Steinmentz-Charles: Primer

Project: Charles Proteus Steinmetz (1865-1923 [memory: high]; chief consulting engineer of the General Electric Company, GE [web: sites.md]) and his theory of transients, surges and "cumulative oscillations" in power systems. (The folder name misspells the surname; the man is Steinmetz.) Three of his books are in the folder:

• Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients, 2nd ed., 1914 (called Discharges below);
• Theory and Calculation of Alternating-Current Phenomena, 5th ed., 1916 (Alternating-Current);
• Theory and Calculation of Transient Electric Phenomena and Oscillations, 3rd ed., 1920 (Transient).

The Investigator added his 1917 book Theory and Calculation of Electric Circuits (Electric Circuits, archive.org), the 1898 Electrical World article on line frequencies, his 1912 GE address, the 1917 US patent 1,230,615, and the abstracts of his 1914 and 1923 AIEE papers (AIEE = American Institute of Electrical Engineers) [web: sites.md].

Source gap (repeated from Step 1 and the arbiter). None for text: all three books have an OCR text layer. But only parts were deep-read: Discharges PDF pp.111-150; Alternating-Current pp.179-215, 379-386, 402-404; Transient pp.104-124, 142-157, 292-316, 318-397, 431-482, 486-545, 602-703, 704-739 (the second set after the quality check, QE). The rest rests on page-by-page digests. The 1914 paper "Instability of Electric Circuits" (Proc. AIEE 33(1), pp.13-29) and the 1923 paper (Trans. AIEE 42, pp.470-477) were seen only as abstracts. The Schenectady County Historical Society notebooks (including "Frequency transformation by third class conductor", 23 March 1922, and 1912 "tests on titanium and carbind") were not seen. The folder holds no document by a critic or a replicator. [read; arbiter]

Measurements in the folder. An earlier version of this primer said the folder holds "no measurement by Steinmetz beyond oscillograms". That was wrong (QE, arbiter "QE revision" section 6). The books record these observations, none of them a test of a contested claim: a hot-wire ammeter reading of 29 A against about 28.8 A calculated for a spark-gap oscillating-current generator [read: Transient p.123]; test curves and oscillograms of a constant-current mercury-arc rectifier [read: Transient pp.306-309]; on "an experimental 10,000-volt, 40-cycle line", after a destructive short-circuiting arc, "the author observed a drop in generator e.m.f. to about 5000 volts" [read: Transient p.156]; observed V-curves of a synchronous motor [read: Alternating-Current pp.355-356]; a transformer inrush oscillogram (510 A calculated against 4.5 A normal) [read: Transient pp.226-227]; the 28-mile line oscillogram (Discharges Fig. 41); and an 1890 oscillogram of an alternating carbon arc (Tobey and Walbridge, reproduced in the 1916 book). They show that he measured and compared. The central contested claims (danger frequencies, the raindrop chain, the radiation numbers, the dielectric explanation) have no measurement in the folder, except for the 1920 statement on transformer windings quoted in section 4, which claims experimental support but names no source.

This is not a suppressed-inventor case. Steinmetz was an insider: GE's senior theorist and an AIEE Fellow, publishing continuously on these subjects from 1898 to his death in October 1923, and later engineers cited his last arc paper (Gilkeson 1933) [web: sites.md; arbiter "Absence-of-evidence analysis"]. The review's question is not "was he silenced?" but "which of his explanations hold up, and what can a learner get from the ones that did not?"

Source tags used below: [read] = from the project's documents (page given); [web] = Internet source, with URL or the review file that holds it; [memory] = recalled general knowledge, with confidence; [calc] = arithmetic done for this primer and reproduced in example.py. Verdicts are the arbiter's (arbiter.md); this primer adds none.

---

1. The idea in one paragraph

When a power line or a transformer is switched, struck by lightning, or touched by a spark to ground, its stored electric and magnetic energy sloshes back and forth as a brief oscillation, a "transient", that normally dies away. Steinmetz spent twenty-five years working out the mathematics of these transients and made three bold points. First, an oscillation that keeps going or keeps growing must be fed continuously, and in a power system the feeder is usually an electric arc that turns the steady power of the generators into high-frequency power; this, not the small stored energy, is what wrecks insulation. Second, the long lines and big transformer coils of a power network behave like tuned circuits, so at the right frequency they can build up voltages hundreds of times the supply. Third, he extended the same circuit thinking to nature and to the field itself: lightning inside a cloud might be a chain of tiny sparks jumping from raindrop to raindrop, and every wire radiates energy because its field spreads out at the speed of light, which shows up as an extra resistance. Most of this is now standard electrical engineering, and much of it he helped make standard. The learning value lies in the places where his explanation was incomplete (how exactly an arc feeds an oscillation, an explanation he corrected himself three years later) and where his worked numbers do not close (the raindrop chain and the radiation resistance).

