CHAPTER I. THE CONSTANTS OF THE ELECTRIC CIRCUIT. 1. To transmit electric energy from one place where it is generated to another place where it is used, an electric cir- cuit is required, consisting of conductors which connect the point of generation with the point of utilization. When electric energy flows through a circuit, phenomena take place inside of the conductor as well as in the space out- side of the conductor. In the conductor, during the flow of electric energy through the circuit, electric energy is consumed continuously by being converted into heat. Along the circuit, from the generator to the receiver circuit, the flow of energy steadily decreases by the amount consumed in the conductor, and a power gradi- ent exists in the circuit along or parallel with the conductor. (Thus, while the voltage may decrease from generator to receiver circuit, as is usually the case, or may increase, as in an alternating-current circuit with leading current, and while the current may remain constant throughout the circuit, or decrease, as in a transmission line of considerable capacity with a leading or non-inductive receiver circuit, the flow of energy always decreases from generating to receiving circuit, and the power gradient therefore is characteristic of the direc- tion of the flow of energy.) In the space outside of the conductor, during the flow of energy through the circuit, a condition of stress exists which is called the electric field of the conductor. That is, the surrounding space is not uniform, but has different electric and magnetic properties in different directions. No power is required to maintain the electric field, but energy 3 4 TRANSIENT PHENOMENA is required to produce the electric field, and this energy is returned, more or less completely, when the electric field dis- appears by the stoppage of the flow of energy. Thus, in starting the flow of electric energy, before a perma- nent condition is reached, a finite time must elapse during which the energy of the electric field is stored, and the generator therefore gives more power than consumed in the conductor and delivered at the receiving end; again, the flow of electric energy cannot be stopped instantly, but first the energy stored in the electric field has to be expended. As result hereof, where the flow of electric energy pulsates, as in an alternating- current circuit, continuously electric energy is stored in the field during a rise of the power, and returned to the circuit again during a decrease of the power. The electric field of the conductor exerts magnetic and elec- trostatic actions. The magnetic action is a maximum in the direction concen- tric, or approximately so, to the conductor. That is, a needle- shaped magnetizable body, as an iron needle, tends to set itself in a direction concentric to the conductor. The electrostatic action has a maximum in a direction radial, or approximately so, to the conductor. That is, a light needle- shaped conducting body, if the electrostatic component of the field is powerful enough, tends to set itself in a direction radial to the conductor, and light bodies are attracted or repelled radially to the conductor. Thus, the electric field of a circuit over which energy flows has three main axes which are at right angles with each other: The electromagnetic axis, concentric with the conductor. The electrostatic axis, radial to the conductor. The power gradient, parallel to the conductor. This is frequently expressed pictorially by saying that the lines of magnetic force of the circuit are concentric, the lines of electrostatic force radial to the conductor. Where, as is usually the case, the electric circuit consists of several conductors, the electric fields of the conductors super- impose upon each other, and the resultant lines of magnetic and of electrostatic forces are not concentric and radial respec- tively, except approximately in the immediate neighborhood of the conductor. THE CONSTANTS OF THE ELECTRIC CIRCUIT 5 In the electric field between parallel conductors the magnetic and the electrostatic lines of force are conjugate pencils of circles. 2. Neither the power consumption in the conductor, nor the electromagnetic field, nor the electrostatic field, are pro- portional to the flow of energy through the circuit. The product, however, of the intensity of the magnetic field, , and the intensity of the electrostatic field, "^ is proportional to the flow of energy or the power, P, and the power P is there- fore resolved into a product of two components, i and e, which are chosen proportional respectively to the intensity of the magnetic field and of the electrostatic field V. That is, putting P = ie (1) we have = Li = the intensity of the electromagnetic field. (2) Mf = Ce = the intensity of the electrostatic field. (3) The component i, called the current, is defined as that factor of the electric power P which is proportional to the magnetic field, and the other component e, called the voltage, is defined as that factor of the electric power P which is proportional to the electrostatic field. Current i and voltage e, therefore, are mathematical fictions, factors of the power P, introduced to represent respectively the magnetic and the electrostatic or " dielectric " phenomena. The current i is measured by the magnetic action of a circuit, as in the ammeter; the voltage e, by the electrostatic action of a circuit, as in the electrostatic voltmeter, or by producing a current i by the voltage e and measuring this current i by its magnetic action, in the usual voltmeter. The coefficients L and (7, which are the proportionality factors of the magnetic and of the dielectric component