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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRD.2016.2569120, IEEE Transactions on Power Delivery

A Proposal for a High Voltage Transmission Line Directional Coupler R. G. Olsen, Fellow, IEEE

1

Zhi Li, Member, IEEE

Abstract—Directional couplers are devices generally used in high frequency transmission lines and waveguides that respond to forward and reverse traveling waves separately. Hence they can be used to either measure standing wave ratio in the steady state or to determine the direction of a propagating transient wave. Here, a design is proposed for a directional coupler to be used on multimode high voltage transmission lines. Its performance is analyzed and several suggestions are made for improving its design. Index Terms— power transmission lines, wave propagation, directional coupler

I. INTRODUCTION AND BACKGROUND voltages and currents on any high voltage THE transmission line consist of the superposition of traveling waves in both the forward and reverse direction. For most power lines it is assumed that the voltage and current are relatively constant along their length because the length of the line is short compared to the wavelength (λ) at operating frequency (i.e., λ = 5000 km at 60 Hz). Nevertheless, it is always possible to write an expression for the voltage (or alternatively, current) of any natural mode along the line as

V z   V  e  j 2z /   V  e  j 2z / 

use of (2) is the traditional method for expressing the voltage for short high voltage transmission lines. Nevertheless, it is always possible to decompose the voltage into forward and reverse traveling waves as in (1). II. THE DIRECTIONAL COUPLER At radio and microwave frequencies it is often important to know the amplitudes of the forward and reverse traveling waves. A commonly used, but simple, device for measuring these is called a “directional coupler.” One type of directional coupler requires a physical length on the order of a wavelength and, and while commonly used for microwave devices is not practical at lower frequencies [1]. Another that is commonly used on radio frequency transmission lines is illustrated in Fig. 1. Here, an electrically short probe wire of length ℓ is placed parallel to the unbalanced (e.g., coaxial cable) transmission line conductors and terminated with impedances Z 0 and Z  at each respective end [2]. This probe wire is both capacitively and inductively coupled to the transmission line.

(1)

where V  is the amplitude of the forward traveling wave and V  is the amplitude of the reverse traveling wave and λ is the wavelength of the mode. Attenuation due to losses has been neglected here. If z   , then the small argument expansion

(i.e., e x  1  x ) can be used for the exponentials in (1) to get









V z   V   V   j 2 V   V  z / 

(2)



  This form yields the total voltage at z = 0 as V  V











and at z = ℓ as this minus the term j 2 V  V  /  where the latter term can be understood as a small inductive voltage drop proportional to some equivalent inductance and the mode









current that is proportional to V  V . The mode current is proportional to this term since the mode current (obtained by solving the transmission line equations) can be written as

I z  

V   j 2z /  V   j 2z /  e  e Z0 Z0



(3)



  and at z = 0, the current is proportional to V  V . The ________________

R. G. Olsen is with the School of EECS, Washington State University, Pullman, WA, USA (e-mail: [email protected]) Zhi Li is with the Electrical & Electronics Systems Res. Div., Oak Ridge National Laboratory, Oak Ridge, TN, USA (e-mail: [email protected])

Fig. 1. Directional coupler geometry in coaxial cable for radio frequencies.

The fundamental principle of operation of this coupler is that capacitively and inductively coupled currents add/subtract at the left/right ends of the probe wire as illustrated in Fig. 2. Here, the inductively coupled currents are represented by a voltage source in series with the probe wire that is proportional to the mutual inductance ( l m ), the total current on the transmission line ( I t ) and the length of the probe wire (ℓ). Capacitively coupled currents are represented by a current source that injects currents into the probe wire and are proportional to the mutual capacitance ( c m ), the total voltage on the transmission line ( Vt ) and the length of the probe wire. It should be clear that the capacitively and inductively induced currents through the terminating impedances add at one end while they subtract at the other. The key to the operation of this directional coupler is that according to (1) and (3), if the ratio of current to voltage for a forward traveling wave is positive, then the ratio of current to voltage for the reverse traveling wave is negative. Hence, if Z o  Z  and these impedances are set so that the voltage

