SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965 POWER TRANSMISSION AT HIGH FREQUENCY F. P. Emery - Darius Irani AiResearch Manufacturing Company* Torrance, California ABSTRACT This paper presents data for power transmission up to a frequency of 5000 cycles per second and enables calculation of the resistance and inductance of a transmission line delivering high frequency power. Classical expressions are involved. Graphs and data are presented from which values can be selected and the transmissIon Iline parameters eas i ly determined. A knowledge of these parameters enables the design of an efficient and acceptable high frequency power transmission system. INTRODUCTION Two undesirable effects are associated with high f requency power t ransm I s s ion: I) A decrease in the current carrying capacity of the cable. The a-c resistance R of a cable increas- es with frequency and the effect of an increase in resistance is to increase transmission losses and reduce current carrying capacity. Since transmis- sion losses are given by I R, the current carrying capacity is reduced by the ratio (R/Ro)l/2 where Ro is the d-c resistance of the cable and is obtained from cable data. 2) Poorer regulation due to an increase in cable voltage drop. The reactance of a cable is given by 21TfL. Thus the higher the frequency the higher the reactance. Actually the inductance L decreases with an Increase of frequency, but this reduction Is small Also L is quite independent of cable size. Thus if high frequency transmission at low voltage intro- duces serious regulation problems, the choice of a larger size cable will not alleviate the situation and it may then be necessary to resort to special cabl e conf I gu rat i ons . High frequency power systems of the type considered are generally low voltage and have short transmis- sion lines. The effect of capacitance C of the line Is negligible. The leakage conductance G of the line can be neglected since the dielectric loss Is quite low in the frequency range considered. ACKNOWLEDGEMENT The work described In this paper was supported by Contract No. NOBS 88475 granted to the Louis Allis Co. by the U.S. Navy Bureau of Ships, Washington D.C. The authors express their thanks to the Commanding Officer, BuShips for permission to publish their findings. EFFECT OF FREQUENCY ON RESISTANCE OF CONDUCTOR D-C OR ZERO FREQUENCY RESISTANCE Ro In a homogenous conductor carrying direct current, the current will be distributed uniformly in the cross section of the conductor. The conductor can *A Division of THE GARRETT CORPORATION be considered as an infinite number of small strands of equal infinitesimal cross-sectional area in paral- lel with each other. In order to have the same vol- tage drop in each of the small strands, the current in each must be the same; in other words, the cur- rent density is the same in all parts of the cross section. Only when a conductor is carrying continu- ous current or current of zero frequency is the re- sistance of the conductor given by the formula. Ro = py, ((I) oA Direct current transmission is general ly accomplished by two parallel wires, with the second wire acting as the return conductor. Since the two wires are carrying continuous current in opposite directions, there is a force of repulsion between them. Since this force is due to the current it may at first seem that most of the current in each of the two wires would be repelled to the side farthest away from the other. However, the voltage drop in each of the infinite number of infinitesimal strands must be the same, hence the current has to be uniformly distributed In each conductor. A parallel conductor, therefore, does not affect the current distribution of a conductor carrying direct current. SKIN EFFECT When a current flows in a conductor it sets up a magnetic field. If the current in the wire is alter- nating,magnetic flux within the conductor induces an e.m.f. The induced e.m.f. is greater at the center of the wire than at the surface because there are more lines of flux surrounding the central part. In each of the infinite number of infinitesimal strands the voltage drop must be the same. Since the drop in this case consists of an induced e.m.f. in addi- tion to the ohmic drop, the ohmic drop in an infini- tesimal strand at the center will be less than alona one at the surface. The e.m.f., therefore, causes the current density to decrease in the interior of the wire and increase at the surface. The unequal current distribution, called the SKIN EFFECT, results in a larger power loss for a given rms alternating current than for the same value of direct current flowing in the wire. Since the induced e.m.f. is directly proportional to the frequency of the alter- nating current, the skin effect becomes more pro- nounced as the frequency is increased. The power loss In a conductor is thus, a function of frequency. The a-c resistance of a conductor is defined as the average power loss in the conductor divided by the mean square current. The res istance rat io R ji5 the quot ient of the a-c Ro resistance divided by the d-c or zero resistance. This ratio is very nearly unity for small wires of non-magnetic material at commercial power frequencies, but increases with the area of cross section of the conductor, the permeability, and the frequency. NOMINAL DEPTH OF PENETRATION As the frequency is increased, the current progres- 402
Transcript
SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
POWER TRANSMISSION AT HIGH FREQUENCY
F. P. Emery - Darius IraniAiResearch Manufacturing Company*
Torrance, California
ABSTRACT
This paper presents data for power transmission upto a frequency of 5000 cycles per second and enablescalculation of the resistance and inductance of atransmission line delivering high frequency power.Classical expressions are involved. Graphs and dataare presented from which values can be selected andthe transmissIon Iline parameters eas i ly determined.