of the electric field, are called the inductance and the capacity of the circuit, respectively. As electric power P is resolved into the product of current i and voltage e, the power loss in the conductor, Ph therefore can also be resolved into a product of current i and voltage et which is consumed in the conductor. That is, P, = iet. 6 TRANSIENT PHENOMENA It is found that the voltage consumed in the conductor, eh is proportional to the factor i of the power P, that is, et = ri, (4) where r is the proportionality factor of the voltage consumed by the loss of power in the conductor, or by the power gradient, and is called the resistance of the circuit. Any electric circuit therefore must have three constants, r, L, and (7, where r = circuit constant representing the power gradient, or the loss of power in the conductor, called resistance. L = circuit constant representing the intensity of the electro- magnetic component of the electric field of the circuit, called inductance. C = circuit constant representing the intensity of the electro- static component of the electric field of the circuit, called capacity. 3. A change of the magnetic field of the conductor, that is, of the number of lines of magnetic force surrounding the conductor, generates an e.m.f. '-3 <•> in the conductor and thus absorbs a power *"-*-<£ (6) or, by equation (2) : = Li by definition, thus : d& , di i r>/ T • di - = L-,and: P' = Lt-, (7) and the total energy absorbed by the magnetic field during the rise of current from zero to i is WM --P'dt (8) = LJidi, that is, ,, WM - (9) THE CONSTANTS OF THE ELECTRIC CIRCUIT 1 A change of the dielectric field of the conductor, ^, absorbs a current proportional to the change of the dielectric field : and absorbs the power or, by equation (3), P»=ei'=e—, (11) (12) and the total energy absorbed by the dielectric field during a rise of voltage from 0 to 6 is WK ==p"dt (13) = cfede, that is *n The power consumed in the conductor by its resistance r is Pr = ieh (15) and thus, by equation (4), Pr = tV. (16) That is, when the electric power P = ei (1) exists in a circuit, it is pr = tfr = power lost in the conductor, (16) WM = l— = energy stored in the magnetic field of the circuit, (9) l Ll W K = — = energy stored in the dielectric field of the cir- £t cuit, (14) 8 TRANSIENT PHENOMENA and the three circuit constants r, L, C therefore appear as the components of the energy conversion into heat, magnetism, and electric stress, respectively, in the circuit. 4. The circuit constant, resistance r, depends only on the size and material of the conductor, but not on the position of the conductor in space, nor on the material filling the space surrounding the conductor, nor on the shape of the conductor section. The circuit constants, inductance L and capacity (7, almost entirely depend on the position of the conductor in space, on the material filling the space surrounding the conductor, and on the shape of the conductor section, but do not depend on the material of the conductor, except to that small extent as represented by the electric field inside of the conductor section. 5. The resistance r is proportional to the length and inversely proportional to the section of the conductor, r = plT' (IV) J\. where p is a constant of the material, called the resistivity or specific resistance. For different materials, p varies probably over a far greater range than almost any other physical quantity. Given in ohms per centimeter cube,* it is, approximately, at ordinary tem- peratures : Metals: Cu .................. ........... 1 . 6 X lO"6 Al. .., ......................... 2.5 X 10-6 Fe .............................. 10 X 10-6 Hg ............................. 94 X 10-6 Gray cast iron .............. up to 100 X 10~6 High-resistance alloys ....... up to 150 X 10~8 Electrolytes: N03H ............. down to 1 .3 at 30 per cent KOH ............. down to 1 . 9 at 25 per cent NaCL . ............ down to 4 . 7 at 25 per cent up to Pure river water ....................... . 104 and over alcohols, oils, etc., to practically infinity. Meaning a conductor of one centimeter length and one square centimeter * section. THE CONSTANTS OF THE ELECTRIC CIRCUIT 9 So-called ' ' insulators" : Fiber about 1012 Paraffin oil about IQ» Paraffin about 1014 to 1016 Mica about 1014 G1ass about 1014 to 1016 Rubber about 1016 Air practically oo In the wide gap between the highest resistivity of metal alloys, about p = 150 X 10~6, and the lowest resistivity of electrolytes, about p = 1, are Carbon: metallic down to 100 X 10~6 amorphous (dense) 0.04 and higher anthracite very high Silicon and Silicon Alloys: Cast silicon 1 down to 0 . 04 Ferro silicon 0.04 down to 50 X 10~a The resistivity of arcs and of Geissler tube discharges is of about the same magnitude as electrolytic resistivity. The resistivity, p, is usually a function of the temperature, rising slightly with increase of temperature in metallic conduct- ors and decreasing in electrolytic conductors. Only with few materials, as silicon, the temperature variation of p is so enor- mous that p can no longer be c.onsidered as even approximately constant for all currents i which give a considerable tempera- ture rise in the conductor. Such materials are commonly called pyroelectrolytes. 