CHAPTER III. INDUCTANCE AND RESISTANCE IN CONTINUOUS- CURRENT CIRCUITS. 20. In continuous-current circuits the inductance does not •enter the equations of stationary condition, but, if e0 = impressed e.m.f., r = resistance, L = inductance, the permanent value of /> current is ia = — • r Therefore less care is taken in direct-current circuits to reduce the inductance than in alternating-current circuits, where the inductance usually causes a drop of voltage, and direct-current circuits as a rule have higher inductance, especially if the circuit is used for producing magnetic flux, as in solenoids, electro- magnets, machine-fields. Any change of the condition of a continuous-current circuit, as a change of e.m.f., of resistance, etc., which leads to a change of current from one value i0 to another value iv results in the appearance of a transient term connecting the current values i0 and iv and into the equation of the transient term enters the inductance. Count the time t from the moment when the change in the continuous-current circuit starts, and denote the impressed e.m.f. by e0, the resistance by r, and the inductance by L. p il = - = current in permanent or stationary condition after the change of circuit condition. Denoting by i0 the current in circuit before the change, and therefore at the moment t = 0, by i the current during the change, the e.m.f. consumed by resistance r is ir, and the e.m.f. consumed by inductance L is di Ldt' where i = current in the circuit. 26 26 TRANSIENT PHENOMENA di Hence, eQ = ir + L — > (1) dt or, substituting eQ = if, and transposing, -i*-i±V This equation is integrated by - -t = log (i - ij - logc, where — log c is the integration constant, or, r i — i^ = ce L . However, for t = 0, i = iQ. Substituting this, gives IQ — il = c, -ft hence, i = il + (i0 - t\) e ' , (3) the equation of current in the circuit. The counter e.m.f. of self -inductance is e^-L^rtf.-*,).^', . (4) hence a maximum for t = 0, thus : «i° = r (i, - i,). (5) The e.m.f. of self-inductance ex is proportional to the change of current (i0 — t\), and to the resistance r of the circuit after the change, hence would be <*> for r = <*> , or when opening the circuit. That is, an inductive circuit cannot be opened instantly, but the arc following the break maintains the circuit for some time, and the voltage generated in opening an inductive circuit is the higher the quicker the break. Hence in a highly inductive circuit, as an electromagnet or a machine field, the insulation may be punctured by excessive generated e.m.f. when quickly opening the circuit. As example, some typical circuits may be considered. CONTINUOUS-CURRENT CIRCUITS 27 21. Starting of a continuous-current lighting circuit, or non-in- ductive load. Let e0 = 125 volts = impressed e.m.f. of the circuit, and tj « 1000 amperes = current in the circuit under stationary condition; then the effective resistance of the circuit is = 0.125 ohm. Assuming 10 per cent drop in feeders and mains, or 12.5 volts, gives a resistance, r0 = 0.0125 ohm of the supply conductors. In such large conductor the inductance may be estimated as 10 mh. per ohm; hence, L = 0.125 mh. = 0.000125 henry. The current at the moment of starting is i0 = 0, and the general equation of the current in the circuit therefore is, by substitution m (3)) i = 1000 (1 - e-1000')- (6) The time during which this current reaches half value, or i = 500 amperes, is given by substitution in (6) 500 = 1000 (1 - s-1"*"), hence e"1000' = 0.5, t = 0.00069 seconds. The time during which the current reaches 90 per cent of its full value, or i = 900 amperes, is t = 0.0023 seconds, that is, the current is established in the circuit in a practically inappre- ciable time, a fraction of a hundredth of a second. 22. Excitation of a motor field. Let, in a continuous-current shunt motor, e0 = 250 volts = impressed e.m.f., and the number of poles = 8. Assume the magnetic flux per pole, 0 = 12.5 megalines, and the ampere-turns per pole required to produce this magnetic flux as $ = 9000. Assume 1000 watts used for the excitation of the motor field gives an exciting current 1000 h =- = 4 amperes, and herefrom the resistance of the total motor field circuit is r = e-? = 62.5 ohms. 28 TRANSIENT PHENOMENA To produce JF = 9000 ampere-turns, with il = 4 amperes, cjr requires — = 2250 turns per field spool, or a total of n = 18,000 l\ turns. n = 18,000 turns interlinked with 0 = 12.5 megalines gives a total number of interlinkages for i1 = 4 amperes of n<&0 = 225 X 109, or 562.5 X 109 interlinkages per unit current, or 10 amperes, that is, an inductance