---

2. What you need to know first

AC circuits and complex numbers (phasors). A sinusoidal voltage v(t) = V cos(ωt + φ) (amplitude V in volts, angular frequency ω = 2πf in radians per second, frequency f in hertz, Hz, phase φ) is written as the complex number V e^{jφ}, with j = √(-1) (engineers write j, not i, because i is current). Resistors, inductors and capacitors then become complex impedances Z = R, jωL, 1/(jωC) (R resistance in ohms; L inductance in henries, H; C capacitance in farads, F), and Ohm's law V = Z I works with complex numbers. The real part of Z decides how much average power is absorbed; the imaginary part ("reactance") only stores energy and gives it back. Steinmetz is the engineer who made this "symbolic method" standard practice in the 1890s [memory: high]. Read: any circuits text, e.g. Nilsson and Riedel, Electric Circuits (the sinusoidal steady-state chapters); Griffiths' E&M does not cover it.

Average power and in-phase / quadrature parts. If current i = A cos ωt flows and voltage v = A(R cos ωt − X sin ωt), the average power is ½RA². Only the part of the voltage in phase with the current carries net energy; the part 90° out of phase (in "quadrature") carries none over a full cycle. This one fact decides the arc question in section 5.

The series RLC circuit and the energy balance of an oscillator. A capacitor C in series with an inductor L and resistance r obeys L di/dt + r i + q/C = 0 (q is the capacitor's charge, i = dq/dt). This is the damped harmonic oscillator: solutions decay as e^{−(r/2L)t} while oscillating at ω₀ = 1/√(LC). Stored energy W = ½Li² + q²/(2C) falls at the rate r i². To keep the oscillation going you must feed in at least r i² on average. Read: Taylor, Classical Mechanics, ch. 5 (oscillations) for the mechanics analogue; any circuits text for the electrical one.

Negative resistance. An ordinary resistor's voltage rises with current (dv/di > 0). Some devices have a "falling characteristic": over some range the voltage drops as the current rises (dv/di < 0). The electric arc is the classic case; tunnel diodes and gas discharge tubes are others. Put a negative incremental resistance −R in the RLC loop and the damping term becomes (r − R)/2L: if R > r the oscillation grows. The energy comes from the DC source that keeps the device biased. Read: Strogatz, Nonlinear Dynamics and Chaos, ch. 7 (limit cycles and the van der Pol oscillator, which was invented to model such circuits).

Transmission lines (the telegrapher's equations). A long pair of wires has, per unit length, a series resistance r and inductance L and a shunt leakage conductance g and capacitance C (the "four constants"). Voltage and current then travel as waves at speed 1/√(LC) (close to the speed of light for overhead lines), with a characteristic or "surge" impedance Z₀ = √(L/C). A wave reflects at any change of impedance. Read: Pozar, Microwave Engineering, ch. 2 (transmission line theory); Griffiths, Introduction to Electrodynamics, ch. 9 for waves in general.

Dielectrics and loss. Insulators (dielectrics) between conductors store energy; real ones also absorb a little each cycle, measured by the power factor or loss tangent tan δ. Read: Griffiths ch. 4 (dielectrics) for the static part; Jonscher, Dielectric Relaxation in Solids (1983) for the frequency dependence.

Radiation from currents. Changing currents radiate because their fields take time (distance/c) to reach distant points; the radiated power appears to the circuit as a "radiation resistance". Read: Griffiths ch. 11 (radiation: dipole radiation and its power).

---

3. The textbook view

3.1 Feeding an oscillation: energy balance and negative resistance (claims 1 and 2)

Take the series loop of section 2 and add an element whose small-signal behaviour is a resistance −R_n (R_n > 0, in ohms). The loop equation is

L d²i/dt² + (r − R_n) di/dt + i/C = 0,

so i grows as e^{+(R_n − r)t/(2L)} when R_n > r. The growth stops only when something nonlinear makes the effective R_n fall to r at large amplitude: that is a stable "limit cycle" (Strogatz ch. 7). The power flows from the DC supply that biases the device; the device converts it.

The electric arc is such a device. Its stationary (slow) volt-ampere characteristic falls: H. T. Ayrton measured it, and W. Duddell (1900) used it to make the "singing arc", an arc shunted by a series L-C circuit that sings at the circuit's natural frequency; V. Poulsen turned it into a radio transmitter [memory: high]. Two textbook facts follow: the frequency is set by L and C, not by the arc; and an arc cannot follow very fast changes, because the hot gas column and the electrodes take time to heat and cool. A slow-responding arc behaves, at high frequency, like a lagged negative resistance, and the lag weakens it (made quantitative in section 5).

3.2 Lines and surges (claims 3 and 4)

For a line with constants r, L, g, C per unit length, voltage v(x,t) and current i(x,t) obey

∂v/∂x = −(r i + L ∂i/∂t), ∂i/∂x = −(g v + C ∂v/∂t).

