CHAPTER XIV DIELECTRIC LOSSES Dielectric Hysteresis 116. Just as magnetic hysteresis and eddy currents give a power component in the inductive reactance, as "effective resistance," so the energy losses in the dielectric lead to a power component in the condensive reactance, which may be repre- sented by an "effective resistance of dielectric losses" or an "effective conductance of dielectric losses." In the alternating magnetic field, power is consumed by mag- netic hysteresis. This is proportional to the frequency, and to the 1.6*'' power of the magnetic density, and is considerable, amounting in a closed magnetic circuit to 40 to 60 per cent, of the total volt-amperes. In the dielectric field, the energy losses usually are very much smaller, rarely amounting to more than a few per cent., though they may at high temperature in cables rise as high as 40 to 60 per cent. The foremost such losses are: leakage, that is, ih loss of the current passing by conduction (as "dynamic current") through the resistance of the dielectric; corona, that is, losses due to a partial or local breakdown of the electrostatic field, and dielectric hysteresis or phenomena of similar nature. It is doubtful whether a true dielectric hysteresis, that is, a molecular dielectric friction, exists. A dielectric loss, propor- tional to the- frequency and to the 1.6*^' power of the dielectric field: P = njD'-^ has been observed in rotating dielectric fields, but is so small, that it usually is overshadowed by the other losses. In alternating dielectric fields in solid materials, such as in condensers, coil insulation, etc., a loss is commonly observed which gives an approximately constant power-factor of the elec- tric energizing circuit, over a wide range of voltage and of fre- quency, from less than a fraction of 1 per cent, up to a few per cent. 150 DIELECTRIC LOSSES 151 Constancy of the power-factor with the frequency, means that the loss is proportional to the frequency, as the current i, and thus the volt-ampere input, ei, are proportional to the frequency. Constancy of the power-factor with the voltage, means that the loss is proportional to the square of the voltage, as the current i is proportional to the voltage, and the volt-ampere input ei thus proportional to the square of the voltage. This loss thus would be approximated by the expression : P = vfD' and thus seems to be akin to magnetic hysteresis, except that at least a part of this dielectric loss is possibly consumed in chemical and mechanical disintegration of the insulating material, while the magnetic hysteresis loss is entirely converted to heat. Leakage 117. The eddy current losses in the magnetic field are the ih loss of the currents flowing in the magnetic material, and as such are proportional to the square of the frequency and of the mag- netic density: P = eyPB^ where 7 = conductivity of the magnetic material. This expression obviously holds only as long as the m.m.f. of the eddy currents is not sufficient to appreciably affect the mag- netic flux distribution. As corresponding hereto in the dielectric field may be con- sidered the conduction loss through the resistance of the dielectric. In a homogeneous dielectric of electric conductivity 7 (usually very low) and specific capacity or permittivity k, if: I = thickness of the dielectric, A = area or cross-section, e = impressed alternating-current voltage, effective value, the dielectric capacity of the material is: JcA ^ ~~ I and the capacity susceptance: 152 ALTERNATING-CURRENT PHENOMENA hence the current passing through the dielectric as capacity- current or "displacement current," is: ^ ^^ 2 7r//cA iQ = eo — 2 TTjCe = — -. — e The conductance of the dielectric is: yA hence, the current, conducted through the dielectric, or leakage current : yA ii = eg = -J- e thus, the total current: eA I = H+jii = -j{y -\-2Trfkj] here the j denotes, that the current component lo is in quadrature ahead of the voltage e. The absolute value of the current thus is: g^ i = ViTTi? = ^Vy' + (2x/A;)2 and the power consumption: or, since the dielectric density D is proportional to the voltage € ... gradient t and the permittivity: D= '^ 4.irvH (where y = 3 X 10^" = velocity of light, see "Theoretical Ele- ments of Electrical Engineering.") Thus: ^ ~ k^ where V = Al = volume The power-factor then is: ^ ei Vy'-h (2 7r/A;)2 DIELECTRIC LOSSES 153 Or, if, as usually the case, the conductivity 7 is small compared with the susceptivity 2 irfk : = — '^- 27r//v that is, the power-factor is inverse proportional to the frequency. The observation of leakage losses and leakage