A knowledge of these parameters enables the designof an efficient and acceptable high frequency powertransmission system.
INTRODUCTION
Two undesirable effects are associated with highf requency power t ransm I s s ion:
I) A decrease in the current carrying capacity ofthe cable. The a-c resistance R of a cable increas-es with frequency and the effect of an increase inresistance is to increase transmission losses andreduce current carrying capacity. Since transmis-sion losses are given by I R, the current carryingcapacity is reduced by the ratio (R/Ro)l/2 where Rois the d-c resistance of the cable and is obtainedfrom cable data.
2) Poorer regulation due to an increase in cablevoltage drop. The reactance of a cable is given by21TfL. Thus the higher the frequency the higher thereactance. Actually the inductance L decreases withan Increase of frequency, but this reduction Is smallAlso L is quite independent of cable size. Thus ifhigh frequency transmission at low voltage intro-duces serious regulation problems, the choice of alarger size cable will not alleviate the situationand it may then be necessary to resort to specialcable conf I gu rat i ons .
High frequency power systems of the type consideredare generally low voltage and have short transmis-sion lines. The effect of capacitance C of the lineIs negligible. The leakage conductance G of the linecan be neglected since the dielectric loss Is quitelow in the frequency range considered.
ACKNOWLEDGEMENT
The work described In this paper was supported byContract No. NOBS 88475 granted to the Louis AllisCo. by the U.S. Navy Bureau of Ships, Washington D.C.The authors express their thanks to the CommandingOfficer, BuShips for permission to publish theirfindings.
EFFECT OF FREQUENCY ON RESISTANCE OF CONDUCTOR
D-C OR ZERO FREQUENCY RESISTANCE Ro
In a homogenous conductor carrying direct current,the current will be distributed uniformly in thecross section of the conductor. The conductor can
*A Division of THE GARRETT CORPORATION
be considered as an infinite number of small strandsof equal infinitesimal cross-sectional area in paral-lel with each other. In order to have the same vol-tage drop in each of the small strands, the currentin each must be the same; in other words, the cur-rent density is the same in all parts of the crosssection. Only when a conductor is carrying continu-ous current or current of zero frequency is the re-sistance of the conductor given by the formula.
Ro = py, ((I)oA
Direct current transmission is general ly accomplishedby two parallel wires, with the second wire actingas the return conductor. Since the two wires arecarrying continuous current in opposite directions,there is a force of repulsion between them. Sincethis force is due to the current it may at firstseem that most of the current in each of the twowires would be repelled to the side farthest awayfrom the other. However, the voltage drop in eachof the infinite number of infinitesimal strands mustbe the same, hence the current has to be uniformlydistributed In each conductor. A parallel conductor,therefore, does not affect the current distributionof a conductor carrying direct current.
SKIN EFFECT
When a current flows in a conductor it sets up amagnetic field. If the current in the wire is alter-nating,magnetic flux within the conductor induces ane.m.f. The induced e.m.f. is greater at the centerof the wire than at the surface because there aremore lines of flux surrounding the central part. Ineach of the infinite number of infinitesimal strandsthe voltage drop must be the same. Since the dropin this case consists of an induced e.m.f. in addi-tion to the ohmic drop, the ohmic drop in an infini-tesimal strand at the center will be less than alonaone at the surface. The e.m.f., therefore, causesthe current density to decrease in the interior ofthe wire and increase at the surface. The unequalcurrent distribution, called the SKIN EFFECT, resultsin a larger power loss for a given rms alternatingcurrent than for the same value of direct currentflowing in the wire. Since the induced e.m.f. isdirectly proportional to the frequency of the alter-nating current, the skin effect becomes more pro-nounced as the frequency is increased. The powerloss In a conductor is thus, a function of frequency.The a-c resistance of a conductor is defined as theaverage power loss in the conductor divided by themean square current.