• Wave speed 1/√(LC). For any line in air, LC = 1/c² (c = 3 × 10⁸ m/s), whatever the wire size and height [memory: high].
• Surge impedance Z₀ = √(L/C): the ratio of voltage to current in a single travelling wave. A line terminated in a resistor equal to Z₀ reflects nothing.
• Distortionless line: if r/L = g/C, all frequencies are attenuated equally, so a pulse keeps its shape (Heaviside, 1880s [memory: high]).
• Quarter-wave resonance: a line of length l open at the far end resonates when l is a quarter wavelength, f = v/(4l), with v the wave speed. A lossless quarter-wave line fed with a fixed voltage gives an infinite open-end voltage; with small loss the voltage gain is about 1/(αl), where α = r/(2Z₀) is the attenuation per unit length (for small g). It turns a constant-voltage source into a constant-current output, which is the basis of the modern "quarter-wave transformer" [memory: high].
• A line discharging to ground through a spark is a quarter-wave resonator of its own length l₀: f₁ = c/(4l₀).
• A junction between two lines with surge impedances Z₁ (incoming side) and Z₂: a wave of voltage E arriving at the junction passes on a voltage 2Z₂/(Z₁ + Z₂) × E and reflects (Z₂ − Z₁)/(Z₂ + Z₁) × E; the current passed on is 2Z₁/(Z₁ + Z₂) times the incoming current. The transmitted voltage is at most 2E, reached at an open end (Z₂ → ∞) [memory: high]. Example: a wave on a line of 346 Ω entering a transformer winding of 1,580 Ω arrives at 2 × 1,580/1,926 = 1.64 times its voltage [calc]. Repeated reflections between several junctions can push the voltage above 2E [memory: high].
• Energy flow between sections. When sections with different losses are joined, the stored energy of the whole circuit decays together, so net power flows across each junction from the low-loss section into the high-loss one. Poynting's theorem (power flow = E × H, Griffiths ch. 8) accounts for it [memory: high].
• Fronts flatten as they travel. A steep wave front contains high frequencies, which a real line attenuates more than low ones (skin effect, earth-return resistance, and above all corona, which adds capacitance to ground during the surge so the crest travels slower than the toe). So a front gets less steep with distance, and steep fronts are dangerous mainly near where they start [web: https://orca.cardiff.ac.uk/107185/, search abstract; arbiter QE revision 3e].

These are textbook transmission-line results; Steinmetz's books contain all of them, and the arbiter rates them Established.

3.3 Dielectric loss and corona (claim 6)

A capacitor C at voltage V (rms) and frequency f absorbs power P = 2πf C V² tan δ. If tan δ is constant, P ∝ f V². Many solid insulators show a loss tangent that is nearly constant over a wide frequency range ("universal dielectric response", Jonscher [memory: medium]). Mechanisms are (i) molecular dipoles that lag the field (Debye relaxation, 1912 onward) and (ii) interfacial or Maxwell-Wagner-Sillars polarization: charge piles up at the boundary between layers of different conductivity [web: https://en.wikipedia.org/wiki/Dielectric_spectroscopy]. One such mechanism gives a loss peak at one frequency; a flat loss tangent over decades needs a spread of time constants.

Air breaks down at about 30 kV/cm peak (21.2 kV/cm rms) at standard density. Around thin high-voltage wires the field near the surface exceeds this and the air glows ("corona"), wasting power. F. W. Peek's empirical law (GE, 1910s) is, per km per phase, P = 241 × 10⁻⁵ (f + 25)/δ × √(r_w/D) × (V − V₀)² kW (δ relative air density, r_w wire radius, D spacing, V and V₀ the operating and onset voltages in kV) [web: https://eepower.com/technical-articles/understanding-corona-discharge-in-high-voltage-transmission-lines/].

3.4 Lightning (claim 5)

The ambient fields measured in thunderclouds (of order 100-200 kV/m, with balloon soundings giving maxima of roughly 100-400 kV/m [memory: medium; the Physicist took 100-200 kV/m from an informal search summary and the Contrarian could not confirm it, finding sources that cite tens of kV/m]) are far below the breakdown strength of air (about 3 MV/m at sea level). The gap of at least ten times is what matters here, and it holds for any of these figures. Modern physics explains how a flash still crosses kilometres with the stepped leader: a hot, conducting channel advances in steps, its tip concentrating the field so that only a small "streamer zone" ahead of it needs breakdown strength [web: https://arxiv.org/pdf/0906.0429, search result, not read in full; physicist.md]. Ice and water particles help initiate the discharge by enhancing the field locally. Estimates of intracloud potential differences run from tens to hundreds of megavolts, one model about 300 MV [web: https://pure.psu.edu/en/publications/production-of-very-high-potential-differences-by-intracloud-light/].

3.5 Radiation resistance (claim 7)

A current element radiates. For a short wire of length l carrying a uniform current at wavelength λ, the radiation resistance is 80π²(l/λ)² ohms; a centre-fed short dipole with the usual triangular current distribution has 20π²(l/λ)² ≈ 197(l/λ)² [web: https://en.wikipedia.org/wiki/Dipole_antenna]. For an infinitely long straight wire carrying a uniform current and no return path, the radiated power per unit length corresponds to a resistance ωμ₀/4 per metre (μ₀ = 4π × 10⁻⁷ H/m) [calc; memory: high]. A pair of wires a distance d apart carrying opposite currents mostly cancels its own radiation; per unit length, the resistance is about ωμ₀(kd)²/8, with k = 2π/λ [calc, arbiter B]. At 100 kHz (λ = 3 km) all of these are tiny for ordinary wire lengths and spacings, which is why low-frequency radio stations need enormous antennas.

