A Theoretical Examination of The Bandwidth Limits For Mixed Mode Electrically Small Antennas Abbas Abbaspour-Tamijani Department of Electrical Engineering, Arizona State University P.O. Box 875706, Tempe, AZ 85287-5706 E-mail:
[email protected] Introduction Electrically small antennas are known to suffer from small radiation resistance, small bandwidth, and low efficiency. Small radiation resistance can be attributed to the small retardation between the field components originating from the different points on the radiator body. Small bandwidth and low efficiency stem from the fact that for a small antenna the “reactive” energy stored in the surrounding space is much larger than the energy radiated in one radian cycle. This ratio, that is quantified by the radiation Q (Qr), has been shown to be inversely proportional to the ratio of the effective volume of the antenna to the radian sphere at the operating frequency. In their classical works, Wheeler [1], Chu [2], Harrington [3], and Collin-Rothschild [4], using various methods have predicted that the lower bound of Qr for a linearly polarized antenna is given by: Qr ≈ ( ka )
−3
(1)
where a is the radios of the smallest sphere enclosing the antenna geometry. This limit is in fact is calculated for the lowest order TE or TM modes which incidentally yield the smallest values of radiation Q, and its application to more complex scenarios is based on the orthogonality of the modes. Although the orthoganility guarantees that the total energy and radiated powers for different modes are additive, a similar prediction about the reactive energy is indeed arguable. In mixed mode TE/TM regimes, two of the basic assumptions of implicit in the Q calculation process cease to apply. These assumptions are: 1) the reactive energy is predominantly in only one of the electric and magnetic fields, and 2) the radiative portion of the stored energy can be directly calculated from the radiated power. In fact, our study of the coupled TE/TM cases reveals a situation where both of these assumptions are violated.
A Basic Observation Before describing the actual electromagnetic problem, let us consider an analogy with the one dimensional problem of a TEM transmission line. Fig. 1(a) shows a straight length of a transmission line with characteristic impedance Z0 connected between a generator and load of the same impedance and distance l. This system is matched over the entire frequency spectrum and is marked with perfect power transmission between the source and load. The total energy stored in the transmission line can be calculated as:
Wtot = PL ⋅ T = PL ⋅ l v
(2)
where PL is the total transmitted power (equal to the available power of the generator) , T
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is the traveling time through the line, and v is the group velocity. Wtot is equally divided between electric and magnetic energies. All of this energy is traveling and none of it is reactive. Now consider the situation depicted in Fig. 1(b) where the source and load are kept at the same distance, but the transmission line is replaced by a winding segment of length l’=N.l . Obviously, the line is still perfectly matched in all frequencies and PL remains unchanged. However, the total stored energy can be calculated as:
Wtot′ = PL ⋅ T ′ = PL ⋅ l ′ v = NWtot
(3)
Again, all of the stored energy is still contained in a traveling wave and there is no reactive component. However, as the total distance between source and load is l, the conventional way of thinking would lead to the conclusion that a transmitted energy is equal to Wtot of (2) and a reactive energy given by:
′ = Wtot′ − Wtot = PL ( N − 1) ⋅ l v Wreac
(4)
which is clearly incorrect. To relate this example to the case of antennas, let the radiation boundary at a far field sphere of radius Rf replaces the load resistance, and the near filed region the distance between the source and load. A single mode in this region can be compared to the straight transmission line segment, while multiple modes can assimilate the case of winding (multi-conductor) transmission line. If this analogy is valid, then it means that when more than one mode is present, the power flow in the near field region is not always outward and therefore the stored energy can increase without adding to the reactive near field.