resistance thus is best made at low frequencies or at direct-current voltage. While, however, in magnetic materials the conductivity 7 is fairly constant, varying only with the temperature, like that of all metals, the very low conductivity of the dielectric is often not even approximately constant, but may vary with the tempera- ture, the voltage, etc., sometimes by many thousand per cent. 118. While in a homogeneous dielectric field, the leakage cur- rent power losses are independent of the frequency and herein differ from the magnetic eddy current losses, which latter are proportional to the square of the frequency, in non-homogene- ous dielectric fields, leakage current losses may depend on the frequency. As an instance, let us consider a dielectric consisting of two layers of different constants, for instance, a layer of mica and a layer of varnished cloth, such as is sometimes used in high- voltage armature insulation. Let 7i = electric conductivity, ki = permittivity or specific capacity, li = thickness and, A 1 = area or section of the first layer of the dielectric, and 72, k-z, h, Ao the corresponding values of the second layer. It is then : yA g = -y- = electric conductance kA C = -J- = electrostatic capacity of the layer of dielectric, hence: 2 irfk A b = 2irfC = — J — = capacity susceptance, and (1) 154 AL TERN A TING-C URREN T PHENOMENA Y = g -\- jh = admittance, thus : Z =y = r — jx = impedance, where: 9 r = ^ = vector resistance (not ohmic resistance, but energy component of impedance, T see paragraph 89.) X = r = vector reactance, and (2) y = y/g^ + &^ = absolute admittance, (z = -y/r^ -\- x^ = absohite impedance.) If then. El = potential drop across the first, E^ = potential drop across the second layer of dielectric, E = El -\- Eo = voltage impressed upon the dielectric. (3) The current i, which traverses the dielectric, partly by con- duction through its resistance, partly by capacity as displace- ment current, then is the same in both layers, as they are in series in the dielectric field, and it is: El = i(ri - jxi) I E^ = ^■(^2 - JX2) j and, by (3) : E = i { (ri + ra) - j{xi + X2) } or, absolute : e = iV{ri + ro)^ + (xi + Xo^ Thus, the current: (4) (6) (6) V(ri + TiY + (Xi + X2)2 the apparent power, or volt-ampere input: Q =^ ei = V(j\ + r2y -\- (xi + x^y the power consumed in the dielectric is: p = p(ri + r.) e^(ri 4- Tj) ~ {ri + r-iY -\- {xi + XiY and the power-factor: p ^P ^ ^1 + r2 Q Vin + r2)2 -j- (xi + x^y (7) (8) (9) (10) 119. Let us consider some special cases: (a) If the conductivity, 71 and 72, of the two layers of dielectric DIELECTRIC LOSSES 155 is so small that the conduction current, ge, is negligible compared with the capacity current, 2irfCe. In this case, ri and r^ are negligible compared with Xi and x^, and it is: e P = V = Xi + Xo {xi + 0:2)- ri + Ti (11) Xi + X2 Substituting now for the impedance quantities Z= r — jx, which have no direct physical meaning in the dielectric field, the admittance quantities Y = g -{- jh, which have the physical meaning that g is the effective ohmic conductance, b the capacity susceptance, it is: g negligible compared with h and y, and b = y. Thus, by (2) : . ebM _ 2x/CiC2e ' ~ 61 + 62 ~ C1 + C2 ^ ^ hence proportional to the frequency /: (61 + b.y (Ci + c,r hence, the loss of power by current leakage in the dielectric in this case is independent of the frequency. 62 .61 C2 . Ci (13) 9i V = 61 + ^^62 ^^c; + ^^ C2 (14) 61 + 62 27r/(Cx+C2) hence, in this case the power-factor is inverse proportional to the frequency. (6) If in both layers the leakage current is large compared with the capacity current, that is, 2iTJCe negligible compared with ge. In this case, Xi and x-i are negligible compared with ri and r-^, and: e (15) V r\ + ri Q e2 ri + r2 P e2 ri + Ti V = 1 156 ALTERNATING-CURRENT PHENOMENA and as in this case ri and r^ are the effective ohmic resistance of the dielectric, all the quantities are independent of the frequency; that is, the case is one of simple conduction. 