Steinmentz-Charles · Source catalogue

Descriptive, mostly non-mathematical companion to Steinmetz's larger 'Theory and Calculation of Transient Electric Phenomena and Oscillations'; the preface says it began as lectures to graduate classes at Union University and 'translates mathematics into English'. It develops transients from the physical idea of stored energy (magnetic and dielectric) readjusting after a change of circuit conditions, illustrated with oscillograms from industrial systems. Contents: Lectures I-II nature of transients, the electric field; III-V single-energy transients (DC circuits, AC circuits including alternator short-circuit, ironclad circuits); VI double-energy transients; VII line oscillations (wavelength, quarter-wave and half-wave oscillation, velocity unit of length); VIII traveling waves; IX oscillations of the compound circuit; X continual and cumulative oscillations; XI inductance and capacity of round parallel conductors. The 1914 edition adds the chapter on continual and cumulative oscillations, which the preface links to the destruction of high-power, high-voltage transformers. The deep read of Lectures VIII-X found: (1) standing waves carry no net power (double-frequency surge only) while traveling waves carry net power; a 'power-transfer constant' s lets low-loss sections of a compound circuit feed high-loss sections during a transient (worked numerical examples: u0 = 800 from sections of u = 900, 100, 900, 1600; s = -100, +700, -100, -800). (2) Spark-gap method for local high-frequency waves: a spark across a 13 microhenry coil gave about 2,000,000 cycles for a 100-kV line (arithmetic checked). (3) Continual or cumulative oscillations need a steady supply of oscillating energy; Steinmetz says the only known source is power fed over an arc ('arcing ground', singing arc); the energy comes from engine or turbine power, not from the stored energy of the line. (4) Steinmetz's mechanism for continual oscillation without dead time is a lag of arc voltage behind arc current, a 'negative energy cycle, or reversed hysteresis loop', shown by Fig. 66, the transient volt-ampere characteristic of a titanium-carbon arc (0 to 4.1 A and back within 1/120 s; voltage higher on rising than falling current; readings from the graph by the deep reader, about +/-5 V). He gives NO equation for the arc's energy supply and says the mechanism 'is still practically unknown in the case of continual oscillations between magnetic and dielectric energy in electric circuits'. (5) The frequency is set by the circuit, not the arc: 'the electric arc has no frequency of its own'. Hunting of synchronous machines is cited as another hysteresis-cycle oscillation. Oscillogram evidence is named but not tabulated (grounded phase on a 154-mile 82-kV line; 22-kV 40-cycle arcing grounds on 10, 21.5 and 32 miles of line; cumulative oscillation in a 2500-kW 100-kV transformer; artificial-line tests by Cunningham 1911). Corrections applied after QE and arbiter (2026-10-08): (a) the 1914 lag wording is not Steinmetz's final account of the arc. The 1917 book 'Theory and Calculation of Electric Circuits' (Ch. X secs. 96-99, printed pp.187-197; read by the Mathematician and Contrarian from OCR, not in this folder) and the 1923 Trans. AIEE paper (abstract only) give a negative-resistance account in which the lag limits amplitude; the sentence 'is still practically unknown' stands for the 1914 book and is repeated in 1917 sec. 85 [web: sites.md; arbiter.md claim 2]. (b) The 1916 Alternating-Current book in this folder already uses the falling arc characteristic e = e0 + c/sqrt(i), but only to explain wave distortion, not as an oscillation energy source (see that entry). (c) The 1914 'danger frequency' statement carries no measurement in the Discharges pages; the 1920 book makes a measurement claim for transformer windings (see the Transient entry). (d) Citation: 'Instability of Electric Circuits', Proc. AIEE 33(1), Jan. 1914, pp.13-29 (IEEE); the pp.2005-2021 on Wikipedia is a Transactions pagination. (e) Cunningham's artificial-line test (Proc. AIEE Jan. 1911) and the oscillograms are Steinmetz's secondary account of others' and company tests; a 130-mile artificial line, z = 589.5 ohm, reproduces [calc, Mathematician].

