Lesson 00

How to use this course

Nikola Tesla is two people in the public imagination. One is the engineer whose 1888 patents on polyphase alternating-current motors were licensed by George Westinghouse and became part of the system that powers the world. The other is a legend: the lone genius who invented radio, free energy and a death ray, and whose papers were hidden by the government. The first is documented in patents, lectures, technical journals and company records. The second grew mostly after his death, from his own late press statements and from biographers and enthusiasts.

This course teaches the first and examines the second. Every technical lesson works through the physics Tesla actually used. Lesson 10 and Lesson 12 take his most famous unfulfilled claims, wireless power around the globe and the weapons of his old age, and test them against physics and the record. Where a story comes only from Tesla’s own autobiographical articles of 1919, the text says so.

Prerequisites for Part II

Circuit theory: phasors, impedance, RLC resonance, mutual inductance.
Electromagnetism: Faraday’s law, magnetic fields of coils, Maxwell’s equations, skin effect.
Ordinary differential equations and basic vector calculus.
For Lesson 10, the Laplacian on a sphere and spherical harmonics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The companion course on Guglielmo Marconi treats spark transmitters, antennas and coupled circuits from the radio side; Lesson 07 here uses the same coupled-circuit model.

Reference

Timeline

1856	Born 10 July in Smiljan, in the Military Frontier of the Austrian Empire (now Croatia), to Milutin Tesla, a Serbian Orthodox priest, and Georgina (Đuka) Mandić.
1870–73	Higher Real Gymnasium in Karlovac; graduates in three years.
1875	Enters the Imperial-Royal Technical College in Graz on a Military Frontier scholarship.
1878	December: leaves Graz without a degree.
1879	17 April: his father dies.
1880	Goes to Prague; audits lectures at the university without enrolling.
1881	Budapest Telephone Exchange under Tivadar Puskás; becomes chief electrician.
1882	Continental Edison Company, Paris (works at Ivry-sur-Seine). By his own later account, conceives the rotating magnetic field this year.
1884	June: arrives in New York; works about six months at the Edison Machine Works.
1885–86	Tesla Electric Light & Manufacturing Company, arc lighting in Rahway, New Jersey; loses control of the company.
1887	April: Tesla Electric Company founded with Alfred S. Brown and Charles F. Peck. 12 October: files the first induction-motor patent application.
1888	1 May: seven patents on motors and AC distribution granted. 16 May: lecture to the AIEE. July: Westinghouse licenses the patents.
1889	Visits the Paris Exposition and learns of Heinrich Hertz’s experiments; begins high-frequency work.
1891	Files the Tesla coil patent (25 April; US 454,622). 20 May: AIEE lecture at Columbia College. 30 July: becomes a US citizen.
1892	3 and 4 February: lectures to the Institution of Electrical Engineers and the Royal Institution in London. Vice-president of the AIEE, 1892–94.
1893	Lectures in Philadelphia and St Louis. Westinghouse lights the World’s Columbian Exposition in Chicago with AC.
1895	13 March: fire destroys his South Fifth Avenue laboratory. 25 August: the Adams station at Niagara Falls begins generating two-phase AC.
1896	X-ray experiments with single-terminal tubes driven by his coil.
1898	Radio-controlled boat shown at Madison Square Garden; US patent 613,809.
1899–1900	Experimental station at Colorado Springs; magnifying transmitter.
1901	March: J. P. Morgan advances $150,000. September: construction begins at Wardenclyffe, Long Island.
1903	Morgan declines further funds.
1906	Wardenclyffe work stops. Bladeless turbine demonstrated on his 50th birthday.
1916	Edison Medal of the AIEE (presented in 1917).
1917	From 4 July: Wardenclyffe tower demolished for scrap.
1919	Autobiographical series “My Inventions” in Electrical Experimenter.
1931	75th birthday; Time cover.
1934	Moves to the Hotel New Yorker; Westinghouse pays him $125 a month. Announces a “death ray”.
1943	Dies 7 January in his hotel room, aged 86. 21 June: US Supreme Court rules on Marconi’s tuning patent.
1960	The SI unit of magnetic flux density is named the tesla.

Part I · Formation

Lesson 01

Smiljan to Graz, 1856–1879

Nikola Tesla was born on 10 July 1856 in the village of Smiljan, in the Lika region of the Austrian Military Frontier, today in Croatia. His family was Serbian. His father, Milutin Tesla, was an Orthodox priest and a writer of articles and poems; his mother, Georgina, known as Đuka, had no schooling but, by her son’s account, had a gift for making household tools and memorizing long epic poems. Nikola was the fourth of five children. His older brother Dane died in a riding accident when Nikola was a child.

In 1862 the family moved to the town of Gospić, where his father became parish priest. In 1870 Nikola went to the Higher Real Gymnasium in Karlovac, where teaching was in German, and finished in three years. He later wrote that the physics demonstrations of his teacher there first drew him to electricity. Back home in 1873 he fell ill with cholera and was bedridden for months. His father, who had wanted him to enter the priesthood, promised to send him to an engineering school if he recovered. In 1874 he avoided conscription by spending time in the mountains.

Graz

In 1875 Tesla enrolled at the Imperial-Royal Technical College in Graz on a Military Frontier scholarship. In his first year he worked very hard and passed many examinations. The lectures on electricity of the physics professor Jakob Pöschl impressed him. By his own account, Pöschl demonstrated a Gramme dynamo used as a motor, sparking at its commutator, and Tesla suggested that a machine without a commutator should be possible; the professor told him this was impossible. The scholarship ended, he began gambling, and in December 1878 he left Graz without a degree. He worked as a draughtsman in Maribor, was sent home to Gospić in 1879 for lacking a residence permit, and lost his father on 17 April of that year.

Lesson 02

Prague, Budapest, Paris, 1880–1884

In January 1880 Tesla went to Prague, intending to study at the university. He arrived too late to enrol and audited lectures without receiving grades. In 1881 he moved to Budapest to work for the telephone exchange being built by Tivadar Puskás, a former associate of Thomas Edison. He became chief electrician of the exchange and improved its equipment, including a telephone repeater or amplifier, which he never patented.