120 (c) If in the first layer the leakage is negligible compared with the capacity current, but is not negligible in the second layer. That is, in a two-layer insulation, one layer leaks badly. Assuming for simplicity that the two layers have the same capacity, C — Ci = C^. If the two capacities are unequal, the treatment is analogous, but merely the equations somewhat more complicated. Let the conductance of the second layer = g, the capacity susceptance 2 irfC = h. It is then: Ti negligible compared with the other quantities. ''- f + 6^ 1 h h X2 = (16) Substituting these values in equations (7) (8) (9) (10) gives: i = g(g' + b') ^ e(g' + (2TfCy) ^ V^ + \h + -g) W^ + \2^ + ^ p ^ e'-(g' + b') _ eHg' + (2^fCy) V (19) gJl 'V-^\2.fC-^ g 1 1 As seen, in this case current, power loss and power-factor depend on the frequency, but in a more complex manner. With changing values of the conductance from low values, where g is negligible compared with the other terms, but the other terms negligible compared with — , up to high conductivity, where 1 . . . ^ . — IS negligible, but the terms with g predominate, the current changes from: DIELECTRIC LOSSES 157 low g: i = TrfCe, proportional to the frequency, to: high g: i = 2wfCe. Again proportional to the frequency, but twice as large, and at intermediate values of g, the current changes more rapidly than proportional to the frequency. The loss oj power changes from: low g: or independent of the frequency, to: high g: p ^(2jrfCre^ g or proportional to the square of the frequency. The power-factor changes from: low g: or inverse proportional to the frequency, to: high g: 2irfC p = -, g or proportional to the frequency. And over a considerable range of intermediate values of conduct- ance, g, the power-factor, therefore, remains approximately con- stant; or, inversely, with changing frequency and constant g and h, the power-factor changes from proportionality with the fre- quency at low frequencies, up to inverse proportionality at high frequencies, and thereby passes through a maximum. The value of g, for which the power-factor in equation (19) is a maximum, is found by differentiating: -j- = 0, as: g =-- 2 V2 tt/C (20) and this maximum power-factor is po = y^s- For C2 > Ci, higher, for C2 < Ci, lower values of power-factor maximum result, where C2 is the leaky dielectric. 158 ALTERNATING-CURRENT PHENOMENA As illustration, Fig. 95 gives the values of power-factor, p, from as abscissae. equation (19), as function of t^ = o^Tr ~ 2 fk A dielectric circuit, in which the power-factor decreases with increasing frequency, for instance, is that of the capacity of the transmission line; a dielectric circuit, in which the power-factor increases with the frequency, is that of the aluminum-cell light- ning arrester. 121. As seen, in the dielectric circuit, that is, in insulators in which the current is essentially a displacement current, the relations between voltage, current, power, phase angle and power- factor can be represented by the same symbolic equations as the relations between voltage, current, power and power-factor in metallic conductors, in which the current flow is dynamic, by the introduction of the effective admittance of the dielectric circuit, or part of circuit: where g is the effective conductance of the dielectric circuit, or the energy component of the admittance, representing the energy consumption by leakage, dielectric hysteresis, corona, etc., and h = 2 tt/C is the capacity susceptance. Instead of the admittance Y, its reciprocal, the impedance Z = r — jx, may be used. The main differences between the dielectric and the electro- dynamic circuit are: In the dielectric circuit, the susceptance, h, is positive, the reactance, x, negative; the current normally leads the voltage, DIELECTRIC LOSSES 159 that is, capacity effects predominate and inductive effects are usually absent. In the dynamic circuit, the reactance, x, usually is positive, the susceptance, b, negative; the current usually lags, that is, inductive effects predominate and capacity effects are usually absent. In the dielectric circuit, the a.dmittance terms, Y — g + j6, have a physical meaning as the effective conductance and the capacity susceptance, 2 t/C, but the impedance terms, Z ^ r —jx, are only derived quantities, without direct physical meaning: the vector resistance, r, is not the effective ohmic resistance of the dielectric, -, but is also depending on the capacity, r = ^ ^ , g, and the vector reactance, x, is not the condensive reactance, r = .