Theory and Calculation of Transient Electric Phenomena and Oscillations — part 11 of 20

ARC RECTIFICATION 271 wave in one, the other in the other transformer coil — is half this value, or | . 4, 1 ae a ee r= —-Y——— sin?0’d# + f [sin 6"+ sin (& — 0,) Fd@”’ 2 rc A, 6; 0 iy/ 1 a . aff) [oso o a+ f2sino’ sin — 0) a0" 2 zr 0, ry 0 . iny/ 1 fs [ ; [ = 2 ——}- +] @ cos9, — ” -86 3 z—0,t2* cos , — sin (2 ) =3V | + 6, cos 6, — sind ; 2%2-6,l2'° °° 3 14? or, substituting . .r—d, ha = feV/ce — 0) $7 40,008 0, — sind] ; i % v(1 2) (1 26, 2. ); = -- + —cos 0, — -sin @,}; 22V2 n m ne , or, substituting 78, 1 5 | in’ m Vy 4, + 26,7 cos On 2 sin Oz 2 2V2 z+0, (x+0.)?> 2+0, 2+ 7x+86, where i = ratio of effective value to mean value of sine wave. 27. An approximate representation by equivalent sine waves, if e, = the mean value of direct terminal voltage, 1, = the mean value of direct current, is therefore as follows: : The secondary generated e.m.f. of the transformer is e= (€, + & + Py) nV2 : OT a "1 + 008 8, ’

272 TRANSIENT PHENOMENA the secondary current of the transformer is o 2. 4 ate tify fy 25 ogg Oo? 2 2/2 T+0, (+O)? +0 z+0 x+h the pulsation of the direct current is vr : v2 = . _— 1 ° , Prey ‘ the anode voltage of the rectifier is . , / 26, — sin 26, eae=eV1—- On ’ . . . . Cute : and herefrom follows the apparent efficiency of rectification, “- the power factor, the efficiency, etc. a, eit tt ttt ttt rrr Ty) vit tt ttt tT Tr TT Vy ey) 2 eee wttt TTT ttt ttt ArT vet tt} TT Tare wet tT Tet tT TT Ae TT

• watt tT ET tT TT fof TT vet {ttt ttt | eet TT welt tt tert TTT riper tT : ro} Pree es ose te EE 01 Ht tt eset TL 9 opel [TTT TTT Trt rr yy rs ot | ATT TTT ost {| | PEAT RT TTT TTT of | tt TT | Pe ee TT atti ttt | ttt | Pee TT SERRE a= Cr a | | ao Fig. 79. E.m.f. and current ratio and secondary power factor of constant- current mercury arc rectifier. From the equivalent sine waves, e and 7, of the transformer secondary, and their phase angle, the primary impressed em. and the primary current of the transformer, and thereby the |

ARC RECTIFICATION 278 power factor, the efficiency, and the apparent efficiency of the system, are calculated in the usual manner. In the secondary circuit, the power factor is below unity essentially due to wave shape distortion, less due to lag of cur-. rent. . As example are shown, in Fig. 79, with the angle of overlap 0, . 2% as abscissas, the ratio of voltages, mt the ratio of currents, a ) 0 ‘ NA the current pulsation, = and the power factor of the secondary 0 ; circuit.

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SECTION III TRANSIENT PHENOMENA IN SPACE |

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TRANSIENT PHENOMENA IN SPACE.

CHAPTER I. .

INTRODUCTION. |

1. The preceding sections deal with transient phenomena in time, that is, phenomena occurring during the time when a change or transition takes place between one condition of a cir- cuit and another. The time, t, then is the independent variable, electric quantities as current, e.m.f., etc., the dependent variables. Similar transient phenomena also occur in space, that is, with space, distance, length, etc., as independent variable. Such transient phenomena then connect the conditions of the electric quantities at one point in space with the electric quantities at another point in space, as, for instance, current and potential difference at the generator end of a transmission line with those at the receiving end of the line, or current density at the surface of a solid conductor carrying alternating current, as the rail return of a single-phase railway, with the current density at the center or in general inside of the conductor, or the distribution of alternating magnetism inside of a solid iron, as a lamina of an alternating-current transformer, etc. In such transient phenom- ena in space, the electric quantities, which appear as functions of space or distance, are not the instantaneous values, as in the preceding chapters, but are alternating currents, e.m.fs., etc., characterized by intensity and phase, that is, they are periodic functions of time, and the analytical method of dealing with such phenomena therefore introduces two independent variables, time ¢ and distance /, that is, the electric quantities are periodic

functions of time and transient functions of space.

The introduction of the complex quantities, as representing the |

alternating wave by a constant algebraic number, eliminates |

277 |

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278 TRANSIENT PHENOMENA

the time ¢ as variable, so that, in the denotation by complex quantities, the transient phenomena in space are functions of one independent variable only, distance J, and thus lead to the same equations as the previously discussed phenomena, with the difference, however, that here, in dealing with space phenom- ena, the dependent variables, current, e.m.f., etc., are complex quantities, while in the previous discussion they appeared as instantaneous values, that is, real quantities.

Otherwise the method of treatment and the general form of the equations are the same as with transient functions of time.