Tesla later wrote that during a walk in a Budapest park in 1882, while reciting Goethe, he suddenly saw how a magnetic field could be made to rotate by alternating currents out of step with one another, and that he drew the idea in the sand. The story comes from his 1919 autobiographical articles, written 37 years after the event. What is certain is that by 1887 he could build and patent such motors, as Lesson 03 shows.

Continental Edison

In 1882 Puskás found him a post with the Continental Edison Company in Paris, which was installing Edison’s direct-current lighting systems in France and Germany. Tesla worked at the company’s works in Ivry-sur-Seine designing and repairing dynamos. He later wrote that he was sent to Strasbourg to fix the lighting plant at the railway station, and that while there he built a crude working model of an induction motor in his spare time. No independent record of that model survives. Charles Batchelor, Edison’s close associate who managed the Ivry works, saw Tesla’s ability and arranged for him to go to the United States. Tesla arrived in New York in June 1884.

Lesson 03

New York and the motor patents, 1884–1888

Tesla worked at the Edison Machine Works in Manhattan for about six months, redesigning dynamos and solving field problems; his diary marks his departure at the turn of 1884–85. His later account says that he was promised $50,000 for improving Edison’s dynamos, and that Edison dismissed the promise as a joke. Historians note that the company had nothing like that sum to spare, and the story has no source but Tesla himself.

In 1885 the patent lawyer Lemuel Serrell introduced him to investors in Rahway, New Jersey. The Tesla Electric Light & Manufacturing Company built an improved arc-lighting system and won Tesla his first patents. The investors then kept the utility business and dropped him, and Tesla spent part of 1886 doing manual labour for about two dollars a day. Late that year he met Alfred S. Brown, a Western Union superintendent, and the lawyer Charles F. Peck, who backed a new company, the Tesla Electric Company, founded in April 1887 with a laboratory at 89 Liberty Street.

The patents and the AIEE lecture

On 12 October 1887 Tesla filed the first of a group of applications on alternating-current motors and distribution. Seven of them were granted on 1 May 1888, among them US 381,968 and 381,969 (“Electro Magnetic Motor”) and 381,970 (“System of Electrical Distribution”). On 16 May 1888 he lectured to the American Institute of Electrical Engineers on “A New System of Alternate Current Motors and Transformers”, at the invitation of William Anthony and Thomas Commerford Martin.

Independently, the Italian physicist Galileo Ferraris had built a rotating-field motor model in 1885 and published his analysis in Turin in March 1888, before Tesla’s lecture but after Tesla’s filing. Ferraris did not patent his work. In July 1888 George Westinghouse licensed Tesla’s patents for $60,000 in cash and stock and a royalty of $2.50 per horsepower of motors sold, with a consulting contract at $2,000 a month in Pittsburgh. Westinghouse also acquired an option on Ferraris’s ideas. Tesla’s year in Pittsburgh was frustrating: his two-phase motors did not run well on the company’s existing single-phase 133 Hz system, and he worked on adapting the system, including a 60-cycle design.

Part II · The physics and engineering

Lesson 04

The rotating magnetic field

For the general reader. A compass needle follows a magnet. If you could make a magnetic field spin, a magnet or a piece of iron placed in it would spin too, with no sliding contacts at all. Tesla’s central idea was that two or three alternating currents, timed a fraction of a cycle apart and sent through coils pointing in different directions, add up to a field of constant strength that turns steadily. Every induction motor in a fan, pump or electric car uses this principle.

Goals

Show that balanced two-phase and three-phase currents produce a uniformly rotating field.
Decompose a pulsating field into two counter-rotating fields.
Relate the field’s rotation speed to frequency and the number of poles.

Two phases

Place two coil pairs at right angles, A along the $x$-axis and B along the $y$-axis, and feed them with currents $i_A=I\cos\omega t$ and $i_B=I\sin\omega t$, a quarter cycle apart. Near the centre the field of each pair is proportional to its current, so

\[ \mathbf B(t)=\kappa I\,(\cos\omega t,\ \sin\omega t). \]

The field has constant magnitude $\kappa I$ and turns at angular velocity $\omega$. This was the arrangement of Tesla’s 1888 two-phase motors.

Theorem (balanced polyphase field)

Let $N\ge 2$ identical windings have axes at angles $\theta_k=2\pi k/N$, $k=0,\dots,N-1$, and carry currents $I\cos(\omega t-\theta_k)$. If $N\ge 3$, or $N=2$ with axes and phases $90^\circ$ apart, the resultant field at the centre is

\[ \mathbf B(t)=\frac{N}{2}\,\kappa I\,(\cos\omega t,\ \sin\omega t), \]

a field of constant magnitude rotating at $\omega$ in the direction of the phase sequence.

Proof. The $x$-component is $\kappa I\sum_k\cos(\omega t-\theta_k)\cos\theta_k$. Using $\cos a\cos b=\tfrac12[\cos(a-b)+\cos(a+b)]$,

\[ \sum_{k}\cos(\omega t-\theta_k)\cos\theta_k=\frac12\sum_{k}\big[\cos(\omega t-2\theta_k)+\cos\omega t\big]=\frac N2\cos\omega t, \]

because $\sum_k e^{-2i\theta_k}=\sum_k e^{-4\pi ik/N}=0$ whenever $N\ge3$ (a geometric sum of roots of unity), so the first sum vanishes. The $y$-component works the same way with $\cos a\sin b=\tfrac12[\sin(a+b)-\sin(a-b)]$ and gives $\tfrac N2\sin\omega t$. The two-phase case was checked directly above. Three-phase, $N=3$, gives $\tfrac32\kappa I$.

Pulsating fields and poles

A single coil with alternating current produces a field $B_0\cos\omega t\,\hat{\mathbf x}$ that pulses along a line. It decomposes as

\[ B_0\cos\omega t\,\hat{\mathbf x}=\frac{B_0}{2}(\cos\omega t,\ \sin\omega t)+\frac{B_0}{2}(\cos\omega t,\ -\sin\omega t), \]

two equal fields rotating in opposite directions. A rotor at rest is pulled equally both ways, so a plain single-phase induction motor does not start by itself. Real single-phase motors add a second winding with a capacitor or resistance to shift its current in phase.

