IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. I , FEBRUARY 1988 13

Predicting VHF/UHF Electromagnetic Noise from Corona on Power-Line Conductors

Abstract-A method for predicting VHF/UHF noise generated by corona on power-line conductors in steady rain has been developed. The corona sources are modeled as electric dipoles which are randomly distributed along the conductors and driven by impulsive currents. Each source induces currents on the conductors which in turn radiate as traveling wave antennas. The receiving antenna then responds to the incoherent superposition of the fields from each source. The model is calibrated by using the Bonneville Power Administration Corona and Field Effects Program (BPACFE) which is based on 75-MHz measure- ments in steady rain. The final formula is a function of the power-line parameters and receiving system characteristics. The most significant result is that the received noise is a strong function of the receiving antenna orientation. The noise is minimized for antennas directed at the line, while it is maximized for antennas directed down the line at an angle dependent on the other problem parameters. Lower frequency (Le., 75 MHz) results are in good agreement with experiment.

Key Words-Power lines, corona, VHF/UHF noise, model, experi- ment.

Index Code-B4e/f.

I. INTRODUCTION LECTROMAGNETIC noise fields caused by electrical E discharges on power lines have historically caused

interference in the radio and television broadcast frequency bands [l]. At low frequencies (i.e., 100-1000 kHz), this noise is reasonably well understood, and sophisticated models for predicting the noise exist [ 2 ] . Recently, there has been interest in expanding the understanding and model development of this noise to other communication bands. This is the task to be addressed here.

Two types of sources are predominantly responsible for this interference. The first is corona which is distributed along the power-line conductor and tower hardware. Below 30 MHz, the amplitude of the spectrum of this corona noise is inversely proportional to the square of the frequency. However, at higher frequencies the amplitude of the spectrum appears to be inversely proportional to the frequency [3]. A second source of noise is spark and microspark discharges between gaps on transmission-line hardware. The spectrum of this ‘‘gap” noise contains more high-frequency components than the corona noise, and its amplitude is inversely proportional to the frequency above 30 MHz. Corona noise has been detected at

Manuscript received January 1, 1987; revised September 9, 1987. R. G . Olsen is with the Department of Electrical and Computer Engineer-

ing, Washington State University, Pullman, WA 99164-2752. Tel. (509) 335- 4950.

B. 0. Stimson is with the Research and Development Department, Meteor Communications Corporation, Seattle, WA.

IEEE Log Number 8718464.

frequencies up to 900 MHz, while gap noise has been detected at frequencies up to 8 GHz [4].

Below 30 MHz, corona noise from conductors is the dominant source of the noise. Above 30 MHz, gaps (if they exist) are usually the dominant source of noise in fair weather [5] . This is especially true for lower voltage lines because these lines exhibit much less corona activity. In foul weather, however, gap sources can be shorted by water. In addition, corona from conductor and tower hardware increases under these conditions. For these reasons, corona noise can be the dominant source of high-frequency noise in foul weather. This behavior is more pronounced for higher voltage lines because corona activity is generally higher for these lines.

Since corona noise can be dominant under some conditions, and because (unlike gap noise) it cannot be eliminated by tightening hardware, it becomes an important design parame- ter. For this reason, this paper is limited to a study of the generation and propagation of electromagnetic noise fields caused by corona. However, certain types of gap sources can be modeled in a similar way to corona. Although these will not be discussed further here, they can be analyzed using techniques identical to those used here. The paper will be limited to a noise model in the frequency range of 30-1000 MHz because this model is significantly different from low- frequency models and because more is known about low- frequency models.

The corona noise is mainly caused by surface perturbations on the conductors and towers formed by water droplets and insects. These cause local increases in the electric field and, hence, electrical breakdown of nearby air molecules. A typical corona distribution is shown in Fig. 1.

It is clear from Fig. 1 that it is important to study the problem of determining fields caused by a discharge in the immediate vicinity of a conductor. This is the basic problem to be considered here and has application to both corona and gap discharges.

The physics and spectrum of these discharges will not be considered, though, because other papers have been written on the subject [6]. Rather, the noise fields will be found for discharges of arbitrary spectrum so that measured spectra can be inserted into the final expressions. Here, an alternative method will be used to find the magnitude of the noise. An empirical prediction formula for measured noise will be used to calibrate the analytic method. This is done by comparing the two methods using the same power-line geometry and mea- surement-receiving system as used to develop the empirical

0018-9375/88/0200-0013$01 .OO 0 1988 IEEE

14 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 1, FEBRUARY 1988

formula. The two results are then fitted together by adjusting the amplitude of the source term. The calibrated analytic method can then be used to predict the noise for other frequencies, power-line geometries, and receiving systems.