Key claims

• A continual (constant-amplitude) or cumulative (growing) electric oscillation can exist only if a steady supply of oscillating energy is fed to the circuit; it cannot come from the energy stored in the system, and the only source Steinmetz knows is power supplied over an arc, especially an unstable arc.
• In an arcing ground or singing arc the energy is taken from the power supply (engine or turbine power) and converted from low-frequency or direct-current form to the high-frequency oscillation.
• Where successive oscillation trains overlap, the arc supplies energy because arc voltage lags arc current, forming a 'negative energy cycle' (reversed hysteresis loop); his Fig. 66 (titanium-carbon arc, 0 to 4.1 A within 1/120 s) shows higher voltage on rising than on falling current.
• The frequency of an arc-fed oscillation is determined by the circuit constants; the arc has no frequency of its own.
• In a standing wave of a uniform circuit the average power flow is zero (only a double-frequency surge); a traveling wave carries net power, and a power-transfer constant s makes low-loss sections of a compound circuit feed power to high-loss sections during a transient.
• At a junction of line sections with different surge impedance, equal stored energy per wavelength makes voltage scale as the square root of z0; a low-voltage high-current wave in a line becomes a high-voltage low-current wave in a transformer.
• The spark-gap method (striking distance across a small inductance, with f = e1/(2 pi L i0)) estimates local transient frequencies up to millions of cycles per second that an oscillograph cannot record.
• Cumulative oscillations are dangerous because recurrence over seconds to hours can destroy insulation, the energy is not limited to stored energy, and arresters often fail to respond; Steinmetz proposes continuous dissipation of the oscillating energy by a resistance (with a series capacity that blocks machine frequency).
• The voltage-scaling rule sqrt(z2/z1) at a junction of line sections holds for the gradual-transition case; for an abrupt junction the 1914 pages acknowledge reflection only partly (p.141), and the 1920 book gives the exact abrupt-junction factors (see the Transient entry).

Relevance: The central source on the arc as an energy supplier to oscillation. Because Steinmetz himself calls the mechanism 'practically unknown', later steps must not credit him with a derivation. Bears on arcing grounds, singing-arc and spark-gap oscillators, and negative-resistance oscillators; also the clearest plain-language statement of his traveling-wave and energy-transfer ideas (s, u) used in the transient book.

Mainstream conflict: None for the circuit theory (traveling and standing waves, reflection, surge impedance are standard transmission-line theory). The arc-as-energy-source idea matches the standard picture of the arc as a negative (falling-characteristic) dynamic resistance, as in the Duddell singing arc [memory: high]; Steinmetz in the 1914 pages emphasises a lag/hysteresis loop and does not use the words 'negative resistance' in the pages read; his 1917 book and 1923 paper do give a negative-resistance account (the 1914 loop is then the quadrature signature of a lagged negative resistance, and the lag limits amplitude) [web: arbiter.md claim 2, from the Investigator's finds]. The deep reader notes that the loop 'area' of v against i has units of watts not joules, so the analogy to a B-H loop is loose; this is the reader's own observation, not a claim of the book. No claim in these lectures conflicts with mainstream physics.