---

4. Where Steinmetz departs

Most of the three books is the textbook view, written by one of the people who made it so. The departures are specific.

Being fair to the starting points. Each departure grows from sound physics. Energy conservation (claim 1) is exactly right, and in 1914 most engineers had not yet absorbed that arcing grounds are fed by the generators, not by the line's stored charge. The Fig. 66 loop (claim 2) is a real measurement of a real lag in the arc. The raindrop chain (claim 5) starts from a correct objection: no cloud can hold breakdown field over a mile of straight air, and modern leader theory begins from the same objection. The radiation number (claim 7) is the exact answer for the idealisation he chose (an infinite wire with no return current); the trouble is applying it to real wires.

---

5. Worked example: the energy ledger of an arc oscillator

Question. Steinmetz's 1914 Fig. 66 shows a titanium-carbon arc whose voltage is "materially higher" on rising than on falling current, a loop he called a "negative energy cycle, or reversed hysteresis loop", and he said this lag supplies the energy of a continual oscillation [read: Discharges printed pp.124-125, PDF pp.147-148, page images]. In 1917 he said the energy comes from the arc's negative resistance, and the lag limits the amplitude. Which is right? We answer with numbers, using his own 1917 arc.

Every step is reproduced by example.py (run command in its header). Everything I assumed beyond Steinmetz's numbers is marked "assumed".

Step 1. Steinmetz's arc and its operating point

Electric Circuits (1917), sec. 97, printed p.191: "The arc characteristic in Fig. 94 is drawn with the equation

e = 35 + 200/√i (e arc voltage in volts, i arc current in amperes),

and for i = I = 3 amp. as mean value ..." [read: page image, archive.org item theorycalculatio00stei, leaf n216 = PDF p.217]. The constants are missing from the OCR text the Investigator saved, so this equation rests on the page image. The primer author read that image and read it again on 2026-10-08; it is not saved in the project folder (it can be fetched from https://archive.org/details/theorycalculatio00stei, printed p.191). The arc is shunted by a series C, L, r circuit (his Fig. 87). The same law, with other constants, is already in the 1916 book (section 4). At rest:

• e₀ = 35 + 200/√3 = 35 + 115.47 = 150.47 V.
• DC power into the arc: I e₀ = 3 × 150.47 = 451.4 W.

Step 2. Its negative resistance

Differentiate: de/di = −200/(2 i √i) = −100 i^{−3/2}. At i = 3 A, √3 = 1.732, so i^{3/2} = 5.196 and

R = −de/di = 100/5.196 = 19.25 Ω.

Steinmetz prints 18.5 Ω [read: p.191]. The difference (4%) is a slip in the printed figure, not in the idea: with his formula the small-swing value is 19.25 Ω [calc]. For a swing between two currents i′ and i″ he uses the chord, R = −(e″ − e′)/(i″ − i′):

• 2 to 4 A: e(2) = 35 + 141.42 = 176.42 V; e(4) = 35 + 100 = 135.00 V; R = 41.42/2 = 20.71 Ω (printed 20.3).
• 1 to 5 A: e(1) = 235.00 V; e(5) = 35 + 89.44 = 124.44 V; R = 110.56/4 = 27.64 Ω (printed 27.5).

A more careful rule for a sine-shaped swing i = I + A sin θ is the "describing function": take the part of the arc voltage that oscillates in step with the current, R = −(1/(πA)) ∫₀^{2π} e(I + A sin θ) sin θ dθ. Numerically this gives 20.33 Ω for A = 1 A (matching his printed 20.3) and 25.03 Ω for A = 2 A [calc]. All versions agree on his key conclusion: R rises with amplitude, so with this static characteristic an oscillation that starts can never settle. "An oscillation, once started, proceeds to the extinction of the arc, and only a 'rasping arc' could be produced" [read: p.191].

Start condition (his sec. 97). A tank whose load resistance r is below R oscillates at once; with r = 22.5 Ω it does not start by itself, but once the swing exceeds 1.5-4.5 A (chord 23.0 Ω [calc], above 22.5) it grows. His statement checks.

Step 3. Where the oscillation power comes from (no lag yet)

Let the current swing be i = 3 + 1·sin θ (A = 1 A; θ = ωt). A large choke in the supply holds the arc's mean current at 3 A, so the swing flows round the tank. Averaging over one cycle (numerically, in example.py):

• Mean arc voltage: ⟨e⟩ = 153.03 V (higher than 150.47 V because the curve is convex: the voltage rises more at low current than it falls at high current).
• Power drawn from the DC supply: I⟨e⟩ = 3 × 153.03 = 459.10 W.
• Power dissipated in the arc: ⟨e i⟩ = 448.94 W.
• AC power handed to the tank: −⟨e · sin θ⟩ × A = ½ R A² = ½ × 20.33 × 1² = 10.16 W.
• Check: 459.10 − 448.94 − 10.16 = 0 (to 10⁻¹⁴ W).

