3.2.1. Circuit Modeling: Loop Impedance A loop antenna can be represented by a lumped circuit when its dimension is small with respect to a wavelength. In this representation, the circuit parameters (generally constant) can be specified by a quasistatic approximation. Power, energy and heat characteristics of the loop can be modeled by an equivalent circuit as An equivalent circuit of a loop antenna. At low frequencies, the capacitive effect is generally negligible. The inductive and resistive effects are dominant. Therefore, a loop impedance is where and. In reality, circuit parameters are frequency dependent. Therefore, determination of the frequency dependent loop input impedance is a fundamental concern for loop design. Generally, can be drawn as a given figure at below. The frequency dependence of the loop impedance. This graphic means that an inductive behavior of the loop impedance can be maintained in the reasonable band with a constant and not big 1. The given frequency dependency is very important for wide-band loop applications. The loop design is based on - number of turns, - wire thickness and electromagnetic parameters, - heat and used isolation materials, - physical and geometrical structure, - skin depth and hysteresis. The effects of the above-given parameters can be modeled over and of the loop. Therefore, we should concentrate for the extraction of these parameters. 1 At near DC, Ω ( 0.86 m Ω)
Loop Loss Resistance, Two types of loop loss resistance are DC (Direct Current) and AC (Alternative Current) resistances. - DC Loop Loss Resistance A coils DC resistance can be expresses as where is the number of turns, is the mean diameter and is the diameter of copper wire [Kesler, 1997]. can be somehow extracted by using an empty cylinder model with a loop thickness because this model is valid at the low frequency regime. - AC Loop Loss Resistance constitutes the resistive part of a small loop antenna input impedance and can be formulated by a simple closed formula using transmission line modeling of the loop as where is the wavenumber, is the loop perimeter. In terms of the loop area, this relation becomes where it should be noted that is almost independent from the radius of the wire. These formulas are accurate for a loop perimeter up to. It was also shown that the reactive (parts) formulas are also accurate for a loop perimeter up to [Awadalla et al, 1984]. Alternatively, it is assumed that for loosely wound loop equals high-frequency loss resistance ( of a straight wire of the same length. Accordingly, for uniform current distribution where is perimeter of wire cross section. However, considering proximity effect, of an -turns circular loop becomes where and are loop and wire radius, respectively., and are surface resistance, ohmic resistance per unit length due to proximity effect and ohmic resistance per unit length due to the skin effect [Smith, 1972a], [Balanis, 1997], [Nikolova 2014]. Loop Capacitance The loop capacitance (Farad) is generally neglected at low frequencies and more effective at high frequencies in which shielding and grounding are used to protect capacitive coupling. The parasitic
capacitance may be temperature dependent. Anyway, tuning capacitors can be used to operate the loop at resonance. Loop Inductance The inductance (Weber/m 2 or Henry) of a multi-turn loop antenna can be approximated as where is the number of turns, is the cross-sectional area. and are the other dimension of the multiturn loop shown above [Moreland, 1999]. It should be noted that is not present in this formula. This formula can also be related to the Wheeler s formula [Wheeler, 1928]. Alternatively, inductance of a single circular loop of radius made of wire of radius can be represented [ ] where it can be noted that self-inductance of a straight wire of length is, for a single loop [Nikolova 2014]. The loop inductance can also be represented as where the area (m 2 ), is an inductor shape factor (effective length/actual length ) [Wheeler, 1947]. According to formula reducing the antenna size, reduces. However, can be increased by increasing the number of turns,. This gives a great flexibility to the loops comparing to the electric dipoles. The inductance can also be extracted from the induced voltage (see the previous section) where is the current [Devore and Bohley, 1977]. Impedance Matching (Tuning) The input impedance of the loop has a real and an imaginary part. Even if the real part is equal to the characteristic impedance of the transmission line (lossless), the imaginary part causes impedance mismatching. This can be eliminated by adding extra capacitors for tuning the coupling loop. Adding extra capacitors for tuning the coupling loop. Unavoidable impedance mismatching from generator to antenna or antenna to load limits the output to a fraction of power input [Wheeler, 1947]. This can depend on the nature of generator or load coupled and principally affects the bandwidth.