With either gap or corona sources there is a distribution of moving charge near the affected conductor. The field from this charge (as distinct from the sources induced on the conductor) can be represented by a multipole expansion [7]. Since the algebraic sum of the total charge for a corona discharge is zero, there will be no point-charge term in the series. In addition, for distances from the space-charge distribution that are large relative to its size, the field due to the dipole term will dominate. Thus, it is reasonable to model these noise sources as electric dipoles driven by impulsive currents where the dipoles are oriented normal to the conductor surface in the direction of the corona discharge.

The noise field predictions using this model will be developed in several steps. First, the currents induced on the wire by a single dipole will be derived. Second, the fields from these induced currents will be calculated. The direct field of the dipole can usually be neglected when compared to the magnitude of the fields from the induced currents [7]. Next, the response of an antenna and receiver system will be expressed in terms of these fields. At this point, the sources are considered to be incoherent and uniformly distributed and excited along each conductor [7]. The total noise is obtained by adding the contribution of each dipole. Finally, an empirical formula for the noise fields is used to calibrate the final result. This formula is based on numerous measurements and is used because the corona spectrum is unknown.

11. THE PROBLEM GEOMETRY

Each discharge is modeled as an electric dipole on one of the conductors. The power line consists of several parallel conductors which are supported by metal towers or wood poles and are above a homogeneous lossy earth. It will be assumed that neither the tower nor any hardware on the pole acts to scatter the electromagnetic waves. When this is the case, the problem may be approximated by neglecting the poles or towers as shown in the cross section in Fig. 2. Here the nth conductor is located at a height d, and a horizontal distance h, from an arbitrary reference axis. The conductors are excited by dipoles distributed randomly along the length of the power line.

111. CALCULATION OF THE INDUCED CURRENTS

Under most conditions, the dominant noise field is from the currents induced on the transmission line [7]. Further, the current induced by a single dipole is approximately indepen- dent of the dipole orientation so long as the dipole is normal to the conductor. For this reason, it may be assumed that all dipoles are oriented vertically.

The temporal and spatial Fourier transform pairs to be used in the field calculations are, respectively,

e(,) = G(t )e - iWf dt - m

1 + m

27r - m G ( t ) = - j e(o)e’u‘ do (1)

J I 4 1

CORONA DISCHARGES 7 K’ RANDOMLY DI STRl BUTED

w Fig. 1 . Typical corona distribution.

0

, 1 1 1 1 1 1 , ‘ ‘.CX

EQ, ucl Fig. 2. Cross section of the power-line geometry.

and

J - m

1 +-

27r - m G ( z ) = - j G(y)e-juz dy.

It is clear that the tilde (”) indicates a spatially Fourier transformed variable, while the caret (-) indicates a tempo- rally Fourier transformed variable.

Using the thin-wire boundary condition on each wire in the spatial (i.e., with respect to z) Fourier transform domain, the following equations for the unknown wire currents caused by the pth corona source on wire n can be written:

l < n < N , (3) m = l

where

is the axial electric field of the dipole source at z,, where

I?:; is the axial electric field of the mth (unknown) wire

OLSEN AND STIMSON: PREDICTING VHFIUHF ELECTROMAGNETIC NOISE 15

current at the location of the nth wire for which a formula will shortly be given.

Zsn = .2,,/(2aa,,)

a wavelength above the earth, then the pole of the third term in the integrand can be ignored and simple saddle-point integra- tion may be used [ l l ] . This derivation is described fully in [12]. The result follows. It is valid for frequencies above approximately 30 MHz for typical power-line heights:

is the unknown nth wire current, and

(4)

is the intrinsic impedance of the nth solid wire where

and u,,, is the conductivity of the nth wire. N, is the total number of wires, eo is the permittivity of free space, and po is the permeability of free space. It is assumed here that only a single corona source at z, exists. Later, the noise from a distribution of sources will be summed. It is also assumed in (3) that any sources which might occur on wires other than the nth wire do not contribute to &.

Before (3) can be solved for the induced wire currents, an appropriate expression for E;; must be found. In the low- frequency case, the Carson approximation can be used [ 9 ] . However, in the high-frequency case, the significant distances (e.g., wire heights) are comparable to, or larger than, a wavelength. Thus, a different expression must be found.

