, but this shrinks toabout 3 ILs with the drive circuit shown in Fig. 4. Theactual switching times depend on the BOSFET terminalcurrent; these values are typical at the maximum 400 mApass currents. The 0.47 ILF speed-up capacitor across the100-0 current limiting resistor boosts the initial currentinto the LEDs. Although high currents are not recom-mended for long durations, LED damage normally occursdue to thermal problems, and pulses shorter than a fewmicroseconds are safe. The turn-off time was only a fewmicroseconds, which is unimportant in, this application.


o 20 40 60 80 100 120 140 160 180 200

(b)

Fig. 5. Tesla coil operation for weak coupling, k=O.072. (a) The oscil-loscope photograph, (b) a computer simulation. This value of k corre-sponds to n= 13. Upper trace: Secondary circuit, 5 V/div. Lower trace:Primary circuit, 2 V/ div, input 8.7 V; 25-mA LED drive current. Thebottom axis numbers are microseconds.

Once the BOSFETs turn on, a low dc ON-resistance ofabout 1.2 fl (for the four units in parallel) is maintained.The OFF-resistance value is not important, but the speci-fications indicate many megohms are typical.

VI. DATA AND ANALYSIS

Experiments were performed to measure the Tesla coilparameters and to provide comparison data for the com-puter simulation. Excellent agreement is seen.

The case of small coupling, where the coils are just be-ginning to overlap, corresponds to k values less than about0.2. Figure 5 shows the oscilloscope traces and the match-ing computer simulations for two primary beats. Exceptnear the zero crossing voltage, the simulations very closelymatched the measured voltages.

According to Eq. (3), the beat period should be 94 ILsfor the k=O.072 case. This value agrees well with the Fig.5 data, even though this calculation is for R=O. Thisshows that resistive damping does not significantly affectthe oscillation frequencies.

Figure 6 shows the case for centered (5=0) coils, whichgives the maximum k value of 0.43. In this case, manybeats occur and the simulation is very sensitive to minutechanges in the coil parameters. The general form of thetraces match surprisingly well, but due to the many zero

Donald G. Bruns 801

(a)

(b)1 0 5 10 15 20 25 30 35 40 45 50

Fig. 6. Tesla coil operation for strong coupling, k=0.43. (a) The oscil-loscope photograph, (b) a computer simulation. This value of k corre-sponds to n =2. Upper trace: Secondary circuit, 10 V/ div. Lower trace:Primary circuit, 2 V/div, input 9.3 V; 29 mA LED drive current. Thebottom axis numbers are microseconds.

crossings, the curves do not match in detail. Here, the beatperiod calculates as 14J-Ls,but a periodicity of about 16 J-Lsis seen. For large coupling, the resistance values are moreimportant in the equations.

The most efficient Tesla coil operation occurs when theprimary circuit stops oscillating near its beating envelopeminima while the total circuit losses are still small. Figure7 illustrates this; the BOSFETs were turned off just as theprimary voltage envelope reached a minimum on its secondenvelope cycle. Some stray pickup causes the small voltageoscillations in the photograph. The secondary amplitudeno longer beats with the primary, but decays normally.

VII. CONCLUSION

This demonstration Tesla coil provides a tool to explorevarious Tesla coil design parameters without dangerousexperiments or high-voltage nonlinearities. Excellent theo-retical agreement results using measured parameters in acomputer simulation. The circuit could use other high-speed switches, but none currently are as convenient and aspractical as the BOSFET. Another possibility for a dem-onstrator uses frequencies so low that ordinary mechanicalrelays could be used. If the relay's response time is repro-ducible to 1ms, then a resonant frequency of 100 Hz couldbe practical. This very low frequency coul9 be obtained byusing a lOO-cmradius primary coil with lop turns and 100


100 120 140 160 180 ,Jo 20 40 60 80

(b)

Fig. 7. Tesla coil operation for weak coupling, k=O.072. (a) The oscil-loscope photograph, (b) a computer simulation. This value of k corre-sponds to n= 13. Upper trace: Secondary circuit, 5 V/div. Lower trace:Primary circuit, 2 V/div, input 8.7 V; 25-mA LED drive current. Thebottom axis numbers are microseconds. BOSFET turn off occurred at72 p,s.

each of I-J-LFnonpolarized capacitors in parallel for CpoThe secondary would be similarly scaled to large dimen-sions. While that approach would be interesting, this solid-state BOSFET design demonstrates the important param-eters in a personal, inexpensive unit. The same BOSFETswitch could temporarily replace the spark gap in a high-power Tesla coil, permitting tuning and some debugging atlower powers. As long as people are fascinated with high-voltage demonstrations, Tesla coils' will be popular, andthis demonstration can help them safely learn about them.