2. Some of the cases in which transient phenomena in space are of importance in electrical engineering are:

(a) Circuits containing distributed capacity and self-induc- tance, as long-distance energy transmission lines, long-distance telephone circuits, multiple spark-gaps, as used in some forms of high potential lightning arresters (multi-gap arrester), etc.

(b) The distribution of alternating current in solid conductors and the increase of effective resistance and decrease of effective inductance resulting therefrom.

(c) The distribution of alternating magnetic flux in solid iron, or the screening effect of eddy currents produced in the iron, and the apparent decrease of permeability and increase of power consumption resulting therefrom.

(d) The distribution of the electric field of a conductor through space, resulting from the finite velocity of propagation

of the electric field, and the variation of self-inductance and mutual inductance and of capacity of a conductor without return, as function of the frequency, in its effect on wireless telegraphy.

(e) Conductors conveying very high frequency currents, as lightning discharges.

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| | CHAPTER II. LONG-DISTANCE TRANSMISSION LINE.

8. If an electric impulse is sent into a conductor, as a trans- mission line, this impulse travels along the line at the velocity of light (approximately), or 188,000 miles per second. If the line is open at the other end, the impulse there is reflected and returns at the same velocity. If now at the moment when the impulse arrives at the starting point a second impulse, of. opposite direction, is sent into’the line, the return of the first impulse adds itself, and so increases the second impulse; the : return of this increased second impulse adds itself to the third . impulse, and so on; that is, if alternating impulses succeed each other at intervals equal to the time required by an impulse to travel over the line and back, the effects of successive impulses add themselves, and large currents and high e.m.fs. may be produced by small impulses, that is, low impressed alternating e.m.fs., or inversely, when once started, even with zero impressed e.m.f., such alternating currents traverse the lines for some time, gradually decreasing in intensity by the energy consumption in the conductor, and so fading out.

The condition of this phenomenon of electrical resonance thus is that alternating impulses occur at time intervals equal to the time required for the impulse to travel the length of the line and back; that is, the time of one half wave of impressed e.m.f. is the time required by light to travel twice the length of the line, or the time of one complete period is the time light requires to travel four times the length of the line; in other words, the number of periods, or frequency of the impressed alternating e.m.fs., in resonance condition, is the velocity of light divided by four times the length of the line; or, in free oscillation or resonance condition, the length of the line is one quarter wave length. . \

279

280 TRANSIENT PHENOMENA If then 7 = length of line, S = speed of light, the frequency of oscillations or natural period of the line is S . to = ran (1) or, with / given in miles, hence S = 188,000 miles per second, it is = ue cycles. (2)

To get a resonance frequency as low as commercial frequencies, as 25 or 60 cycles, would require / = 1880 miles for f, = 25 cycles, and / = 783 miles for f, = 60 cycles.

It follows herefrom that many existing transmission lines are such small fractions of a quarter-wave length of the impressed frequency that the change of voltage and current along the line can be assumed as linear, or at least as parabolic; that is, the line capacity can be represented by a condenser in the middle of the line, or by condensers in the middle and at the two ends of the line, the former of four times the capacity of either of the two latter (the first approximation giving linear, the second a para- bolic distribution).

For further investigation of these approximations see ‘‘Theory and Calculation of Alternating-Current Phenomena,” 4th edition, pages 225 to 233.

If, however, the wave of impressed e.m.f. contains appreciable higher harmonics, some of the latter may approach resonance frequency and thus cause trouble. For instance, with a line of 150 miles length, the resonance frequency is f, = 313 cycles per second, or between the 5th harmonic and the 7th harmonic, 300 and 420 cycles of a 60-cycle system; fairly close to the 5th har- monic. .

The study of such a circuit of distributed capacity thus _ becomes of #mportance with reference to the investigation of | the effects of higher harmonics of the generator wave.

In long-distance telephony the important frequencies of speech probably range from 100 to 2000 cycles. For these fre- quencies the wave length varies from ; = 1880 miles down to 4 miles, and a telephone line of 1000 miles length would thus

LONG-DISTANCE TRANSMISSION LINE 281 contain from about one-half to 11 complete waves of the im- . pressed frequency. For long-distance telephony the phenomena : occurring in the line thus can be investigated only by consider- . ing the complete equation of distributed capacity and inductance as so-called ‘‘wave transmission” and the phenomena thus essentially differ from those in a short energy transmission line.

4. Therefore in very long circuits, as in lines conveying alter- nating currents of high value at high potential over extremely long distances, by overhead conductors or underground cables, , or with very feeble currents at extremely high frequency, such as telephone currents, the consideration of the line resistance, which consumes e.m.fs. in phase with the current, and of the line reactance, which consumes e.m.fs. in quadrature with the current, is not sufficient for the explanation of the phenomena taking place in the line, but several other factors have to-be taken into account.

In long lines, especially at high potentials, the electrostatic capacity of the line is sufficient to consume noticeable currents.

The charging current of the line condenser is proportional to the difference of potential and is one-fourth period ahead of the e.m.f. Hence, it either increases or decreases the main current, according to the relative phase of the main current and the e.m.f.