of the motor field circuit L = 562.5 henrys. The constants of the circuit thus are e0 = 250 volts; r = 62.5 ohms; L = 562.5 henrys, and i0 = 0 = current at time t = 0. Hence, substituting in (3) gives the equation of the exciting current of the motor field as ' i = 4 (1 _ e-o-m") (7) Half excitation of the field is reached after the time t = 6.23 seconds; 90 per cent of full excitation, or i = 3.6 amperes, after the time t = 20.8 seconds. That is, such a motor field takes a very appreciable time after closing the circuit before it has reached approximately full value and the armature circuit may safely be closed. Assume now the motor field redesigned, or reconnected so as to consume only a part, for instance half, of the impressed e.m.f., the rest being consumed in non-inductive resistance. This may be done by connecting the field spools by two in multiple. In this case the resistance and the inductance of the motor field are reduced to one-quarter, but the same amount of external resistance has to be added to consume the impressed e.m.f., and the constants of the circuit then are: e0 = 250 volts; r = 31.25 ohms; L = 140.6 henrys, and i0 = 0. The equation of the exciting current (3) then is i = 8 (1 - e" °'22220, (8) that is, the current rises far more rapidly. It reaches 0.5 value after t = 3.11 seconds, 0.9 value after t = 10.4 seconds. An inductive circuit, as a motor field circuit, may be made to respond to circuit changes more rapidly by. inserting non- inductive resistance in series with it and increasing the im- CONTINUOUS-CURRENT CIRCUITS 29 pressed e.m.f., that is, the larger the part of the impressed e.m.f. consumed by non-inductive resistance, the quicker is the change. Disconnecting the motor field winding from the impressed e.m.f. and short-circuiting it upon itself, as by leaving it con- nected in shunt with the armature (the armature winding resistance and inductance being negligible compared with that of the field winding), causes the field current and thereby the field magnetism to decrease at the same rate as it increased in (7) and (8), provided the armature instantly comes to a stand- still, that is, its e.m.f. of rotation disappears. This, however, is usually not the case, but the motor armature slows down gradually, its momentum being consumed by friction and other losses, and while still revolving an e.m.f. of gradually decreas- ing intensity is generated in the armature winding; this e.m.f. is impressed upon the field. The discharge of a motor field winding through the armature winding, after shutting off the power, therefore leads to the case of an inductive circuit with a varying impressed e.m.f. 23. Discharge of a motor field winding. Assume that in the continuous-current shunt motor dis- cussed under 22, the armature comes to rest tl = 40 seconds after the energy supply has been shut off by disconnecting the motor from the source of impressed e.m.f., while leaving the motor field winding still in shunt with the motor armature winding. The resisting torque, which brings the motor to rest, may be assumed as approximately constant, and therefore the deceler- ation of the motor armature as constant, that is, the motor speed decreasing proportionally to the time. If then S = full motor speed, S (l - - j is the speed of the motor at the time t after disconnecting the motor from the source of energy. Assume the magnetic flux 3> of the motor as approximately proportional to the exciting current, at exciting current i the magnetic flux of the motor is <&= ^<&0, where 4>0= 12.5 mega- lines is the flux corresponding to full excitation it = 4 amperes. 30 TRANSIENT PHENOMENA The e.m.f. generated in the motor armature winding and thereby impressed upon the field winding is proportional to the magnetic flux of the field, , and to the speed $ (l J, V and since full speed S and full flux 0 generate an e.m.f. e0 = 250 volts, the e.m.f. generated by the flux <£ and speed S 1 1 - -\> that is, at time t is and since we have -«-r / 4 e = ir (l - -J; (10) or for r = 62.5 ohms, and tl = 40 seconds, we have e = 62.5^ (1 - 0.025 t). (11) Substituting this equation (10) of the impressed e.m.f. into the differential equation (1) gives the equation of current i during the field discharge, henC6< rtdt di integrated by where the integration constant c is found by hence, t = 0, i = iv log cii = 0, c = - , 2 (15) CONTINUOUS-CURRENT CIRCUITS 31 This is the equation of the field current during the time in which the motor armature gradually comes to rest. At the moment when the motor armature stops, or for it is rtl i2 = v~^. (16) This is the same