If the stator windings are repeated $p$ times around the circumference ($p$ pole pairs), the field pattern advances by one pole pair per cycle, so it turns at the synchronous speed

\[ \Omega_s=\frac{\omega}{p},\qquad n_s=\frac{60f}{p}\ \text{rpm}. \]

Exercise 4.1In a three-phase motor, two supply leads are interchanged. Show that the field reverses its direction of rotation.

Show solution

Interchanging phases 1 and 2 means winding $k$ now carries $I\cos(\omega t+\theta_k)$, since $\theta_2=-\theta_1$ modulo $2\pi$. Repeating the proof with $\omega t$ replaced by $-\omega t$ gives $\tfrac32\kappa I(\cos\omega t,-\sin\omega t)$, a field rotating clockwise. This is how three-phase motors are reversed in practice.

Exercise 4.2Two windings at right angles carry $I\cos\omega t$ and $I\cos(\omega t-\varphi)$ with $0<\varphi<\pi$. Decompose the field into forward and backward rotating parts and find their amplitudes.

Show solution

Write the field as $\mathrm{Re}[(X,Y)e^{i\omega t}]$ with $(X,Y)=\kappa I(1,e^{-i\varphi})$. A forward field is $\mathrm{Re}[a(1,-i)e^{i\omega t}]$ and a backward one $\mathrm{Re}[b(1,i)e^{i\omega t}]$. Solving $a+b=1$, $-ia+ib=e^{-i\varphi}$ gives $a=\tfrac12(1+ie^{-i\varphi})$, $b=\tfrac12(1-ie^{-i\varphi})$, so $|a|=\cos(\tfrac\pi4-\tfrac\varphi2)$ and $|b|=|\sin(\tfrac\pi4-\tfrac\varphi2)|$, times $\kappa I$. At $\varphi=\pi/2$ the backward part vanishes. For other phase shifts the field traces an ellipse, and the backward part produces a braking torque.

Exercise 4.3Find the synchronous speed of a 4-pole motor at 50 Hz and at 60 Hz, and of a 12-pole motor on the 25 Hz Niagara supply of 1895.

Show solution

Four poles means $p=2$: $n_s=1500$ rpm at 50 Hz and 1800 rpm at 60 Hz. Twelve poles means $p=6$: $n_s=60\times25/6=250$ rpm.

Lesson 05

The induction motor

For the general reader. In Tesla’s motor the rotating field drags the rotor along by induction. If the rotor turned exactly as fast as the field, its conductors would see no change and carry no current, and there would be no force. So the rotor always lags slightly behind. That small lag, called slip, sets the current in the rotor and hence the torque. The motor has no brushes or commutator, which is why it is so durable.

Goals

Define slip and find the frequency and size of the rotor currents.
Derive the torque–slip curve from a simple equivalent circuit.
Show that the maximum torque does not depend on rotor resistance.

Slip

Let the field turn at $\Omega_s$ and the rotor at $\Omega_m$. The slip is $s=(\Omega_s-\Omega_m)/\Omega_s$, equal to 1 at standstill and 0 at synchronous speed. Relative to the rotor the field turns at $s\Omega_s$, so the induced rotor voltages have frequency $sf$ and, by Faraday’s law, amplitude $sE$, where $E$ is the standstill value. If the rotor circuit has resistance $R_r$ and leakage inductance $L_r$, its reactance at frequency $sf$ is $sX$ with $X=\omega L_r$. The rotor current is

\[ I_r=\frac{sE}{\sqrt{R_r^2+s^2X^2}}=\frac{E}{\sqrt{(R_r/s)^2+X^2}}. \]

The second form says the rotor behaves, seen from the stator, like a load resistance $R_r/s$ in series with $X$. This is the transformer view of the motor: a transformer whose secondary rotates.

Torque–slip relation (simplified equivalent circuit)

Neglect stator resistance, stator leakage and magnetizing current, and let $V$ be the per-phase voltage (referred to the rotor). For a three-phase machine,

\[ P_{\text{gap}}=3I_r^2\frac{R_r}{s},\qquad P_{\text{mech}}=(1-s)P_{\text{gap}},\qquad T(s)=\frac{P_{\text{gap}}}{\Omega_s}=\frac{3V^2}{\Omega_s}\,\frac{R_r/s}{(R_r/s)^2+X^2}. \]

The torque is largest at $s_*=R_r/X$, where $T_{\max}=\dfrac{3V^2}{2\Omega_sX}$, independent of $R_r$.

Derivation. The power crossing the air gap is dissipated in the equivalent resistance $R_r/s$. Of this, the real rotor loss is $3I_r^2R_r=sP_{\text{gap}}$, and the rest, $(1-s)P_{\text{gap}}$, becomes mechanical work. Since $\Omega_m=(1-s)\Omega_s$, the torque is $P_{\text{mech}}/\Omega_m=P_{\text{gap}}/\Omega_s$. With $u=R_r/s$, the function $u/(u^2+X^2)$ is maximal at $u=X$, giving $s_*$ and $T_{\max}$.

Two consequences shaped motor design. Near synchronous speed, $T\approx3V^2s/(\Omega_sR_r)$: torque grows linearly with slip, so speed stays nearly constant under load. And the rotor efficiency can never exceed $1-s$, so efficient motors run at small slip. Tesla’s first motors used wound rotors; the squirrel-cage rotor, with bars shorted by end rings, was introduced by Mikhail Dolivo-Dobrovolsky in 1889 and became universal.

Exercise 5.1A 4-pole, 60 Hz motor runs at 1746 rpm under full load. Find the slip, the frequency of the rotor currents, and an upper bound on the efficiency.

Show solution

$n_s=1800$ rpm, so $s=54/1800=0.03$. The rotor currents have frequency $sf=1.8$ Hz. The rotor loss is $3\%$ of the air-gap power, so the efficiency is at most $97\%$, before stator and mechanical losses.

Exercise 5.2Show that $T/T_{\max}=2/(s/s_*+s_*/s)$ (Kloss’s formula). For $R_r=0.2\ \Omega$ and $X=1\ \Omega$, find the starting torque as a fraction of $T_{\max}$. What happens if $R_r$ is doubled?