As a consequence the current changes in intensity, as well as in phase, in the line from point to point; and the e.m.fs. con- sumed by the resistance and inductance, therefore, also change in phase and intensity from point to point, being dependent upon the current.

Since no insulator has an infinite resistance, and since at high potentials not only leakage but even direct escape of electricity into the air takes place by “brush discharge,’”’ we have to rec- ognize the existence of a current approximately proportional and in phase with the e.m.f. of the line. This current represents consumption of power, and is therefore analogous to the e.m.f. consumed by resistance, while the condenser curfent and the e.m.f. of inductance are wattless or reactive.

Furthermore, the alternating current passing over the line pro- duces in all neighboring conductors secondary currents, which react upon the primary current and thereby introduce e.m.fs.

Of mutual inductance into the primary circuit. Mutual induc- _ tance is neither in phase nor in quadrature with the current, |

282 TRANSIENT PHENOMENA

. and can therefore be resolved into a power component of mutual inductance in phase with the current, which acts as an increase of resistance, and into a reactive component in quadrature with the current, which decreases the self-inductance.

This mutual inductance is not always negligible, as, for instance, its disturbing influence in telephone circuits shows. |

The alternating potential of the line induces, by electrostatic influence, electric charges in neighboring conductors outside of the circuit, which retain corresponding opposite charges on the line wires. This electrostatic influence requires the expenditure of a current proportional to the e.m.f. and consisting of a power component in phase with the e.m.f. and a reactive com- __ ponent in quadrature thereto.

_ The alternating electromagnetic field of force set up by the line current produces in some materials a loss of power by mag- netic hysteresis, or an expenditure of e.m.f. in phase with the cur- rent, which acts as an increase of resistance. This electro- magnetic hysteresis loss may take place in the conductor proper

’ if iron wires are used, and may then be very serious at high fre- quencies such as those of telephone currents.

The effect of eddy currents has already been referred to under “ mutual inductance,” of which it is a power component.

The alternating electrostatic field of force expends power in dielectrics by what is called dielectric hysteresis. In concentric cables, where the electrostatic gradient in the dielectric is com- paratively large, the dielectric hysteresis may at high potentials

. consume considerable amounts of power. The dielectric hystere- sis appears in the circuit as consumption of a current whose component in phase with the e.m.f. is the dielectric power current, which may be considered as the power component of the charging

~ current.

Besides this there is the apparent increase of ohmic resistance due to unequal distribution of current, which, however, is usually not large enough to be noticeable at low frequencies.

, Also, especially at very high frequency, energy is radiated into space, due to the finite velocity of the electric field, and can be represented by power components of current and of voltage respectively.

5. This gives, as the most general case and per unit length of line,

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LONG-DISTANCE TRANSMISSION LINE 288

E.mJs. consumed in phase with the current, I, and = rl, repre- senting consumption of power, and due to resistance, and its apparent increase by unequal current distribution; to the power component of mutual inductance: to secondary currents; to the power component of self-inductance: to electromagnetic hysteresis; and to electromagnetic radiation.

E.mfs. consumed in quadrature with the current, I, and = cl, reactive, and due to self-inductance and mutual inductance. ;

Currents consumed in phase with the e.mf., E, and = gE, representing consumption of power, and due to leakage through . the insulating material, including brush discharge; to the power component of electrostatic influence; to the power component of capacity, or dielectric hysteresis, and to electrostatic radiation.

Currents consumed in quadrature with the e.mf., E, and = bE, being reactive, and due to capacity and electrostatic influence.

Hence we get four constants per unit length of line, namely: Effective resistance, 7; effective reactance, x; effective conduc- tance, g, and effective susceptance, b = — b; (6. being the absolute value of susceptance). These constants represent the coefficients per unit length of line of the following: e.m.f. consumed in phase with the current; e.m.f. consumed in quadra- ture with the current; current consumed in phase with the e.m.f., and current consumed in quadrature with the e.m.f.

6. This line we may assume now as supplying energy to a recewer circuit of any description, and determine the current and e.m.f. at any point of the circuit.

That is, an e.m.f. and current (differing in phase by any desired angle) may be given at the terminals of the receiving circuit. To be determined are the e.m.f. and current at any point of the line, for instance, at the generator terminals; or the impedance, Z, = r, — jz,, or admittance, Y, = g, + 7),, of the receiver circuit, and e.m.f., Z,, at generator terminals are given; the current and e.m.f. at any point of circuit to be deter- mined, etc.