value which the current would have with the armature permanently at rest, that is, without the assistance of the e.m.f. generated by rotation, at the time t = — • The rotation of the motor armature therefore reduces the decrease of field current so as to require twice the time to reach value i2, that it would without rotation. These equations cease to apply for t > tv that is, after the armature has come to rest, since they are based on the speed equation S ( 1 J , and this equation applies only up to t = tv but for t > tj_ the speed is zero, and not negative, as given by S ( 1 - -j • That is, at the moment t = ^ a break occurs in the field discharge curve, and after this time the current i decreases in accordance with equation (3), that is, L \ ) /-i >7\ I = 12£ > (17) or, substituting (16), i = v"1 '*'*'. (18) Substituting numerical values in these equations gives : for t < tv i= 4 £-0.001388*'. (19) for t = tl = 40, i = 0.436; (20) for t > tv i = 4 r °-mi (<-2o). (21) 32 TRANSIENT PHENOMENA Hence, the field has decreased to half its initial value after the time t = 22.15 seconds, and to one tenth of its initial value after t = 40.73 seconds. 40 seconds 5 10 15 20 Seconds Fig. 5. Field discharge current. Fig. 5 shows as curve I the field discharge current, by equations (19), (20), (21), and as curve II the current calculated by the equation i = 4£-. The magnetic flux ® varies with the magnetizing current i by an empirical curve, the magnetic characteristic or saturation curve of the machine. This can approximately, within the range considered here, be represented by a hyperbolic curve, as was first shown by Frohlich in 1882 : *--' (22) where = magnetic flux per ampere, in megalines, at low density. — = magnetic saturation value, or maximum magnetic flux, in megalines, and £.!+« (23) can be considered as the magnetic exciting reluctance of the machine field circuit, which here appears as linear function of the exciting current i. Considering the same shunt-wound commutating machine as in (12) and (13), having the constants r = 62.5 ohms = field resistance; 0 = 12.5 megalines = magnetic flux per pole at normal m.m.f.; SF = 9000 ampere-turns = normal m.m.f. per pole; n = 18,000 turns = total field turns (field turns per pole = 18'QQQ = 2250), and ^ = 4 amperes = current for full 8 excitation, or flux, 4>0 = 12.5 megalines. Assuming that at full excitation, 4>0, the magnetic reluctance has already increased by 50 per cent above its initial value, that 34 TRANSIENT PHENOMENA ampere-turns i is. that the ratio - . _ — > or — , at

= 12.5 mega- magnetic flux lines and i = il = 4 amperes, is 50 per cent higher than at low excitation, it follows that 1 + bi, = 1.5, or b = 0.125. (24) Since i = iv = 4 produces $ = $0 = 12.5, it follows, from (22) and (24) <£ - 4.69. That is, the magnetic characteristic (22) of the machine is approximated by - Let now ec = e.m.f. generated by the rotation of the arma- ture per megaline of field flux. This e.m.f. ec is proportional to the speed, and depends upon the constants of the machine. At the speed assumed in (12) and (13), $0 = 12.5 megalines, e0 = 250 volts, that is, ec = -^ = 20 volts. $0 Then, in the field circuit of the machine, the impressed e.m.f., or e.m.f. generated in the armature by its rotation through the magnetic field is, e = ec being given in megalines, eQ in volts.) CONTINUOUS-CURRENT CIRCUITS 35 The differential equation of the field circuit therefore is (1) (26) n Since this equation contains the differential quotient of 4>, it is more convenient to make 4> and not i the dependent variable; then substitute for i from equation (22), which gives or, transposed, 100 ~dt ' 100 n (27) (28) (29) This equation is integrated by resolving into partial fraction by the identity ec- r) - bec resolved, this gives hence, A B ec - r - bec -r)- (Abec * - B *); (30) ec — r — be, This integrates by the logarithmic functions 100 1 4 r (31) (32) n ec(ec - r} log(ec-r-bec$>)+C. (33) 36 TRANSIENT PHENOMENA The integration constant C is calculated from the residual magnetic flux of the machine, that is, the remanent magnetism of the field poles at the moment of start. Assume, at the time, t = 0, 4> = 4>r = 0.5 megalines = residual magnetism and substituting in (33), and herefrom calculate C. C substituted in (33) gives 100 1 <}> <$> r ec-r- bec3> n or, substituting and n em = where em = e.m.f. generated in the armature by the rotation in the residual magnetic field, n ( e (bec — r — be (36) This, then, is the relation between e and t, or the equation of the building up of a continuous-current generator from its residual magnetism, its speed being constant. Substituting the numerical values n = 18,000 turns; (f> = 4.69 megalines; b = 0.125; ec = 20 volts; r = 62.5 ohms; 4>r = 0.5 megaline, and em = 10 volts, we have t = 26.8 log $ - 17.9 log (31.25 - 2.5 $) + 79.6 (37) and t = 26.8 log e - 17.9 log (31.25 - 0.125 e) - 0.8. (38) CONTINUOUS-CURRENT CIRCUITS 37 Fig. 6 shows the e.m.f. e as function of the time t. As seen, under the conditions assumed here, it takes several minutes before the e.m.f. of the machine builds up to approximately full value. 