Show solution

Divide: $\dfrac{T}{T_{\max}}=\dfrac{2X(R_r/s)}{(R_r/s)^2+X^2}=\dfrac{2}{(R_r/s)/X+X/(R_r/s)}=\dfrac{2}{s_*/s+s/s_*}$. Here $s_*=0.2$, so at $s=1$, $T/T_{\max}=2/(0.2+5)=0.385$. Doubling $R_r$ gives $s_*=0.4$ and $T/T_{\max}=2/(0.4+2.5)=0.69$: more starting torque, same maximum, but more slip and loss at full speed. Wound rotors with external resistance switched out after starting exploit this.

Exercise 5.3Why can an induction motor never reach synchronous speed by itself, and what happens if it is driven faster?

Show solution

At $s=0$ the rotor sees no changing flux, $I_r=0$ and $T=0$; any friction makes it slow down until a small slip supplies the needed torque. If an external engine drives it above $\Omega_s$, then $s<0$, $R_r/s$ is negative and $P_{\text{gap}}<0$: power flows back into the supply, and the machine acts as an induction generator.

Lesson 06

Polyphase power and transmission

For the general reader. Electric power is voltage times current, but the heat wasted in a wire depends on the current alone. Sending power at high voltage and low current therefore saves energy, and only alternating current can be stepped up and down easily, by transformers. Tesla’s polyphase system added two further advantages: motors that start by themselves, and a total power flow that does not pulse. The great demonstration was Niagara Falls in 1895.

Goals

Derive the dependence of line loss on transmission voltage.
Prove that balanced polyphase power is constant in time.
Compare the conductor needed by single-phase and three-phase lines.

Line loss

A load drawing real power $P$ at RMS voltage $V$ and power factor $\cos\varphi$, through a line of total resistance $R$, causes a line loss

\[ P_{\text{loss}}=I^2R=\frac{P^2R}{V^2\cos^2\varphi}. \]

Raising the voltage tenfold cuts the loss a hundredfold. An ideal transformer with turns $N_1,N_2$ gives $V_2/V_1=N_2/N_1$ and $I_2/I_1=N_1/N_2$. It works only because the core flux changes in time, so it needs alternating current. Lucien Gaulard and John Gibbs, and then the Budapest engineers Zipernowsky, Bláthy and Déri, developed practical transformers in the early 1880s; William Stanley built an AC lighting system for Westinghouse in Great Barrington in 1886. Tesla’s contribution was the motor, which made AC useful for power as well as light.

Theorem (constant polyphase power)

In a balanced three-phase system with phase voltages $v_k=\sqrt2V\cos(\omega t-2\pi k/3)$ and currents $i_k=\sqrt2I\cos(\omega t-2\pi k/3-\varphi)$, $k=0,1,2$, the total instantaneous power is constant:

\[ p(t)=\sum_{k=0}^{2}v_ki_k=3VI\cos\varphi. \]

Proof. Each product is $VI[\cos\varphi+\cos(2\omega t-4\pi k/3-\varphi)]$. The three angles $4\pi k/3$ are equally spaced modulo $2\pi$, so the oscillating terms cancel. A single-phase load instead receives $VI[\cos\varphi+\cos(2\omega t-\varphi)]$, a power that pulses at twice the line frequency. Constant power means constant torque in motors and generators, and less vibration.

Niagara

The Cataract Construction Company, led by Edward Dean Adams, chose alternating current for the Niagara Falls power station after an international commission had studied the options. Tesla advised Adams that a two-phase system would be the most reliable, and Westinghouse won the generator contract. The Adams station began generating on 25 August 1895 with Westinghouse two-phase generators of 5000 horsepower at 25 Hz, and power reached the city of Buffalo late in 1896. Three-phase transmission had already been shown over 175 km from Lauffen to Frankfurt in 1891, by Dolivo-Dobrovolsky and Oskar von Miller. The polyphase system of today descends from all these.

Exercise 6.1A plant sends 10 MW at unity power factor through a line of total resistance 5 Ω. Find the fraction lost at 10 kV and at 100 kV.

Show solution

At 10 kV, $I=1000$ A and $P_{\text{loss}}=10^6\times5=5$ MW, half the power. At 100 kV, $I=100$ A and $P_{\text{loss}}=50$ kW, or 0.5%. Direct current at the 110 V of an Edison system would have been hopeless at this scale.

Exercise 6.2Show that a balanced two-phase system (two phases $90^\circ$ apart) also delivers constant power, and find it.

Show solution

Phase A gives $VI[\cos\varphi+\cos(2\omega t-\varphi)]$. Phase B, shifted by $\pi/2$, gives $VI[\cos\varphi+\cos(2\omega t-\pi-\varphi)]=VI[\cos\varphi-\cos(2\omega t-\varphi)]$. The sum is $2VI\cos\varphi$, constant.

Exercise 6.3Compare the copper needed to deliver power $P$ over distance $\ell$ with the same line-to-line voltage $V$ and the same total loss, using a single-phase two-wire line and a three-phase three-wire line.

Show solution

Single-phase: $I=P/V$, loss $2I^2\rho\ell/A_1=2P^2\rho\ell/(V^2A_1)$, copper $2A_1\ell$. Three-phase: $I=P/(\sqrt3V)$, loss $3I^2\rho\ell/A_3=P^2\rho\ell/(V^2A_3)$. Equal losses require $A_3=A_1/2$, so the copper is $3A_3\ell=1.5A_1\ell$, which is 75% of the single-phase amount.

Lesson 07

The Tesla coil: a resonant transformer

For the general reader. A Tesla coil makes very high voltages without an iron core and without a large turns ratio. A capacitor discharges through a spark gap into a few turns of thick wire. Nearby stands a tall coil of many turns of fine wire, tuned to the same frequency. Energy sloshes from the first circuit into the second over several cycles, the way a swinging pendulum can set a second, loosely linked pendulum swinging. Because the second circuit has a tiny capacitance, the same energy appears there as an enormous voltage.

Goals

Find the normal modes of two tuned, loosely coupled circuits.
Derive the energy-transfer time and the voltage gain.
Understand what limits the output voltage in air.

Tesla learned of Hertz’s experiments at the Paris Exposition of 1889 and began building high-frequency apparatus. He filed his patent on 25 April 1891 (US 454,622, “System of Electric Lighting”, granted 23 June 1891) and showed the coil in his lecture to the AIEE at Columbia College on 20 May 1891. Elihu Thomson developed the same circuit independently at about the same time.