7. Counting now the distance, /, from a point 0 of the line which has the e.m_f.

BE, =e, + je,’ and the current I, = a, + iy,

284 TRANSIENT PHENOMENA . and counting / positive in the direction of rising power and negative in the direction of decreasing power, at any point J, in the line differential di the leakage current is Kg dl and the capacity current is — jEb dl; . hence, the total current consumed by the line differential di is dl = E(g — jb) dl | = EY dl, or = YE. . (1) In the line differential di the e.m.f. consumed by resistance is Ir dl, the e.m.f. consumed by inductance is -jied; hence, the total e.m.f. consumed by the line differential dl is dE = | (r — jx) dl = 12d, E or sd = ZI. (2) These fundamental differential equations (1) and (2) are sym- metrical with respect to / and F. Differentiating these equations (1) and (2) gives PI. dE Ca vd (3) | eh ,at an ae = al ? |

LONG-DISTANCE TRANSMISSION LINE 285 and substituting (1) and (2) in (3) gives the differential equa- tions of E and I, thus: ;

CE ge 7 Y2e (4) PI and z= YZI. (5) These differential equations are identical, and consequently I and E are functions differing by their integration constants or by their limiting conditions only. These equations are of the form PU * ae 7 2 and are integrated by U = Ae", where ¢ is the basis of the natural logarithms, = 2.718283. Choosing equation (5), which is integrated by I = Ae" - 6) and differentiating (6) twice gives ‘ Pi waa . | Gp 7 VA and substituting (6) in (5), the factor Ae’’ cancels, and we have V? = ZY, or __ V = VZY, (7) hence, the general integral, [= At) — Ago". (8) By equation (1), ° 1 dj ad and substituting herein equation (10) gives V Bm Shae de" f, @)

286 TRANSIENT PHENOMENA or, substituting (7), Z B-V2N agent Ag f (10) | The integration constants A, and A, in (8), (9), (10), in general, are complex quantities. The coefficient of the exponent, V, as square root of the product of two complex quantities, also is a complex quantity, therefore may be written V =a — jf, (11) and substituting for V, Z and Y gives (a — 73)? = (r — jx) (g — jd), or (a? — |F) — 2ja3 = (rg — xb) — j (rb + gz), ’ and this resolves into the two separate equations a — FP =rg — xb LP Dada ion.) (12) since, when two complex quantities are equal, their real terms as well as their imaginary terms must be equal. Equations (12) squared and added give | (a? + fF = (rg — 2b) + (rb + 29)? =(P+r)97 +) : = zy; hence, a’? + PF = zy, (13) and from (12) and (13), ; a = Vi(zy + 1g — xb) and fl (14) B = V4(zy — rg + xb). Equations (8) and (10) now assume the form I = A,e toi _ Aye ‘4 ~ and sO (15) E V2 A,e* (e180 + Aje (4-8 ‘ |

| U LONG-DISTANCE TRANSMISSION LINE 287 Substituting for the exponential function with an imaginary _ exponent the trigonometric expression e+ — eos Bl + j sin Al, (16) equations (15) assume the form I = Aye*“(cos Bl — jsin Bl) — Aye~“(cos Bl + j sin Bl) and Z (17) E= v2 A,e*“(cos Bl —j sin Al) + A,¢~ “(cos Bl +] sin Al) where A, and A, are the constants of integration. . The distribution of current J and voltage E along the circuit, therefore, is represented by the sum of two products of expo- nential and trigonometric functions of the distance /. Of these terms, the one, with factor Ae*™, increases with increasing dis- tance /, that is, increases towards the generator, while the other, with factor A,-“, decreases towards the generator and thus increases with increasing distance from the generator. The phase angle of the former decreases, that of the latter increases towards the generator, and the first term thus can be called the main wave, the second term the reflected wave. At the point / = 0, by equations (17) we have . I, =4,- A, - Z Ey = v2h4, + A, ., and the ratio A, . —4= m (cost +j sin 1), . A, where z may be called the angle of reflection, and m the ratio of amplitudes of reflected and main wave at the reflection point. 8. The general integral equations of current and voltage dis- tribution (17) can be written in numerous different forms. Substituting — A, instead of + A,, the sign between the terms reverses, and the current appears as the sum, the voltage as <dlifference of main and reflected wave.

288 TRANSIENT PHENOMENA Rearranging (17) gives [= (Ae** 7 Ay“) cos Bl — 7 (Aet*+ Ay~“) sin Al and _ (18) RAVES cette “Seospl- ide Again al} Substituting (7) gives V2 ZV Yy = V = y’ (19) . and substituting a = B, and S = B,, or fs = C, and 2 =C, changes equations (17) to the forms, I=V B,e*™ (cos Bl — j sin Bl) —B,e~% (cos fl + sin Bl) . and (20) E=Z | Be**(cos lj sin Bl) +B,¢% (cos fl +7 sin Bl), | or I1=Y Ce +4! (eos Bl—]j sin Bl) —C,e~~ (cos Bl +7 sin Bl) | and (21) E=V t cye*(con Bl—]j sin Bl) +C,e “(cos Bl + j sin fl) | |