6. The inductance L is proportional to the section and inversely proportional to the length of the magnetic circuit surrounding the conductor, and so can be represented by L = ^ (18) where /* is a constant of the material filling the space surround- ing the conductor, which is called the magnetic permeability. As in general neither section nor length is constant in differ- ent parts of the magnetic circuit surrounding an electric con- 10 TRANSIENT PHENOMENA ductor, the magnetic circuit has as a rule to be calculated piecemeal, or by integration over the space occupied by it. The permeability, /*, is constant and equals unity or very closely fj. = 1 for all substances, with the exception of a few materials which are called the magnetic materials, as iron, cobalt, nickel, etc., in which it is very much higher, reaching sometimes and under certain conditions in iron values as high as fjL = 6000. In these magnetic materials the permeability /* is not con- stant but varies with the magnetic flux density, or number of lines of magnetic force per unit section, (B, decreasing rapidly for high values of (B. In such materials the use of the term /* is therefore incon- venient, and the inductance, L, is calculated by the relation between the magnetizing force as given in ampere-turns per unit length of magnetic circuit, or by "field intensity," and magnetic induction (B. The magnetic induction (B in magnetic materials is the sum of the "space induction" 3C, corresponding to unit permeability, plus the "metallic induction" (B', which latter reaches a finite limiting value. That is, (B = 3C + (&'. (19) The limiting values, or so-called "saturation values," of (B' are approximately, in lines of magnetic force per square centi- meter: Iron .20,000 Cobalt 12,000- Nickel 6,000 Magnetite 5,000 Manganese alloys up to 4,000 The inductance, L, therefore is a constant of the circuit if the space surrounding the conductor contains no magnetic material, and is more or less variable with the current, i, if magnetic material exists in the space surrounding the conductor. In the latter case, with increasing current, i, the inductance, L, first slightly increases, reaches a maximum, and then decreases, approaching as limiting value the value which it would have in the absence of the magnetic material. THE CONSTANTS OF THE ELECTRIC CIRCUIT 11 7. The capacity, C, is proportional to the section and inversely proportional to the length of the electrostatic field of the con- ductor: ^ *A G = T, (20) where K is a constant of the material filling the space surround- ing the conductor, which is called the "dielectric constant," or the " specific capacity/' or " permittivity." Usually the section and the length of the different parts of the electrostatic circuit are different, and the capacity therefore has to be calculated piecemeal, or by integration. The dielectric constant K of different materials varies over a relative narrow range only. It is approximately: AC = 1 in the vacuum, in air and in other gases, K = 2 to 3 in oils, paraffins, fiber, etc., K = 3 to 4 in rubber and gutta-percha, K = 3 to 5 in glass, mica, etc., reaching values as high as 7 to 8 in organic compounds of heavy metals, as lead stearate, and about 12 in sulphur. The dielectric constant, /c, is practically constant for all voltages e, up to that voltage at which the electrostatic field intensity, or the electrostatic gradient, that is, the "volts per centimeter," exceeds a certain value d, which depends upon the material and which is called the "dielectric strength" or "disruptive strength" of the material. At this potential gradient the medium breaks down mechanically, by puncture, and ceases to insulate, but electricity passes and so equalizes the potential gradient. The disruptive strength, d, given in volts per centimeter is approximately : Air: 32,000. Oils: 250,000 to 1,000,000. Mica: up to 4,000,000. The capacity, (7, of a circuit therefore is constant up to the voltage e, at which at some place of the electrostatic field the dielectric strength is exceeded, disruption takes place, and a part of the surrounding space therefore is made conducting, and by this increase of the effective size of the conductor the capacity C is increased. 12 TRANSIENT PHENOMENA 8. Of the amount of energy consumed in creating the electric field of the circuit not all is returned at the disappearance of the electric field, but a part is consumed by conversion into heat in producing or in any other way changing the electric field. That is, the conversion of electric energy into and from the electromagnetic and electrostatic stress is not complete, but a loss of energy occurs, especially with the magnetic field in the so-called magnetic materials, and with the electrostatic field in unhomogeneous dielectrics. The energy loss in the production and reconversion of the magnetic component of the field can be represented by an effective resistance rr which adds itself to the resistance r0 of the conductor and more or less increases it. The energy loss in the electrostatic field can be represented by an effective resistance r" , shunting across the circuit, and consuming an energy current i" , in addition to the current i in the conductor. Usually, instead of an effective resistance r", its reciprocal is used, that is, the energy loss in the electro- static field represented by a shunted conductance g. In its most general form the electric circuit therefore contains the constants : 1. Inductance L, storing 'the energy, -— , ft 2. Capacity C, storing the energy, - — > & 3. Resistance r = r0 + r', consuming the power, tfr = ? 