So the 10 W that sustains the oscillation is DC supply power converted by the arc. The arc even draws more DC power when it oscillates. Nothing here needs a lag; this is the 1917 account, and Steinmetz's 1914 energy principle (claim 1) holds exactly.

Step 4. Add a lag and see what it does

Now let the arc voltage follow its characteristic only after a delay: a first-order lag with time constant τ (seconds; assumed, it stands for the time the arc stream and electrodes take to heat and cool, Steinmetz's own sec. 98 explanation). For small swings of angular frequency ω this turns the arc's impedance into

Z = −R/(1 + jx), with x = ωτ.

Separate it into real and imaginary parts by multiplying top and bottom by (1 − jx):

Z = −R(1 − jx)/(1 + x²) = −R/(1 + x²) + j R x/(1 + x²).

• The real (in-phase) part −R/(1 + x²) is still negative, so the arc still gives power, but it is smaller than R by the factor 1/(1 + x²).
• The imaginary (quadrature) part +Rx/(1 + x²) is positive, like an inductor: it stores and returns energy and supplies none.

Now compute the two things Steinmetz's argument mixes up, for a current swing δi = A cos ωt and voltage swing δv = A[Re Z cos ωt − Im Z sin ωt]:

• Energy delivered per cycle = −∫ δv δi dt over one period T = 2π/ω. Only cos² survives the averaging: ∫cos² ωt dt = π/ω. So W = π A² R / [ω(1 + x²)], and the average power is P = W/T = ½ A² R/(1 + x²).
• Area of the loop drawn in the (current, voltage) plane = ∮ δv d(δi). With d(δi) = −Aω sin ωt dt, only the sin² term survives: area = π A² R x/(1 + x²).

Note the units: the loop area is volts × amperes, not joules. A loop in the current-voltage plane is not an energy diagram, unlike the B-H loop of magnetic hysteresis, where field H and flux density B are energy-conjugate variables (∮H dB is energy per unit volume). For a circuit the energy-conjugate pair is voltage and charge (v dq = v i dt), not voltage and current.

Numbers with Steinmetz's printed R = 18.5 Ω and a 1 A swing [calc]:

(At ω = 1 rad/s the energies per cycle are 58.1, 46.5, 29.1, 11.6, 2.2 J, the figures the reviewers quote.)

The table is the whole answer. With no lag the arc delivers the most power and draws no loop at all. As the lag grows the power falls steadily, while the loop first grows and then shrinks. The loop measures the lag; the power comes from the falling slope.

Which way round is the loop? Where the current passes its mean value while rising (cos ωt rising through zero, sin ωt = −1), δv = +ARx/(1 + x²) > 0; while falling, it is negative. So the voltage is higher on rising current: exactly the sense of Fig. 66 [calc]. A falling characteristic plus thermal lag explains both features of Fig. 66, the falling mean curve and the loop, with the lag taking power away rather than supplying it.

Step 5. A time-domain check with a real tank

Assumed tank: L = 10 mH, C = 2.5 µF (natural frequency f₀ = 1/(2π√(LC)) = 1,007 Hz), load r = 5 Ω, arc R = 18.5 Ω. Rough rule: the oscillation grows if R/(1 + (ω₀τ)²) > r.

For ω₀τ = 1, example.py integrates the circuit for 20 ms after a 10 mA kick and keeps an energy ledger: energy from arc to tank 8.281 mJ = energy burnt in r 4.054 mJ + rise in stored energy 4.227 mJ, with an error of 3 × 10⁻⁸ mJ. The lag is the reason the oscillation stops growing at high enough frequency, which is Steinmetz's 1917 role for it (and why real singing arcs only work up to some frequency). The small-signal model grows without limit; a real arc would go out or saturate, as he says.

What the example shows

• Steinmetz's 1914 energy principle is right, and the arc is the converter: Established.
• His 1914 mechanism, "the lag supplies the energy", gets the roles reversed: the in-phase negative resistance supplies, the lag subtracts and draws the loop. He fixed this himself in 1917. Hence the arbiter's verdict on the 1914 wording: Historically superseded but instructive.
• His 1917 printed numbers are close but not exact (18.5 Ω printed for 19.25 Ω).

Try it yourself. Read Fig. 66 by eye (deep read: about 95 V on rising and 72-75 V on falling current at 1 A; stationary curve about 80 V at 1 A and 61 V at 3 A [read: deep read of page image p.148]). Estimate the stationary slope and the loop width, and ask what ωτ would make the loop as wide as it is. (Fig. 66 is a large swing, 0 to 4.1 A, so the small-signal formulas are only a guide.)

---

6. Evidence table

Verdicts are the arbiter's; "tested?" means a controlled test of the claim as Steinmetz stated it.

