New Results for Minimum Q, Maximum Gain, and Polarization Properties of Electrically Small Arbitrary Antennas Professor David M. Pozar Electrical and Computer Engineering University of Massachusetts Amherst Amherst, MA USA

Outline 1. Minimum Q of electrically small antennas a. Definition of radiation Q b. Early results c. Statements from the literature d. Two types of circular polarization e. Some derivations f. Summary of results 2. Maximum gain of an arbitrary (non-supergain) antenna a. Statements from the literature b. LP and CP c. Degrees of freedom and directionality d. Summary of some results e. Cross-polarization of optimal gain antennas

Definition of Q for a resonant circuit: Definition of radiation Q for an antenna: Definitions of Q W e and W m are the total stored electric and magnetic energies, P rad is the radiated power. W e and W m are the stored energies modified to exclude the (infinite) energy of the antenna radiation field.

Reasoning: When the electric and magnetic stored (non-radiative) energies are not equal, the antenna is not resonant. Presumably, an external matching element will be used to cancel the non-zero antenna reactance, and this will increase the stored energy of the smaller of (electric or magnetic) energy, making the total energy equal to twice that of the larger of the original (magnetic or electric) energy. Example: A small electric dipole has a large capacitive reactance, and its stored electric energy is larger than its stored magnetic energy. Matching can be done with a series inductor, which will increase the magnetic energy of the system to equal the stored electric energy. Note: other matching techniques are possible, but these may further increase stored energy. small dipole capacitive reactance small dipole inductively matched

Impact of this definition: It is this definition of radiation Q that leads to different values of Q for different electrically small antennas. Right now we know of only two possible values of minimum Q for small antennas: or The first value results for electric or magnetic dipoles (TM or TE modes, one or more), while the smaller value results when using a superposition of TM and TE modes.

Early Results Chu (1948): Used spherical modes to derive results for minimum Q and maximum gain (without supergain) for small arbitrary omnidirectional antennas (LP, CP-Type 2). Approximations limit accuracy for multi-mode cases. Harrington (1958, 1959): Used a similar analysis to derive results for the minimum Q and maximum gain for directional antennas (LP only). Collin and Rothschild (1964): Improved rigor of previous analyses by eliminating infinity caused by stored energy of radiating field. Fante (1969), McLean (1996): Simplified method for properly accounting for infinite energy in radiated field, and allowed calculations to be accurately extended to multiple TM and TE modes

From the recent literature (relative to the minimum Q of an electrically small arbitrary antenna)... “...the lowest achievable Q for a circularly polarized antenna (using TM and TE modes)... is one-half of the radiation Q associated with the TM mode acting alone” – from an article in APS May “... the radiation Q of a small circularly polarized antenna is about one half of the Q of a small linearly polarized antenna” – from an article in APS May 2005 “Another paper (the above reference) claims that orthogonal TE and TM modes produce a gain of 3 and a Q half that of either mode. Both are in error; the input power and the peak power density are both doubled, leaving the gain at 1.5 and the Q that of one mode.” – from a book published in “...(for) collocated, time quadrature electric dipoles... the calculated Q... (is) less than that of a single dipole” – from an article in Proc. IEEE Int. Conf. on Electromagnetic Compatibility, (later papers by these authors claim that the Q may approach zero for some configurations) “It is shown that the results obtained by (the above authors) are based on incorrect hypothesis and are wrong”, from an article in JEWA 1998.