0 20 40 60 80 100 120 140 760 ISO 200 Sec. Fig. 6. Building-up curve of a shunt generator. The phenomenon of self-excitation of shunt generators there- fore is a transient phenomenon which may be of very long duration. From equations (35) and (36) it follows that ec - r e = 250 volts (39) is the e.m.f. to which the machine builds up at t = o>, that is, in stationary condition. To make the machine self-exciting, the condition fa - r > 0 must obtain, that is, the field winding resistance must be r < fa or, (40) (41) r < 93.8 ohms, or, inversely, ec, which is proportional to the speed, must be r *<>$' or, er> 13.3 volts. (42) 38 TRANSIENT PHENOMENA The time required by the machine to build up decreases with increasing ec, that is, increasing speed; and increases with increasing r, that is, increasing field resistance. 25. Self-excitation of direct-current series machine. Of interest is the phenomenon of self-excitation in a series machine, as a railway motor, since when using the railway motor as brake, by closing its circuit upon a resistance, its usefulness depends upon the rapidity of building up as generator. Assuming a 4-polar railway motor, designed for e0 = 600 volts and iv= 200 amperes, let, at current i = i1= 200 amperes, the magnetic flux per pole of the motor be 4>0 = 10 megalines, and 8000 ampere-turns per field pole be required to produce this flux. This gives 40 exciting turns per pole, or a total of n = 160 turns. Estimating 8 per cent loss in the conductors of field and armature at 200 amperes, this gives a resistance of the motor circuit r0= 0.24 ohms. To limit the current to the full load value of ^ = 200 amperes, with the machine generating eQ= 600 volts, requires a total resistance of the circuit, internal plus external, of r = 3 ohms, or an external resistance of 2.76 ohms. 600 volts generated by 10 megalines gives ec= 60 volts per megaline per field pole. Since in railway motors at heavy load the magnetic flux is carried up to high values of saturation, at i± = 200 amperes the magnetic reluctance of the motor field may be assumed as three times the value which it has at low density, that is, in equation <22>' . 1 + W, - 3, °r' 6 - 0.01, and since for i = 200, r = 1 megaline, hence em = ec4>r= 60 volts, and substituting in equation (36) gives n = 160 turns; = 0.15 megaline; b = 0.01; ec= 60 volts; r = 3 ohms; 4>r= 1 megaline, and em = 60 volts, t = 0.04 log e - 0.01333 log (600 - e) - 0.08. (44) This gives for e = 300, or 0.5 excitation, t = 0.072 seconds; and for e = 540, or 0.9 excitation, t = 0.117 seconds; that is, such a motor excites itself as series generator practically instantly, or in a small fraction of a second. The lowest value of ec at which self-excitation still takes place is given by equation (42) as ec = ^ = 20> that is, at one-third of full speed. If this series motor, with field and armature windings connected in generator position, — that is, reverse position, — short-circuits upon itself, r = 0.24 ohms, we have t = 0.0274 log e - 0.00073 log (876 - e) - 0.1075, (45) that is, self-excitation is practically instantaneous : e = 300 volts is reached after t = 0.044 seconds. Since for e = 300 volts, the current i = = 1250 amperes, the power is p = ei = 375 kw., that is, a series motor short- circuited in generator position instantly stops. Short-circuited upon itself, r = 0.24, this series motor still builds up at ec = - = 1.6, and since at full load speed ec= 60, 9 ec = 1.6 is 2.67 per cent of full load speed, that is, the motor acts as brake down to 2.67 per cent of full speed. It must be considered, however, that the parabolic equation (22) is only an approximation of the magnetic characteristic, 40 TRANSIENT PHENOMENA and the results based on this equation therefore are approximate only. One of the most important transient phenomena of direct- current circuits is the reversal of current in the armature coil short-circuited by the commutator brush in the commutating machine. Regarding this, see " Theoretical Elements of Elec- trical Engineering," Part II, Section B.