Also kept in the record (no verdict attached): his 1922 laboratory lightning generator, "a discharge of 10,000 amperes at over 100,000 volts ... lasting for a hundred-thousandth part of a second" (AP story, New York Herald, 3 March 1922 [web: https://www.loc.gov/resource/sn83045774/1922-03-03/ed-1/?sp=6]); the figures are self-consistent (10⁹ W peak, at most 10 kJ [calc]). It tested arresters, not the cloud model.

Also kept: contemporary records found by the Investigator that bear on the claims without testing Steinmetz's mechanisms. - Commonwealth Edison trouble report, 1919. Groups of alternators oscillated against each other, with the lag of field flux treated as damping (trouble of 18 September 1919) [web: https://babel.hathitrust.org/cgi/pt?id=uiuo.ark:/13960/t0ks6mb3n, full view, searched in text, not read in full]. It is a field case of machine oscillation, and it bears on the open question of whether the 1914 "lag" wording came from machine hunting (section 7). - J. Slepian and J. F. Peters, "Surges on Transmission Lines" (Westinghouse), presented in the same April 1923 AIEE session as Steinmetz's third-class-conductor paper [web: https://www.worldradiohistory.com/Archive-AIEE/Journal-Of-The-AIEE-1923-04.pdf, programme]. It is a contemporary treatment of the same phenomena by a competitor of GE, so it is an interested party's account. It has not been read. - E. E. F. Creighton's arcing-ground suppressors (GE) and the debate on grounded against isolated neutrals, J. AIEE May and June 1923 [web: WorldRadioHistory J. AIEE 1923-05, 1923-06; sites.md]. They show that arcing grounds were a real operating problem in 1923, and that industry answered them by grounding the neutral. They do not test his mechanism.

Printed slips found by the reviewers (none changes a principle): 300,000 V should be 150,000 V (750 A × 200 Ω) in the short-circuit interruption example; glass-plate gradients 8.4 and 35.7 kV/cm should be 11.4 and 45.5 kV/cm from his own formula (the air gap sparks over either way); "7.86 wavelengths" in the arrester example should be about 1.97; seven misprints in the sil/col appendix; in the 1916 hard-carbon arc, the printed 388 W is probably 338 W (only 338 W fits his 77% power factor; example.py Part 5) [calc; arbiter C and QE revision; deep reads].

---

7. What a learner should take away

What is solid. - Energy conservation applied to oscillations: a sustained or growing oscillation needs an energy source, and in a power network the source is the generators, reached through a nonlinear element such as an arc. This is the foundation of every self-excited oscillator, from the singing arc to the transistor oscillator. - The transmission-line toolkit: four constants per unit length, surge impedance, reflections, the distortionless condition, quarter-wave resonance, and f₁ = c/(4l₀) for a line discharging to ground (dependent only on length). Steinmetz had the frequency-length table in print in 1898. - The practical warnings: arcing grounds, cumulative oscillations, and very high voltages on resonant lines are real hazards. His protective patent (filed 1913) used a resistance equal to the transformer's surge impedance. - Surges at junctions and along lines: a wave entering a station or a winding can arrive at up to twice its voltage; energy flows across junctions from low-loss into high-loss sections; steep fronts flatten as they travel, so they are most dangerous near where they start.

What is wrong, and why the mistake is instructive. - The arc's "reversed hysteresis loop" as the energy source (1914). A loop in the current-voltage plane measures the quadrature (reactive) response, not energy. The lesson is general: before reading a loop's area as energy, check that the axes are energy-conjugate (H and B, force and displacement, voltage and charge). Steinmetz corrected himself in 1917, and his willingness to write "practically unknown" in 1914 is a model of honest uncertainty. - The 24:1 raindrop chain. With his own constants the series arm is capacitive at 100 MHz, so no wave propagates and the gap voltages decay along the chain (sum 1.46 × the end voltage). The printed attenuation and phase constants look swapped. The lesson: check that a printed intermediate result actually follows from the stated inputs; a striking ratio can come from a transcription slip. The qualitative effect (out-of-phase gap voltages summing to more than the end voltage) is real in a propagating chain with a net-inductive series arm; it just does not occur with these numbers. - 0.2 Ω per metre of radiation resistance. The number is exactly right for an infinitely long wire carrying a uniform current with no return, a configuration that cannot exist (charge would pile up). Real circuits close, and the return current cancels most of the radiation. The lesson: an exact answer to an idealised problem can be thousands of times wrong for the real one. He says himself that the no-return condition has meaning only above about 10⁴ cycles [read: Transient printed p.430]. The error does not stay in one chapter: in 1920 the same radiation term sets his 20-100 kHz "danger band" for lines, which moves up several times once the term is corrected. A wrong coefficient early in a chain of reasoning travels into every later result that uses it.