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2

across Z o is zero for a forward traveling wave, then the voltage across Z  ( Z 0 )

is zero for a reverse traveling

wave. Given this result the voltage across Z   ( Z 0 ) represents the amplitude of the forward traveling wave while the voltage across Z o represents the amplitude of the reverse traveling wave and the total current and voltage have been decomposed into forward and reverse traveling waves.

lightning protection. In fact, some utilities use spark gaps in this way in order to reduce losses from induced currents on shield wires [4]. V. PROBLEM GEOMETRY AND PARAMETERS The problem to be solved is illustrated in Fig 3a (end view) and Fig 3b (view from above). It consists of a two wire horizontal transmission line with a single shield wire (of finite length ℓ) above and to the left. The two wire transmission line is excited by a distant current source and, in general, carries currents and voltages in both common and differential modes as well as forward and reverse traveling waves. The shield wire is coupled to the transmission line in a way that will be described more fully shortly.

Fig. 2. Model for calculating capacitively and inductively coupling from the transmission line into the directional coupler.

III. APPLICATION TO MULTICONDUCTOR POWER LINES In principle, the same idea can be used for high voltage transmission lines that support multiple propagation modes; however, a practical design for such a device has not yet been identified. Preliminary work on such a device was reported by Li [3]. However, the performance of the device proposed there was not sufficiently good to propose its actual use. Here the basic theory of a directional coupler for a high voltage transmission line is introduced in the context of a two wire transmission line above earth to illustrate some of the challenges introduced when extending the directional coupler concept to multiconductor transmission lines. This simplification is made since the inherent symmetry allows the theory for multiple mode transmission lines to be illustrated without to being obfuscated by the mathematics. If a device can be designed for this case, it can likely be extended to the case for practical high voltage transmission lines (three or more phases with multiple subconductors). If such a device was available, the voltage at power system operating frequencies, could be inferred if a measurement of current is known or vice versa. In addition, there are clear applications for such a device to measure the amplitude of higher frequency traveling waves during faults. More specifically, knowledge of the direction of travel and amplitude of traveling waves during faults is useful for fault location.

(a)

IV. DESCRIPTION OF THE PROPOSED DESIGN The fundamental idea is to use a transmission line’s own shield wire as a sensor since it is a finite length of conductor coupled to the power line. It should be noted, however, that if a shield wire either does not exist or cannot be used for this purpose, a separate parallel wire can be installed for use as a directional coupler. A single span of shield wire is isolated from other sections of the shield wire by replacing the connection to the shield wire at the adjacent spans with a spark gap. Doing this isolates the segment of wire, but does not negate the use of the shield wire for its intended purpose of

(b) Fig. 3. View of the power line with grounded shield wire to the left side. a) Cross sectional view b) view from above. The shield wire is isolated from shield wires on other sections.

The upper half space (i.e., y > 0) is free space and is characterized by permittivity and permeability  1   0 and



1   0 respectively while the lower half space (i.e., y < 0) is a linear, homogeneous, isotropic, lossy material characterized by conductivity, permittivity and permeability  2 ,  2   r 2 0 and  2   0 respectively, where  r 2 is the relative permittivity of the half space.

3

 c2

(9)  J 0, h  a, h   J c d , h  a, h      y11  y12 z iw k 02 1  c ln( 2hs /(ad ))  

 d2 

(10)  J 0, h  a, h   J c d , h  a, h      y11  y12 z iw k 02 1  c ln 2hd / as   

VI. INCIDENT FIELDS To solve this problem, reciprocity theory will be used [5]. The first step in the process is to determine the space potential and the “longitudinal electromotive force” sometimes called the “longitudinal electric field” (LEF) which is related to the mutual impedance of the two phase conductors in the absence of the shield wire. These are called the “incident” fields. The “incident” fields (in the absence of the shield wire) can be determined using the geometry shown in Fig. 4. Here the two conductor power line is excited by a current injected into conductor #1 at z = 0. This excitation simulates current injected by a phase-to-ground fault current or lightning.







s  2h 2  d 2

and

1/ 2

.