The res istance rat io R ji5 the quot ient of the a-cRo
resistance divided by the d-c or zero resistance.This ratio is very nearly unity for small wires ofnon-magnetic material at commercial power frequencies,but increases with the area of cross section of theconductor, the permeability, and the frequency.
NOMINAL DEPTH OF PENETRATION
As the frequency is increased, the current progres-
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SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
sively uses smaller portions of the wire cross sec-tion until, at radio frequencies, the current in awire of moderate size Is concentrated in a thin skinat the surface.
By analysis It can be shown that in the case of con-ductors whose cross section Is large compared to thethickness of the "skin", the current density variesexponentially Inward from a maximum value at thesurface. The distance in which the current densitydecreases to l/e or 0.368 of the maximum value iscalled the nominal depth of penetration 6, where, 8is given by
6 = ImetersV/F p. (2)
The free space value of p is 4 1r x 10 7 henry/meter
For a copper conductur
8 =6.64 2.61 Inches~-ff ft (3)
For an upper limit of frequency of 5000 cycles persecond
8 = -61 0.037 inches
if a round conductor has a radius equal to the nomi-nal depth of penetration 6, the high frequency re-sistance of the isolated conductor is the same asthe d-c resistance, or in other words the ratioRisolated/Ro = 1. More generally, a conductor canbe replaced by a ficitious hollow conductor of thesame surface shape but with a wall thickness equalto the nominal depth of penetration. Then the a-cresistance of the actual conductor is preciselyequal to the d-c resistance of the fictitious hol-low conductor.
A-C RESISTANCE OF AN ISOLATED CYLINDRICAL CONDUCTOR
The derivation of the formulas for the a-c resis-tance and internal inductance of an Isolated cylin-drical conductor are given in detail in Reference 1,pages 71 - 80 and Reference 2, pages 54 - 62.
The effective resistance of an isolated cylindricalconductor is given by:
Rp ber q bei'q - _bei q berq ohmsRisolated = /2TTa8 (beilq)0+ (ber'q)0 meter
.f29 (4)whe re q= 8
and ber and bei are the real and imaginaryBessel funct ion
ber' and bei' are derivatives of berand bei, respectively
As the frequency approaches zero q-.P.6 and the aboveexpression reduces to:
Ro = T ohms/meter (d-c resistance) (5)
The resistance ratio is given by:
Risolated _q ber q bel'q - bei q ber(qR - 2 (beitq)z + (bertq)2 (6,
The ratio Risolated/Ro is plotted In Figure I as afunction of the ratio of radius to nominal skindepth, a/A. When a/6 is large:
R -= R a-e- ohms/meter (7)I solIated ~2TTa6 - 26
R1 x T ohms/meterisolated - R0 rTa (8)
Figure 2 is a plot of visolate s where RdcRioae -Rheec d
Is the resistance in ohms per 1000 feet. For sizeof wire selected, Rdc can be readily obtained froma handbook.
For those who have an apprehension of picking fig-ures from graphs, Table I gives the skin effect re-sistance and Internal inductance ratios for differ-ent values of "q".
2 505 1 0 20 - 40 50
0 .30
.16 A-I""
Figure 1. Skin Effect Resistance Ratio for an Iso-lated Cyl1 indrical Conductor
PROXIMITY EFFECT
When the two parallel conductors of an a-c trans-mission line are close together, the current distri-bution is no longer symmetrical about the axis ofthe wire due to the fact that the current in one con-ductor induces eddy currents in the other. This in-fluence of one wire on another is called the PROXIM-ITY EFFECT.