ACKNOWLEDGMENT

The author would like to thank his father, Arnold G.Bruns, for stimulating a:life-long interest in Tesla coils bybuilding the author's childhood home on top of the samesmall hill in Colorado Springs where Tesla's lab was lo-cated some 60 years earlier.

IN. Tesla, Colorado Springs Notes. 1899-1900. Nolit, Terazije 27, Bel-grade, Yugoslavia, with commentary by A. Marincic (1978).

2M. Cheney, Tesla: Man Out of Time (Prentice-Hall, Englewood Cliffs,NJ, 1981).

3The Tesla Journal, published annually by the Tesla Memorial Society,453 Martin Road, Lackawanna, New York 14218, and Tesla CoilBuilders Association, RD3 Box 181, Glens Falls, NY 12801.

Donald G. Bruns 802

4D. C. Cox, Modern Resonance Transformer Design Parameters (Reso-nance Research Inc., Baraboo, WI, 1984).

5C. C. Lauritsen and R. Crane, "A Combined Tesla Coil and VacuumTube," Rev. ScL Inst. 4, 497-500 (1933).

6D. H. Sloan, "A Radiofrequency High-Voltage Generator," Phys. Rev.47, 62-71 (1935).

7J. B. Kelly and L. Dunbar, Sr., "The Tesla Coil," Am. J. Phys. 20,32-35 (1952).

8E. Robberson, "How to Build a Tesla Coil," Pop. Sci., pp. 190-196,234 (August 1954).

9R. A. Kawcyn and T. C. Marshall, "The Design and Construction ofVacuum-Tube Tesla Coils," Radio Tel. News, pp. 45, 98 (August1954).

IOK. Richardson, "Tesla's Trickery," Pop. Electron. pp. 72-76 (May1960).

IIC. Caringella, "Big TC," Pop. Electron. pp. 29-32, 76 (July 1964).12E. N. Kaufman, "Li'l TC," Pop. Electron. pp. 33-35 (July 1964).13K. A. Wright and W. Kern, "Building Tesla's Famous Coil," Electron.

Hobbyist, pp. 19-22, 116 (1968).14R. K. Golka, "Long Arc Simulated Lightning Attachment Testing

Using a ISO kW Tesla Coil," 2nd Int. Pulsed Power Conference, pp.136-141 (June 1979).

15E. A. Abramyan, "Transformer Type Accelerators for Intense NeutronBeams," IEEE Trans. Nuc1. Soc. NS-18, 447-455 (1971).

16N. Tesla, Experiments with Alternate Currents of High Potential and

High Frequency (McGraw-Hill, New York, 1904) (Reprinted byOmni, Hawthorne, CA in 1979).

17MicroelectronicRelay Designer's Manual (International Rectifier, EISegundo, CA, 1990), third edition.

18C.R. J. Hoffman, "A Tesla Transformer High-Voltage Generator,"Rev. Sci. Instrum. 46, 1-4 (1975).

19W. Heise, "Tesla-Transformatoren," Elektrotechn. Zeit. A 85, 1-8(1964).

2'1I. Matsuzawa and S. Suganomata, "Design Charts for Tesla-transformer-type Relativistic Electron Beam Generators," Rev. Sci In-str. 53, 69~96 (1982).

21W. H. Press, B. P. Flannery, S. A. Teukelsky, and W. T. Vetterling,Numerical Recipes (Cambridge U. P., New York, 1989), Chap. IS.

22V.G. Welsby, The Theory and Design of Inductance Coils (Macdonald& Co., London, 1960), second edition, p. 41.

23Reference 22, p. 46.24Reference 22, p. 148.25H. B. Dwight and F. W. Grover, "Some Series Formulas for Mutual

Inductance of Solenoids," Electr. Eng. 56, 347-353 (1937). This ref-erence uses inductance in abhenries, but the formula is printed as nano-henries in Eq. (12).

26F. E Terman, Radio Engineers Handbook (McGraw-Hill, New York,1943), p. 71. Note that Gray's formula, Par IS, has the k3 sign reversedand the last term should be r.

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