4. Conductance g} consuming the power, ezg, where r0 is the resistance of the conductor, r' the effective resist- ance representing the power loss in the magnetic field L, and g represents the power loss in the electrostatic field C. 9. If of the three components of the electric field, the electro- magnetic stress, electrostatic stress, and the power gradient, one equals zero, a second one must equal zero also. That is, either all of the three components exist or only one exists. Electric systems in which the magnetic component of the field is absent, while the electrostatic component may be consider- able, are represented for instance by an electric generator or a battery on open circuit, or by the electrostatic machine. In such systems the disruptive effects due to high voltage, there- THE CONSTANTS OF THE ELECTRIC CIRCUIT 13 fore, are most pronounced, while the power is negligible, and phenomena of this character are usually called " static." Electric systems in which the electrostatic component of the field is absent, while the electromagnetic component is consider- able, are represented for instance by the short-circuited secondary coil of a transformer, in which no potential difference and, there- fore, no electrostatic field exists, since the generated e.m.f. is consumed at the place of generation. Practically negligible also is the electrostatic component in all low-voltage circuits. The effect of the resistance on the flow of electric energy in industrial applications is restricted to fairly narrow limits: as the resistance of the circuit consumes power and thus lowers the efficiency of the electric transmission, it is uneconomical to permit too high a resistance. As lower resistance requires a larger expenditure of conductor material, it is usually uneco- nomical to lower the resistance of the circuit below that which, gives a reasonable efficiency. As result hereof, practically always the relative resistance, that is, the ratio of the power lost in the resistance to the total power, lies between 2 per cent and 20 per cent. It is different with the inductance L and the capacity C. Of the two forms of stored energy, the magnetic — and electro- e*C static , usually one is so small that it can be neglected com- fk pared with the other, and the electric circuit with sufficient approximation treated as containing resistance and inductance, or resistance and capacity only. In the so-called electrostatic machine and its applications, frequently only capacity and resistance come into consideration. In all lighting and power distribution circuits, direct current or alternating current, as the 110- and 220-volt lighting circuits, the 500-volt railway circuits, the 2000-volt primary distribution circuits, due to the relatively low voltage, the electrostatic energy - ^ is still so very small compared with the electro- 2 magnetic energy, that the capacity C can for most purposes be neglected and the circuit treated as containing resistance and inductance only. 14 TRANSIENT PHENOMENA Of approximately equal magnitude is the electromagnetic energy —— and the electrostatic energy - ^ in the high-potential Zi iL long-distance transmission circuit, in the telephone circuit, and in the condenser discharge, and so in most of the phenomena resulting from lightning or other disturbances. In these cases all three circuit constants, r, L, and C, are of essential impor- tance. 10. In an electric circuit of negligible inductance L and negligible capacity C, no energy is stored, and a change in the circuit thus can be brought about instantly without any disturb- ance or intermediary transient condition. In a circuit containing only resistance and " capacity , as a static machine, or only resistance arid inductance, as a low or medium voltage power circuit, electric energy is stored essentially in one form only, and a change of the circuit, as an opening of the circuit, thus cannot be brought about instantly, but occurs more or less gradually, as the energy first has to be stored or discharged. In a circuit containing resistance, inductance, and capacity, and therefore capable of storing energy in two different forms, the mechanical change of circuit conditions, as the opening of a circuit, can be brought about instantly, the internal energy of the circuit adjusting itself to the changed circuit conditions by a transfer of energy between static and magnetic and inversely, that is, after the circuit conditions have been changed, a transient phenomenon, usually of oscillatory nature, occurs in the circuit by the readjustment of the stored energy. These transient phenomena of the readjustment of stored electric energy with a change of circuit conditions require careful study wherever the amount of stored energy is sufficiently large to cause serious damage. This is analogous to the phenomena of the readjustment of the stored energy of mechanical motion: while it may be harmless to instantly stop a slowly moving light carriage, the instant stoppage, as by collision, of a fast railway train leads to the usual disastrous result. So also, in electric systems of small stored energy, a sudden change of circuit con- ditions may be safe, while in a high-potential power system of very great stored electric energy any change of circuit conditions requiring a sudden change of energy is liable to be destructive. THE CONSTANTS OF THE ELECTRIC CIRCUIT 15 Where electric energy is stored in one form only, usually little danger exists, since the circuit protects itself against sudden change by the energy adjustment retarding the change, and only where energy is stored electrostatically and magnetically, the mechanical change of the circuit conditions, as the opening of the circuit, can be brought about instantly, and the stored energy then surges between electrostatic and magnetic energy. In the following, first the phenomena will be considered which result from the stored energy and its readjustment in circuits storing energy in one form only, which usually is as electro- magnetic energy, and then the general problem of a circuit storing energy electromagnetically and electrostatically will be considered.