LONG-DISTANCE TRANSMISSION LINE 289 Substituting in (17) A, A —L = D,and— = D. VY v1 VY 73 gives 1=VY }D4** (cospl—jsin A) —Dg-(cossl+jsin at} and (22) E=VZ } D,e (cosfl —jsin Bl) + D,¢“ (cosfl+ jain Bl) . Reversing the sign of /, that is, counting the distance in the opposite direction, or positive for decreasing power, from the generator towards the receiving circuit, and not, as in equations (17) to (22), from the receiving circuit towards the generator, exchanges the position of the two terms; that is, the first term, or the main wave, decreases with increasing distance, and lags; the second term, or the reflected wave, increases with the dis- tance, and leads. Equations (17) thus assume the form I = Aye“ (cos Al + j sin fl) — Ae*# (cos fl — j sin Al) and 3 (23) B= V2 4 go (cosp + ising) +A +“ (cos fl —jsin fl) ; and correspondingly equations (18) to (22) modify. 9. The two integration constants contained in equations (17) to (23) require two conditions for their determination, such as current and voltage at one point of the circuit, as at the generator or at the receiving end; or current at one point, voltage at the other; or voltage at:one point, as at the generator, and ratio of voltage and current at the other end, as the impedance of the "receiving circuit. . . Let the current and voltage (in intensity as well as phase, that is, as complex quantities) be given at one point of the circuit, and counting the distance / from this point, the terminal con- ditions are | -L=90, . T=I1,=% + jt, (24) . and E = E, =e, + je,. .

290 TRANSIENT PHENOMENA | Substituting (24) in (17) gives : I,= A, - A, and Z E, = V2(4, + A,); . hence, ° 1 y¥ ‘ A=3h1,4+ 8 rt and 1 Y A, = -3\l. - EV 54, and substituted in (17) gives 1 Y\ 4a - ; 1 =54(Fo+ BV z)e** (cos Al — j sin a | Y\ 4 a +(1- E, ZIé (cos Al + j sin Al) and _ (25) . B=-4S(B, + 1,2) (cos al — j sin al Z\ a -

• (Ba — 1aV Ge (cos fl + isin 9. J If then /, and E, are the current and voltage respectively at the receiving end or load end of a circuit of length /,, equations (25) represent current and voltage at any point of the circuit, from the receiving end / = 0 to the generator end / = 1,. If 7, and £, are the current and voltage at the generator terminals, since in equations (17) / is counted towards rising power, .in the present case the receiving end of the line is repre- sented by / = — 1,; that is, the negative values of / represent the distance from the generator end, along the line. In this case it is more convenient to reverse the sign of /, that is, use equations (22) and the distribution of current and voltage at distance / from the generator terminals. J/,, E, are then given by

LONG-DISTANCE TRANSMISSION LINE 291 1 Y\ a os I=5 {(iot BV 5)e-* (cos pl + jsin Al) Y) 4a «a

• (14 — BV z)e** (cos At — j sin a} and ‘ (26) 1 yz a .. E =5 { (B+ I, ze (cos fl + 7 sin Al) Z\ sal . +(B,- LV et (cos fl jsin pt. ° ° Y,

10. Assume that the character of the load, that is, the impe- dance, Bs =Z,=17,—-jz,, oF admittance, 4! -¥,-7 = 9,+90,, 71 1 1 of the receiving circuit and the voltage E, at the generator end of the circuit be given. Let J, = length of circuit, and counting distance / from the generator end, for 1 = 0 we have E=E,; this substituted in equation (23) gives E, = v2a, + A,). (27) . However, for 1 = 1,, | 774 | substituting (23) herein gives ZV 2 Ait (008 Bla + jsin fl) + Aye*% (cos fly j sin Bly). ' YAe-% (cos fl, + jain Al,)— A,e*% (cos Al, — j sin £l,) , hence, substituting (19) and expanding, . As ~ “(cos Bl, +jsinfl,) VZ,+2Z | Agt (cos fl, —jsin fl.) VZ,— 2’ VZ,-Z _ a or A,= Atz aa 2b (cos 2 Bl, + j sin 2 £l,), i

292 TRANSIENT PHENOMENA | and denoting the complex factor by VZ,-Z - C= Wa.4z°. “*(cos 2 Al, + j sin Bl,), (28) | which may be called the reflection constant, we have A, = CA,, and by (27), _ | 4-2 yt “14+ C°2Z (29) . and 4-2, 72 1 + C Z ? hence, substituted in (22), . [= Ee VE e+ (cools isin) — Ceo 6l—j sin pp} and E 725 jes (cos fl + 7 sin Bl) + Ce** (cos fl —j sin Bl) . (30)

11. As an example, consider the problem of delivering, in a . three-phase system, 200 amperes per phase, at 90 per cent power . factor lag at 60,000 volts per phase (or between line and neutral) and 60 cycles, at the end of a transmission line 200 miles in length, consisting of two separate circuits in multiple, each consisting of number 00 B. and S. wire with 6 feet distance between the conductors. ,

Number 00 B. and S. wire has a resistance of 0.42 ohms per mile, and at 6 feet distance from the return conductor an inductance of 2.4 mh. and capacity of 0.015 mf. per mile.