It is well known that the exact expression for the axial electric field of an infinitely long wire above earth [ l o ] is

- - E ;;= p :"I", (6) As= { sin e;,,. where

F , : inn =- 1 r2 bh2)([rrn, , ) - - j im (E-- 2ki

It is possible to identify the terms of (8) as a direct wave (ray) from a line source at (x, y ) = (h,,, d,) plus a wave which is reflected from the earth and multiplied by the appropriate

4 W E o a --oo uo uo+u, Fresnel reflection coefficient for a plane wave [13] . This interpretation is described in detail in [ 121.

It should be noted that in (8) the Hankel function is left as is because when rn = n its argument is [a,,, and the asymptotic expansion (and hence the ray interpretation) for this term is usually not valid.

+ k i y ) e- uo(dm + dn 1 e - 0 (hm - hn ) d A]

where HL.)(~) is the Hankel function of the second find of order 0, and

kiu,+ kiuo

[ = ( k i - y 2 ) 1 / 2 , Im ( [ ) > O V. THE INDUCED CURRENT

r,,, = ((h, - + (d , - dm)2)1i2 Equation (3) can now be recast in matrix form as

u ~ = ( A ~ + y * - k i ) ~ / ~ , Re (uo)>O ldZll?,l = - \ f i d l e + J y z w (9)

~ , = ( A ~ + y ~ - k i ) ~ / ~ , Re (u,)>O where

k g = W ( p o E r g E O - j p o u ~ / W ) 1 / 2 , Re (k,)>O -I 1 i i y = 4WEo - [ [2Hf)(rrn, , ) where erg and ug are, respectively, the relative dielectric constant and conductivity of the earth.

Equation (6) may be obtained from [9, eq. ( l ) ] through the - ( + > 1 / 2 [[g(As, Y ) - ( k i + y 2 ) C O S e ~ n

use of the identity d A s , Y) + COS

j r2 m e - uo(d, + dn 1 e - j X ( h , - h,, 1

UO e --I < r h n , j n/4 1 1 2 k i y 2 COS e;,, + rzHh2)(rr;,,) = - i dA (7) H - m

kig(As, y) + k i r COS where r;,, = ( ( A m - + ( d , + d n ) 2 ) 1 / 2 .

IV. ASYMPTOTIC EXPANSION FOR THE AXIAL FIELD - 6 m n g s n

The integral portion of (6) can be evaluated by saddle-point integration. As long as the wires are more than approximately

and 6,,, is the Kronecker delta function. At this point, the matrix M, is a function of y in a relatively

complicated way. Thus, the solution for the induced current will be a function of y as well. This would create computa- ' E, should be multiplied by - 1 in [9, eq. ( I ) ] .

16 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. I , FEBRUARY 1988

tional problems if a numerical method is used to invert the spatial Fourier transform.

Under most circumstances at VHF and above, however, the conductors of power lines can be considered as weakly coupled [12]. Physically, this means that the current induced in the conductor that has the corona source will be signifi- cantly larger than the currents induced on other conductors by the same source. Hence, the secondary induced currents can be ignored when calculating the fields. Assuming weak coupling, the Fourier transform of the induced conductor current on wire n due to a dipole with unit current at znp is

If the angle between the incident ray and the conductor is less than 5 " , the multiwire coupling terms become large enough to be significant, and the approximation for the induced wire currents is invalid. It must also be recognized that the asymptotic expansions for the fields which will be presented shortly are not valid for this case. This is because the simple saddle-point integration is not valid when there are singularities in the y plane very close to the saddle point, as happens for field points near the conductor [7], [ l l ] . This fact reinforces the assertion that this theory is probably in error if too much of the received noise is incident from grazing angles relative to the conductor.

VI. EXPRESSIONS FOR THE FIELD COMPONENTS-SPATIAL TRANSFORM

Once the induced wire currents are known, it is possible to calculate any component of the noise field. Here, explicit formulas will be given for all components of the electric field because they will be used to calculate the noise induced in an antenna and receiver system.

The electric field components due to the induced wire currents can be written

Eiw = 1 f e i 11 f w I ( 1 1)

where the F matrix is a row matrix of field factors for each wire and the subscript i (i.e., x , y , or z ) designates the field component. The x and y electric field components can be written as [14]

ex

H y ) ( p ) is the Hankel function of the second kind and order one, and

rn = ( (x- h,,)2+ ( y - d n ) 2 ) 1 / 2

ri = ( ( x - h n ) 2 + ( y + d n ) 2 ) 1 / 2 .

Fez can be obtained by using (6) with r,,,, d,, and h, replaced by r,,, y, and x , respectively.

