11. CAPACITY AND CONDENSERS 51. The charge of an electric condenser is proportional to the impressed voltage, that is, potential difference at its terminals, and to its capacity. A condenser is said to have unit capacity if unit current exist- ing for one second produces unit difference of potential at its terminals. The practical unit of capacity is that of a condenser in which 1 amp. during one second produces 1 volt difference of potential. The practical unit of capacity equals 10~9 absolute units. It is called a farad. One farad is an extremely large capacity, and therefore one millionth of one farad, called microfarad, mf., is commonly used. If an alternating e.m.f. is impressed upon a condenser, the charge of the condenser varies proportionally to the e.m.f., and CAPACITY AND CONDENSERS 55 thus there is current to the condenser during rising and from the condenser during decreasing e.m.f., as shown in Fig. 26. That is, the current consumed by the condenser leads the impressed e.m.f. by 90 time degrees, or a quarter of a period. Denoting / as frequency and E as effective alternating e.m.f. impressed upon a condenser of C'mf. capacity, the condenser is charged and discharged twice during each cycle, and the time of one complete charge or discharge is therefore j^- Since E \/2 is the maximum voltage impressed upon the con- denser, an average of CE \/2 10~6 amp. would have to exist during one second to charge the condenser to this voltage, and FIG. 26. — Charging current of a condenser across which an alternating e.m.f. is impressed. to charge it in j^ seconds an average current of 4 fCE \/2 10~6 amp. is required. effective current TT Since average current 2\/2' the effective current is I = 2-irfCE 10~6; that is, at an impressed e.m.f. of E effective volts and frequency /, a condenser of C mf. capacity consumes a current of 1 = 2 irfCE 10~6 amp. effective, which current leads the terminal voltage by 90 degrees or a quarter period. Transposing, the e.m.f. of the condenser is 106/ 106 The value z0 = fn is called the condensive reactance of the ^ 7T/C condenser. 56 ELEMENTS OF ELECTRICAL ENGINEERING Due to the energy loss in the condenser by dielectric hysteresis, the current leads the e.m.f. by somewhat less than 90 time de- grees, and can be resolved into a wattless charging current and a dielectric hysteresis current, which latter, however, is generally so small as to be negligible, though in underground cables of poor quality, it may reach as high as 50 per cent, or more of the charging or wattless current of the condenser. 52. The capacity of one wire of a transmission line is i.nxio-6x/ . C = - — ~-i - , in mf., where Id = diameter of wire, cm.; 18 — distance of wire from return wire, cm.; I = length of wire, cm., and 1.11 X 10~6 = reduction coefficient from electrostatic units to mf . The logarithm is the natural logarithm; thus in common loga- rithms, since loge a = 2.303 logio a, the capacity is 0.24 X 10~6 X I i ^ t>s logio -7- I'd . , , in mf . The derivation of this equation must be omitted here. The charging current of a line wire is thus 1 = 2 7T/CE 10~6, where / = the frequency, in cycles per second, E = the difference of potential, effective, between the line and the neutral (E — y^ line voltage in a single-phase, or four-wire quarter-phase sys- tem, — -i=. line voltage, or Y voltage, in a three-phase system). V 3 EXAMPLES 53. In the transmission line discussed in the examples in 37, 38, 41 and 45, what is the charging current of the line at 6000 volts between lines, at 33.3 cycles? How many volt-amperes does it represent, and what percentage of the full-load current of 44 amp. is it? The length of the line is, per wire, I = 2.23 X 106 cm. The distance between wires, ls = 45 cm. The diameter of transmission wire, Id = 0.82 cm. Thus the capacity, per wire, is C = - . 0.26 mf. 1 <£ ts loglo -• IMPEDANCE OF TRANSMISSION LINES 57 The frequency is / = 33.3, The voltage between lines, 6000. Thus per line, or between line and neutral point, E = = 3460 volts; hence, the charging current per line is Jo = 2 irfCE 10 ~6 = 0.19 amp., or 0.43 per cent, of full-load current; that is, negligible in its influence on the transmission voltage. The volt-ampere input of the transmission is, 3 IQE = 2000 = 2.0 kv-amp.