Standard AC engineering text, dedicated to the memory of his father Carl Heinrich Steinmetz. The fifth edition split the old 746-page fourth edition into three volumes (this one plus 'Theory and Calculation of Electric Circuits' and 'Theory and Calculation of Electrical Apparatus'). The preface stresses complex quantities (the symbolic method with j = sqrt(-1), developed from his 1893 International Electrical Congress paper) with calculus only where needed. Contents: Section I methods (sine waves, integral values, Ohm and Kirchhoff, symbolic method, topographic method, polar coordinates); II circuits and effective constants (impedance, admittance, hysteresis, eddy currents, dielectric losses and corona, distributed capacity, double-frequency power); IV induction apparatus (transformer, induction motor and generator); V synchronous machines; VI general waves and higher harmonics; VII polyphase systems; appendix on complex algebra. The deep read covered Chapters XIV-XVI. XIV: dielectric loss modelled as effective conductance; a true dielectric hysteresis is 'doubtful', the commonly observed constant power factor implies loss P = n f D^2; a two-layer (leaky/non-leaky) dielectric can reproduce a near-constant power factor; admittivity v = gamma - j beta; air breaks down at 30 kV/cm maximum (21 kV effective); worked glass-plate example; 'energy distance' explanation of corona onset; corona power formula P = a (f+c)(e-e0)^2 attributed to F. W. Peek Jr. XV: distributed capacity in long lines, with numerical tests of when a single lumped condenser suffices (50 km: charging current 2.5 per cent of main current; 150 km: 22 per cent; 300 km: nearly 90 per cent) and the exponential line solution E = A1 e^(+sqrt(ZY) l) + A2 e^(-sqrt(ZY) l). XVI: power as a double-frequency vector, torque as 'imaginary power'. Deep-reader arithmetic checks found printed figures that do not follow from the book's own formulas: the glass-plate gradients (printed 8.4 and 35.7 kV/cm; the printed formula gives 11.4 and 45.5), and the corona formula as printed gives megawatts per kilometre unless a factor of 10^-5 (recalled from Peek, [memory] medium, unverified) is missing. The conclusion of the glass-plate example is unchanged. Second deep read (PDF pp.379-386 and 402-404; printed pp.350-357 and 373-375). The alternating arc is introduced as a 'synchronously pulsating resistance' (para 239-242): with R = r(1 + eps cos 2 beta) the current i = I sqrt2 sin beta gives a third harmonic in the voltage, and a power factor below unity with no phase displacement; the worked Fig. 176 example (I = 12, r = 3, eps = 0.9) prints E = 25.5, P = 244, p = 0.796, where the printed formulas give P = 237.6, p = 0.77 (about 3 per cent off) [calc]. The arc characteristic is stated as e = e0 + c/sqrt(i) (voltage in the arc flame 'decreases with increase of current, approximately inversely proportional to the square root of the current'), with a residual-vapour branch e = r i near zero current; the lower of the two curves is followed [read: p.383, page image]. Hard-carbon example (Fig. 178): e0 = 30 V, c = 40, r = 70 ohm, current i = 13 cos phi + 2 cos 3 phi, effective current 9.30 A, voltage 47.2 V, 439 VA, 'power-factor, 77 per cent'; softer carbons (r = 30 ohm) give 85 per cent. The printed '388 watts' fits no power factor of 77 per cent and is probably a misprint for about 338 [calc]. For metal arcs the starting peak 'frequently reaching values of several thousand volts' (asserted, no measurement cited) [read: p.386]. Fig. 177 (experimental oscillogram) is credited to Tobey and Walbridge, Trans. AIEE 1890, p.376 (Stanley alternate-arc dynamo). The book uses the falling characteristic ONLY to explain wave distortion, a power factor below unity and dangerous arcing-ground voltage peaks; nowhere in pp.379-386 is it called a negative resistance or an energy source for oscillation [read: all of pp.379-386]. Long-line harmonic resonance (para 252-255): a 60-cycle step-up transformer plus line (resistance drops 1 and 10 per cent, reactance 5 and 20 per cent, charging current 20 per cent) gives a voltage rise a = 1/sqrt(1 - 0.059856 m^2 + 0.0009 m^4) for harmonic m; printed table 1.03, 1.36, 3.76, 2.18, 0.70, 0.38 for m = 1, 3, 5, 7, 9, 11, maximum 14.4-fold at m = 5.77 (346 Hz); the printed formula gives 1.37, 3.89, 2.09 for m = 3, 5, 7 (up to 3.5 per cent from the table) while the 14.4 and 346 Hz are reproduced exactly [calc]. Steinmetz judges that 'the danger of resonance in high-potential lines is frequently overestimated' and that harmonics below 7 per cent of the fundamental cannot double the voltage; no field measurement is given. Experimental material in the book: observed V-curves of a synchronous motor (Fig. 163, printed pp.355-356 [read: digest]).