VII. ASYMPTOTIC EXPANSIONS FOR THE FIELDS

The asymptotic expansion of (6) is available as (8) where the parameters are changed, as mentioned after (13). Asymp- totic expansions for the other field (i.e., (12) and (13)) components can be obtained in an identical manner and are given in [12].

These expansions are for spatial Fourier transforms of the fields. In order to find the fields in the space domain, the inverse Fourier transform must be evaluated.

To evaluate these transforms, steepest descent integration may be used again as described in [ 121. The axial field is

where the asymptotic expansion of the Hankel function

H f ) ( w ) ( 2 / 7 r ~ ) ~ / ~ e ~ ( ~ - " / ~ ) , for w 1

has been used, and

R , = (ri + (Z - Z n p ) 2 )

~ ~ ~ = ( r n 1 2 + ( ~ - ~ ~ ~ ) 2 ) 1 / ~

Y ~ I = ko COS d n p

ys2 = ko COS dip

g(As, ys2)= - j ( k i sin2 d i p sin2 OiP+ki cos d;p-k: )1 /2 ,

Im (g)<O. (15)

The angles are defined in Fig. 3. The transverse fields are

OLSEN AND STIMSON: PREDICTING VHFlUHF ELECTROMAGNETIC NOISE

~

17

Fig. 3. Definition of angles and distances (z, assumed = 0).

and

Equations (14), (16), and (17) are valid if k&, 9 1 and @np is not too close to zero. The latter condition occurs because the transform of the current has singularities in the y plane, as discussed in [ l l ] and [12].

The electric field expressions listed above are equivalent to the fields of an infinitely long horizontal antenna above a lossy earth with some distribution of current. It is possible to obtain this current distribution by finding the inverse spatial Fourier transform for the solution of (9). However, this step is not necessary as only the Fourier transform is needed to calculate the fields. That this is true should not be surprising because the far field of an antenna is the Fourier transform of the current distribution on the antenna [15].

Since the extent of the induced current is generally much larger than the size of the corona (dipole) source (usually electrically short), the fields of the induced current will be much larger than the direct fields of the dipole. This is consistent with the fact that the fields of a short dipole are proportional to the product of dipole length and current. For this reason, the direct fields of the dipole have been neglected here [7].

It is shown in [ 121 that (14), (16), and (17) can be given ray optic interpretations. The total field at the field point can be thought of as a direct ray (terms containing Rnp) plus a reflected ray (terms containing R ' ) in which the appropriately polarized components are multiplied by the appropriate Fresnel reflection coefficients. This interpretation is illustrated in Fig. 4.

"4

"" w'RE2 1 - Fig. 4. Ray interpretation for the fields (znD assumed = 0).

VIII. RADIATION PATTERN OF THE INDUCED CURRENTS

Using the electric field equations, (14), (16), and (17), the radiation pattern of the current induced on a conductor by a single dipole can be calculated. Fig. 5 shows this pattern for both the horizontal and vertical field components. It should be noted that the radiation pattern does include earth reflection effects, but this effect is slight.

As seen in Fig. 5, the radiated fields are zero when the field point is perpendicular to the conductor at the dipole, as has been noted previously [7], [8]. This observation is of particular interest when interpreting antenna azimuth profile characteristics in the final noise prediction.

The currents induced on the conductor by a single dipole can be considered to be decaying traveling waves: one traveling in the positive z direction, and one traveling in the negative z direction. Hence, one would expect the radiation pattern in Fig. 5 to resemble the radiation pattern of two traveling wave antennas of opposite orientation. In fact, the two compare quite closely, as seen in [13].

IX. RECEIVING SYSTEM GEOMETRY AND GAIN

When measuring low-frequency radio interference, a stand- ard antenna (i.e., loop or rod) in a standard orientation is used.

18 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. I , FEBRUARY 1988

,-DIPOLE LOCATION A vector which is normal to the vertical plane is /

CONDUCTOR

VERTICAL FIELD PATTERN HORIZONTAL FIELD PATTERN

Fig. 5. Relative polar plot of radiation pattern (IE 2, of the current induced by a single dipole on a conductor (R = 53 m, a = 2.0 cm,f = 75 MHz, d = 17.7 m, y , = 1 m, erg = 2.3, ug = 0.01 Wm).

_- DIRECT RAY

HORIZONTAL PLANE VERTICAL PLANE

PATH OF MAXIMUM ANTENNA GAIN (BEAM AXIS)

ElVlNG

REFLECTED RAY

Fig. 6 . Receiving antenna geometry.

