Since the component of radiation which reaches the fare field of an antenna in free space is in the ratio of 120z : 1 E to H, it would seem expedient to follow the proposal of Hately and Kabbery and mix the Fields in the physical aperture of the antenna in precisely this ratio. Also, because the intensity of the fields around the physical aperture are not constant, it would seem reasonable to take an average value of each field in the area where the fields interact and to see if this value has physical meaning.

Another assumption made is that the resulting antenna structure is small compared to the wave length of radiation and so phase related phenomena pertaining spacifically to physical size of the antenna can be ignored during the derivation. The validity of this approach is given as a proof by comparison with the accepted equation for characteristic impedance of twin balanced lead transmission line, which is derived by the procedure outlined above. The theory is developed around the conductive geometry as shown in figure 1.

The general case for any dielectric media is: E) j Z7--I J Taking this as air we have: E 120r Hav = Equation2To begin with the average value of electric field, (see figure 6) It can be shown that: sm972 v-v f sit-JL 0/2 tiav cJav--c sJ where v Scaler potential between the straight conductors of figurel s Depth of the aperture as depicted in figure 4 If we define a parameter k such that: v 0 v/d Where d Separation of the two straight conductors depicted in figurel Equation 2 represents the ratio of the average value of electric field to the max value of electric field, we can therefore write: The magnetic field external to the coil can be written as: o-Equation 11 1--r r 2 where it Number of turns on the coil : Current in the coil Length of the coil Radial distance from the center of the coil The equivalent current in a solid conductor with radius equal to the coil Radius of coil The first term on the right of equation 4 is the standard definition of the magnetic field strength a distance r from a cylindrical coil carrying current i. The second term on the right of equation 4 is the definition of magnetic field strength a radial distance r from the center of a solid conductor. In this instance however the Field vector is taken as being rotated by 90 degrees to bring it parallel to the axis of the conductor.

The average value of the H field is then given by: I'°+s 7~ TO 0 1Which evaluates to: I H Equation 8 av 2n If we now define a characteristic impedance: .-u 1 This represents the characteristic impedance of the transmission line formed by the outside of the cylindrical coil and the two conductors. If we assume that the average electric and magnetic field over the region of interaction is in a ratio of 120 we can take this value and multiply it by the aspect ratio of antennas physical aperture to form equation 10. The general principle here can be proven using field mapping.

We can now state that: -, -'av ,.- Equation 10 a I I Equation 8 can then be written as: vu av 240,, "240' Taking the ratio as stated in equation 2: zip 120g Hav 240' 240yz2ds o Giving us a match condition onto free space of : - . I')/ir--I, kas 0 It can be seen from equation 13 that the depth of the aperture s as shown graphically in figure 4 is finite and that in fact it is a small fraction of ro. If that is the case we can proceed to an good approximation for the input impedance to the antenna derived from equations 6 and equation 9 as follows.

The input impedance of the antenna is given by the ratio of the terminal voltage to the terminal current as: Equation in From equation 6 and equation 9 we have that: Equation 2407z2d From which we have : r2 z < t, Section 1.2, Significance of the capacitor depicted in figure 1 Figure 5 shows the antennas equivalent circuit. The inductance represents the reactance due to the fields inside the coil. The series reactances of the inductor and the capacitor can impose any relative phase between the electric and the magnetic vectors.

In the case of air dielectric the required phase is around zero and the required condition is produced by: Equation 17 L-C Section 1.3, Example of dielectric media other than air (sea water) Evaluating equation 1 for water at 100MHz : I Zant-4 \.'7 From here the derivation proceeds as section 1. 1 we arrive again at equation 13 however, equation 10 now reads: v v Za-v--Eav I Equation 15 now reads: I nzrO_ 1 And the input impedance is given by: z {14arg-45"} , i7. I/Equation Because the input impedance of the antenna is designed as the conjugate of the intrinsic impedance of water, the actual impedance seen at the terminals when the antenna is loaded by this media is then simply given by: 2 . - mu", 2 Equation nw-12 (14) Section 1. 3, Influence of the factor W The material inside the coil could have tir >1 without affecting the physics of the derivations as presented, other than to lower the frequency of radiation by a factor given by: f ç / Al ° Equation 23This is simply because the phase is established by the series resonance depicted in figure 5.

Section 1.4, Long wire antennas and the proof of the theory developed in section 1. 1 Figure 7 shows a section of a long twin lead transmission line. It will be demonstrated is that a contour integral between these two conductors taking the ratio of average electric and average magnetic fields and setting this ratio to 120 results in the equation for the characteristic impedance of balanced twin lead in free space.