The circuit

The primary has inductance $L_1$ and capacitance $C_1$, charged to $V_1$ before the gap fires. The secondary has inductance $L_2$ and a small capacitance $C_2$, made up of its self-capacitance and the top terminal. The two are tuned, $L_1C_1=L_2C_2=1/\omega_0^2$, and coupled by a mutual inductance $M=k\sqrt{L_1L_2}$ with $k$ typically 0.05 to 0.2. Neglecting losses,

\[ L_1\ddot q_1+M\ddot q_2+\frac{q_1}{C_1}=0,\qquad L_2\ddot q_2+M\ddot q_1+\frac{q_2}{C_2}=0. \]

Normal modes and transfer

The tuned system oscillates in two modes of frequency $\omega_\pm=\omega_0/\sqrt{1\pm k}$. For $k\ll1$, starting with all energy in the primary, the primary energy varies as $\cos^2(k\omega_0t/2)$ and the secondary energy as $\sin^2(k\omega_0t/2)$. The first transfer is complete after

\[ t_1\approx\frac{\pi}{k\omega_0},\quad\text{that is about } \frac{1}{2k}\ \text{cycles}. \]

If all the energy $\tfrac12C_1V_1^2$ reaches the secondary, its peak voltage is

\[ V_2=V_1\sqrt{\frac{C_1}{C_2}}=V_1\sqrt{\frac{L_2}{L_1}}. \]

Derivation. Trying $q_j\propto e^{i\omega t}$ gives the determinant condition $(1/C_1-\omega^2L_1)(1/C_2-\omega^2L_2)=\omega^4M^2$. Dividing by $L_1L_2$, $(\omega_0^2-\omega^2)^2=k^2\omega^4$, so $\omega^2=\omega_0^2/(1\pm k)$. The two modes start in phase and drift apart at the beat rate $\omega_--\omega_+\approx k\omega_0$; the energy moves to the secondary when they are half a beat apart. The voltage gain follows from energy conservation, and the second form uses the tuning condition $C_1/C_2=L_2/L_1$. In the exact problem, complete transfer at one instant happens only for special values of $k$; builders choose the coupling so that the spark gap stops conducting (quenches) near the first energy minimum of the primary, trapping the energy in the secondary.

The gain does not come from the turns ratio, as in an ordinary transformer, but from resonance. The secondary is long enough that it behaves partly as a distributed resonator, with current largest at the base and voltage largest at the top. Output voltages range from tens of kilovolts in small coils to several million volts in the largest.

Exercise 7.1A primary capacitor of 20 nF is charged to 20 kV. The secondary capacitance is 20 pF. Find the energy per discharge and the ideal peak secondary voltage.

Show solution

$W=\tfrac12C_1V_1^2=\tfrac12\times2\times10^{-8}\times4\times10^8=4$ J. $V_2=V_1\sqrt{C_1/C_2}=20\text{ kV}\times\sqrt{1000}\approx630$ kV. Real coils lose a good part of this to spark-gap and corona losses.

Exercise 7.2Both circuits are tuned to 200 kHz with $k=0.1$. How long does the first energy transfer take, and how many cycles is that?

Show solution

$t_1\approx\pi/(k\omega_0)=\pi/(0.1\times2\pi\times2\times10^5)=25\ \mu$s, which is 5 cycles at 200 kHz, as $1/(2k)=5$ predicts.

Exercise 7.3Air breaks down at about 3 MV/m. The top of the coil is a smooth sphere of radius $r$, far from other conductors. How large must $r$ be to hold 630 kV without discharge?

Show solution

An isolated sphere at potential $V$ has surface field $E=V/r$. So $r\ge V/E_b=6.3\times10^5/(3\times10^6)=0.21$ m. This is why large coils carry big toroids or spheres: they raise the capacitance and lower the surface field, and the discharge breaks out only at a chosen point.

Lesson 08

High-frequency phenomena

For the general reader. Between 1891 and 1893 Tesla’s lectures in New York, London and Paris made him famous. He lit glass tubes held in the hand with no wires attached, passed high-frequency currents over his body without harm, and spoke of a future in which light and power would be distributed through space. The demonstrations were real. The explanations of some of them, including the popular claim that high-frequency current only flows on the skin, were wrong.

Goals

Derive the skin depth in a conductor.
Explain wireless gas-discharge lamps by displacement current.
Correct the skin-effect explanation of high-frequency shocks.
Relate the voltage of an X-ray tube to its shortest wavelength.

Skin effect

In a conductor of resistivity $\rho$ and permeability $\mu$, with displacement current negligible, Maxwell’s equations reduce to a diffusion equation $\nabla^2\mathbf J=(\mu/\rho)\,\partial\mathbf J/\partial t$. A field oscillating at $\omega$ at a plane surface penetrates as $e^{-z/\delta}\cos(\omega t-z/\delta)$.

Skin depth \[ \delta=\sqrt{\frac{2\rho}{\omega\mu}}\qquad(\text{valid when } \omega\varepsilon\rho\ll1). \]

Substituting $J=J_0e^{\gamma z+i\omega t}$ gives $\gamma^2=i\omega\mu/\rho$, so $\gamma=\pm(1+i)/\delta$; the decaying root gives the result. At high frequency current crowds into a thin layer of a metal wire, raising its resistance. Tesla and his contemporaries used stranded conductors and copper tubes for this reason.

Lamps without wires

Tesla placed gas-filled tubes between, or near, large plates connected to his oscillators. A tube has no conducting path to the plates, but the glass wall acts as the dielectric of a capacitor. The current through a capacitance $C$ at voltage amplitude $V$ is $I=\omega CV$, so at high frequency a small capacitance passes a useful current, enough to ionize the gas. This is the same physics as the fluorescent tube that glows near a high-voltage line.

Shocks and the skin-effect myth

Tesla let currents from his coils pass through his body and lit lamps in his hand. It became common to say that high-frequency current is harmless because the skin effect confines it to the surface. Body tissue is a poor conductor, with resistivity of order ohm-metres, a hundred million times that of copper, so its skin depth at hundreds of kilohertz is tens of centimetres or more. The current goes through the body. It does not cause pain or muscle spasm because nerves do not respond to stimulation much above about 10 to 20 kHz. Large currents still heat and burn tissue, and Tesla coil arcs can be dangerous.