ANTENNA

Since the antenna is electrically small, the output of the antenna is directly proportional to the electric field (rod antenna) or the magnetic field (loop antenna). However, antennas used for measuring high-frequency interference are neither standard nor electrically small. The output of the measuring antenna cannot be determined simply by knowing the field at the center of the antenna. This is because the origin of the incident field must be known to account for the directional gain of the antenna. In addition, because each corona source has a unique location, it is impossible to translate field measurements into antenna output without full specification of the antenna. Further, there is no standard noise-measuring antenna for all frequencies in the VHFKJHF range. For these reasons, the antenna and receiving system characteristics will be considered as unspecified and, thus, inputs to the problem.

The noise field components from (14), (16), and (17) are expressed in terms of the power-line geometry, as illustrated in Figs. 2 and 3. To determine how the noise fields interact with the receiving antenna, they must first be decomposed into a component in the vertical plane of the antenna and to another component in the horizontal plane of the antenna, as shown in Fig. 6. However, since the direct ray and reflected ray are incident to the antenna at different angles, they are treated separately. Thus, the incident field can be written

Einc = Ei + E; (1 8)

where

(19) d,, = COS 6.4 dz - Sin 6.4 ax.

Thus, the part of the direct incident field E; in the vertical plane of the receiving antenna is

(20) ~- -

E, =E; - ( E ; * dnu)dnu.

Similarly, the part of the reflected incident field E; in the vertical plane of the receiving antenna is

- . E ; = E ; - ( E , a,,)fi,,. (21)

A vector normal to the horizontal plane is

d n h = d y cos 6 E + dx sin 6 E Cos 6, + dz Sin 6~ Sin 6.4. (22)

Thus, the part of E; in the horizontal plane of the receiving antenna is

- - - (23) E - - h - E i - ( E j . c f , h ) d n h

and the part of E; in the horizontal plane of the receiving antenna is

A -

E ' = E ' - ( E ' ( h l , . nh ) - a n h . (24)

E; = Exax + Eyay + Ezdz

Therefore, given

E; = E ; a x + E ; a y + E p z (25)

whose components are defined in (14), (16), and (17), expressions for the fields in both planes are defined.

The vertical and horizontal components of the direct field are then

E u = [ C O S 2 6AEx+cOs 6.4 Sin 6 ~ E z ] d x + E y d y

+ [sin2 0,412~ + cos 0.4 sin 6 ~ 1 ? ? ~ ] d ~ (26)

and

& = [(I - sin2 6, cos2 o A > E ~

- cos 6.4 Sin 6 E cos 6~a!?~

- Sin2 6~ Sin 6,4 COS 6.4 EZ] dx

+ [Sin2 - Sin 6~ COS 6~ COS 6,4 Ex

- sin eE cos OE sin 6,4 Ez] dY

+ [(I - sin2 dE sin2 6,)$

- Sin2 6, Sin 6.4 COS 6,4 Ex - Sin 6~ COS 6~ Sin 6.4 EY] dz . (27)

The field components due to the reflected term are identical to (26) and (27) but with primed variables.

With the components of the incident fields known, the response of the receiving system can now be calculated. As in

Einc total incident field, I?; E:

incident field direct from source, and incident field from the reflected ray path.

- .

[7], the corona sources are considered to be incoherent and generated by current pulses of the form id(f - T , , ~ ) where idt) is unknown and rnp is a random variable. Transforming the

Each of the fields Ei and E; is transverse to its corresponding direction of arrival, as is characteristic of far fields.

OLSEN AND STIMSON: PREDICTING VHFlUHF ELECTROMAGNETIC NOISE 19

where

Qnp= i l,e-Jk'RnP/lin d v = Q ~ p ; , r I , + Q ~ p a h + Q ~ p ~ r "

and Q:, is :he component of Q, in the direction of propagation. Qip is identical except that R , is replaced by Rip. In (31) R , and R' (in the amplitude term) are assumed constant over the receiving antenna and are measured from the

voc

?P

Fig. 7. Receiving system.

current to the frequency domain yields

1

corona streamer to the center of the receiving antenna.

impedance is Now, the voltage across the real part of the receiver input

(32) Vooc(U)RRx zu + ZRx

V R x ( w > = fd(w)eJwr w . (28)

The receiving system can be modeled as in Fig. 7 where Z R ~ is the input impedance to the receiver, 2, is the input impedance to the antenna, and V,, is the open-circuit voltage of the antenna.

