Diameter 0.148" 0.375 cm. Curves of impedance and permeability observed in a galvanized iron wire No. 9, B. W. G. drawn into one straight loop 2690 cms. long and 10 cms. interaxial distance. Specific gravity of wire, 7.76. Impedance of loop compared with a loop of german silver wire of diam. 0.28 cm. 1,000 cms. long and 7 cms interaxial distance by differential dynamometer. E. M. F. of generator very closely sinusoidal. Frequency 140 Temp. of wire under test 10° C. Resistance of iron wire per linear cm. 208 microhms. Resistivity 12.09 microhms at 10° C. ~. Curve I., Impedance factor as function of current; II., Permeability as function of H; III., Reluctivity as function of H. where GS is the arithmetical product, but G+ Sa vector sum. This case could also be dealt with by the rule for joint imped where the vector reciprocals are added vectorially. All continuous current corollaries from Ohm's law thus become applicable to harmonic currents, when the sums and differences of impedances or their reciprocals are treated vectorially, and if all alternating currents were strict sinusoids, all inductances non-ferric, there would be almost as little difficulty in reckoning their quantitative relations as in continuous current circuits. But Kirchoff's laws, while true instantaneously, are only applicable to alternating current circuits, when the currents are all in step, and the vector sums degrade into arithmetical sums. Algebraically, all, and more than all the foregoing statements concerning impedance combinations are contained in the following: Any combination of resistances, non-ferric inductances, and capacities, carrying harmonically alternating currents, may be treated by the rules of unvarying currents, if the inductances are considered as resistances of the form pl-1, and the capacities as resistances of the form 1 k p 1, the algebraic operations being then performed according to the laws controlling "complex quantities." A valuable application of the principles of impedance is to the determination of the limits of error in measuring apparatus-such as voltmeters-intended for both alternating and continuous current circuits. For example, a particular sample of alternating and continuous current voltmeter, consisting of a dynamometer without iron, in circuit with a resistance coil, has a resistance of 2,085 ohms, and an inductance that varies with the relative position of the dynamometer coils, but which does not exceed 91.5 millihenrys = 0.0915 henry. At a frequency of 140 which is about the maximum in ordinary use, the inductance speed of this instrument will not be greater than 0.0915 × 6.283 x 140 80.5. Constructing the right-angled triangle with base 2085, and perpendicular 80.5, we find the hypothenuse to be 20852 + 80.52 = 2086.5 which is the impedance of the instrument at this frequency, so that 50 volts E. M. F. harmonically alternating would force less current through the dynamometer than 50 volts continuous E. M. F. in the 2085 ratio of and would therefore under-indicate by about 2086.5 0.075 per cent. A similar instance is afforded by the Thomson wattmeter. The armature of this instrument has a certain non-ferric inductance and is connected across the supply mains through a resistance. The equality of the meter's record for both alternating and continuous current circuits depends upon the equality of current strength through the armature under equal alternating and continuous voltages, and this is again dependent on the impedance of the armature circuit. In a particular instrument of 1,500 watts capacity (50 volts and 30 amperes) the total resistance of this circuit is 455.5 ohms, and its total inductance 25 millihenrys (0.025 henry). This represents at 140~ an inductance-speed of 22, and an impedance of 456.0 ohms W √ (455.5)2 + (22)2 so that the current in the armature will be about one-ninth of one per cent. less, on a supply pressure of 50 volts, alternating harmonically, than on 50 volts continuous. Strictly speaking the accuracy of a wattmeter depends not only upon the non-diminution of current strength through fine wire circuit in virtue of inductance, but also upon the lag in phase that is also there established. It is generally safe to assume, however, that when the error due to impedance is very small, the error due to lag is very small also. = An important application of impedance principles is to conductors for the distribution of alternating currents. Let the common case be considered of two parallel overhead wires running from a central station to a consumption point one mile distant, each wire of copper, No. 4, a. w. g., 0.2043 in. (0.519 cm.) in diameter, fixed on pole insulators, at a distance between their axes of 30 inches. At a temperature of 15° C this wire of standard conductivity, would have a resistance of 1.285 ohms per mile of 5,280 feet, and if the wire actually possessed this conductivity, the resistance of the loop would be 2.57 ohms. unvarying current of 10 amperes would establish in this wire a An TABLE I. Impedance factors for pairs of parallel copper wires carrying simple harmonic currents of 140 ~ factor. factor. cms. D. D. factor. factor. cms. 1.351 200, 1.736 1.658 150. 1.756 270. 2.151 175. 2.778 200. 3.528 2.847 240. 3.622 270. 225. 4.40 250. 5.383 2.904 a 999 320. 280. 3.699 315. 4.617 3.766 360. 3.826 405. 4.518 300. 5.530 350. 5.654 4.704 400. 5.675 4.780 450. 5.858 600. 1.773 300. 2.333 350. "drop" of 12.85 volts on each conductor, or 25.7 volts in the circuit. But if 10 amperes are supplied harmonically at a frequency of 120~, the "drop" will no longer be 12.85 volts on each wire, but 19.27, or almost 50 per cent. more, making the total "drop" in the loop 38.54 volts. The impedance of the loop is one and onehalf times its resistance, or the impedance factor is in this case 1.5. The additional drop in the wires is owing to the inductance of the loop. The magnetic flux from the current moves back and forth through this loop at each pulse of the current, and establishes a back pressure or counter E. M. F. that compounds with the ordinary er drop, rectangularly. No attention is here paid to the static capacity of the line, which is indeed of minor inportance in overhead wires of only a few miles in length, but whose influence on the impedance factor increases with the capacity of the line, the frequency, and the E. M. F. In order to reduce the impedance of our pair of overhead wires as far as possible, we must bring them close together, so that the loop may hold less flux, and the counter E. M. F. from the oscillation of this, will be correspondingly lessened. A safe practical distance might be ten inches (25.4 cms.) apart. By bringing them to this distance their impedance would be reduced to 34.7 ohms for the whole loop, and the impedance factor from 1.5 to |