X-rays

After Wilhelm Röntgen announced X-rays in December 1895, Tesla made X-ray photographs in early 1896 with vacuum tubes having a single electrode, driven by his coils at very high voltage. He reported damage to skin and attributed it to ozone and other products rather than to the rays. He also held, incorrectly, that X-rays were longitudinal waves. Claims that he had made X-ray images before Röntgen rest on an ambiguous photograph and are not established.

Exercise 8.1Find the skin depth of copper ($\rho=1.7\times10^{-8}\ \Omega\,$m, $\mu=\mu_0$) at 60 Hz and at 100 kHz.

Show solution

$\delta=\sqrt{2\rho/(2\pi f\mu_0)}$. At 60 Hz: $\sqrt{3.4\times10^{-8}/(4.74\times10^{-4})}=8.5$ mm. At 100 kHz: $\sqrt{3.4\times10^{-8}/0.790}=0.21$ mm. A 1 cm copper rod uses its whole section at 60 Hz but only a thin skin at 100 kHz.

Exercise 8.2A gas tube couples to an electrode through about 10 pF of glass. The electrode swings 10 kV at 100 kHz. Find the current through the tube. What would it be at 60 Hz?

Show solution

$I=\omega CV=2\pi\times10^5\times10^{-11}\times10^4=63$ mA in amplitude, enough to keep a discharge lit. At 60 Hz the current would be $10^5/60\approx1700$ times smaller, about 38 $\mu$A, which explains why high frequency was essential to Tesla’s wireless lamps.

Exercise 8.3Use the skin-depth formula with $\rho=2\ \Omega\,$m to estimate $\delta$ in tissue at 100 kHz. Why is the formula only a rough guide here?

Show solution

$\delta=\sqrt{4/(2\pi\times10^5\times4\pi\times10^{-7})}=\sqrt{4/0.79}\approx2.2$ m. The condition $\omega\varepsilon\rho\ll1$ is only marginally satisfied in tissue, whose relative permittivity is very large at these frequencies, and tissue is far from uniform, so the number is an order-of-magnitude estimate. Either way the depth far exceeds the scale of skin, and the current flows through the body.

Exercise 8.4An X-ray tube is driven at a peak of 100 kV. Find the shortest X-ray wavelength it emits.

Show solution

An electron gives at most its full energy $eV$ to one photon (the Duane–Hunt limit): $\lambda_{\min}=hc/(eV)=1.24\ \mu\text{m}\cdot\text{eV}/10^5\ \text{eV}=12.4$ pm.

Lesson 09

Tuning, radio control and the patent record

For the general reader. In 1898 Tesla steered a model boat around a tank at Madison Square Garden by radio signals, and patented the method of controlling vehicles at a distance. His patents of 1897–1900 described tuned circuits at both the sending and receiving ends of a wireless system. These are real contributions to radio. But the popular claim that the US Supreme Court declared Tesla the inventor of radio in 1943 misreads the decision.

Goals

Quantify the selectivity of a tuned receiver.
Understand Tesla’s “individualized” signalling as a logical AND of channels.
Read the 1943 Supreme Court decision accurately.

The telautomaton

US patent 613,809 (1898), “Method of and Apparatus for Controlling Mechanism of Moving Vessels or Vehicles”, described a boat with a receiver based on a coherer, a sensitive detector of radio waves. Each received signal advanced a mechanism that set the rudder and propeller. Tesla called the device a telautomaton and saw it as the beginning of machines that act at a distance. The US Navy showed little interest.

Tuned circuits

A receiver tuned to $\omega_0$ with quality factor $Q$ responds to a signal at $\omega$ with relative power

\[ \frac{P(\omega)}{P(\omega_0)}=\frac{1}{1+Q^2x^2},\qquad x=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega}\approx\frac{2\,\Delta\omega}{\omega_0}. \]

Tesla’s patents 645,576 (applied for in 1897, granted 20 March 1900) and 649,621 (granted 15 May 1900) described transmitting and receiving apparatus each built from tuned primary and secondary circuits, intended mainly for the transmission of energy. In 1903 he patented “individualized” signalling (US 723,188 and 725,605): the transmitter sends on two or more frequencies at once, and the receiver acts only when tuned circuits for all of them are excited together.

The 1943 decision, stated precisely

In Marconi Wireless Telegraph Co. of America v. United States (decided 21 June 1943), the Supreme Court held the broad claims of Marconi’s US tuning patent No. 763,772 invalid because they were anticipated, mainly by John Stone Stone’s patent No. 714,756, with Oliver Lodge’s variable tuning inductance. The opinion described Tesla’s patent 645,576 as prior art that disclosed a four-circuit tuned arrangement, but its holding rested on Stone. It did not declare anyone the inventor of radio, and it acknowledged Marconi as the first to achieve successful radio transmission.

A related misunderstanding concerns the physics. Tesla’s wireless system was meant to send energy mainly by conduction through the Earth, and for about two decades he held that Hertz had misidentified his own waves. Marconi’s system, and all later radio, used the radiated electromagnetic waves Tesla discounted. Tesla’s tuned circuits were sound and important; his theory of what his transmitters did was not.

Exercise 9.1A receiver at 50 kHz has $Q=100$. An unwanted station at 51 kHz arrives with the same field strength. By how much is its received power reduced?

Show solution

$x\approx2\times1/50=0.04$, $Qx=4$, and the power ratio is $1/17$, about $-12$ dB. A single tuned stage gives limited protection; that is why multiple tuned circuits mattered.

Exercise 9.2Noise triggers each of two independent tuned detectors by mistake with probability $10^{-2}$ per second. If the receiver acts only when both trigger within the same second, how often does it act falsely? What does this illustrate?

Show solution

Assuming independence, $10^{-2}\times10^{-2}=10^{-4}$ per second, about once every three hours instead of once every 100 seconds. Tesla’s individualized signalling is a logical AND, and the same idea of combining independent channels underlies coding and security in modern communication.