Starting with Electric Field, we have the average Electric field as: Equation The average Magnetic Field is: T + y 2res to Equation 25 evaluates to: I-8 H po av = ^ +) Equation 26Since we require a fare field ratio defined by the intrinsic impedance we can write: -- Hav If we now define a characteristic impedance, which by definition, must conform to the ratio of voltage to current all along the matched (onto free space) transmission line, we can write the following equation: Equation 0 Hav Equation 29 Z, Now, taking the ratio as stated in equation 27 we have that: 120 Equation 30 Hav v Z, _ Eav _ v I sWhich reduces to: Equation 'o With a change of variable into standard form, equation 31 then reads: r -) r, =1201n (-This equation was derived on the basis of forming a match onto free space but has resulted in the definition of characteristic impedance of twin lead which represents the load impedance of a transmission line as seen at the terminals. It was proven during practical experimentation that taking Zo=75 Q in equation 32 and arranging the twin lead geometry accordingly does indeed provide a 75Q match onto free space. QED Section 2. 1, Discussion of configurations and design equations Configurations 1-2 Configurations 1-2 are connected as per figure 1 and produces radiation which is long in wavelengths compared to the physical size of the antenna. The configuration produces an omnidirectional radiation pattern. To a close approximation the antenna radiation pattern will therefor be that of a Hertzian dipole. Directivitv will be 1. 5 numerical, or 1. 76dBi balanced and 4. 76dBi unbalanced with the E vector normal to ground. The physical geometry of the antenna is determined by equation 16 as 240Z2nr f,.- in l2 dWhere 11 Number of turns on the coil ro Radius of coil <BR> <BR> <BR> <BR> <BR> <BR> d Seperation <BR> <BR> <BR> of the E- field rods 7 Circumference of antenna In this instance the length parameter in the denominator is the circumference of the antenna structure.

The approach to constructing these antennas is to pick a reasonable value for the number of turns of the coil and to place a reasonable gap between the turns consistent with keeping the feed through capacitance to a minimum (a reasonable value to pick is around 10-20 turns). The number of turns is also consistent with achieving the smallest possible structure. The coil is then bent around to form a toroid. Input impedance is chosen as the system characteristic impedance and from there the remainder of the parameters of the above equation can be solved. The assembly is electrically connected as shown in the electrical schematic of configuration 1. The E- field plates are depicted as C1. The unwanted part of the coils reactance is trimmed out using C2 leaving an in phase E and H field with the radiation resistance of the antenna across the input terminals as required. to meet all conditions for radiation.

It is to be pointed out that equation 16 was derived on the basis of reasonable approximations, however the equation gives results which are consistent with the requirements of the antennas industry which is fundamentally founded on trimming any antenna structure to meet a final specification by the use of network analysers.

Many Electromagnetic phenomena cannot be taken into account in the mathematical analysis of a practical radiating structure making the above procedure accepted practice rather than the exception.

Configurations 3-4 Configurations 3-4 are opened out version of configuration 1 and is exactly as depicted in figure 1. Directivity will be slightly higher than Configuration 1 at around 6dBi balanced 9dBi unbalanced respectively. The exact figure will depend on final chosen geometry.

The approach for construction is the same as for configurations 1-2 only this time the denominator of equation 16 is simply taken as the length of the antenna.

Configurations 5-6 Configurations 5-6 are the magnetic version of configuration 1 and will have the same electromagnetic properties. The antennas are configured as a series resonant circuit with an inductor followed by a capacitor followed by a second inductor. The two coils are arranged so that the H field is cumulative in the physical aperture. Directivity will be 1.76dBi balanced and 4.76dBi unbalanced. Fundamentally however the principle of operation is the same with the coils reactance being cancelled by the capacitors reactance which then brings the E and H fields into the required phase and orientation and a real radiation resistance is observed at the terminals of the antenna.

The governing equation for these configurations is not derived in section 1 but is simply stated here as: Equation in G 33Where n Number of <BR> <BR> <BR> turns on one coil i Radius of the coils Separation of the two conductive surfaces of the coils | Length of each coil The approach for the construction is to choose reasonable values for the number of turns, the appropriate input impedance, and then to solve the remainder of the parameters in equation 33 consistent with small size as outlined in the description of configuration 1-2 Configurations 7-8 Configurations 7-8 are again an opened out version of configuration 3 and will have 3dB higher directivity than the balanced or unbalanced antennas depicted in configurations 5-6. The antenna will produce a figure of eight pattern in the H-plane with a gain of 4.77dBi when run balanced and 7.77dBi when run unbalanced. The design equation and construction approach are the same as configuration 5-6 however the presence of the ground for unbalanced operation will mean a little network analyser work will invariably be required to match the antenna.