When two parallel conductors carry currents in oppo-site directions as In a single phase transmissionline, the magnetic flux density is greatest in theregion between the conductors and the current willtend to concentrate on the surfaces close to eachother. Similarly, when two parallel conductors carrycurrent in the same direction, the current will tendto concentrate on the surfaces which are farthestfrom each other.
The proximity effect resistance ratio R isRisolatedthe quotient of the a-c resistance when the conduc-tors are near each other, divided by the a-c resist-ance of the isolated conductor. The a-c resistanceR, thus includes the increase in resistance due toskin effect, and also the increase in resistance dueto the proximity of a parallel conductor or conduc-tors. The a-c resistance of the isolated wireRisolated includes only the increase in resistance
due to the skin effect.
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SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
3.8
2.5
20
1010 _ _ ~__,____
0 100 200 300
,7k- (R4c IN PER 1000 FT)
Figure 2. Skin Effect Resistance Ratio for an Iso-
lated Cylindrical Conductor.Plotted as a Functionof
,/_,/ d_
TABLE I
R/R0 AND Li/Li0 RATIOS FOR ISOLATED CYLINDRICAL
CONDUCTORS AS A FUNCTION Q /2 a/b
In setting up an equation defining current distribu-tion due to the proximity of the conductors, it be-comes apparent that the solution of the equation re-
quires a massive amount of mathematical manipulation.
The paper by H. B. Dwight entitled "Proximity Effectin Wires and Thin Tubes" Reference 3, is a classicand is the only known comprehensive paper on Prox-
Rimity Effect which gives the ratio R The
R isolatedformulae for the ratio for single phase and
R isolatedthree phase circuits of various configurations are
extremely involved, requiring an enormous amount ofcalculation of such a nature as to restrict the use.
The curves in Figure 3 show the proximity effect oftwo wires in a single phase circuit for differentvalues of the ratio D/2a where D is the axial spac-
ing and 2a the diameter of the conductor, both fac-tors in the same units.
These curves are obtained from Reference 4, and are
based on tests conducted with great precision andaccuracy. The proximity effect ratio is plotted as
a function ofRdc
By the use of Figure 3 and interpolating for differ-ent values of D/2a, the proximity effect ratio canbe determined with sufficient accuracy. By the use
of Figure 2 the skin effect ratio R_ can be
Risolatedobtained. The product of these two ratios will givethe overall ratio R/Ro to a fair degree of accuracy.
Tests indicate that Figure 3 can also be used in de-termining the proximity effect ratio for three phasecables of equilateral triangular configuration. Here"D" is the spacing between any two wires.
Figure 4 Is a curve of spacing vs frequency and is
plotted for a constant ratio R/Ro of 1.1. At a fre-quency of around 95 cycles the skin effect alone
gives a ratio of R/Ro of 1.1, but as the two equal
copper wires of I cm radius are brought closer to-gether, the frequency at which R/Ro is 1.1 steadilydecreases, quantitatively demonstrating the effectof proximity.
EFFECT OF STRANDING AND SPIRALING OF THE STRANDS
If a cable was made from solid round conductors only,it would be extremely inflexible in the larger sizes.The cable is made flexible by stranding the conduc-tors. The greater the number, and thus, the smallerthe size of the strands., the more flexible is thecable. A superior mechanical construction is ob-tained by spiraling the stranding, successive layersbeing alternately spiraled.
The problem of determining the a-c resistance ofstranded conductors is more complex than determiningthe a-c resistance of solid conductors because spiral-ing of the strands induces voltages in each of thelayers. These induced voltages are different for thedifferent layers, and affect the current distribu-tion in addition to the distortion produced by skineffect and proximity effect in the individual strands.