Finally, the Carson integral is J c x  x' , y, h  



2 k 22

 u   e

  y  h 

cos  x  x'd

(11)

0

where k 0    0 0 , u   2  k 22 ,

 0  0 r 2  j 2 /    exp j / 4  0 2 ,  2

k2  

  0 r 2 .

and

 k a  J (k a) z iw ( )  rdc  w  0 w  2  J 1 (k w a)

(12)

where rdc 

1

(13)

 wa 2

is the resistance per unit length of the conductor at dc (i.e., zero frequency). σw is the wire conductivity, J o (k w a) and Bessel functions of argument J o (k w a ) are

Fig. 4. A current injected into conductor #1 at z = 0.

A current I f is assumed to be injected at z = 0 as shown in

kwa   j 0 w 1 / 2 a and order zero and one respectively

Fig. 4. The details of the solution for currents propagating on the conductors can be found in [5]. The result is that the total current on conductors #1 and # 2 consists of both differential and common modes and is

where the conductor has been assumed to be non-magnetic since  w (the conductor permeability) =  0 . Note that displacement currents in the conductor can be neglected (i.e.,  0 rw   w /  ). Hence it is not necessary to specify  w , the dielectric constant of the conductor. If the frequency is either low enough or high enough then (12) can be replaced respectively by an equivalent that is (13) augmented by an inductive term or an impedance that is consistent with the skin effect [5]. The voltage of each conductor with respect to the earth as a function of z can be determined by using

Iˆ1 z  

If 4 y11

 y

11

 y12 e  j c z   y11  y12 e  j d z , z > 0 (4)



 y

11

 y12 e  j c z   y11  y12 e  j d z , z > 0 (5)

and

Iˆ2 z  

If 4 y11

where

y11 



j 2 0  2h  ln   ln 2hd /(as )  ln 2hs /(ad )   a 

y11  y12

j 2 0 ,  ln 2hs /(ad ) 

y11  y12 

j 2 0 , ln 2hd /(as ) 

Vˆ z  y

1

(6)

A  Iˆz   ˆ I z    z j z

 

 

(14)

where (7)

A  (8)

1 2 0

  2h   s  ln    ln  a   d  ,   ln  s  ln  2h      d   a  

(15)





Z  y  y12 e  j c z  Z 0 D  y11  y12 e  j d z Vˆ1 z   I f 0C 11 4 y11

and



Z  y  y12 e Vˆ2 z   I f 0C 11

4

The space potential at x, y , z  is then1

The specific results (for z > 0) are

 Z 0 D  y11  y12 e 4 y11

 j c z

 (16)

 j d z

 (17)

where the characteristic impedances for the “common” and “differential” transmission line modes are defined respectively as

Z 0C 

z11  z iw  z12  z11  z12    y11  y12   y11  y12 

(18)

Z 0D 

z11  z iw  z12  z11  z12    y11  y12   y11  y12 

(19)

and

ˆ z   r ' Vˆsp x, y, z   1 ln  1 2 0  r1

x  x   y  y  ' 2 i

where ri 

' 2 i

 ˆ  2 z   r2'  ln   2 0  r2   ' , ri 

   

(26)

x  x   y  y  ' 2 i

' 2 i

and x1 , y1   d / 2, h  , x 2 , y 2    d / 2, h  . (26) reduces to (16)

and

(17)

x, y    d / 2, h  a 

for

x, y   d / 2, h  a 

and

respectively. Extrapolation to more conductors involves calculating an inverse of the potential coefficient matrix and is not as straightforward. Second, the portion of the electric field associated with inductive coupling and relevant to the reciprocity coupling problem (here superscripted as the LEF) is equal to the sum of the mutual impedance from each conductor to a point (x,y,z) in space multiplied by the appropriate current. Hence, LEF  x, y , z   Eˆ inc





j M x  d / 2, y, h Iˆ1 z   M x  d / 2, y, h Iˆ2 z 

Here,

z11  z iw  z12  z iw  j 0 2

  2h   (20) s ln    ln    J c 0, h  a, h   J c d , h  a, h  a d      

where M i x  xi , y, y i  

where it has been assumed that a 0) are





According to [5], a first order coupling model can be used if the frequency is less than about 100 kHz. Using this model, as shown in Fig. 5, the currents induced into the two ends of the shield wire are

(22)

Now that the currents and voltages have been determined, the space potential and the LEF relevant to the reciprocity coupling problem should be determined. First, using (15) the space potential at any point (x,y) due to the currents on these conductors can be determined if the line charge density on each conductor is determined from

If



VII. THE COUPLING MODEL

z11  z12  z iw  z iw 

ˆ 1 z  



0 ln ri' / ri  J c x  xi , y, y i  . (28) 2

A. General Case

z11  z12  z iw  z iw  j 0 2

(27)



(25) 1 It is assumed here that all transverse distances are small compared to the length over which the current on the transmission line changes significantly.