Configurations 9-10 Configurations 9-10 result directly from the proof of the validity of taking averages of electromagnetic fields and in this form they are little more than Beverage antennas with the unique feature that there is no load resistor terminating the end of the antenna.

The antenna structure assumes that several cycles of phase are present over the length of the antenna and under these circumstances placing a load resistor on the end of the antenna actually reduces the gain of the antenna. As long as the above condition is met the antenna's input impedance is fixed and does not vary with frequency. The only parameter which does change with frequency is the directive gain of the antenna which is given by: D o ) Equation 34 = 1 OL6Where 7 Length of line <BR> <BR> <BR> <BR> <BR> Wave length<BR> <BR> A of operation And the input impedance of the antenna is given by : Zin 20t 35 These equations relate to the balanced form of the antenna and the symbols for equation 35 are the accepted definition for the characteristic impedance of twin lead.

The unbalanced form of the antenna has design equations: Equation 36 The input impedance of the antenna is given by : Zin= 60 ln 37 Equation 37 is the accepted equation for the characteristic impedance of unbalanced circular microstrip.

Configuration 11-14 Configurations 11-14 are slow wave version of configuration 5, constructed to prove that the directivity of the long wire antennas is proportional to the number of normalise wavelengths in the meander or delay line as opposed to the length of the antenna. This antenna can also be run in balanced and unbalanced modes. The directivity of the unbalanced mode being 3dB above balanced. The antenna dimensions and input impedance are governed by the above equations for configurations 9-10 as long as the gain equations are calculated with the total electrical length of line. The effect is to reduce the length of the antenna while maintaining gain.

Configurations 15-16 Configurations 15-16 represent the next logical step in the slow wave concept. The unique feature of this antenna configuration however is that the phase length of the line is imposed on the transmission line by employing phase shifters with an input and output impedance equal to the characteristic impedance of the transmission line. This configuration gave very high directivity. The directivity of the unbalanced mode being 3dB above balanced. The practical form of the antenna used pi network phase shifters as shown in diagrams for configurations 15-18. The advantage offered here is that the phase delay of a pi phase shifter operated with input and output impedance equal to the load impedance is that a fixed 90° of phase occurs across the network and with many such shifters low frequency operation and high gain becomes possible.

Configuration 13-14 can produce gains in excess of 18dBi at HF frequency with a physical size of less than 0.5 by. 035 by. 010 meter. This antenna can also be run in balanced or unbalanced modes.

Design equations for the balanced antenna are: D10 Equation 38 360 Where n Number of- phase shifters 7 Length of line i, Wavelength of operation ZiM 39 The design equation for unbalanced antenna are: 0 D ) ) 360 zill =10LConfigurations 17-18 Configurations 17-18 are basically. the same as configuration 9-10 but is curled round to form an omnidirectional antenna, radiation-pattern. This antenna can also be run in balanced and unbalanced modes. The directivity of the unbalanced mode being 3dB above balanced. Directive gain of the antenna will be 1.76dBi balanced and 4.76dBi unbalanced. Input impedance to the antenna will be given by: Balanced 7 42 iyz n 1-P-quat,. onUnbalanced Z,,, ) " =601n ( ? i/Configuration 19, Minimum antenna configuration This antenna configuration has a field system which is not clearly defined being dependent on a very variable antenna geometry. This fact makes the radiation pattern and input impedance undefined, however the configuration represents the absolute minimum component and materials aerial presented. The final form of this antenna must be iteratively defined using a network analyser. Gain will be approximately 4.76dBi. The first step in the design procedure is to assume configuration 8 and related design equations hold then the antenna will have to be iteratively altered from this point to meet specification.

Section 2.2, General statement on the subject of characteristic impedance With all the long wire antennas (configurations 9-14) the equations stated as input impedance are identical to the equations for characteristic impedance for the stated type of line. The phenomena described ie radiating balanced or microsrip lines (straight, meander, or delay) is general. Any balanced or microstrip line of any geometry (ie flat or circular or rectangular) separated with any dielectric material what so ever will radiate and operate as described. The equations presented need only be altered to take into consideration the effective phase length of the line.