Even if the spiraling effect is neglected, the mathe-matical solution for the a-c resistance of strandedconductors breaks down when the effect of proximityis considered. It has been shown by very accurate
SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
7 I Ia < s t. r
2.0!2_ / |
1.8 T 1Tri.6FT:=4 =
. L < <t | ~~~~~~~~NOTE:VALUES ON CURVIES
:~~~~~~~~~~~~~~~~~R D/2a0 50 100 200
/fIR-, (R IN OHMS PER 1000 FTdc dc
Figure 3. Single Phase Proximity Effect Ratio for Cylindircal Conductors
FREQUENCY A-100
Figure 4. Proximity Effect Interaxial Spacing vsFrequency for R/Ro = 1.1 for two equal Cu Wires ofRadius 1.0 cm Each
measurements (References 5 and 6) that the a-c re-sistance of a stranded conductor, if it is not spi-raled, is about the same as that of a sol id conduc-tor if the two have equal cross-sectional areas.
The stranding of a conductor and spiraling of thestrands generally results in a lesser increase ofresistance ratio R/Ro than in the case of a sol idconductor, and depends on the degree of spiralingand contact resistance between strands.
In the theoretical calculation of R/Ro, the effectof stranding and spirality is neglected in this paper.
SAMPLE CALCULATIONS FOR R/Ro
The resistance ratio R/Ro for solid conductor 0000cable will be calculated for three interaxial spac-ings of 0.9 cm, 2.4 cm and 60 cm respectively. Dataon this cable for the spacings selected is publishedin the discussion to Reference 4 and is based ontests conducted with great precision for scientificrather than engineering accuracy.
The interaxial spacing is the distance between con-ductors hence:
D = Interaxial Spacing + 2a (9)
The method of calculation is straightforward. Thediameter "2a" and resistance Rdc in ohms per 1000feet are determined from a handbook of cable data.D/2a is then calculated for the three separationdistances of 0.9 cm, 2.4 cm, and 60 cm. For the pur-
pose of this sample calculation, five frequencies ofIKC, 2KC, 3KC, 4KC and 5KC were chosen and ftiFdcdetermined in each case. At the several values of
405
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LLJLo
.L4
:r
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300
A-10029
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SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
/f7W-dc, the skin effect ratios RIsolated/Ro were
determined from Figure 2. At the various values of/V7dc and D/2a the proximity effect ratios
R/RlsoIated were determined from FIgure 3. The pro-
duct of these two ratios is the calculated resist-ance R/Ro. It is seen from Table II, that the cal-culations agree closely with the values obtainedfrom Reference 4.
The values obtained at a frequency of IKC from a
test method developed by the authors is also shownin Table II. This test method will be the subjectof a subsequent paper.
From Cable Handbook
Rdc for 0000 cable = 0.05 ohms/1000 ft at 200C
Diameter of solid conductor 0000 cable,2a = 0.460 inches
Calculation for D/2a
D = 0.46 + 0.9/2.54 = 0.814 in. D/2a = 1.765
D = 0.46 + 2.4/2.54 = 1.405 in. D/2a = 3.05
D = 0.46 + 60/2.54 = 24.06 in. D/2a = 52.5
TABLE II
CALCULATION OF R/Ro FOR 0000 CABLE
are spaced relatively close to each other and thereduction In Internal inductance should be taken in-to account.
INTERNAL INDUCTANCE OF A CYLINDRICAL CONDUCTOR
The internal Inductance of a cylindrical conductoris given by:
wp bei q bei'q + ber q ber'q ohms
T I 1 -7 ~~~meterLi Ta; (eIq)Z + (ber'q) mtr (0
As the frequency approaches zero, q-_o and the aboveexpression reduces to
Li = -4 henrys/meter
Lio is the low frequency internal inductance.
Li 4 bei q bei'q + ber q bertL io q (be i Iq)' + (berI q)
when
(I I)
(1 2)
a
8 is large
L L 2-6 henrys/meter'2TTaw8 a(13)
The ratio Li is plotted in Figure 5 as a function
a/6. Table I gives the internal inductance ratiofor different values of q.
.0c
o.
0.01o10 too
R-10032
VARIATION OF INDUCTANCE WITH FREQUENCY
The total Inductance of a conductor is the sum ofthe external and internal inductances.
The inductance caused by the flux which Is externalto the conductors is not affected by frequency.
