Introduction:

Recently, the City Council passed and an amendment to the zoning ordinance "to permit and regulate installation of wireless telecommunications equipment and facilities" which purports to minimize the number of cell phone antennas (CPAs) in Hoboken subject to the constraints that the cell towers should preferentially be located as far as possible from residential areas and that all cell phones should be adequately serviced by the CPAs.

Unfortunately, no local or state government can regulate the placement of CPAs based solely on health effects as long as the electromagnetic (EM) emissions from such CPAs do not exceed Federal Communications Commission (FCC) maximum thresholds, even though there is mounting evidence that FCC thresholds are way too lax, perhaps 100 to 1000 times higher than the radiation levels at which negative health effects, including cancer, have been observed.

The CPA ordinance thus regulates based upon criteria of aesthetics rather than health.  However, the criteria may still be used to minimize negative health impacts.

The purpose of the following calculation is to determine conditions under which the placement criteria from this CPA ordinance can be used to mitigate negative health impacts as well.
 

Statement of the problem:

For the purposes of this calculation, we will model the distribution of CPAs as follows:

Suppose we wish to service a minimum of NPmin cell phones at any time with a maximum of N CPA sites within an area H.  Each site radiates a hemispherical signal with total power P:

P == n * p                                                                             [1]

where n == number of cell phones serviced by each CPA site and p == power radiated per cell phone serviced.   The total power flux from each CPA site at distance R is given by:

flux(P,R) == P / (2*pi*R**2)                                                [2]

The "coverage area" of a CPA site is a circle with radius D, subject to the "minimum flux" constraint that at the edge of the coverage area, the signal power flux must be strong enough to service a cell phone, i.e.:

flux(p,D) == p / (2*pi*D**2)  >=  flux_min                          [3]

where flux_min == minimum flux needed to operate a cell phone.  In addition we want to insure that the CPA site does not create health effects, so we limit the total power flux produced at distance A == distance from the CPA site to the nearest neighbor to be less than flux_max == maximum flux beyond which negative health effects occur:

flux(P,A) == (n * p) / (2*pi*A**2)  <=  flux_max                  [4]

Since N such CPA sites services area H and a minimum of NPmin phones, we have also:

H = N * pi * D**2                                                              [5]

NPmin  <=  N*n                                                                [6]

We will assume that H, A, NPmin, flux_min, and flux_max are given, and use the above relations to determine N, n, and D.
 

Estimation of given "imput" parameters H, A, NPmin, flux_min, and flux_max:

For our Hoboken problem, H represents the area to be serviced which is taken to be the area of Hoboken, about 1 sq.mile.   A is the distance to the nearest neighbor, below we will consider two scenarios:  (1.) the nearest neighbor is on the top floor across the street from the CPA site, at a distance of about 50 feet (this assumes that the dwellings underneath the antenna within 50 feet are shielded from the radiation);  (2.) the nearest neighbor is 250 feet from the CPA site.   NPmin will be estimated to be about 1000, and flux_min and flux_max are determined as follows:

One of the AT&T CPA sites approved by the zoning board is intended to reach a cell phone 610 m away.  From http://www.spectrum.ieee.org/publicfeature/aug00/prad.html: "Analog hand-held phones radiate 600 mW or less of time averaged power, and many digital models produce 125 mW".  This suggests that a cell phone radiating between 125 mW and 600 mW can communicate with a CPA  at distance D == 610 m away, so that the roof mounted CPA shouold be able to do the same -- i.e. 0.6 w from the CPA site should be sufficient to power a phone 610 m away.Then:

flux_min  <=  flux(0.6w, 610m) = 0.6 w / pi * [610 m]**2  = 5.1*10**-7 w/m**2                [7a]

flux_min may be less than flux(0.6w, 610m), but we can be sure that if we set flux_min to flux(0.6w, 610m), then the cell phone will be adequately powered by the CPA.  So we estimate:

flux_min  ~=  flux(0.6w, 610m) =  5.1*10**-7 w/m**2                                                         [7b]

To estimate flux_max, we use the value recommended by Dr. Neil Cherry of Lincoln University in Canterbury, New Zealand, which he based on the large number of studies worldwide indicating serious damage from long-term exposure to low level radio and microwave. Dr. Cherry recommended a maximum exposure flux standard of 0.1 microwatt per square centimeter "if cancer risk is to be reduced." He recommended an even lower level of 0.01 of a microwatt per square centimeter "if miscarriage risk, sleep disruption, children's performance impairment and chronic fatigue symptoms are to be reduced."  Cherry concluded that transmitters should be kept away from schools and residences "by such a distance that the intensity of the microwaves, when averaged over a year, does not exceed 0.1 of a microwatt per square centimeter." More recently he has tightened his recommendations to a "mean chronic public exposure" of 0.01 of a microwatt per square centimeter with an outside limit not to exceed 0.1 of a microwatt per square centimeter.  We will use, then:

flux_max ~= 0.1 microwatt/cm**2 = 10**-3 w/m**2                                                            [8]
 

Derivation of Constraints on N and n:

Combining [3] and [4] gives:
 

flux_min  <=  p / (2*pi*D**2)  <=  flux_max  * (A / D)**2  / n                 [9a]

==>    (A / D)**2  >=  n * flux_min  /  flux_max                                        [9b]

Using [5] to eliminate D in [9b]:

n / N   <=   Rmax(A)                                                                                  [10a]

where:

Rmax(A) == ( flux_max  /  flux_min ) * (pi * A**2) / H                         [10b]

combining  [10a]  with  [6]  gives:

NPmin  <=    N * n    <=   N**2  *  Rmax(A)                                           [10c]

so that:

N  >=  squareroot { NPmin  /  Rmax(A)  }                                              [10d]

and:

NPmin / N   <=    n    <=   Rmax(A) * N                                                 [10e]
 

Case (1.):  A = 50 feet:

Rmax(A=50ft) =  ( 10**-3 w/m**2  / 5.1*10**-7 w/m**2 )  *  [ pi * (50 ft)**2 ]  /  (5280 ft)**2  =    0.5524                [11a]

N  >=  squareroot { NPmin  /  Rmax(A=50ft)  }  =  squareroot { 1000 /  0.5524  }  =  42.55                                    [11b]

Since N and n must be integers, then N >= 43, and n is an integer that must solve [10e]:

1000 / N   <=   n   <=   0.5524 * N                                                                                                                                  [11c]

The following table gives the smallest two values of N, and the corresponding inequalities from [11c]:

N                      n
43        23.26 <=  n  <=  23.75
44        22.73 <=  n  <=  24.30

N = 43 does not permit an integer solution for n, but {N= 44, n = 23, N*n = 1012}, {N= 44, n = 24, N*n = 1056} are both permitted solutions.  The range of each CPA site is:

D = sqroot {H / [N*pi])  = 5,280 ft / sqroot[44*pi] = 450 ft.
 

Case (2.):  A = 250 feet:

Rmax(A=250ft) = ( 10**-3 w/m**2  / 5.1*10**-7 w/m**2 )  *  [ pi * (250 ft)**2 ]  /  (5280 ft)**2   =   13.81                [12a]

N >= squareroot { NPmin  /  Rmax(A=250ft)  }  =  =  squareroot { 1000 /  13.81  }  =  8.51                                     [12b]

Since N and n must be integers, then N >= 9, and n is an integer that must solve [10e]:

1000 / N   <=   n   <=    13.81 * N                                                                                                                                    [12c]

Substituting N = 9 into [12c] gives the inequality that must be solved by integer n:

for N = 9:        111.11 <=  n  <=  124.29

N = 9 permits an integer solutions for n between 112 and 124 cell phones serviced per CPA site, permitting service to between 1008 and 1116 cell phones.  The range of each CPA site is:

D = sqroot {H / [N*pi])  = 5,280 ft / sqroot[9*pi] = 993 ft.
 

Discussion

We have solved for two scenarios for CPA site placement which permit the servicing of at least 1000 cell phones with power flux levels never rising above a threshold of .001 w/m2, which is considered a conservatively safe exposure level.

In the first scenario, where CPA sites are placed on roofs in residential neighborhoods, within 50 feet of the nearest neighbor, there must be at least 44 cell sites, each servicing between 23 and 24 cell phones (for a total service of between 1012 and 1056 cell phones).

In the second scenario, where CPA sites are placed 250 feet away from the nearest neighbor, there must be at least 9 cell sites, each servicing between 112 and 124 cell phones (for a total service of between 1008 and 1116 cell phones).

The first scenario reflects the approach being used by cell phone providers in Hoboken, i.e. putting CPA sites on residential rooftops, except that the number of phones serviced per site can be as high as 6000 phones [i.e. 3600w @ .6w/phone] as far as 2000 ft away from the CPA site, causing levels of flux to be 3600w/(2*pi*15m*15m) = 2.55 w/m**2, about 2500 times the safe level.  Our calculation show that safe level can be achieved in this rooftop scennario, but only if many (44) short range (D = 450ft)  CPA sites are used.

The second scenario could be used for CPA sites on the periphery of Hoboken.  In this scenario, up to 124 phones per site can be serviced, but the sites must each be > 250 feet from the nearest neighbor, i.e. removed by 250 feet from residential neighborhoods.

Our study is crude:  it does not take into account the fact that the site ranges must overlap, that there are cumulative effects from all CPA sites in Hoboken (not just the closest one), and that CPA sites may focus their EM radiation more narrowly than the hemisphere contemplated above.  Nevertheless, it suggests that aesthetics, economy, and safety could all be insured by locating CPA sites > 250 feet from residential neighborhoods, as would be permitted under the new Hoboken zoning amendments.

An obvious caveat is that, in order to minimize the number of CPA sites, the energy and cost of the sites, and the harm to the public, cell phone service providers should not only colocate their CPA sites (as is encouraged by the zoning amendment ordinance), they should actually share the sites, i.e. the same CPA equipment should be used to communicate with cell phones serviced by all providers.