What is genuinely open. - Did Steinmetz measure the in-phase negative resistance of his Ti-C arc? The 1912 "tests on titanium and carbind" and the 1922 "frequency transformation by third class conductor" notebooks at Schenectady could show it; nobody in this review has seen them. - What experimental investigations stood behind the 1920 statement that transformer windings oscillate at 20-100 kHz ("experimental investigations have shown"), and behind "frequently been observed" (1917 sec. 101)? Who ran them, on which transformers? AIEE Transactions and the GE Review of about 1910-1920 [memory: medium] and General Electric test records are the places to look. - Does the 1/√(distance) front-flattening law match later measurements of front steepness on real lines, with and without corona? - With a corrected radiation term and earth-return loss added (Carson 1926 [web: https://ieeexplore.ieee.org/document/6768096]), where does the line "danger band" really fall? The arbiter's rough estimate is 0.14-0.19 MHz for ground return and 0.9-1.2 MHz for 6-ft spacing. - Does a real transformer winding, rung down alone and then connected to a line, show the decay change and junction power flow that the compound-circuit equations predict? - Was the 1914 "lag" wording carried over from synchronous-machine hunting, where a lag in the restoring force really does supply energy (negative damping)? The full 1914 "Instability" paper would tell. The 1919 Commonwealth Edison trouble report (section 6), where alternator groups oscillated against each other and the lag of field flux was treated as damping, shows engineers of the time reasoning about lag in machine oscillation. - Could a drop-to-drop chain with realistic cloud constants propagate at all? Redoing his Sec. 47 calculation with measured drop spacings is an inexpensive test of the idea; VHF lightning mapping data would be the observational one. - How do modern readers who build on him (Eric P. Dollard, from 1982; the "Steinmetz Decoded" site) read his finite-velocity radiation model? Dollard and his publishers have a stake in that reading, and mainstream engineering does not accept Dollard's wider claims [memory: high]; both are recorded here as leads, not evidence.

---

8. Check yourself

1. An arc biased at 3 A follows e = 35 + 200/√i. What is its small-signal negative resistance, and what is the largest tank load resistance that lets an oscillation start by itself?
2. An arc's voltage response to a current swing is Z = −R/(1 + jωτ). Show that the average power it delivers is ½A²R/(1 + (ωτ)²). What happens to it as τ grows? What happens to the area of the voltage-current loop?
3. Why is the area of a B-H loop an energy, while the area of a voltage-current loop is not?
4. A 1,000 km overhead line, open at the far end, is struck by a spark to ground at the near end. Roughly what frequency does it ring at, and why does the wire size not matter?
5. Steinmetz's formula gives 0.197 Ω of radiation resistance per metre of wire at 100 kHz. A short centre-fed 1 m dipole at the same frequency has about 2.2 × 10⁻⁵ Ω. What idealisation explains the factor of almost 10,000?

---

9. Further reading

Mainstream. - D. J. Griffiths, Introduction to Electrodynamics: ch. 4 (dielectrics), ch. 7 (electrodynamics, inductance), ch. 9 (electromagnetic waves), ch. 11 (radiation). - D. M. Pozar, Microwave Engineering: ch. 2 (transmission line theory, quarter-wave transformer). - S. H. Strogatz, Nonlinear Dynamics and Chaos: ch. 7 (limit cycles, van der Pol oscillator). - A. Greenwood, Electrical Transients in Power Systems (Wiley): switching surges, arcing grounds, transformer winding transients. - C. A. Balanis, Antenna Theory: radiation resistance of dipoles and loops. - V. A. Rakov and M. A. Uman, Lightning: Physics and Effects (Cambridge, 2003): leaders and intracloud discharges. - A. K. Jonscher, Dielectric Relaxation in Solids (1983): constant loss tangent and Maxwell-Wagner polarization. - F. W. Peek, Dielectric Phenomena in High Voltage Engineering (McGraw-Hill, 1915): corona law, from Steinmetz's GE colleague. - W. Duddell, on the singing arc, Journal of the IEE, 1900 [memory: medium for the exact citation]. - J. R. Carson, "Wave propagation in overhead wires with ground return", Bell System Technical Journal 5(4), Oct 1926, pp.539-554 [web: https://ieeexplore.ieee.org/document/6768096]: the earth-return loss that Steinmetz's 1920 model lacks.

Contemporary records (see section 6). - Commonwealth Edison Company, report on the trouble of 18 September 1919 (alternator groups oscillating) [web: https://babel.hathitrust.org/cgi/pt?id=uiuo.ark:/13960/t0ks6mb3n]. - J. Slepian and J. F. Peters (Westinghouse, an interested party), "Surges on Transmission Lines", AIEE, April 1923 [web: J. AIEE April 1923 programme, https://www.worldradiohistory.com/Archive-AIEE/Journal-Of-The-AIEE-1923-04.pdf]. - E. E. F. Creighton, arcing-ground suppressors, discussed in J. AIEE May and June 1923 [web: WorldRadioHistory; sites.md].