The tendency of alternating currents is to flow withgreater density near the outside of the conductors.As the current flow moves to the outside of the con-
ductors, the internal flux is dimini.shed and thereis a reduction in that part of the inductance whichIs caused by the internal flux. For non-magneticconductors spaced many radii apart, the majority ofinductance is caused by flux external to the con-
ductors, and in such cases the total inductance is
not greatly reduced. However in the case of powertransmission by cable, the non-magnetic conductors
Figure 5. Skin Effect Internal Inductance Ratio foran Isolated Cylindrical Conductor
INDUCTANCE OF PARALLEL WIRE LINES
The external inductance of a cylindrical conductoris given by:
Le = 2 x 10 7 log De a (14)
For a single phase line, "D" is the spacing betweenthe two wires, and for a three phase line of equi-lateral triangle configuration,"D" is the spacingbetween any two wires. For a transposed three phaseline with unsymmetrical spacing, D = 3/Dab Dbc Dcawhere, as indicated by the subscripts, Dab, Dbc andDca are the distances between individual conductorsa, b and c.
406
Proximity Skin Effect Test RatioEffect Ratio Ratio Calc. Test Ratio R/RR/Rlsit Ri0 Atd' Re R/R B0fiso0la ted 150la ted o Res . Byo
f /77 From From Ratio from AuthorsD/2a (Kc) dc Figure 3 FIgure 2 R Ref. 4 Method
SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
The total inductance is the sum of the external andinternal inductances
L = [2 log + uc Lil x 107 henry/meter (15)oge a' 2 uo Li0
where uc is the absolute permeability of the conduc-tor and uc/uo is the relative permeability of theconductor (approximately unity for non-magneticmaterials). The above formula for inductance isbased on the assumption that the wire diameter 2a ismuch smaller than the spacing between wires D.
At high frequencies with a well developed skin effect,a more exact formula for inductance is
L = 2 x 10-7 Cosh 1 a henrys/meter2a
(16)
This formula does not make the assumption that D isvery much greater than 2a.
INDUCTANCE AS A FUNCTION OF CONDUCTOR SIZE
The Inductance per unit length of each phase of a
three phase cable employing non-magnetic conductorsis given by:
L = 1F2 loge D+ Li x 10 7 henry/meter (17)L a 2 Li0]
Referring to Figure 6, D = 2 (a+t) where t is theinsulat ion.
L = 2 (a+t)/a + Li 10 7 henry/meter2
Figure 6. Cross Section of Three Phase Cable WithEqui lateral Spacing
Table III shows various values of t for standard wiresizes. The values of t were obtained from a cablehandbook. The inductance L per unit length is cal-culated on the assumption that the skin effect isnegligible, I.e., Li = Lio.
Table III shows that the variation of inductancewith cable size is of a small order, from 0.25 to0.35 microhenry/meter.
As an average value, the inductance per phase ofthree phase cable from Table III is 0.3 microhenrys/meter.
Now Reactance, X = 2TrfL
= 6.28 x 0.3 x 12/39.4
= 0.575 micro-ohms/cycle for eachfoot of length
TABLE III
CALCULATION OF INDUCTANCE PER UNIT LENGTHFOR VARIOUS SIZES OF CABLE
EXAMPLE OF HIGH FREQUENCY TRANSMISSION LINE DESIGN
Consider 100 KVA,3 phase transmission at 450 volts,5000 cycles for a length of 200 ft.
current = x 103 128 amps
Using the average value of reactance:
Reactance drop per phase = 128 x 0.575 x 10x6x5000 x 200
= 74 volts/phase
- 128 volts line-to-line
The voltage drop in the transmission line is a func-tion of load power factor. High frequency loads gen-
erally require a low lagging power factor. A line-to-line reactance drop of 128 volts would thus beunacceptable. Three conductor, No. 0 cable with 600volt, high temperature insulation has a maximum con-
tinuous current capacity in the neighborhood of 175amps depending on the ambient temperature.
From wire tables, Rdc = 0.098 ohms/1000 ft at 200C
v'fRdc = 5000/0.098 = 226
and from Table III, D/2a = (aet)/a = 1.48. FromFigure 2 Risolated/Ro = 2.5, and from Figure 3R/Rlsolated = 1.35 estimated
R/Ro = 2.5 x 1.35 = 3.38
The current capacity is reduced _-- = 54 percentwhich also is unacceptable. 3.38
If, however, we design a 24 conductor cable with 8groups of 3 conductors, and if the size of each con-
ductor is No. 10 AWG, then the area per phase is8 x 10380 = 83040 circular mils compared to copperarea per phase of 105,500 circular mils for No. 0,
three phase cable.