The two circuits in multiple give, at 60 cycles, the following line constants per mile: r = 0.21 ohm, L = 1.2 x 107° henry, and C = 0.03 x 107° farad; hence,

x = 2afL = 0.45, Z = 0.21 — 0.45), z = 0.50, and, neglecting the conductance (g = 0), b = 22fC = 11Xx10-*, Y = — 11 x 10-°j, y = 11 x 107°,

LONG-DISTANCE TRANSMISSION LINE 293 and a = 0.524 x 1073, B = 2.285 x 107°, — BD V = (0.524 — 2.285 7) 107°, Y ov ag Vv; “377 (4.53 — 0.9 7) 10 JZ Vv and \V F 7 p 7 (0.208 + 0.047 7) 10**. Counting the distance / from the receiving end, and choosing the receiving voltage as zero vector, we have l=0, E = E, = e, = 60,000 volts, . and the current of 200 amperes at 90 per cent power factor, T=1,=%, + jt, = 180 + 87), and substituting these values in equations (25) gives ] =(226 + 14.4 j) e+” (cos Bl—j sin Bl) — (46 —72.6 7) e~@

. (cos Bl + 7 sin Bl), in amperes, . and (32) E= (46.7 + 13.3 j)e*™(cos Bl—]j sin Bl) + (13.3 — 13.3 j) «~

(cos fl + j sin fl), in kilovolts, where a and # are given by above equations (31). From equations (32) the following results are obtained. Receiving end of line, 1=0 1 = 180 + 877 i = 200 amp. tan 6’ = 0.483 6, = 26° E= 60 x 10 e = 60,000 volts 6,= 0 fi , 0.90 Middle of line, ’ t= 100 Power factor 198 I = 177 + 18] i = 178 amp. tan0,=+0.102 6,= 6° E= (66.2 —6.97) 10° ¢ = 66,400 volts tané,——0.104 6,=——6° 6,-6,=0 = 12° power factor, cos 0 = 0.979 lag. Generator end of line, l = 200 1 = 165.7 — 567 t = 175 amp. tan 6, =— 0.338 6, = —19° E= (69 — 157) 10° e = 70,700 volts tan 0#,=—0.218 06, = —12° 0,-6<0—=-7° power factor, cos ? = 0.993 lead.

294 TRANSIENT PHENOMENA

As seen, the current decreases from the receiving end to the middle of the line, but from there to the generator remains prac- . tically constant. The voltage increases more in the receiving | half of the line than in the generator half. The power-factor is practically unity from the middle of the line to the generator.

, 12. It is interesting to compare with above values the values | derived by neglecting the distributed character of resistance, | inductance, and capacity.

From above constants per mile it follows, for the total line of 200 miles length, r, = 42 ohms, z, = 90 ohms, and b, = 2.2 x 107 * mho; hence,

Z, = 42 — 90j and Y, = — 2.27 10-*. (1) Neglecting the line capacity altogether, with J, and E, at the receiver terminals, at the generator terminals we have I,=1, and E, = E, + 214; ; hehce, I, = 180 + 877 i, = 200 amp. tan 0,= 0.483 6, =+ 26° E,= (75.4 —12.67)10 e, = 76,400 volts tané@,=—0.167 6,=— 9° 6,—6,=0 =+ 34° power factor, cos § = 0.83 lag.

These values are extremely inaccurate, voltage and current at generator too high and power factor too low.

(2) Representing the line capacity by a condenser at the generator end, that is, adding the condenser current at the generator end,

I,=1,+ Y,, and . BE, = E, + 2,14; hence, | I, = 152 — 897 i, = 176 amp. tan 0, =— 0.585 6, = —30° E,= (75.4 — 12.67)10° e, = 76,400 volts tan 6, = —0.167 6,—= — 9° 6,-6,=6 =— 21° | power factor, cos 6 = 0.93 lead. |

LONG-DISTANCE TRANSMISSION LINE 295 As seen, the current is approximately correct, but the voltage | is far too high and the power factor is still low, but now leading. (3) Representing the line capacity by a condenser at the receiving end, that is, adding the condenser current at the load, I, =1,+ Y,k, , and E, = EB, + Z)1,; hence, I, = 180 — 457 i, = 186 amp. tan 0, =— .250 0, = —14° E,= (63.5 — 18.17) 10° e, = 66,000 volts tan 6, =— .285 6, = —16° 0,-6,-0 = + 2 power factor, cos? = 1.00 In this case the voltage e, is altogether too low, the current . somewhat high, but the power factor fairly correct. (4) Taking the average of the values of (2) and of (3) gives I, = 166 — 677 t;=179 amp. tan 0, =— 0.403 6, =— 22° E,= (69.4 — 15.37) 10° e, = 71,100 volts tan é@,=—0.220 6,=— 12° 0, — 0,=0 =— 10° power-factor, cos 9 = 0.985 lead. As seen by comparing these average values with the exact result as derived above, these values are not very different, but constitute a fair approximation in the present case. Such a close coincidence of this approximation with the exact result can, however, not be counted upon in all instances. 18. In the equations (17) to (23) the length 27 ly = (33) B . . . 2 is 8 complete wave length, which means that in the distance 7 the phases of the components of current and of e.m.f. repeat, and that in half this distance they are just opposite. Hence, the remarkable condition exists that in a very long line at different points the currents are simultaneously in oppo- site directions and the e.m.fs. are opposite.