Exercise 9.3A quarter-wave vertical antenna is the natural resonant length for a grounded radiator. Find its height at 100 kHz and at 10 MHz.

Show solution

$\lambda=c/f$: 3 km at 100 kHz, so $\lambda/4=750$ m; 30 m at 10 MHz, so $\lambda/4=7.5$ m. Low-frequency stations of Tesla’s and Marconi’s era used tall masts with large top loads because a full quarter wave was impractical.

Lesson 10

Earth resonance: claims and physics

For the general reader. At Colorado Springs in 1899 Tesla built his largest oscillator and watched thunderstorms with sensitive receivers. He concluded that the Earth itself could be set ringing like a bell, so that energy pumped in at one point could be drawn out anywhere on the planet. The Earth does have natural electromagnetic resonances, discovered half a century later. But they are weak, lossy and very low in frequency, and they cannot carry useful power around the world.

Goals

Derive the resonant frequencies of the cavity between the Earth and the ionosphere.
Estimate the radiation resistance of a tower at these frequencies.
Assess global wireless power quantitatively.

What Tesla claimed

From May 1899 to January 1900 Tesla worked at a station on the edge of Colorado Springs, with power supplied by the local electric company. His magnifying transmitter used a primary about 51 feet (16 m) in diameter. He estimated its output at several million volts and reported discharges over 100 feet long; these figures come from his own notes and statements. He wrote that lightning showed the Earth to be in electrical vibration with stationary waves, and that he had lit lamps without wires at distances of several hundred metres. Later claims of lamps lit 26 miles away come from a biographer, not from his notebooks. He also reported regular signals that he thought might come from another planet; they were most likely natural or terrestrial radio noise.

The Earth–ionosphere cavity

The conducting ground and the conducting lower ionosphere, about 60 to 90 km up, form a thin spherical shell. Winfried Otto Schumann predicted its resonances in 1952, and they were measured in 1960. Lightning strikes, about 50 per second worldwide, keep them faintly excited.

Schumann resonances (ideal cavity)

For a thin shell between perfectly conducting spheres of radius about $a$, the lowest transverse-magnetic modes have frequencies

\[ f_n=\frac{c}{2\pi a}\sqrt{n(n+1)},\qquad n=1,2,3,\dots \]

With $a=6371$ km this gives 10.6, 18.4 and 25.9 Hz. The observed peaks are near 7.8, 14 and 20 Hz, lower because the ionosphere is a lossy, imperfect conductor; the quality factor of the cavity is only about 4 to 6.

Derivation sketch. When the shell is thin compared with a wavelength, the lowest modes have a nearly vertical electric field $E_r$ independent of height. Maxwell’s equations then make $E_r$ satisfy the two-dimensional Helmholtz equation on the sphere, $\nabla_S^2E_r+(\omega/c)^2E_r=0$. The eigenvalues of the spherical Laplacian are $-n(n+1)/a^2$, so $\omega^2=c^2n(n+1)/a^2$. The mode $n=0$ is excluded because it would need net charge.

Why global wireless power fails

Admirers point to the Schumann resonances as vindication of Tesla’s idea. The resemblance is partial. Tesla pictured currents conducted through the body of the Earth, not waves guided between ground and ionosphere. More important, resonance does not remove losses. In a cavity with $Q\approx5$, stored energy decays within a few cycles, so a transmitter must constantly supply the losses of the whole globe. A tower is also tiny compared with the wavelength of about 38,000 km, so it couples very weakly to the cavity. And any receiver collects only the field in its own neighbourhood. Modern analyses agree that the scheme could not have delivered useful power at a distance.

Exercise 10.1Estimate the radiation resistance at 8 Hz of a grounded vertical antenna 57 m high (the height of the Wardenclyffe tower), using $R\approx395(h/\lambda)^2\ \Omega$ for a short monopole. What does this mean for efficiency?

Show solution

$\lambda=3\times10^8/8=3.75\times10^7$ m, $h/\lambda=1.5\times10^{-6}$, and $R\approx395\times2.3\times10^{-12}\approx9\times10^{-10}\ \Omega$. Ground and conductor losses of even a fraction of an ohm would absorb essentially all the input power. A large top load raises this only by a modest factor.

Exercise 10.2Suppose 10 MW could somehow be spread uniformly over the Earth’s surface ($5.1\times10^{14}$ m²). What power density results, and how large a collector would be needed to run a 100 W lamp?

Show solution

$10^7/5.1\times10^{14}=2\times10^{-8}$ W/m². A 100 W lamp would need $5\times10^9$ m², about 5000 km², of perfect collector. Broadcasting power to everyone at once spreads it too thin to use, whatever the propagation mechanism.

Exercise 10.3Check the ideal value $f_1$ and compute the ratio $f_2/f_1$. Compare with the observed ratio $14.3/7.8$.

Show solution

$c/(2\pi a)=3\times10^8/(4.00\times10^7)=7.49$ Hz, so $f_1=7.49\sqrt2=10.6$ Hz. Ideally $f_2/f_1=\sqrt{6/2}=\sqrt3=1.73$; observed $14.3/7.8=1.83$. The ratios agree roughly, and the overall lowering of all frequencies reflects the losses in the ionosphere.

Part III · Life and legacy

Lesson 11

Wardenclyffe and after, 1899–1934

In the 1890s Tesla was a celebrity. He became a US citizen on 30 July 1891, served as vice-president of the AIEE from 1892 to 1894, and dined in New York society. The fire of 13 March 1895, which destroyed his laboratory on South Fifth Avenue with its notes and models, was a heavy blow. In 1897 Westinghouse, under financial pressure, bought out the motor patents for $216,000; Tesla had already given up the per-horsepower royalty in 1891. The story that he tore up a contract worth millions out of loyalty to Westinghouse is a later embellishment of this real concession.