Reactance drop per phase = 128 x (0.575 x 10-6/8)x5000 x 200 5000 x 200
SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
The reactance drop is now acceptable and can beaccounted for in the system design.
Rdc = 1.0 ohm/1000 ft at 200C for No. 10 AWG
vfYRdYc 500= 70.7
and from Table III, D/2a = (a+t)/a = 1.92. FromFigure 2 Risolated/Ro = 1.07 and from Figure 3R/Risolated = 1.05.
R/Ro 1.07 x 1.05 = 1.12
and the current capacity is reduced by 5.5 percent.
If still better regulation Is desired, the number ofparallel circuits can be increased, and the size ofthe conductor reduced.
SPECIAL CABLE CONFIGURATIONS
It is seen that the effects of frequency and conduc-tor size on the inductance of a polyphase cable arerelatively minor and that the inductance is deter-mined primarily by cable configuration. Thus, If thereactance drop Is a serious problem, special cableconfigurations will have to be considered. Thesespecial configurations at the same time reduce theresistance ratio R/Ro.
SIX CONDUCTOR CABLE
In the previous example it was shown how a multi-conductor cable with several groups of three con-ductors could reduce both the resistance ratio andthe inductance of the cable.
The simplest multiconductor cable is one in whichthere are two groups of three conductors. Thiscable is shown in Figure 7 and consists of a regularhexagon arrangement of six equal conductors.
The conductors diametrically opposite each otherbelong to the same phase, and are connected in par-allel. Each side of the hexagon is of length D.
Assuming uniform current density over the entirecross section of each phase, it can be shown thatthe inductance per phase is given by
L - Floge (/3D)/(2 d5) x 10-7 henry/meter (19)
where d5 ae 1/ for a circular conductor0
L =(]log +r0.1) X 10O7 henry/meter (20)
Figure 7. Six Conductor Cable
The inductance per unit length of a three conductorcable is
L = (2 log D/a + 2) x 10i 7 henry/metere 2 hny/ee ( 2 "
Thus the inductance of the six conductor cable isless than half the inductance of the three phasecable.
The resistance ratio R/Ro can be determined by usingthe value given for Rdc of the individual conductor,the conductor diameter "2a" the separation distanceD between adjacent conductors, and by neglecting theproximity effect of the two parallel circuits.
Six conductor cable in the larger sizes are used ex-tensively in 400 cycle power systems.
MULTICONDUCTOR CABLE
A typical multiconductor cable is shown in Figure 8.The conductors are divided into groups of three, anda cable with n groups has n conductcrs in paral lelper phase. Twisting of the conductors in a groupconfines the magnetic field to the group and reducesthe interaction between groups.
In such a case the inductance of the cable per phaseis approximately I/n times the inductance of oneconductor in a group, and the d-c resistance Ro perphase is 1/n times the d-c resistance of one conduc-tor. The ratio R/Ro can be determined as before byusing the conductor diameter "2a", the separationdistance "D" between adjacent conductors in one group,and Rdc for the individual conductor.
Figure 8. Multiconductor Cable
The use of multiconductor cable results in an in-crease in weight due to the full voltage insulationrequired between each group. In addition the con-nections Involving multiconductor cables are tediousand complicated.
TRIAXIAL CABLE
A triaxial cable is a three phase cable involvingconcentric tubular conductors as shown in Figure 9.This arrangement gives a low reactance drop and re-duced loss from skin effect.
Figure 9. raxi bINSULATION
Figure 9. Triaxial Cable
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The triaxial cable has a decided advantage In thetransmission of unbalanced three phase power in thatthe magnetic field associated with the currents isconfined to the cable, thereby greatly reducing theinductive Interference with neighboring communica-tion circuits.
