The book History of Induction by Charles Page assessed the later improvements to the induction coil as a series of advances in each of it's component parts. Btu crucially, it's power and thus versatility came from advances in the circuit breaker. So page itemizes a set of twelve, which he overwhelmingly he attributed to himself. By his own count he invented 15 different kinds of circuit breakers before 1840. I'll list them fully because it's actually pretty important.
First: The first mechanical circuit-breaker used with an induction coil (M. Masson)
Second: The first automatic or self-acting circuit-breaker used with an induction coil (C. Page)
Third: The first circuit-breaker used with an induction coil having a primary and secondary circuit. (C. Page)
Fourth: The first independent automatic circuit-breaker used with an induction coil (C. Page)
Fifth: The first attached automatic circuit-breaker used with an induction coil (J. McGauley)
Sixth: The first adjustable automatic circuit-breaker used with an induction coil (C. Page)
Seventh: The first mercurial circuit-breaker used with an induction coil (C. Page)
Eighth: The first automatic attached circuit-breaker, in which the retractile force of the hammer was adjustable(C. Page).
Ninth: The first spark-arresting circuit-breaker used with an induction coil (C. Page)
Tenth: The first attached circuit-breaker which combined the means of adjusting the retractile force of the hammer and its distance from the magnet. (C. Page)
Eleventh: The first automatic circuit-breaker with an induction coil for remedial purposes (C. Page)
Page certainly thought highly of his own contributions. But he was very keen on crediting McGauley and Masson (but not Callan or anyone else). There is a reason for this. Page wrote the History of Induction in 1867. In 1864 Ruhmkorff was awarded the Volta Prize a 50,000-franc prize by Napoleon III for the most important discovery in the application of electricity - the Induction Coil. Page was furious and righteously; his book was an act of vengeance. When he wrote the history he was trying to certify his own significance but also to shame Ruhmkorff, and those that awarded him. Ruhmkorff's act of invention was a myth. The induction coil had already been available at retail for a decade when he "invented" it. In 1872 Jean Baptiste Alexandre Baille wrote his own history of the Induction coil and concluded:
Ruhmkorff patented his first induction coil in 1851. It used long copper windings and could produce a 2 inch spark. In 1859 a Mr. Gassoit published in France, an account of the Edward Samuel Ritchie coils. Then Ruhmkorff had the opportunity to examine a superior induction coil developed by the American inventor Ritchie in 1860. A Professor McCulloh of Columbia College brought a Ritchie Coil to Paris. Ruhmkorff examined it and that led him (understandably) to incorporate some of Ritchie's improvements and to change his induction coil. His improvements included glass insulation, larger coils, a current reverser (Rheotrope) to change the direction of the primary circuit, and he also added a capacitor."With the exception of Fizeau's condenser, the electrostatic coil is an American invention, and the so-called Ruhnikorff coil was commenced and perfected in the United States, Professor Page developing the electrostatic properties, and Edward S. Ritchie, of Boston, consummating the perfection of the instrument, to an extent far exceeding anything known in Europe."
The improvements were all from page, and Ritchie except for two: The Rheotrope was an invention of M. Masson from 1834. The capacitor (condenser) has two different etymologies. It was French physicist, Armand Hippolyte Louis Fizeau recommended using the capacitor. But that development also was not wholly original. In 1853 Jonathan Nash Hearder exhibited an induction coil with a capacitor. Though little information is available, presumably that capacitor resonated the secondary coil producing yet more voltage. Fizeau developed that improvement separately and patented it in 1853. The awarded coil could produce sparks over 12 inches long.
Why does all this matter to radio? The history of the induction coil is the history of the spark gap transmitter. The devices are synonymous. In 1895 Marconi's first transmitter consisted of an induction coil connected between a wire antenna and ground, with a spark gap across it. The most basic spark-gap transmitter is a spark gap connected across an oscillatory circuit consisting of a capacitor and an inductor in series or parallel. I'll cite an obscure article by Frank Butler, the Chief Assistant to Dr. Lee De Forest wrote an article in 1924 titled Making Wireless History for Radio Broadcast Magazine.
"The apparatus for sending was a Ruhmkorff induction coil with a vibrator on one end. Direct current was used in the coil and the vibrator converted it into alternating current of slow oscillations as compared with those used to-day. The power used then to send six miles would to-day send almost six thousand."
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