Using reciprocity, the antenna open-circuit voltage is [16]

where R~ = Re (zRx)* Using (3 l ) gives

c,Qx (w) = f i n p (a) fd(d(w)eiwr nP ( 3 3 )

where

I 1 ^ I ^ - I - h ! where P is the receiving antenna polarization state, Iin is the

transmitting antenna, and 1; is the current density on the receiving antenna when used as a transmitting antenna. Note that the polarization state P of the antenna is determined by the currents lt which flow on it. Thus, for example, a vertically oriented wire is a vertically polarized receiving antenna.

The incident field from a single corona source as given by (18) is the sum of two plane waves, one the direct ray and the other the reflected ray. This field can be written

Einc = f d ( ~ ) e J u r n ~ [ ~ n p e - J k ' R n p / R nP +F ' np e-Jk.R&/RiP] (30)

input current to the receiying antenna when used as a + y ( F i p Q i L + F t p Q n p ) * (34) R n p

The spectral density of (33) can be shown to be [71

(SR,(W)) = ( ~ R ~ ( o ) P:x(w))

= Iib.(w)Iz C f i n p ( ~ ) A ~ q ( ~ ) ( e J w ( r ~ ~ - r m q ) ) ( 3 5 ) n,p m,q

where ( ) means the ensemble average. For w %= lo4, (35) reduces to 171

where

while F=Pipau+P;pdh

and Ft = i u f n +ph'a

np u np h .

F and F' represent the radiation pattern of currents induced on the wire by a dipole.

Here, the unit vectors du and Oh are transverse to the direction of propagation R , in the vertical and horizontal planes of the receiving antenna, respectively, as defined in (26) and (27). There is no field in the direction of propagation because far-field conditions have been assumed. Similar statements can-be made about the reflected term.

Assuming F and E' to be constant over the receiving antenna,

V,,(W, P, EinC) = &(w)eiwrnp

N w m

(SRx(w))= Ifd(w)I2 Ifinp(w)lz- (36) n = l p = - m

Weeks [ 171 reports that the gain of a receiving antenna for an incident wave coming from z , is

where vo is the impedance of free space and R, = Re (Z,) is the radiation resistance of the receiving antenna.

It is assumed here that Edjp is the electric field incident from a distant electric dipole source at z , which has a dipole moment of 1 . Clearly, this formula shows that the gain of the antenna is a function of direction and polarization. Now, if the dipole is located at the pth corona source point on wire n and oriented in the vertical plane of the receiving antenna and normal to the direction of propagation, then

. - F~~ . Q, +A FiP . QiP] (31 ) Similar expressions can be written for horizontally polarized waves or waves which originate at the image point. Using (38) [R', = - R w

20 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATlBlLITY, VOL. 30, NO. 1, FEBRUARY 1988

in (37), the vertically polarized gain can be written

(39)

Again, a similar expression can be written for gains for horizontally polarized waves or waves which have been reflected from the earth.

At this point (34) will be used in (36) to find an expression for the spectral density:

Now it will be assumed that the receiving antenna is predominantly vertically polarized or horizontally polarized. This is reasonable because many antennas are linearly polar- ized. For vertically polarized antennas

If the phases of Q& and Q;; are not too different or if one or the other dominates (the first case is reasonable for broad- beam antennas while the second is reasonable for narrow- beam antennas), then Q;p and Q;; can be replaced by

(47~0Ra) 'I2 ( G ;p 1 'I2

UP0

and

(47voRo) (G

UP0

respectively. Thus,

The horizontally polarized case can be added to this (since it has been assumed that one or the other dominates):

Equation (43) represents the spectral density of the noise as a function of the radiation patterns of the induced currents and the receiving antenna, relevant distances, and the input impedances of the receiver and receiving antenna.

The noise power introduced into the receiver is the spectral density (square of the voltage per hertz) divided by RRx and multiplied by the receiver bandwidth. This noise power is

pR = ( S R (a))IRRx (44)

where B is the receiver bandwidth in hertz. The equivalent noise voltage at the receiver is then

Veq= K ~ K ~ J P R R R X (45)

where Kp and K, are factors which account for preamplifier gain andlor cable and connector losses if any. Both factors reduce to 1 if these gainsllosses can be ignored.