Let the inside and outside diameters of the tubesfrom the smallest to largest be di, d2, d3, d4, d5and d6.
There is no neutral current. Ia + Ib + Ic = 0
The reactive drop of the inner tube which carriesIa amperes is given by:
J29a d \ 2\in- 219 dz[\d d )'7 ( dln4dd/- b(d2 d2)
I)2I - 2 d d
3 n ) ( 4 ) (in (d4))I (22)+ 7 2+ (dzd d \4 / \ d2/]
40 32 25 d4 ())j
-1-2~ dz - dIn
))-d4 - d2 in volItS/cm
The reactive drop of the middle tubeIb amperes is given by:
which carries
n ( )
J)21[I d' d (3d2 d')[b 3( )( 3 4]/l(7 3/d- 4(d
i) (n (d ))j voItS/cm
The reactive drop of theIc amperes Is given by:
outer tube which carries
a b
2( n 6S )
n) volIts/cmT6'0- (d2 ~dY ) 4(d' d')'
6 5/ 5/ 6 5J
(24 )
RECTANGULAR CONDUCTOR
The advantage of a rectangular shape over a cyl in-drical shape is two fold: First the conductor hasa large surface or skin area resulting in a reducedloss from skin effect, and second the conductorscan be placed closer to each other resulting In a
low inductance, and thus a low reactance drop.
A three phase rectangular cable is shown In Figure10.
The Inductance of each of the three conductors will
differ, and for long distance transmission, the
Figure 10. Rectangular Cable
three lines would have to be transposed to providefor equal reactance drop.
CONCLUSION
The ratio R/Ro gives the effect of frequency on re-sistance, and from an analytical viewpoint can bestudied as three separate effects: (I) The skineffect, (2) the proximity effect, and (3) the effectof stranding and spirality. In a theoretical calcu-lation of R/Ro, the effect of stranding and spiralityis neglected. The skin effect can be determinedfrom Figures 1, and 2,the proximity effect from Fig-ure 3 (it being assumed that the proximity effectfor 3-phase is the same as the proximity effect forsingle phase) Figure It shows the maximum conductordiameter for R/Ro = as a function of frequency.With the thickness of insulation normally employed,the proximity effect is neglible. Thus, with theconductor diameter, at a particular frequency, lessthan or equal to the value of Figure II, the currentcapacity of the cable will not be affected. Themaximum conductor diameter of Figure 11 is twice thenominal depth of penetration 8.
The inductance of a cable can be easily calculatedby Formulas 15 and 16. High frequency power trans-mission at low voltage could introduce serious regu-lation problems, and it may be necessary then to re-sort to special cable configurations.
000 2000 3000 4000 5000
FREQUENCY (CYCLES/SECOND Ix
Figure II. Maximum Conductor Diameter as a Functionof Frequency for R/Ro =
409
SUPPLEMENT TO IEEE TRANSACTIONS ON AEROSPACE / JUNE 1965
GLOSSARY OF TERMS
= Radius of conductor in meters
= Area of cross section
= Self geometric mean distance of one conductor
Frequency cycles per second
= Length of conductor
/2 a
= D-C resistance in ohms per 1000 feet
= D-C zero frequency resistance
= A-C resistance
=/E = Nominal depths of penetration in meters.
5 Resistivity In ohm-meters
= Permeability in henrys per meter
REFERENCES
1. Transmission Lines and Networks, Walter C.Johnson, McGraw-Hill Book Company
2. Principles of Electric Power Transmission, L.F.Woodruff, John Wiley and Sons, Inc.
3. Proximity Effect in Wires and Thin Tubes, H. B.Dwight, AIEE Transactions, June 1923, pp.850-59.
4. Skin Effect and Proximity Effect in TubularConductors, H. B. Dwight, AIEE Transactions,February 1922, pp. 189-98.
5. Experimental Research on Skin Effect in Conduc-tors, A.E. Kennelly, F.A. Laws and P.H. Pierce,AIEE Transactions, Vol. 34, 1915, pp. 1953-2021.
6. Reactance and Skin Effect in Tubular Conductors,H.B. Dwight, AIEE Transactions, July 1942, pp.5 13-18.
