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From: dalyb@godzilla.UUCP (Brian Daly)
Newsgroups: rec.ham-radio,sci.electronics
Subject: Re:  Theoritical Equations for Antennas
Keywords: Antennas
Message-ID: <481303aa.1423f@godzilla.UUCP>
Date: 16 Jan 90 19:41:18 GMT
Organization: gte
Lines: 112


Subject: Re: Need Theoritical Antenna Info
Newsgroups: rec.ham-radio,sci.electronics
Keywords: Antennas


John Moore (NJ7E) posted a request for information on theoritical formulas
for effective aperture, input impedance, and radiation resistance for short/
long dipoles, short and long quarter wave ground planes, and magnetic dipole
antennas. Here's a brief summary of equations:

Reference: Balanis, "Antenna Theory: Analysis and Design", Harper and Row.
	   This text is available from the Arizona State University
           bookstore.

           Kraus, "Antennas", McGraw Hill

 Short Dipole:  (wavelength/50) <= length <= (wavelength/10)

     Effective Aperture  Ae  =  3(wavelength)**2 / ( 8 * pi)
                                         
     Radiation Resistance:  Rr  = (20 * pi**2) * (length/wavelength)

     Input Impedance: For a small dipole of length l and wire radius a:

                 Reactance: Xin  = -120 [ln(l/a)-1] /  tan(kl)

                            where k = (2 * pi) / wavelength
                                   
                 Resistance:  Rin = Rr as given above

     Note that if the antenna length is slightly less than 1/2 wavelength,
     the reactance will go to zero. For l of a halfwavelength, the reactance
     is approximately zero and the resistance is close to 50 ohms. (42.5 ohms
     for wire of small radius).

  Long Dipole:

     The analysis for a finite length dipole gets a little complicated.
     Balanis (reference above) provides a FORTRAN program in his text
     to compute radiation resistance, directivity, and input resistance
     for a finite length dipole based on the following equations:

      Radiation resistance Rr = 60 { C + ln(kl) - Ci(kl) + (1/2)sin(kl)
    
                 * [Si(2kl) - 2Si(kl)] + (1/2)cos(kl)

                 * [C + ln(kl/2) + Ci(2kl) - 2Ci(kl)] }

      where Ci,Si are cosine and sine integrals
            C = Euler's constant = 0.5772

      Input resistance:  Rin =  Rr / (sin (kl/2) **2)

      Input reactance:  Xin = Xm / (sin(kl/2) **2)

        where Xm = 30 { 2Si(kl) + cos(kl)[2Si(kl) - Si(2kl)]
                     - sin(kl)[2Ci(kl) - Ci(2kl) - Ci(( 2k * {a**2})/l)]}

      Effective Aperture (maximum):

               Aem = {(wavelength **2) / (4 * pi)} * (directivity)

        Balanis figure 4.8 plots directivity versus dipole length for
        0 to 3 wavelengths. The calculation for the directivity is
        rather complicated; refer to the reference for details of the
        calculations.

   Magnetic Dipole:

      By applying the principle of duality, a magnetic dipole of magnetic
      moment Iml is equivalent to a small electrical loop of radius a and
      constant electric current Io provided that:

                 Iml = jSwuIo  where S = area of the loop = pi * a**2
                                     w = radian frequency
                                     u = permeability
                                  
       For a small electric loop of radius a:

         Radiation resistance:  Rr = 20 * (pi)**2 * {C/wavelength}**4
                 where C = circumference of the loop

         Effective Aperture (maximum)  Aem = 3 * (wavelength)**2 / (8 * pi)
       

  Short/Long (what is a short or long??) Quarter Wave Ground Plane:

    The analysis of a quarter wave ground plane antenna is essentially
    a vertical electrical dipole above a ground plane.

    When the length of the antenna is a quarter wave, and the height above
    the ground plane is zero, then:

      Input impedance = 36.5 + j21.25 (note that this is one half the input
                                       impedance of a half wave dipole)

     
      Effective aperture (maximum) Aem = 0.13 * (wavelength)**2

    
I hope this helps answer some of your questions. I refer you to the 
references I mentioned for more detailed information. The Balanis
text does contain many FORTRAN programs, which may be of some use.


Brian K. Daly WB7OML
AG Communication Systems, P.O. Box 52179, Phoenix, Az. 85072-2179
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Phone: (602) 582-7644   FAX: (602) 582-7111