Electro-Magnetic Dipole Antenna To clarify the above conjecture, we will examine the energy flow around a mixed mode dipole element. This element that is shown in Fig. 2, is composed of a short electric dipole Gand a small loop antenna and can be viewed as the superposition of an electric G dipole p and a perpendicular magnetic dipole m . The near field of the electric dipole is predominantly electric, while that of the magnetic dipole is predominantly magnetic. The magnitude of the magnetic dipole is chosen as η = 377 times that of the electric dipole, to enforce a proper impedance ratio between the electric and magnetic fields. G Fig. 3(a) shows the direction of the real part of the Poynting vector Pr calculated over the sphere ka = 0.1 for the idealized electro-magnetic dipole model. It is apparent that the power flows outward on the upper part of the sphere and inward on the lower part. In fact the outward flow region is lightly larger than the inward flow region, and G the ratio increased with radius. Power flow lines can be calculated by integrating the Pr vector, and is shown in Fig. 3(b). The power flow lines clearly show the rotational nature of near field, agreeing with the proposed transmission line analogy of Fig. 1(b). The radiation pattern of the electro-magnetic dipole antenna also has interesting properties. It is omnidirectional in azimuth, has a relatively wide beam in the positive z direction and a null on the backside. The radiation pattern as a function of θ is plotted in G Fig. 4. The total radiate power is equal to 20 ⋅ k 2 ⋅ | p |2 which is twice that of each individual dipole. The directivity is also found to be 3 (4.77 dB) that is twice that of a single dipole.
G The electric and magnetic dipoles are dual of each other and in order for | m | to be equal G to η | p | , the loop voltage and dipole current must be kept proportional. One way to
achieve this is by coupling the two antennas through a gyrator circuit. Gyrator is a nonreciprocal device and must be implemented using ferrites or active circuits. An example implementation using FET transistors of the gyrator based feed network is shown in Fig. 5. The frequency band where this circuit operates as an ideal gyrator device is limited by the cutoff frequency (fT) of the transistors. The general behavior of the electromagnetic dipole with the proposed feed network can be simulated using simple circuit models. Fig. 6 shows the simulated S11 for an electromagnetic dipole compose of a 1 cm-long electric dipole and a circular loop of diameter 1 cm. For these antennas, the radiation impedance and admittance can be calculated from the closed form formulas [5]. The reactive portion of the impedance/admittance can then be estimated by applying (1). Simulation shows that a very wideband matching is possible at the input port with fT = 60 GHz. Although the simulated bandwidth extends to beyond 10 GHz, the antenna dimensions become comparable to wavelength after a few GHz and therefore the antenna models used in this simulation become invalid.
Conclusion The classical limits on the radiation Q and bandwidth have been reexamined for the case electrically small antennas with mixed TE/TM modes. A comparison with the case of transmission lines suggests that some of the underlying assumptions of the classical model may not be applicable in the cases where both TE and TM modes exist. The near filed energy flow has been studied in the case of an electro-magnetic dipole pair, showing the rotational nature of the fields. A possible implementation based on gyrators has been considered as an example which shows a very wideband input matching.
References [1] H. A. Wheeler, "Fundamental Limitations of Small Antennas," Proceedings of the IRE, vol. 35, pp. 1479-1484, 1947. [2] L. J. Chu, "Phyisical Limitations of Omni-Directional Antennas," Journal of Applied Physics, vol. 19, pp. 1163-1175, 1948. [3] R. F. Harrington, "Effect Of Antenna Size On Gain, Bandwidth, and Efficiency," Journal of Research of National Bureau of Standards-D. radio Propagation, vol. 64D, pp. 1-12, 1960. [4] R. E. Collin and S. Rothschild, "Evaluation of Antenna Q," IEEE Transactions On Antennas and Propagation, vol. 12, pp. 23-27, 1964. [5] W. L. Stutzman and G. A. Thiele, Antenna Theory And Design, 1998.
Z0
Z0
l
l Z0 Z0
(b) (a) Fig. 1. Matched transmission line model: a single straight segment (a), a meandered segment (b)
Fig. 2. Electro-magnetic dipole antenna
Fig. 4. radiation pattern of the electro-magnetic dipole
(a)
(b) Fig. 3. Power flow in vicinity of the electro-magnetic dipole: The real part of the pointing vector over the sphere ka = 0.1 (a), and power flow lines (b)
Fig. 5. The topology of active feed network for coupling electric and magnetic dipoles
Fig. 6. Simulated return loss of the electromagnetic dipole with active feed network.