Wardenclyffe

In 1899 John Jacob Astor IV invested $100,000 in the Nikola Tesla Company, which paid for Colorado Springs. In June 1900 Tesla published “The Problem of Increasing Human Energy” in The Century Magazine, describing a world system of wireless communication and power. In March 1901 J. P. Morgan agreed to provide $150,000 for a transatlantic station in return for 51% of the patents. Construction began in September 1901 at Shoreham, Long Island, on a laboratory designed by Stanford White and a tower about 57 m tall, topped by a large hemispherical cupola. In July 1901 Tesla had already proposed a much bigger plan than the one Morgan had funded. In December 1901 Marconi’s transatlantic signal (see his course) undercut the commercial case. In 1903 Morgan wrote that he was not disposed to make further advances. Work stopped by 1906. Tesla mortgaged the property to George Boldt, of the Waldorf-Astoria, to cover hotel debts, and lost it by foreclosure; the tower was dynamited for scrap from 4 July 1917.

Turbines, lawsuits, honours

On his 50th birthday in 1906 Tesla demonstrated a bladeless turbine, in which fluid drags on closely spaced flat discs by viscosity; it was tested at the Waterside power station in New York in 1910–11 but never adopted for power generation. He patented a one-way “valvular conduit” without moving parts (US 1,329,559, 1920), now studied as the Tesla valve. From 1915 to 1917 he was paid as an expert witness for Telefunken in the patent litigation over its Sayville station, in which Ferdinand Braun was also engaged. In 1917 he received the AIEE’s Edison Medal, awarded for 1916. In 1919 he published “My Inventions” in Electrical Experimenter, the source of most stories about his youth. He consulted for Allis-Chalmers in Milwaukee from 1919 to 1922 and patented a vertical take-off aircraft in 1928. He moved from hotel to hotel, often leaving unpaid bills, and became known for feeding pigeons in the parks of Manhattan. On his 75th birthday in 1931 he appeared on the cover of Time.

DiscussionMorgan funded a transatlantic telegraph station; Tesla wanted a world power system. Using Lessons 09 and 10, explain what a well-informed engineer of 1903 could have concluded about each goal, and why the commercial argument changed after December 1901.

Lesson 12

Last years, death and the myth, 1934–1943

In 1934 Tesla moved to the Hotel New Yorker, and Westinghouse began paying him $125 a month as a consulting fee, together with his rent. At his birthday press conferences in the 1930s he announced inventions that he never demonstrated or documented publicly: a motor running on cosmic rays (1932), a “death ray” or “teleforce” particle-beam weapon (1934 onward), and a mechanical oscillator that, he said, had caused an earthquake in Manhattan in the late 1890s and could bring down the Empire State Building. In the autumn of 1937 he was struck by a taxi and broke three ribs; he refused a doctor.

Tesla died alone in his room at the Hotel New Yorker on the night of 7 January 1943, aged 86, of coronary thrombosis. A maid found him the next day. Because his nephew Sava Kosanović was a Yugoslav political figure, the FBI asked the Office of Alien Property to seize his belongings. The electrical engineer John G. Trump examined them and reported nothing that would be a hazard in unfriendly hands; a box said to be part of the death ray held an old resistance box. His funeral was held at the Cathedral of St. John the Divine. In 1952 his papers and effects were sent to Belgrade, where the Nikola Tesla Museum holds them, and his ashes followed in 1957. In 1960 the General Conference on Weights and Measures named the SI unit of magnetic flux density the tesla.

Documented and claimed

Documented	Polyphase induction motors and distribution (patents of 1888); independent invention alongside Ferraris; licensing to Westinghouse; role at Niagara.
Documented	Resonant air-core transformer (1891) and pioneering high-frequency lighting and X-ray experiments.
Documented	Radio remote control (1898); tuned transmitting and receiving circuits in patents of 1900; multi-frequency signalling (1903); bladeless turbine; Tesla valve.
Disputed	“Invented radio”: tuned-circuit patents were cited as prior art in 1943, but the decision rested on Stone, and Tesla rejected radiated waves as the mechanism.
Unsupported	Global wireless power; earthquake oscillator; death ray; cosmic-ray motor; signals from Mars. No working device or independent test exists.
Anecdotal	The Budapest park vision, the $50,000 Edison bonus, the Strasbourg motor model: from Tesla’s own later accounts.

Legacy

Engineering: induction motors convert a large share of the world’s electricity into motion; polyphase AC is the standard of power systems.
High frequency: resonant transformers remain standard in physics teaching, and his radio-control patent anticipated an entire field.
Reputation: nearly forgotten at his death outside engineering circles, he became from the 1970s onward a popular icon, celebrated in Serbia and Croatia alike, and the subject of a large literature in which documented work and myth are often mixed.

Final projectPick one popular claim about Tesla. Find the earliest source for it, decide whether it belongs in the documented, disputed or unsupported row of the table above, and write a two-page account that uses at least one calculation from Part II to support your judgement.

Reference

Sources and further reading

Biography

Nikola Tesla, Wikipedia (with extensive references).
W. B. Carlson, Tesla: Inventor of the Electrical Age, Princeton University Press, 2013.
M. Cheney, Tesla: Man Out of Time, Prentice-Hall, 1981.
N. Tesla, My Inventions, Electrical Experimenter, February–October 1919; reprinted, ed. B. Johnston, 1982.
Wardenclyffe Tower, Wikipedia.
World Wireless System, Wikipedia, for Colorado Springs and later assessments.
List of Nikola Tesla patents, Wikipedia.

Physics and engineering

N. Tesla, A New System of Alternate Current Motors and Transformers, Transactions of the AIEE 5 (1888).
N. Tesla, US patents 381,968 (1888), 454,622 (1891), 613,809 (1898), 645,576 and 649,621 (1900).
N. Tesla, Experiments with Alternate Currents of High Potential and High Frequency, lecture to the IEE, London, 3 February 1892.
G. Ferraris, Rotazioni elettrodinamiche prodotte per mezzo di correnti alternate, Atti della Reale Accademia delle Scienze di Torino (1888).
Marconi Wireless Telegraph Co. of America v. United States, 320 U.S. 1 (1943).
Tesla coil and Induction motor, Wikipedia.
W. O. Schumann, Über die strahlungslosen Eigenschwingungen einer leitenden Kugel, die von einer Luftschicht und einer Ionosphärenhülle umgeben ist, Zeitschrift für Naturforschung A 7 (1952).
S. J. Chapman, Electric Machinery Fundamentals, McGraw-Hill, chapters on induction machines.
T. P. Hughes, Networks of Power: Electrification in Western Society, 1880–1930, Johns Hopkins University Press, 1983.