Efficient small antenna operation requires tuning to the desired operational frequency. Therefore, a tuner can be used between antenna and generator or load as shown at below ( and are the tuner and radiation power factor of a loop antenna, respectively. is the coupling coefficient) [Wheeler, 1947] Tuning of the antenna with a generator or a load by a tuner. Here, it is assumed that a generator or load are so coupled to the tuner. In the case of a full coupling, a maximum power can be delivered between antenna and generator or load. This maximum power is called available power. To increase coupling efficiency, should be increased or should be decreased. Lesser bandwidth can be obtained by decreasing the coupling or decreasing the antenna size and greater bandwidth can be obtained by increasing the coupling [Wheeler, 1947]. In order to operate the loop at resonance tuning capacitor is can be connected in parallel to the loop. The value of the In this case, becomes a real number [Nikolova 2014]. Loop Induced Voltage An induced voltages of a turns loop placed in a uniform free space magnetic field is where is the magnetic induction flux where is the loop area and is the angle between the magnetic field lines and the loop plane surface. becomes maximum when means that is perpendicular to the loop plane ( ). For monochromatic sources of angular frequency, and its amplitude are where is the free space magnetic permeability. In a common usage, can be related to an electric field over impedance relation of a plane polarized wave such as where is the free space light velocity [Laurent and Carvalho, 1962]. Alternatively, using the Lorentz condition for the relation, can be simply related to the vector potential as
where the only is present due to the symmetry and is the loop radius [Islam, 1963]. Loop Losses a) Wire (winding) resistance (ohmic) losses: be minimized by a proper choice of wire size and type. They are or and related to the penetration depth of the wires which is related to the applied (source) currents. b) Core (loading) losses: This loads extra magnetic and dielectric loss associated with the core material. The frequency and temperature dependency of the magnetic loss were fairly well investigated. According to this, the magnetic loss increases with frequencies below 100 MHz and there is an optimum frequency range for them [Snelling, 1969]. They can be minimized by selecting lossless ferrites at desired frequency. c) Ground losses: The coil proximity to a ground plane causes a loss (in the ) 2. In some case of 1/4 inch from the plane results db loss [Laurent and Carvalho, 1962]. d) Phase shift losses: To obtain different antenna patterns such as omnidirectional, phase shift can be applied in one of the antenna. This causes extra losses. This also degrades ratio (3 db in best) [Laurent and Carvalho, 1962]. e) Chassis (conducting) losses: usually limits the performance [Pettengill et al, 1977]. The loss resistance originates from skin effect (ohmic loss) and proximity effect because of non-uniform current distribution due to multiturn structures. Close spacing of the turns leads to the proximity effect can be larger than the skin effect [Smith, 1972b]. If the loop is connected at the any circuit terminals, extra losses are - Unmatched network losses from source and load, - Stray losses in transmission lines, - Spurries tuned-circuit losses. Self-Resonances Bandwidth The reduction of the antenna size is a fundamental limitation on the bandwidth. It means that the larger antennas are generally more efficient for wideband applications [Wheeler, 1947]. Noise Figure (Thermal Noise) and Signal to Noise Ratio Noise figure is based on the thermal noise of the antenna 3. The thermal noise produced by the antenna is where is a Boltzman constant, is absolute room temperature, is the material density of winding wire. and are the average antenna diameter and loading material diameter, respectively. is the equivalent winding mass of the antenna. 2 It also detunes the resonance frequency [Carlos and Carvalho, 1962]. 3 The antenna efficiency is also reflected in corresponded noise figure [Steward, 1957].
An antenna volume of a fully wound antenna for desired to diameter ratio as ratio can be calculated for optimum length where is the receiver noise factor, is the bandwidth and is the relative permittivity of the ferrite. is minimum electric field strength for desired ratio. This value can be obtained from diagrams relating to the atmospheric noise [Pettengil, 1977].