^> but also depends on the conductance, x = „ , , ♦ ' ^2 _|_ ^2 In the dynamic circuit, the impedance terms, Z = r -\- jx, have a direct physical meaning, as effective ohmic resistance, r, and as self-inductive reactance, 2irfL, while the admittance terms, Y = g — jb, are derived quantities, and the vector conductance, g, is not the reciprocal of the resistance, r, the vector susceptance, b, not the reciprocal of the reactance, x, as discussed in preceding chapters. Physically, the most prominent difference between the dielec- tric circuit and the dynamic circuit is that for the displacement current of the dielectric circuit, that is, for the electrostatic flux, all space is conducting, while for the dynamic current, most materials are practically non-conductors, and the dynamic circuit thus is sharply defined in the extent of the flow of the current, while the dielectric circuit is not. The dielectric circuit thus is similar to the magnetic circuit ; for the magnetic circuit all space is conducting also, that is, can carry magnetic flux. An unin- sulated submarine electric circuit would be more nearly similar, in the distribution of current flow, to the dielectric and the mag- netic circuit. In the electric circuit, the conductor through which the cur- rent flows is generally sharply defined and of a uniform section, which is small compared with the length, and the conductor thus can be approximated as a linear conductor, that is, the cur- rent distribution throughout the conductor section assumed as uniform. With the dielectric and the magnetic circuit this is 160 ALTERNATING-CURRENT PHENOMENA rarely the case, and such circuits thus have to be investigated from place to place across the section of the current flow. This brings in the consideration of dielectric current density or dielec- tric flux density, and corresponding thereto magnetic flux den- sity, as commonly used terms, while dynamic current density, that is, current per unit section of conductor, is far less frequently considered. Thus, in the dielectric circuit, instead of admittance Y = g + jh, commonly the admittance per unit section and unit length of the dielectric circuit, or the admittivity , v = y — jl3, has to be considered, where 7 = conductivity of the dielectric (or effective conductivity, including all other energy losses), and /3 = 2 irfk = susceptivity, where k = permittivity or specific capacity of the material. We then have: (20) y^^+{2irfkYdA 122. With the extended industrial use of very high voltage, the explicit study of the dielectric field has become of importance, and it is not safe merely to consider the thickness of the insulation in relation to the voltage impressed upon it. In an ununiform electric conductor, the relation of the voltage to the length of the conductor does not determine whether the conductor is safe or whether locally, due to small cross-section or high resistivity, unsafe current densities may cause destructive heating, but the adaptability of the conductor to the current carried by it must be considered throughout its entire length. So in the dielectric field, the thickness of the dielectric may be such that the voltage impressed upon it may give a very safe average voltage gradient or average dielectric flux density, and the dielectric nevertheless may break down, due to local concen- tration of the dielectric flux density in the insulating material. Thus, for instance, in the dielectric field between parallel con- ductors, at a voltage far below that which would jump from conductor to conductor, locally at the conductor surface the concentration of electrostatic stress exceeds the dielectric strength of air, and causes it to break down as corona. In solid dielectrics, under similar conditions, the breakdown due to local over-stress DIELECTRIC LOSSES 161 often may change the flux distribution so as to gradually extend throughout the entire dielectric, until puncture results. Corona 123. — In the magnetic field, with increasing magnetizing force, /, or magnetic field intensity, H, the magnetic flux density, B, increases, but for high field intensities the flux density ceases to be even approximately proportional to the field intensity, and finally, at very high field intensities, H, the "metallic magnetic induction," Bo — B — H, reaches a finite limiting value, which with iron is not far from Bo = 20,000, the so-called "saturation value." In the dielectric field, with increasing voltage gradient, g, or dielectric field intensity, K, the dielectric flux density, D, increases proportional thereto, until a finite limiting field intensity, Kq, or voltage gradient, gfo, is reached, beyond which the dielectric cannot be stressed, but breaks down and becomes dynamically conduct- ing, that is, punctures, and thereby short-circuits the dielectric field. The voltage gradient, go, at which disruption of the dielectric occurs is called the "disruptive strength" or "dielectric strength" of the dielectric. With air at atmospheric pressure and temperature, it is go = 30 kv. per centimeter. Thus under alternating electric stress, air punctures at 21 kv. effective per / 30 \ centimeter I ~-y= I . The dielectric strength of air is over a very wide range proportional to the air density, and thus proportional