296 TRANSIENT PHENOMENA | The difference of space phase z between current J and e.m.f. E at.any point / of the line is determined by the equation m (cos t + j sin )-4, (34) where m is a constant. Hence, t varies from point to point, oscillating around a | medium position, t,,, which it approaches at infinity. | This difference of phase, t,, towards which current and | e.m.f. tend at infinity, is determined by the expression . . m (cos tT, + jsin t,) = [4 , . tl me or, substituting for F and / their values from equations (23), and since e~“ = 0, and A,«“ (cos fl — j sin Bl) cancels, wo Z _ V _ ae 78 : m (cos t, + j sin t,) = Y YY "gop _ (ag + fb) + j (ad — Bg). . b+ g ? hence, tant, = Ee . (35) 14. This angle, c, = 0; that is, current and e.m.f. come more and more in phase with each other when ab — Bg =.0; that is, a+ypBp =g+b,or ek Gf -¥. 208 29d ’” substituting (12) gives . . gabe _ g -¥. gt+br = 2gb ’ hence, expanding, r+zr=g+b; (86) that is, the ratio of resistance to inductance equals the ratio of leakage to capacity.

LONG-DISTANCE TRANSMISSION LINE 297

This angle, r,, = 45°; that is, current and e.m.f. differ by one-eighth period if + ab — fg = ag + fb, or

a bt+g.

B b-g which gives "rg + 2b =0, (37) which means that two of the four line-constants, either g and x or g and b, must be zero.

The case where g = 0 = z, that is, a line having only resistance ; and distributed capacity but no self-inductance, is approxi- mately realized in concentric or multiple-conductor cables, and in these the space-phase angle tends towards 45 degrees lead for infinite length.

16. As an example are shown the characteristic curves of a transmission line of the relative constants, r:z:g:6 = 8:32:1.25 x 10:25 x 10% and e = 25,000,

7 = 200 at the receiving circuit, for the conditions

(a) Non-inductive load in the receiving circuit, Fig. 80.

(b) Wattless receiving circuit of 90 time-degrees lag, Fig. 81. °

(c) Wattless receiving circuit of 90 time-degrees lead, Fig. 82.

These curves are determined graphically by constructing the topographic circuit characteristics in polar codrdinates as explained in “The Theory and Calculation of Alternating- Current Phenomena,” fourth edition, Chapter VII, paragraphs 42 to 44, and deriving corresponding values of current, potential difference, and phase angle therefrom.

As seen from these diagrams, for wattless receiving circuit, current and e.m.f. oscillate in intensity inversely to each other, with an amplitude of oscillation gradually decreasing when passing from the receiving circuit towards the generator, while the space-phase angle between current and e.m.f. oscillates between lag and lead with decreasing amplitude. Approximately maxima and minima of current coincide with minima and maxima of e.m.f. and zero phase angles.

For such graphical constructions, polar codrdinate paper and two angles a and @ are desirable, the angle a being the angle between current and change of e.m.f., tan a = == 4, and the

| | 298 TRANSIENT PHENOMENA 0 af SOT sR ET Pp] Pr ET ese tt pL ft PTT yy tT tet tt tT Ty PT TT TTT TT tt yt yg Pi eye Pr TE TAA SRR eee BRR Zea PEt TTT TT pact tel PET TTT TT tery TT pop ee | tte ptt | tt srs | ft tt tt | PT iT (da Tt Ep>Z ane - ane Une Fig. 80. Current, e.m.f. and space-phase angle between current and e.m.f. ; in a transmission line. Non-inductive load. wee ti ttt ty ttt tt tt ) AN weit ETT PAT TTT TT yt Stee 4a fff LN ZS || law of carrend | iA EEE Sleey7 eens Ll, «LENE CE SN . | tt -4 LAIN Z ‘ens aann ay Eee) Seen =< A505 | | ly Fanwae <a Annnnn PCO EY, NOIZE one ; PM TTT ttt ei et, Fig. 81. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Inductive load, |

: LONG-DISTANCE TRANSMISSION LINE 299 angle 6 the angle between e.m.f. and change of current, tan d = : = 20 in above instance. With non-inductive load, Fig. 80, these oscillations of intensity have almost disappeared, and only traces of them are noticeable in the fluctuations of the space-phase angle and the relative values of current and e.m.f. along the line. Towards the generator end of the line, that is, towards rising power, the curves can be extended indefinitely, approaching more and more the conditions of non-inductive circuit. Towards decreasing power, however, all curves ultimately reach the conditions of a wattless receiving circuit, as Figs. 81 and 82, at the point where the total energy input into the line has been consumed therein, and at this point the two curves for lead and for lag join each other as shown in Fig. 83, the one being a prolongation of the other, and the power in the line reverses. Thus in Fig. 83 energy flows from both sides of the line towards the point of zero power marked by 0, where the current and e.m.f. are in quadrature with each other, the current being leading with regard to the power from the left and lagging with regard to the power from the right side of the diagram. 16. It is of interest to investigate some special cases of such circuits of distributed constants. (A) Open circuit at the end of the line. Assuming a constant alternating e.m.f. £, impressed upon a circuit at one end while the other end of the circuit is open. Counting the distance / from the open end of the line, and denoting the length of the line by /,, for 2 = 0, I = I o~ 0, and for / = l,, E=-E,; . hence, substituting in equations (17), _ 0= A, a A, V2 4 sete sin al.) 4 Agent - . B,-V F} 4e***(c0s Bly—j sin ly) +Aye-*(cos fly + js A) ¢ hence, A,=A,=A |