5

Note that a forward traveling wave has been assumed in (29) and (30). For a reverse traveling wave, the current at z = ℓ would equal zero because the LEF would be in the opposite direction to that given in (33) if Z sw0  Z sw . C. Common Mode

Fig. 5. Equivalent Circuit for the result of (31) – (35) in the case for which the impedance of the capacitances between the conductor and earth are much larger than Z swo or Z sw .

For the common mode,  d is replaced by  c , the two log terms in (26) and two mutual inductance terms in (27) are added rather than subtracted and (as for the differential mode) ( x, y, z )  (d sw / 2, hsw  a sw ,  / 2) . Given the fact that the common mode return current now flows in the earth, there is less cancellation than for the differential mode (e.g., the two Carson integrals no longer approximately cancel). In this case, (29) can be shown to be equal to zero for this common mode condition on the transmission line if

Setting (31) equal to zero results in Z sw

LEF  d sw / 2, hsw ,  / 2 Eˆ inc  ˆ jc swV sp  d sw / 2, hsw ,  / 2 

0  c sw (34)

  r 'r '  0 ln  1 2   r1 r2

B. Differential Mode If it is now assumed that only a differential mode exists on the transmission line, the space potential in (26) and (32) and LEF in (27) and (33) respectively can be reduced to Iˆ z  d  r1' r2   j d z Vˆsp x, y, z   1 ln e 2 0  r1 r2'  since  jˆ  ( z )  Iˆ1 z  / z by current continuity and

 c  r1' r2' ln  0  r1 r2

(35)

LEF  x, y , z   Eˆ inc (36) ˆ jI z M x  d / 2, y, h   M x  d / 2, y, h e  j d z

d Z sw  

 0  2hsw   Ohms ln  2 d  a sw 

(37)

Although this value can be confirmed to produce a zero voltage at z = 0, it is difficult to design a direction coupler with this exact impedance because it depends on the ground conductivity. Hence, since the propagation constant of the differential mode is close to that of free space, the approximation  d  k 0 is suggested. Using this results in d Z sw  

 2h   0  2hsw    60 ln  sw  Ohms, ln    a  2  a sw   sw 

(38)

where  0 = 120π Ohms is the impedance of free space. This is a much easier termination to design since it is purely real.

(39)

    J c x  d / 2, y, h   J c x  d / 2, y, h    

If the same approximation (i.e.,  c  k 0 ) used for the differential mode is used here, (37) becomes (40). c Z sw  

  r 'r ' ln  1 2   0  0   r1r2 c sw

1

Using (35) and (36) with ( x, y, z )  (d sw / 2, hsw  a sw ,  / 2) in (34) and noting that many terms are common to both numerator and denominator and assuming that the two Carson integrals are approximately the same and cancel, it is possible to solve for a value of Z sw that results in zero current at z = 0. In this case, (34) reduces to

 c  Z sw   

    J c x  d / 2, y, h   J c x  d / 2, y, h     (40)  r 'r '  ln  1 2   r1 r2   

Clearly, this is different from (38) and indicates that the proper terminating impedances needed to realize a perfect directional coupler are mode dependent. VIII. POSSIBLE MEASUREMENTS As mentioned in Section III, there is a clear application for a directional coupler to measure the amplitude and direction of traveling waves during faults. This knowledge could be useful in fault location. However, the directional coupler must be effective in the presence of more than one mode propagating on the transmission line. This aspect of the problem will be investigated in the next section. Another possibility is a non-contact measurement of power frequency voltage if current is known. To understand how this could be done, consider the following expressions for current and voltage on a two conductor transmission line that consists of both forward and reverse traveling waves

