to the barometric pressure and inverse proportional to the abso- lute temperature. Air is one of the weakest dielectrics, and liquids and still more solids show far higher values of dielectric strength, up to and beyond a million volts per centimeter. < 124. If then in a uniform dielectric field, such as that between parallel plates A and B as shown in Fig. 96, the voltage is gradu- ally increased, as soon as the voltage maximum reaches a gradi- ent of ^0 = 30 kv. in the gap between the metal plates, the air in this gap ceases to sustain the voltage, a spark passes, usually followed by the arc, and the potential difference across this gap drops from gol — where I is the distance between the metal plates A and B — to practically nothing, and the electric circuit thereby ceases to include a dielectric field. 11 162 ALTERNATING-CURRENT PHENOMENA Assuming now that the gap between the metal plates does not contain a homogeneous dielectric, but one consisting of several layers of different dielectric strength and different permittivity. For instance, we put two glass plates, a and h, of thickness U into the gap, as shown in Fig. 97, thereby leaving an air space, c, of I — 21q. The dielectric flux density in the field is still uniform \ ^ m Fig. 96. )J J 'T*" * ' "^'o V ' m Fig. 97. throughout the field section, but the voltage gradient in the different layers, a, b and c, is not the same, is not the average gra- 6 dient, g = -j ,of the gap, but is inverse proportional to the permit- tivities: where ko is the permittivity of the layers, a and b, ki the permit- tivity of the layer c ( = 1, if this layer is air). The potential drop across a and b thus is Ugo, across c is (I — 2 lo)g], and the total voltage thus: e = 2 lo go + {I - 2h)gi, DIELECTRIC LOSSES 163 or, substituting go = —, — gives : 0 hence: e eko 2/o(--l) + Z 2lo{k,-ko)-\-lko and k, ( go ko eki \ko I Depending on the values of k\ and fco, either ^o or g\ may be higher than the average gradient e Q I To illustrate on a numerical instance: Let the distance between the metal plates A and Bhel = 1 cm. With nothing but air at atmospheric pressure and temperature between the plates, the gap would break down by a spark dis- charge, and short-circuit the circuit of Fig. 96; at e = 30 kv. maximum, and at e = 25 kv., no discharge would occur. Assuming now two glass plates, a and b, each of 0.3 cm. thick- ness and permittivity /co = 4, were inserted, leaving an air-gap of 0.4 cm. of permittivity ki = 1. At e = 25 kv. the gradients thus would be, by above equation : In the glass plates: Qi = 8.4 kv. per cm. In the air-gap: go = 35.7 kv. per cm. The air would thus be stressed beyond its dielectric strength, and would break down by spark discharge. This would drop the gradient in the air down to practically g'o = 0, and the gradient in the glass plates thus would become : g\ = ^ = 41.7 kv. per cm. Thus the insertion of the glass plates would cause the air-gap to break down. The dynamic current which flows through the air-gap in this case would not be the short-circuit current of the 164 AL TERN A TING-C URREN T PHENOMENA electric circuit, as would be the case in the absence of the glass plates but it would merely be the capacity current of the glass plates; and it would not be followed by the arc, but passes as a uniform bluish glow discharge, or as pink streamers — corona. 125. If the dielectric field is not uniform, but varying in density as, for instance, the field between two spheres or the field between two parallel wires, then with increasing voltage the breakdown gradient will not be reached simultaneously throughout the en- tire field, as in a uniform field, but it is first reached in the denser portion of the field — at the surface of the spheres or parallel wires, where the lines of dielectric force converge. Thus the dielectric will first break down at the denser portion of the field, and short- circuit these portions by the flow of dynamic current. This, however, changes the voltage gradient in the rest of the field, and may raise it so as to break down the entire field, or it may not do so. Fig. 98. Fig. 99. h'5 For instance, in the dielectric field between two spheres at distance I from each other, as shown in Figs. 98 and 99, with in- creasing potential difference, e, finally the breakdown gradient of the air, go = 30 kv. = cm., is reached at the surface of the spheres, and up to a certain distance 5 beyond it, and in this space 8 the air breaks down, becomes conducting, and the space up to the distance 8 is filled with corona. As the result, the conducting terminals of the