Key claims

• The dielectric loss commonly observed in solid insulation gives an approximately constant power factor over a wide range of voltage and frequency, so the loss is proportional to frequency and to the square of the dielectric flux density (P = n f D^2); a true molecular dielectric hysteresis is doubtful.
• A non-homogeneous dielectric (a leaky layer in series with a non-leaking layer) gives a power factor that rises with frequency, passes a maximum and falls, and so is approximately constant over a range.
• Air at atmospheric pressure breaks down at a maximum field of about 30 kV/cm (21 kV/cm effective), proportional to air density.
• Inserting solid dielectric plates into an air gap raises the field in the air (flux density is continuous, so field is inversely proportional to permittivity) and can make air that would be safe break down as corona or glow rather than arc.
• Corona on a wire begins not when the surface field reaches 30 kV/cm but when that field extends over a finite 'energy distance', so breakdown needs a finite volume of over-stressed air.
• Corona power loss on transmission lines is proportional to (f + 25) and to the square of the excess of voltage over the critical voltage e0 (Peek formula as quoted).
• For lines up to roughly 150 km a single lumped condenser (or condensers at ends and middle) represents distributed capacity adequately; at 300 km the exact exponential solution is needed.
• Alternating-current power is a quantity of double frequency, and torque can be handled as imaginary power in the double-frequency vector scheme.
• The voltage across an alternating arc falls with increasing current approximately as e = e0 + c/sqrt(i), joined near zero current by a residual-vapour resistance branch e = r i; a hard-carbon arc (e0 = 30 V, c = 40, r = 70 ohm) has power factor 77 per cent without any phase displacement.
• The falling arc characteristic makes the alternating arc a 'synchronously pulsating resistance' that distorts the wave (third harmonic, double-peaked voltage), and with metal arcs gives starting peaks frequently of several thousand volts; a double-peaked voltage wave on an oscillogram signals an arc or arcing ground in the circuit.
• On a long line fed through step-up transformers, partial resonance with a higher harmonic can raise the voltage up to 14.4-fold (at 346 Hz in his example; 3.76-fold at the fifth harmonic), but the danger is 'frequently overestimated' and harmonics below 7 per cent of the fundamental cannot double the voltage.

Relevance: Background for the circuit-theory side of the project: effective constants, dielectric losses, corona, distributed line capacity. These are the base for the transient book's four-constant line equations. It also supplies Steinmetz's own statements on losses 'due to radiation' folded into effective r and g. Pages 379-386 and 402-404 were also deep-read: they date Steinmetz's use of the falling arc characteristic e = e0 + c/sqrt(i) to this 1916 book, used for wave distortion and not yet as the energy source of an oscillation (the point of claim 2). Other pages (transformer, motors, synchronous machines other than Fig. 163, polyphase) rest on the digests only.