X. RECEIVING ANTENNA CHARACTERISTICS

The complete characteristics of a receiving antenna are usually not specified by the manufacturer. For this reason, the model used in this paper will be somewhat crude. If this is done, then only readily available information about the antenna will be required. It should be recognized, however, that a full specification of the receiving antenna pattern can be used in principle. Here, the receiving antenna is assumed to have a Gaussian radiation pattern of the form

(46) ce- o Z / B ~

where G is the gain, Bw is the half-beamwidth of the antenna, and 0 is the angle (corresponding to OG or 0; in Fig. 6) between the main beam direction and the arrival angle of the incident field. It should be noted that the gain and beamwidth of an antenna are related [12]. Equation (46) is used with either vertically or horizontally polarized antennas by assuming the opposite polarization response to be zero.

The antenna pattern side lobes, if they exist, are modeled by a linearly decaying function which starts at some specified level on the antenna main beam and decays to zero at an angle of a12. Though this is a rough approximation, a more exact representation for the side lobes of the particular receiving antenna can be used if desired.

XI. CALIBRATION

With the corona current and spectra unknown, it is impossible to obtain a totally theoretical noise field prediction for a single corona source. An empirical formula does exist, however, which relates the noise power from a full distribu- tion of corona on a test line in steady rain to several geometrical, electrical, and environmental factors [I], [4]. This formula represents the equivalent noise electric field (Eeq) received by a quasi-peak (QP) receiver connected to a horizontal biconical antenna oriented to pick up maximum noise. It is the one used in the Bonneville Power Administra- tion Corona and Field Effects Program (BPACFE) and is reproduced here as (47)':

The equivalent noise field of (47) is related to the equivalent noise voltage at the receiver given in (45) by the antenna factor 141.

OLSEN AND STIMSON: PREDICTING VHFIUHF ELECTROMAGNETIC NOISE 21

Eeq=20 loglo (75/f) + 120 loglo (E/16.3)

+ 40 log,, (0/30.4) + q/300 + W ( X ) + C

dB/1 pV/m Qp (47) in a 150-kHz bandwidth (Le., NM-30 receiver) wherefis the noise frequency in megahertz, E is the maximum surface gradient on the subconductors in kilovolts per meter, D is the subconductor diameter in millimeters, and q is the altitude in meters. The term w(x) is a noise decay characteristic (not given explicitly here) which was developed using both theoretical ideas and measurements. C is a constant (equal to zero in the Bonneville Power Administration (BPA) program) which will be used to calibrate the new model. The maximum surface gradient can be found using techniques described in [I]. The BPA formula was developed for a particular transmission line but has been used to predict the noise for a variety of lines. QP measurements are used because they correlate well with subjective measurements of television picture quality [18]. The difference between QP and rms measurements depends on the design of the QP detector and the nature of the noise. For a standard NM-30 receiver, QP measurements of VHF corona noise are larger than rms by 8-9 dB, while for the NM 37/57 used in [4], QP measurements are 4-5 dB larger than rms [19].

The goal of this paper is to combine (47) with the propagation model developed here. In essence, this means that w(x) is replaced by an algorithm based on the new propagation model. Further, C is chosen to ensure a best fit between the new model and the BPA formula in the case for which the formula was developed [12]. This constant accounts for the corona source amplitude in rainy weather and the fact that the new formula is calibrated to predict QP voltages rather than rms voltages. The method used to determine C will be discussed later. Its value is - 201.5 dB. The remainder of the terms (frequency, subconductor diameter, surface gradient, and altitude) in (47) are assumed to characterize the source and remain in the new model. The advantage of the new method is that different line designs will be accounted for theoretically and that the effect of receiving antenna characteristics, location, and orientation can be understood.

To obtain an accurate noise field prediction formula, the maximum electric field gradient for each phase bundle is first calculated [I]. Next, the noise field components of a particular dipole on the first phase bundle are found using (14), (16), and (17), assuming a unit dipole moment. These fields are converted to an equivalent noise based on the receiving system gain and geometry, as discussed previously. A similar process is performed for all significant corona sources on the first phase bundle, and the total noise power is the summation of their individual contributions. The noise voltage is then calculated as in (45) and converted to decibels. (This is the process used to replace w(x).) Next, the first four terms of the BPA empirical formula are calculated for the first phase bundle and added to the noise ~ o l t a g e . ~ This calculation is used to account for the corona source term. This process is

Also, a term - 6 dB was added to the BPA formula to convert equivalent noise field to equivalent noise voltage since the antenna factor for the biconical dipole used in the BPA measurements is - 6 dB.