dielectric field are not the original spheres, but the entire space filled by the corona, that is, the terminals are in- creased in size, and the convergency of the dielectric flux lines, that is, the voltage gradient at the effective terminals, is reduced. At the same time the gap between the effective terminals is re- duced by 25, and the average voltage gradient thereby increased. DIELECTRIC LOSSES 1G5 If the latter effect is greater — as is the case with large spheres at short distance from each other — the air becomes over-stressed at the edge 8 of the corona formed by the original field, the corona spreads farther, and so on, until the entire field breaks down, that is, no stable corona forms, but immediate disruptive discharge. Inversely, with small spheres at considerable distance from each other, the formation of corona very soon increases the size of the effective terminals so as to bring the voltage gradient at the edge of the corona down to the disruptive gradient, go, and the corona spreads no farther. In this case then, with increasing voltage, at a certain voltage, eo, corona begins to form at the terminals, first as bluish glow, then as violet streamers, which spread farther and farther with increasing voltage, until finally the disruptive spark passes between the terminals. In this case, corona pre- cedes the disruptive discharge. Experience shows that the voltage, e„, at which corona begins at the surface is not the voltage at which the breakdown gradient of air, go = 30, is reached at the sphere surface, but e^ is the vol- tage at which the breakdown gradient, go, has extended up to a certain small but definite distance the "energy distance" from the spheres. That is, dielectric breakdown of the air requires a finite volume of over-stressed air, that is, a finite amount of di- electric energy. As the result, when corona begins, the gradient at the terminal surface, gv, is higher than the breakdown gradi- ent, go, the more so the more the flux lines converge, that is, the smaller the spheres (or parallel wires) are. 126. With the development of high-voltage transmission at 100 kv. and over, the electrical industry has entered the range of voltage, where corona appears on parallel wires of sizes such as are industrially used. Such corona consumes power, and thereby introduces an energy component into the expression of the line capacity, a corona conductance. The power consumption by the corona is approximately proportional to the frequency, its power factor therefore inde- pendent of the frequency. The power consumption by the corona is proportional to the square of the excess voltage over that voltage, eo, which brings the dielectric field at the conductor surface up to the breakdown gradient, go. However, corona does not yet appear at the voltage, eo, which produces the breakdown gradient, go, at the conductor surface, 166 ALTERNATING-CURRENT PHENOMENA but at the higher voltage, e„, which has extended the breakdown gradient by the energy distance from the conductor surface. Then the corona power begins with a finite value, and in the range between eo and e^ it is indefinite, depending on the surface condition of the conductor. The equations of the power consumption by corona in parallel conductors are: P = a(/ + c)(e - eo)2 where : P = power loss in kilowatts per kilometer length of single- line conductor; e = effective value of the voltage between the line conductor and neutral in kilovolts;^ / = frequency; c = 25; and a is given by the equation : where : ^ = W: r = radius of conductor in centimeters; s = distance between conductor and return conductor in centimeters; 8 = density of the air, referred to 25°C. and 76 cm. barometer; A =241; and: Co = effective disruptive critical voltage to neutral, given in kilovolts by the equation (natural logarithm) s 1 ^ €■0 = Wogfo 5r log — where : ^0 = 21.1 kv. per centimeter effective = breakdown gradient of air; nio = surface constant of the conductor. It is: mo = 1 for perfectly smooth polished wire; mo = 0.98 to 0.93 for roughened or weathered wire; * = }4 the voltage between the conductors in a single-phase eircuit,l/v3 times the voltage between the conductors in a three-phase circuit. DIELECTRIC LOSSES 167 decreasing to: mo = 0.87 to 0.83 for7-strand cable (r being the outer radius of the cable). ^ Materially higher losses occur in snow storms and rain. For further discussion of the dielectric field and the power losses in it, see F. W. Peek's "Dielectric Phenomena in High- voltage Engineering." 1 "Dielectric Phenomena in High-voltage Engineering," by F. W. Peek, Jr., page 200.