Mainstream conflict: None for the textbook material. Dielectric loss modelling, corona onset and the line equations are standard for the period. Caveat: the corona formula is quoted from Peek and its printed form appears to lack a scale factor; the book's printed glass-plate numbers contain an arithmetic slip (deep read, own calculation). Neither affects any contested claim. The 1916 arc characteristic and harmonic-resonance results are standard arc and line theory; no conflict. Two printed slips are recorded: Fig. 178 '388 watts' (probably 338) and table values for m = 3, 5, 7.

Steinmetz's main, mathematical treatise on transients, dedicated to the memory of his friend and teacher Rudolf Eickemeyer. It began as a course for Union University seniors. Structure: Section I transients in time (circuit constants, DC and AC circuits, R-L-C in series, surges, mutual inductance, saturation and hysteresis, alternator short circuits); II periodic transients (Tirrill regulator, mechanical and mercury-arc rectifiers); III transients in space (long-distance lines, natural period, high-potential transformer coils, multigap lightning arrester, skin effect in laminae and conductors, velocity of propagation of the electric field, high-frequency conductors); IV transients in time and space (general telegraph-type equations, special cases, impulse currents, standing and traveling waves, reflection at transition points, complex circuit, inductive discharges); V variation of circuit constants (frequency dependence, attenuation, 'degeneration' of waves, flattening of steep fronts); Appendix with tables of sil, col, expl, coll, sill. The third-edition preface says earlier theory took r, L, C and g constant, which disagrees with experience at very high frequency or steep fronts, and that radiation is discussed as effective series and shunt resistance. Deep-read findings. Section IV: starting from p = e i, the four constants r, L, g, C per unit length give d e/dl = r i + L di/dt and d i/dl = g e + C de/dt, hence d2i/dl2 = r g i + (rC+gL) di/dt + LC d2i/dt2 (the telegrapher's equation; algebra confirmed with SymPy). The single stated assumption is constant r, L, g, C; no radiation term appears. Exponential solution gives a^2 = (bL - r)(bC - g), surge impedance z = sqrt(L/C), constants u = (r/L + g/C)/2 (dissipation), m = (r/L - g/C)/2 (distortion), s (energy transfer or wave-front constant), h and k (space attenuation and wave constants), q (frequency constant). Impulses are non-periodic (s^2 > m^2) or periodic in space; the distortionless circuit has m = 0, i.e. r/L = g/C; the wave decrement e^(-ut) along a travelling wave is independent of frequency; a DC analogue uses sqrt(r/g) as the 'surge impedance' of a leaky conductor. The text has no worked numerical example for these chapters. Misprint found: 'h^2 > LC s^2' on p.520 should read h^2 < LC s^2. Section III: quarter-wave resonance f = S/(4 l) (about 47,000/l cycles with l in miles; 783 miles for 60 cycles); worked 200-mile line with exact distributed solution against lumped approximations; quarter-wave line as constant-voltage to constant-current converter, with a transformer-coil example reaching 283 times the supply voltage and a design example transmitting 60,000 kW over 700 miles at 60 cycles (design study only, not stated to be built); natural frequencies of a line discharging to ground f1 = 7.5 x 10^9 / l0 (l0 in cm), independent of wire size and height, tabulated from 4700 cycles at 10 miles to 470 at 100 miles; high-potential transformer coils have 'danger frequencies' of 10,000-30,000 cycles (circuit) and 30,000-100,000 (individual coils), stated in Section III ch. IV with no test cited (see the Section V reading below, where a measurement claim without a reference is made). Multigap-arrester chapter: series-capacity chain model, with a numerical example (300 cylinders, 30,000 V applied, gap voltages 2280-2560 V, sum about 730,000 V, 'one twenty-fourth'). Steinmetz proposes that lightning flashes in clouds are discharges in such a series-capacity circuit between rain drops, so a flash of miles needs only a 'few hundred million volts'; he calls this 'probable' and gives no cloud measurements. Deep-reader checks flagged: the 7.86-wavelength count and the printed alpha and beta values do not match his own eq. (10), and with the stated constants the chain is below series resonance (x = 12.6 < xc = 39.7 ohm), which gives pure exponential decay and a summed gap voltage near 44 kV, not 730 kV (to be confirmed by the Mathematician); in the short-circuit-interruption example the printed 300,000 V should be 150,000 V (pi/4 not pi/2). Chapter VIII of Section III (first summarised by the appendix reader, then deep-read in full in the QE round, see below): inductance, capacity and radiation resistance of conductors computed from a single-delay model of the field (delay l/S), giving sil and col functions; resistance of No. 00 wire radiating at 1,000,000 cycles about 1.95 ohm per metre, 19.5 kW per metre at 100 A. Appendix: sil, col, expl, coll, sill are the cosine-, sine- and exponential-integral family (col = pi/2 - Si, sil = -Ci, expl = -Ei) tabulated from Glaisher (Phil. Trans. 