Steinmetz's own key works. - Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients, 2nd ed., McGraw-Hill, 1914: Lectures VIII-X (travelling waves; compound circuits; continual and cumulative oscillations, Fig. 66). - Theory and Calculation of Electric Circuits, McGraw-Hill, 1917: Ch. X "Instability of Circuits: The Arc", secs. 81-101 (the negative-resistance account) [web: https://archive.org/details/theorycalculatio00stei]. - Theory and Calculation of Transient Electric Phenomena and Oscillations, 3rd ed., 1920: Section III (transients in space, quarter-wave line, arrester chain, radiation and the sil/col appendix); Section IV (the general line equations; chs. VIII-XI on compound circuits, power transfer and transition points); Section V (variation of circuit constants with frequency, danger frequencies, wave-front flattening). - Theory and Calculation of Alternating-Current Phenomena, 5th ed., 1916: Ch. XIV (dielectric losses, corona). - "The Natural Period of a Transmission Line and the Frequency of Lightning Discharges Therefrom", Electrical World, 27 Aug 1898 [web: research/files/]. - "Instability of Electric Circuits", Proc. AIEE 33(1), Jan. 1914, pp.13-29 [web: https://ieeexplore.ieee.org/document/6661036]. - "Frequency Conversion by Third Class Conductor and Mechanism of the Arcing Ground and Other Cumulative Surges", Trans. AIEE 42 (1923), pp.470-477 [web: https://ieeexplore.ieee.org/document/5060887]. - US Patent 1,230,615, "Protective device" (filed 1913, granted 1917, GE) [web: https://patents.google.com/patent/US1230615A/en].

---

Answers

1. R = −de/di = 100 i^{−3/2} = 100/5.196 = 19.25 Ω at 3 A (Steinmetz prints 18.5 Ω). Any load below that (in Steinmetz's terms, r < R) starts oscillating by itself; above it, the arc runs steady until a large enough swing pushes the chord value above r.
2. Real part of Z is −R/(1 + x²) with x = ωτ; average power is ½A² × |Re Z| = ½A²R/(1 + x²). It falls monotonically as τ grows: the lag removes supply. The loop area πA²Rx/(1 + x²) is zero at x = 0, largest at x = 1, and falls again: it tracks the lag, not the power.
3. H and B are energy-conjugate: the work done on a magnetic material per unit volume is ∫H dB, so ∮H dB over a cycle is energy lost per unit volume. For a circuit element the work is ∫v dq = ∫v i dt; voltage and current are not conjugate, so ∮v di has units of V·A (watts) and measures the quadrature part of the response, not energy.
4. f₁ = c/(4l₀) = 3 × 10⁸ / (4 × 10⁶) = 75 Hz. The wire size and height cancel because, for any line in air, the product of inductance and capacitance per unit length is 1/c²; only the length sets the transit time.
5. The 0.197 Ω per metre (= ωμ₀/4) is for an infinitely long wire carrying a uniform current with no return path. A real wire is finite and its current returns (or, in a dipole, falls to zero at the ends); the fields of the returning current cancel most of the radiation, leaving a resistance proportional to (l/λ)², tiny when the wire is 1/3000 of a wavelength.

---

Notes for the orchestrator (no verdict changes)

• New check made for this primer, not in the earlier files: Steinmetz's printed 1917 equation e = 35 + 200/√i (Electric Circuits printed p.191, read as a page image from https://archive.org/details/theorycalculatio00stei, PDF p.217) gives a small-swing R of 19.25 Ω, not the printed 18.5 Ω; the 1-5 A chord gives 27.64 Ω (printed 27.5); the printed 20.3 Ω for 2-4 A matches the sine-swing describing function (20.33 Ω) better than the chord (20.71 Ω). The reviewers and arbiter quote "18.5" correctly as Steinmetz's figure; their lag-model numbers (58.1 / 29.1 / 2.2 and 0 / 29.1 / 11.2) use R = 18.5 and are reproduced here. No verdict depends on the difference.
• Fig. 66 page reference: the text is printed p.124 (PDF p.147) and the figure printed p.125 (PDF p.148); the cycle time reads 1/120 s on the page image [read: page image p.147].
• QE remediation (2026-10-08), applied from qe.md and the arbiter's "QE revision". Corrections to this primer's earlier text:
• Row 4b had "No measurement cited". That was wrong for the 1920 book, which claims "experimental investigations" without giving a reference (Transient PDF pp.674, 676).
• The source gap said "no measurement by Steinmetz beyond oscillograms". That was incomplete; see "Measurements in the folder".
• "Energy-transfer constant: Not testable as stated" is now split. Flow in a compound circuit is Established, eq. 46 is an approximation, and the uniform-line naming stays Not testable as stated.
• The flat "340 ×" for radiation is now given as a range that depends on frequency, spacing and convention.
• The 100-200 kV/m cloud-field figure is retagged [memory: medium].
• Additions: the 1916 arc row (with example.py Part 5), junction doubling, the flattening law, the derived line band, the constant-parameter scope, the sec. 69 radiation estimate, and three restored Investigator finds.
• The 1917 page image (leaf n216) was fetched to a scratch folder outside the project to re-check e = 35 + 200/√i, R = 18.5 / 20.3 / 27.5 Ω and the 1.5-4.5 A, 22.5 Ω passage. All match. The primer author may write only primer.md and example.py, so the image was not saved to research/files. The orchestrator may save it if wanted.