300 TRANSIENT PHENOMENA wh LIANE wert TAT TTT Pe TTT | a Ne ANS PTT tt iV | | YET | NUT ot tH TT TAT A TT TT Pr PTA TT | TSF tT TE TT o( Yi ttt] ttt ttt tT ely, 4RR ERR tt LT TTT TT TT tT TAT ZT Pt TTT tT TT PAY AT P| tT | Pe AA TAS TTT Pt TT wert ST YT TT CT pe Hee XT VNE SAT TTT TT, PK LA | NA oislanct ef [fT TT PITT TTT TTT Tite tT i, Fig. 82. Current, e.m.f. and space-phase angle between current and e.m.f. . ; in a transmission line. Anti-inductive load. paasesran Prec eee eels Ty INT Ela TT | ET TTT | 4 fT Nee tT Pres Seni Cee oi Gt he 4A SReeene ; Vit tt | | ACER | i } | | SaSRGGBRe\o7SRRure Seaee eneer a } | | -|e] 8} | PERN Te “(fe We 4 In| | CH Ue ar anctes } | | | KY Tat L = me ft bf AN eee Fig. 83. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line, . : |

LONG-DISTANCE TRANSMISSION LINE 301 and _ ayy A = [yal (cos Bl, — jin Bl,) +e (cos Bl, —] sin Aly 7 ave = ete + e-) cos Bl, —j (e™* —€ ) sin fl,’ hence, substituting in, (17), 1-E rer — e~%) cos Al — j (e+ + e~%) sin Bl ° “IV Z (et%+e~%) cos Bl,—j (e+ — e~%) sin fl, and (38) E<E (e+ + e~) cos Bl — j (e** — e7™) sin Al “STV etal 4. @— 2) cog Bl, — j (et — e-) sin fl, At I = 0, or the open end of the line, by equations (38), I, =0 and | ~ oO K,g : By = (ere + 2%) cos Bl, ~ 7 (et — e~%) sin Bl, (39) The absolute values of J and E follow from equations (38) | and (39): | l=E yy ee — e~%)? cos? Bl + (e+% + e-“)? sin? Al | Wz T (ete + 6%)? cos? Bl, + (ete — 2%)? sin? Bl,’ which expanded gives 1=B' /y et? + e-? — 2 cos 2 Bl tN g ete 4 ¢— 20 + 2 cos 2 Al, and ee (40) EWE /et?t + e~2% 4 2082 Al tN ¢t2el 4 ¢— 200 + 2 cos Al, and 2E £, = -——-... (41) Ve * 2a + 2h + 2 cos 2 Bl,

. | 302 TRANSIENT PHENOMENA | As function of J, the e.m.f. # or the current J is a maximum or minimum for a 420, ,—2al . a\t +e + 2cos 2 £l)= 0; : | hence, . a (et? — ¢- 24) = 4+ 2 Asin 2 fl. (42)

For J = 0, and since « is a small quantity, the left side of (42) also is small, and for values of sin 2 6l approximating zero, that is, in the neighborhood .of / = zB or where fl is a multiple of a quadrant, equation (42) becomes zero. At fl = 2 ne , or the even quadrants, EF is a maximum, / a minimum, at Bl = (2n - 1) + or the odd quadrants, F is a minimum, J a maximum.

The even quadrants, therefore, are nodes of current and wave crests of e.m.f., and the odd quadrants are nodes of e.m.f. and crests of current.

A maximum voltage point, or wave crest, occurs at the open end of the line at / = 0, and is given by equation (41). As func- tion of the length J, of the line this is a maximum for

S (e+e + e~ 7% + 2cos 2 Bl.) = 0, dl, or a (e+?e% — ¢— 20) — 2 Bsin 2 l,, or approximately at nz | Bl, = 3 | fon 2. (43) on 2 B , that is, when the line is a quarter wave length or an odd multiple | thereof. | | |

| LONG-DISTANCE TRANSMISSION LINE 803 a Substituting in (41), Al, = 5 gives (44) 2E E. = ——_4 21 OO 0 Vette 4g tae 9 2E = Fah yah (45) Since 1 et eo — 1 tal, + gall,’ + 53 alle’ + +..., for small values of al, we have ete _ eo = 2al, and ; E,

E, = al,’ (46) which is the maximum voltage that can occur at the open end of a line with voltage Z, impressed upon it at the other end.

Since, approximately,

B= V2xb = 22fVvLC, by (44) we have 1 = ——==: 47 I-4 1, VIC * a7) the frequency which at the length of line J, produces maximum voltage at the open end.

For the constants in the example discussed in paragraph 11 we have J, = 200 miles, r = 0.21 ohm, L = 1.2 x 10-* henry, C = 0.03 x 10-° farad, g = 0, f = 208 cycles per sec., x = 1.57 ohms, z = 1.58 ohms, b = 39 x 10-* mho, a = 0.53 X 10~, and E, = 9.3 E,.

(B) Line grounded at the end.