1870, vol. 160); the deep reader found seven formula misprints (e.g. sign in the expl asymptotic series, coll scaling rule, constant in col(+-jx), q = 0.37249680 against 0.37250741) and several table errors (e.g. Table VI col 65 and col 120), none changing a Chapter VIII result. Against the exact cylindrical-wave (Hankel function) result, his single-delay model agrees for the radiation resistance at the wire surface but gives a different far-field fall-off (1/kl against 1/sqrt(kl)). QE remediation deep reads (2026-10-08). SECTION V, Variation of Circuit Constants (PDF pp.652-703). The 3rd-edition preface and sec. 1 state the limitation themselves: constant r, L, g, C 'is not even approximately true for very high frequencies and for very sudden circuit changes, as steep wave front impulses' [read: p.15, 652]. Four reasons from 'experience' are given, none with a reference: spark-gap tests show high gradients only near the origin of a disturbance; oscillograms do not show rectangular waves; high-frequency oscillations die out faster than constant-parameter theory says; they show little tendency to become cumulative. The constants become functions of frequency in four ranges (low up to 10^3 Hz; medium 10^3-10^5, skin effect r1 = x1 proportional to sqrt(f); high 10^5-10^7, magnetic radiation r3 = 8 pi^2 f^2 l' l / S x 10^-9 plus a shunt 'dielectric radiation' resistance r_c = 2 l' S / l x 10^-9; extremely high above 10^7, sil/col forms). His skin-effect constants reproduce the modern formula exactly at 100 kHz (No. 00 copper: 2.743 ohm/km) [calc]. A general solution is called 'not feasible'; he gives each Fourier component its own decay constant u_n = u(f_n). Tables IV-VI (No. 00 copper, 6 ft or 30 ft above ground): the duration N = f/u in cycles peaks at 41.4 cycles at 20 kHz (ground return) and 64.9 cycles at 100 kHz (6 ft line to line), against 9 to 12 cycles at 1 kHz; all checked entries satisfy N = f/u [calc]. DANGER FREQUENCIES (pp.674-676): 'It would thus appear, that this range of frequencies, of 20,000 to 100,000, represents what may be called the danger frequencies of transmission systems. It is interesting to note, that experimental investigations have shown, that the natural frequency of oscillation of the high voltage windings of large power transformers usually is within this range of danger frequencies', and 'This accounts for the not infrequent disastrous experience with such transformers, before this matter was realized.' So the line half is a derivation from his attenuation tables (the band follows from u = m1 sqrt(f) + m2 f^2, f* = (m1/2m2)^(2/3): 27 kHz for the ground-return line, 127 kHz for 6 ft [calc]); the transformer half is a statement that measurements exist, with no experimenter, paper, date, transformer or number named. It is documented in this one place in the 1920 book (plus the 1917 'frequently been observed', sec. 101, per the Investigator); no second source with data has been found in the folder. The band position is set by the radiation coefficient m2 (claim 7): if m2 is reduced 340-fold (the 100 kHz ratio, used as a single scaling factor, a simplification), his own formula moves the peak to about 6.2 MHz (6 ft) and 1.3 MHz (ground) [calc]. The 'about one-half of its energy' retained by a 60 kHz wave after 100 km is closer to 30 per cent for that component (the 70 per cent compares a surviving sine with the original rectangle) [calc]. WAVE-FRONT FLATTENING (Ch. III-IV): the equivalent frequency f = (de/dt)/(2 pi e); the maximum gradient G of a rectangular front falls as 1/sqrt(distance) (eqs. 27-28: G = 8100 sqrt((1/(l' l)) log10(l'/l_r)) V/m at 60 kV, i.e. about 360/sqrt(l) V/m for ground return, l in km), valid to about 30 km and fairly to 100 km, and approaches 1/distance^2 only at great distance (eq. 42). The law reproduces his Tables IX and XI and an independent numerical integral to a few per cent out to 30 km [calc]. Practical conclusions he draws: a close return conductor carries a steep front farther; the most effective way to damp high frequencies is to separate the conductors; iron conductors give 10 to 20 times the attenuation at 1-10 kHz but 'practically no advantage' at very high frequency, so a short iron wire at the station gives little protection against nearby lightning fronts. Real lines also flatten fronts by corona and earth-return resistance (Carson 1926) which his constants omit [memory: high for the mechanisms]; field measurement of front flattening began in the later 1920s [memory: medium]. Preface discrepancy recorded: the preface calls Section V 'six new chapters', this copy has four. SECTION III chs. VIII-IX (pp.431-482). Sec. 69 gives the phase-lag argument: at 1 MHz the 6-ft line has a lag of 2.2 degrees, cos 92.2 deg = 3.8 per cent of the reactance, so r_rad = '630 times the ohmic resistance' (ground return, 22 degrees: '12,600 r0'). The arithmetic is right (635 and 12,400) [calc], but it conflicts with his own exact-integral table of the same chapter: 48 ohm/km (6 ft) and 480 ohm/km (ground) at 10^6 Hz [read: p.444], 3.4 and 6.7 times lower than 162 and 3,240 ohm/km; the cause is that sec. 69 applies the lag at the far edge to all the flux, overstating r/x by about ln(l'/l_r). Eq. 23 (1.97 ohm/m at 1 MHz for an infinitely long uniform-current wire with no return) equals omega mu0/4 exactly [calc]; the two-wire eq. 40 is about 67 times the standard per-conductor value at d = 1.82 m (33 times the loop value of 1.43 ohm/km; the arbiter's factor of about 340 is the ratio at 100 kHz, 0.48 against 1.4e-3 ohm/km, and the ratio changes with frequency because the standard pair result goes as f^3 and his as f^2) [calc]. Tables I-III (resistance, reactance, power factor of No. 00 copper, ribbons and pipes from 1 Hz to 10^8 Hz) were extracted from page images; Table III copper wire 6 ft at 10^5 (r = 0.0052, 0.69 per cent) disagrees with Table I and sec. 91 (0.0032, 0.43 per cent). Claims drawn: iron is almost as good as copper at extreme frequency (supported by his tables: impedance ratio iron/copper 20 at 60 Hz, 1.13 at 10^6, 1.04 at 10^7 for wire, 6 ft); a lightning rod 'probably' behaves like the no-return case, radiating 'more than a hundred times' the ohmic loss as 'induced lightning stroke' (hedged: 'probably', 'approximately'; a hundred-fold holds for the ribbon at 10^4-10^5, not the wire). SECTION IV chs. VIII-XI (pp.602-648). Compound circuit: each section has its own u; the whole loop has one common decrement u0 and each section an energy-transfer constant s = u0 - u, so low-loss sections feed high-loss ones. Worked example (line r1 = 52 ohm, L1 = 0.21 H, C1 = 1.6 microfarad; transformer r2 = 5 ohm, L2 = 3 H, C2 = 0.3 microfarad; 35,000 J stored): u0 = 52.2, s1 = -83.8, s2 = +51.2; the line dissipates 1,800 kW, of which its own field supplies 690 kW and the transformers 1,110 kW; the transformers dissipate 22 kW [read: p.626-628]; the printed table is arithmetically correct within the model [calc]. The approximate common decrement u0 = sum(lambda u)/Lambda (eq. 46; 'Whether this expression (46) is more general is still unknown', p.626) is exact for matched distortionless sections, but for this strongly mismatched loop (surge impedances 362 and 3,162 ohm) an exact eigenvalue calculation gives a different u0 for each of 116 modes, from 8.6 to 114.9 1/s with a mean near 50; the direction of power flow (transformers feed the line) holds in every mode, its size does not (about 240 to 1,015 kW instead of 1,110) [calc, Opus deep reader; scratch scripts not kept]. Reflection and refraction (Ch. X): the 1920 book gives the abrupt-junction factors: voltage transmission 2c2/(c1+c2), current transmission 2c1/(c1+c2), reflection (c2-c1)/(c2+c1), a single-junction maximum of twice the incoming voltage at an open end ('This explains the frequent increase to destructive voltages, when entering a station from the transmission line') [read: p.631-634]; QE wrote the voltage factor as 2c1/(c1+c2), which is the current factor [page image p.634]. The 'law of refraction' tan(angle2)/tan(angle1) = c1/c2 (currents) or c2/c1 (voltages) is continuity of V and I at the junction written for the standing-wave phase, an analogy to Snell's law only in form [calc, memory high]. Steinmetz does not discuss repeated reflections between several junctions. The inductive discharge into a line (Ch. XI) has a sec. 98 table whose open-line fundamental is missing. SECTION I ch. VIII Low frequency surges (pp.142-157): lumped-circuit theory of surges from opening a loaded line or interrupting a short circuit; the 'experimental 10,000-volt, 40-cycle line, when a destructive e.m.f. was produced by a short-circuiting arc, the author observed a drop in generator e.m.f. to about 5000 volts' (p.156). What was observed: the destruction and the voltage drop (by Steinmetz, no date, place, witness or record). What was calculated: i = 707 A, frequency 40 sqrt(20,000/10) = 1,789 Hz, amplitude 316,200 V, from estimated x = 10 ohm and x_c = 20,000 ohm [calc agrees]; no oscillogram or measurement of the 316 kV or the 1,800 Hz is given, so the example is a consistency check, not an independent test. Conclusions 1-4 (p.157): opening under load and interrupting a short circuit are dangerous, 'the most dangerous phenomenon is a low resistance short-circuit in open space', and interruption near zero current gives moderate voltages. SECTION I ch. VI Oscillating currents (pp.104-124): the condenser-discharge oscillation as a frequency source with the spark gap as the switch; frequency independent of kVA; the upper limit about 2 x 10^9 Hz (wavelength 6 cm) with visible light needing spheres 'approaching molecular dimensions' (p.110-111) [calc agrees]. Worked example (sec. 52-53): 15,000 V supply at 0.3735 A (5.6 kVA) feeding a Tesla-transformer circuit with an iron arc (e1 = 22,300 V, C = 0.02 microfarad, r1 = 5 ohm estimated, 833 discharges per second) gives 28.88 A calculated; 'A hot wire ammeter in the discharge circuit in this case showed 29 amp.' (p.123). This is one sentence in one source (Steinmetz himself), with no instrument, date, place or reader named; it checks the energy bookkeeping P = f0 C e1^2/2 = r1 i1^2 [calc: 28.79 A] but not the frequencies (143,000 and 562,000 Hz were not measured), and r1 = 5 ohm is an estimate, so the match is partly a fit. Sec. 54 (p.124): in a 25-cycle 30,000 kW 11,000 V cable system an arc can act as an oscillating-current generator (one sentence, no measurement). SECTION II ch. IV Arc rectification (pp.292-316): the mercury-arc rectifier modelled by a constant counter e.m.f. (13-18 V) with on/off conduction; performance curves 'from tests' (Fig. 66 of that chapter, not the Discharges Fig. 66) and oscillograms Figs. 67-77; at the end of the overlap of two arcs the sudden change of L di/dt rings the circuit capacity at 'a few thousand cycles', and 'if the second half wave of the oscillation reaches below the zero value of the e.m.f. wave ab, the rectifying arc is blown out and a disruptive discharge may result' (stated as a conditional; Fig. 77 shows the ringing, no failure record is given). The chapter shows a discontinuity-driven ringing, not the sustained arc-fed oscillation of claim 1 and not the lag mechanism of claim 2; the frequency is set by the circuit, consistent with 'the arc has no frequency of its own'. Recorded tests and observations in the folder, correcting 'no measurement by Steinmetz beyond oscillograms': the 29 A hot-wire reading (Transient p.123), the rectifier performance curves and oscillograms (p.306-309), the 10,000 V 40-cycle line observation (p.156-157), the observed synchronous-motor V-curves (AC p.355-356) and the transformer inrush oscillogram (p.226-227, 510 A calculated against 4.5 A normal). None tests a contested claim; they show that he measured and compared. Misprints found in the book (no verdict changes): eq. 43 of Ch. XI lacks a factor 10^-8, eq. 45 numbered twice, a '10^11' for 10^10 V/s on p.678, Table III copper wire entry at 10^5, exponent slips in the sec. 76 table, 2.83 A in the 2 x 10^9 Hz sphere example not reproducible, and the p.659 range boundaries disagree with the p.661 prose.