VCNP PROGRAM -- a "1 0 - BPA CFE PROGRAM

W

ooL &---- 40 i 60 80 1 --: loo

LATERAL DISTANCE, m

Fig. 8. Calibration lateral profile (75 MHz). Noise voltage in decibels at the 50-0 receiver relative to 1 pV-QP, 150-kHz bandwidth, 346-kV rms I-/, horizontal line--17.7-m height, 10-m spacing, 4.07-cm-radius single conductor, wire conductivity 3.5 X l o7 S/m; earth conductivity-0.01 S / m; altitude-0 m; biconical dipole-97.3" beamwidth, 2.15-dB horizontal gain, - 100-dB vertical gain, Z , = 50 0, 0" elevation angle; receiver antenna azimuth varied for maximum noise at each field point.

repeated for each phase bundle. Then the total noise voltage is found using rms addition. Since the field equations (22), (24), and (25) account for the field point geometry (i.e., radial distance and incident angles), the new prediction method will work for any line geometry and receiving system even though the source term is based on an empirical formula. However, the noise prediction method must be calibrated by choosing the constant C. This is done by adjusting C to give agreement between the formula given here and measured data. This is described in more detail in [ 12).

XII. RESULTS The VHF corona noise prediction program (VCNP) devel-

oped here is shown to correspond well with experimental data at lower VHF frequencies. A comparison of the receiving antenna lateral profile for the VCNP program and the BPACFE program is shown in Fig. 8. This is done for the same case in which the empirical formula in both programs was derived and for which the VCNP program was calibrated [ 121. The corona field effect (CFE) result has a discontinuous slope at 40 m because w(x) decays as lld for the first 40 m and lld2 beyond. As can be seen from Fig. 8, the VCNP program developed in this paper compares closely to the CFE program noise prediction.

Results from the VCNP program also compare closely to the measured noise (near 75 MHz) of several different transmission-line geometries, as found in [l]. Details of the line designs and measurement locations are not included here ,because each is different. Table I compares the measured results with noise predictions from the VCNP program. All results compare closely with the measured noise except for line configuration 6. The discrepancy may occur because line configuration 6 is a 240-kV line and lower voltage lines tend to be noisier than predicted by the CFE program [19]. The BPACFE program exhibits a similar increase in error for this particular line configuration [ 11.

In the receiving antenna azimuth profile of Fig. 9, a comparison of results from measured noise taken near a 1200- kV line, [20], and results found using the VCNP program with the same line configuration are given. The QP noise data from

2 2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 1, FEBRUARY 1988

TABLE I

VERSUS DIFFERENT MEASURED RESULTS FROM SEVERAL LINE VCNP PREDICTED QUASI-PEAK NOISE AT THE RECEIVER TERMINALS

CONFIGURATIONS [I] Noise field converted to noise voltage by adding the - 6-dB antenna factor for

a biconical dipole.

large degree) by the radiation pattern of the corona source (Fig. 5) since line towers are not included in the model. Further experimental work is needed to quantify the relative importance of the two effects.

Line Noise frequency Measured Predicted noise Difference

5 70 15 13.7 + 1.3

Configuration (MH.?) Noise dB/luv VCNP dB/luv Measured-Predic. (u

6 75 15.1 7.9 t 7.2

7 75 17.1 18.1 - 1.0 8 75 28 30.4 - 2.4 9 75 18 20.3 - 2.3

NOISE , dB / I pv / m QP

VCNP PROGRAM EXPERIMENTAL RESULTS

I I I I

-80 -60 -40 -20 0 20 40 60 80 AZIMUTH ANGLE, deg

Fig. 9. Comparison of 75-MHz azimuth profile to experiment. Noise voltage in decibels at the receiver relative to 1 jtV-QP. 1200-kV rms / - I delta line-53.8-111 center phase height, 35.5-111 outside phase height, 11-m horizontal spacing, eight subconductors of 4.07-cm radius and 0.23-111 spacing, wire conductivity 3.5 x lo7 S/m; earth conductivity-4.01 S/m; altitude-0 m; Yagi antenna-39.6” beamwidth, 13.0-dB horizontal gain, - 100-dB vertical gain, 2, = 50 Cl, 0” elevation angle, position (x, y , z) = (40, 3, 0) m as in Fig. 1; receiver-0.12-MHz bandwidth, 16.3-dB preamp gain, ZRx = 50 Cl.

[20] are plotted with a constant factor of 20 dB added to account for the attenuator setting on the receiver. The VCNP prediction data agree reasonably well qualitatively and quanti- tatively with the measured data even though transmission-line towers are not included in its noise model. This is quite significant because it has been speculated that the increase in noise as the receiving antenna is rotated towards the line direction is due to the presence of additional noise sources on a tower toward which the antenna beam moves. The model presented here indicates that this effect is caused (at least to a

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