I I Iˆ1 z   c e  j c z  c e  j c z  d e  j d z  d e  j d z (41) 2 2 I I Iˆ2 z   c e  j c z  c e  j c z  d e  j d z  d e  j d z (42) 2 2



Vˆ1 z  

 A11  A12  c I c 



e 2  A11  A12  d I d 2

 j c z

e

 c e

 j d z

 j c z

 d e

6



 j d z

resolved using perturbation theory to account for the second mode. (43)



IX. RESULTS

 A  A12  c I c  j c z Vˆ2 z   11 e  c e  j c z 2 (44)  A  A12  d I d  j d z  11 e  d e  j d z 2 where I c and I d are unknown amplitudes of the common







and differential modes of propagation,



the values of A11

and A12 can be found from (15), c  Vc / Vc and

A. Individual Modes To determine the quality of a directional coupler using (38), the isolation for a forward traveling wave defined as  Vˆ      (51) isolation (dB )  20 log 10   ˆ  V ( 0)    is plotted in Fig. 6 for frequencies between 60 Hz and 10 kHz.

d  Vd / Vd represent ratios of reverse to forward traveling waves while for each mode the remaining parameters have been defined earlier. For simplicity consider (41) – (44) at z=0. The results are: I I Iˆ1 0  c 1  c   d 1  d  2 2 I I Iˆ2 0   c 1  c   d 1  d  2 2

(45) (46)

 A  A12  c I c 1  c  Vˆ1 0   11 2  A  A12  d I d 1  d   11 2  A  A12  c I c 1  c  Vˆ2 0   11 2  A  A12  d I d 1  d   11 2

(47)

(48)

Next, suppose that measurements of Iˆ1 0 and Iˆ2 0 are known. Note that the relative phase of these two terms can be measured. Given this along with the successive addition and subtraction of (45) and (46), results can be found for I c 1  c  and I d 1  d  respectively. If, from these results,

Fig. 6. Isolation between voltages at z = ℓ and z = 0 for the parameters, h = 15 m, hsw = 20 m, d = 5 m, xsw = -5 m, a = asw = 0.01 m, σw = 3.5 x 107 S/m, σ = 0.01 S/m, εr = 5, and ℓ = 100 meters. An incident differential or common mode wave is assumed incident from z < 0.

For the differential mode, the isolation is always above 20 dB and increases to above 40 dB at the higher frequencies since  d is closer to k 0 at higher frequencies. However, for the common mode, the isolation is only approximately 10 dB. Clearly if the common mode is an important component of the current, this may not be acceptable. B. General Incident Current

it can be inferred that I c  I d , then it can be assumed that a measurement of the total forward and reverse traveling waves at the two ends of the directional coupler are V d and Vd respectively. Hence

d  Vˆd / Vˆd

(49)

Now, solving for I a in terms of the measured currents by subtracting (44) from (43) and using (45) and (46),

 A  A12  d 1  d  ˆ Vˆ1 0  Vˆ2 0  11 I (0)  Iˆ2 (0) 1  d  1 2





(50)

Of course, the accuracy of this result depends on the assumptions that the direction coupler is a perfect measurement and that the effect of the additional mode is negligible. The former question will be examined in the next section and made above and the latter could possibly be

Fig. 7. Isolation between voltages at the ends of the directional coupler which is placed at a distance z from the point of current injection on conductor #1 at z = 0. The incident current consists of both differential and common modes. The other parameters are the same as those in Fig. 6.

To illustrate the response to an incident wave such as the one resulting from an injected current as shown in Fig. 4 (i.e.,



7

both differential and common modes), the isolation is plotted for the incident current (and corresponding voltage) given in (23) – (27) due to a injected current on conductor #1 at z = 0 with a directional coupler placed a distance z from this injected current. The result is shown in Fig. 7. Clearly, for directional couplers placed close to the injection point, the isolation is dominated by that of the common mode and is probably not sufficient for a practical device. This issue must be resolved prior to implementation of a direction coupler for high voltage transmission lines that support multiple modes of propagation.