Key claims

• The behaviour of any electric circuit, from a lightning stroke to a house bell, is described by four constants per unit length (r, L, g, C) assumed constant, giving the telegraph-type equations d2i/dl2 = r g i + (rC+gL) di/dt + LC d2i/dt2. Steinmetz himself states that constant constants are valid only for machine and moderately high frequencies and replaces them by frequency-dependent r, L, g, C in Section V (3rd-ed. preface, p.15; sec. 1, p.652).
• Transients decompose into two components in time, e^(-(u-s)t) and e^(-(u+s)t); the first drops energy along the line and the second picks it up and carries it ('energy transfer constant' s, also the 'wave-front constant'); the decrement e^(-ut) of a traveling wave is independent of frequency.
• A distortionless circuit requires r/L = g/C (m = 0): voltage and current impulses then have the same shape and no time displacement.
• A line one quarter wave long (f = S/(4 l)) with negligible loss converts constant voltage to constant current and can give extremely high open-circuit voltages; a transformer-coil example gives 283 times the supply voltage.
• The natural frequency of a line discharging to ground depends only on its length, f1 = 7.5 x 10^9 / l0 with l0 in cm, not on wire size or height. (The f1 formula was already in print in Electrical World, 27 Aug 1898, per the Investigator; it is not first given in the 1920 book.)
• High-potential coils of large transformers have the lowest attenuation of any industrial circuit and danger frequencies of 10,000-30,000 cycles (whole winding) and 30,000-100,000 cycles (single coils) (Section III ch. IV; no test cited there).
• Lightning flashes in clouds are probably oscillatory discharges between rain drops in a circuit with distributed series capacity, and the sum of local gap voltages can then be many times the end-to-end voltage (a 24:1 ratio in his arrester example), so a flash of miles needs only a few hundred million volts.
• The finite velocity of the electric field makes a conductor radiate; the radiation appears as effective resistance proportional to frequency for a wire without return conductor (about 0.2 ohm per metre of No. 00 wire at 100,000 cycles), computed with the sil and col functions. Sec. 69 of the same chapter gives r_rad = 630 r0 at 1 MHz for a 6-ft line (12,600 r0 with ground return), 3.4 times (6.7 times) above his own table value of 48 (480) ohm/km on p.444; the 0.2 ohm per metre is the exact value omega mu0/4 for an infinitely long uniform-current wire with no return, not a real conductor.
• A direct-current circuit with distributed leakage has surge impedance sqrt(r/g); a terminating resistance equal to it gives no reflection.
• The oscillation duration N = f/u of a transmission-line oscillation is greatest at 20,000-100,000 cycles (41 cycles at 20 kHz for ground return, 65 at 100 kHz for a 6-ft line, against 9-12 at 1 kHz), so this band is the 'danger frequencies of transmission systems' where an arc-fed oscillation is most likely to become stationary or cumulative (Section V ch. II, p.674-676).
• 'Experimental investigations have shown' that the natural frequency of the high-voltage windings of large power transformers is usually within 20,000-100,000 cycles, which accounts for the 'not infrequent disastrous experience with such transformers' (Section V, p.674-676; no experimenter, paper, date or number given; single source in the folder).
• The maximum voltage gradient of a rectangular wave front falls inversely as the square root of the distance travelled (G = 8100 sqrt((1/(l' l)) log10(l'/l_r)) V/m at 60 kV, about 360/sqrt(l) V/m for a ground-return line with l in km), valid to about 30 km and fairly to 100 km, and approaches 1/distance^2 at great distance; a short iron wire at a station gives practically no protection against steep fronts (Section V chs. III-IV, eqs. 27-28, 42).
• At a single transition point between sections of surge impedance c1 and c2 the transmitted voltage is 2c2/(c1+c2) times the incoming voltage (current 2c1/(c1+c2)), at most twice at an open end, which 'explains the frequent increase to destructive voltages, when entering a station from the transmission line'; the sections obey a 'law of refraction' tan(angle2)/tan(angle1) = c1/c2 for currents (Section IV ch. X, p.631-638).
• In a compound circuit the sections ring down at a common decrement u0 and exchange power as Ps = 2 s W per section: in the worked example the transformers (u = 1) transfer 1,110 kW to a line that dissipates 1,800 kW and whose own field supplies 690 kW; the approximate u0 = sum(lambda u)/Lambda is flagged by Steinmetz himself as of unknown generality (Section IV ch. IX, p.626-628).
• An experimental 10,000-volt 40-cycle line was destroyed in the author's observation by a short-circuiting arc, with the generator voltage dropping to about 5,000 volts; the calculated oscillation is about 1,800 cycles at nearly 316,000 volts (Section I ch. VIII sec. 70, p.156-157; the 316 kV is a calculation from estimated constants, not a measurement).
• The hot-wire ammeter in the discharge circuit of the Tesla-transformer and iron-arc example read 29 A against 28.88 A calculated from stored energy times discharge rate (Section I ch. VI, p.123; single sentence, no instrument or date; r1 = 5 ohm is estimated).

Relevance: The mathematical backbone of the project: the general line equations, the s/u/m constants used in the discharge lectures, the cloud-lightning series-capacity hypothesis, and the radiation/finite-velocity chapters, which are the only place Steinmetz discusses field propagation beyond circuit theory. Deep reads now cover the derivation (Section IV chs. I-III and VIII-XI), the line and lightning chapters (Section III I-V), the radiation and high-frequency conductor chapters (III VIII-IX), Section V in full, Section I chapters VI and VIII, the mercury-arc rectifier chapter (II IV) and the appendix. Section V is where the radiation resistance of claim 7 does real work: it fixes the 20-100 kHz danger band and the flattening coefficients. Remaining chapters of Sections I-III rest on digests only.

Mainstream conflict: The circuit theory (telegrapher's equations, surge impedance, quarter-wave resonance, distortionless line) is standard and was confirmed algebraically by the deep reader. Points of possible tension, stated for the reviewers: (1) the cloud-lightning model is a hypothesis labelled 'probable' with no measurements, and the deep reader's recalculation suggests the worked 24:1 example does not follow with the stated constants at 100 MHz; (2) the single-delay radiation model agrees with the cylindrical-wave solution only near the wire (far field differs); (3) several printed numerical slips (150,000 vs 300,000 V; wavelength count; appendix misprints). No claim here contradicts established experiment as read; the cloud mechanism is not the mainstream account of lightning [memory: medium, not checked by the reader]. Update after the QE deep reads: (4) Section V imports the radiation coefficient of claim 7 into r and g; the arbiter found the two-wire coefficient about 340 times too large for real lines, so the high-frequency side of Tables IV-VI, the position of the 20-100 kHz band and the coefficients of eqs. 27-28 do not survive that correction (the skin-effect side, the qualitative flattening and the junction factors are standard) [calc, deep reader]; (5) the constant-parameter assumption is not a hidden assumption but Steinmetz's stated scope, lifted in Section V by frequency-dependent constants used harmonic by harmonic (a legitimate frequency-domain method for a linear line [memory: high]); (6) eq. 46 for u0 is exact only for matched distortionless sections and is a mode-average otherwise [calc]; (7) the transformer-winding half of the danger-frequency statement does not depend on the radiation model and is consistent in kind with the published winding-resonance literature the Contrarian cited (tens of kHz to MHz) [web: https://www.ipstconf.org/papers/Proc_IPST2015/15IPST031.pdf, search snippet only].