enough to be useful. The reasons for this include the multimode nature of propagation on these transmission lines and the difficulty of specifying an impedance with a value that depends on propagation constants which are, in turn dependent on earth conductivity.  Suggestions have been made for improving the design. These include using multiple sensor wires and connections in the earth between towers. XII. REFERENCES [1]

X. SUGGESTIONS FOR IMPROVEMENT OF THE DESIGN The common and differential mode responses could be separated if a pair of directional couplers was used such as illustrated in Fig. 8. From the symmetry, it should be clear that the voltages induced on the ends of the two probe wires by the common mode are identical. However, the voltages induced on the ends of the two probe wires by the differential mode are equal in magnitude, but opposite in sign. Hence, adding and subtracting the voltages from the two probe wires should result in a separation of common mode induced and differential mode induced voltages.

[2] [3]

[4]

[5]

[6]

S Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd Ed., New York, 1994. C. R. Paul, Introduction to Electromagnetic Compatibility, Wiley, New York, 1992 Li, Zhi, “Sensors for Electromagnetic Measurement of Electric Power Transmission Lines,” PhD Dissertation, Washington State University, Pullman, WA, 2011. M. W. Tuominen, “500 kV Shield Wires; Sectionalize or Ground Everywhere,” Proc. American Power Conference; (United States); Volume: 54:2; Chicago, IL , pp. 13-15 April 1992 R. G. Olsen, High Voltage Overhead Transmission Line Electromagnetics, Volume. I, ISBN: 978-1507848043 Free download at http://eecs.wsu.edu/~olsen/Book_Files and in printed form at http://www.amazon.com/, 2015. R. G. Olsen, High Voltage Overhead Transmission Line Electromagnetics: Volume II, ISBN: 978-1508626367, 2015. Available for free download at http://eecs.wsu.edu/~olsen/Book_Files and in printed form at http://www.amazon.com/

XIII. BIOGRAPHIES

Fig. 8. View of the power line with grounded shield wires symmetrically placed on the right and left sides.

Another possible improvement in the design could come from embedding a wire in the earth between the two towers. This may make the result less dependent on the specific grounding impedances at the ends of each probe wire. XI. CONCLUSIONS  A measurement system for either inferring power frequency voltage from a measurement of current and/or for separating forward from reverse traveling waves on multiconductor high voltage transmission lines has been described. The idea for the system is based on the design of the directional coupler that has been used for many years on two conductor high frequency and microwave transmission lines.  Preliminary calculations show that while the system has promise, its performance (as it is proposed) may not be good

Robert G. Olsen (S’66, F’92) received the BSEE degree from Rutgers University, New Brunswick, NJ in 1968 and the MS and Ph.D. degrees from the University of Colorado, Boulder, CO in 1970 and 1974 respectively. He has been with Washington State University since 1973. Other positions include Senior Scientist at Westinghouse Georesearch Laboratory, NSF Faculty Fellow at GTE Laboratories, Visiting Scientist at ABB Corporate Research and EPRI and Visiting Professor at the Technical. University of Denmark. His research interest is the application of electromagnetic theory to high voltage power transmission systems. His work has resulted in approximately 85 and 150 publications in refereed journals and conferences respectively. He is one of the authors of the AC Transmission Line Reference Book – 200 kV and Above which is published by EPRI. He is a Life Fellow of the IEEE, an Honorary Life member of the IEEE Electromagnetic Compatibility (EMC) Society. He is also past USNC representative to CIGRE Study Committee 36 (Electromagnetic Compatibility) and past chair of the IEEE Power Engineering Society AC Fields and Corona Effects Working Groups. In addition, he is past Associate Editor of the IEEE Transactions on Electromagnetic Compatibility and Radio Science. Zhi Li (S’01, M’11) received the BSEE and MS degrees in electrical engineering from Tsinghua University, Beijing, China, in 2000 and 2003, and the Ph.D. degree in electrical engineering from Washington State University, Pullman, WA, in 2011. He was with Oak Ridge National Laboratory, Oak Ridge, TN, as a postdoctoral research associate since 2012 and has become a R&D staff member since 2015. His research interests include electromagnetics for power transmission lines, sensors for electromagnetic (EM) field measurement, magnetic circuit modeling and simulation for power transformers, finite element method (FEM) based simulation, and high voltage insulation and engineering.