Questions... a. Which small antenna configurations have a minimum Q of and which have a value half of this ? b. Do all small CP antennas have a lower value of Q than single mode LP antennas ? c. Does reduced Q depend on polarization ? d. Is it possible to have a small antenna with reduced Q (Q 0 /2), and directionality (G = 3) ? Short Answers: a. TM or TE; TM and TE; b. no; c. no; d. yes

z x JzMz Type 1 Type 2 Jx Jy x z Type 1 uses two orthogonal electric (or magnetic) dipoles with 90º phasing (TM or TE) Equivalent Elementary Dipole Sources for Two Types of Circular Polarization Type 2 uses collinear electric and magnetic dipoles (TM and TE)

Examples of Type 1 CP Antennas (TM or TE only) Crossed dipoles (or turnstile antennas) Microstrip patch with orthogonal feeds (probes, apertures, or lines) Crossed slots

Examples of Type 2 CP Antennas (TM and TE) Helix AntennaDipole – Loop Antenna Monopole-slot (Clavin element)

The same categories (Type 1 – either TM or TE modes, or Type 2 – TM and TE modes) can also be applied to linearly polarized antennas. Depending on orientation and phasing, it is possible to obtain linear, circular, or elliptical polarization from combinations of electric dipoles (TM modes), magnetic dipoles (TE modes), or electric and magnetic dipoles (TM and TE modes). Type 1 and Type 2 Also Apply to Linear Polarization

Spherical Vector Potentials for Arbitrary TM and TE Dipole Modes Jx Jy Jz Mx My Mz

Derivation of Q for Jz and Mz (Type 2: CP) Fields for TM 01 Mode (Jz):Fields for TE 01 Mode (Mz): Obtain CP by multiplying TE 01 fields by ±jη and adding to TM 01 fields.

Compute Non-Propagating Stored Energies Non-radiating electric and magnetic energies are equal in this case. Note: The radiated energy density is subtracted from the total energy density. It is incorrect to subtract the radiated field from the total field.

Compute Radiated Power and Q For small antennas, this is approximately equal to

Derivation of Q for Jx and Jy (Type 1: LP or CP) Fields for TM Modes (Jx and Jy with arbitrary amplitudes):

Compute Non-Propagating Stored Energies Non-radiating electric energy is greater than the magnetic energy in this case. Note that the results are independent of β.

Compute Radiated Power and Q For small antennas, this is approximately equal to - twice the previous result. This result is independent of β.

Summary of Q for TM and TE Modes (lowest order modes) ResonantTM + TE InductiveTE CapacitiveTM ReactanceSources Note that if we used, all three cases would have the same Q.

From the recent literature (relative to the maximum gain of an arbitrary finite non-supergain antenna)... “It is interesting to note that in the special case of all the spherical modes adding in phase in the far field, the antenna must be linearly polarized” – from an article in APS November “The maximum gain of 3 can be obtained only for linear polarization.” – from an article in APS May 2005 “Another paper claims that orthogonal TE and TM modes produce a gain of 3 and a Q half that of either mode (above reference). Both are in error; the input power and the peak power density are both doubled, leaving the gain at 1.5 and the Q that of one mode.” – from a book published in Maximum Gain of an Arbitrary Antenna

Questions... a. Does the maximum gain of an optimum arbitrary (non-supergain) antenna depend on polarization ? b. How do we obtain maximum gain from a CP antenna ? c. What can we say about the polarization properties of an optimal maximum gain arbitrary antenna ? Short Answers: a. no; b. use TM and TE sources; c. cross-pol is identically zero

Omni-DirectionalBi-DirectionalDirectional Three Types of Optimum Patterns for Maximum Gain Arbitrary Antennas

Maximum Gain for These Optimal Antennas Electrically small: Arbitrary size: Bi-DirectionalDirectionalOmni-Directional

Degrees of Freedom, Maximum Gain, and Polarization No degrees of freedom... A single dipole is linearly polarized, and has a maximum gain of 1.5 One degree of freedom... Two orthogonal TM (or TE) dipoles may be phased to produce LP or CP, but will produce a bi-directional pattern with a gain of 1.5 or... A TM dipole and a TE dipole can produce an LP directional (cardioid) pattern, with a gain of 3.0 Two degrees of freedom... Two orthogonal TM dipoles and two orthogonal TE dipoles can be phased to produce CP and a directional pattern, with a gain of 3.0

Conclusions regarding maximum gain and polarization: The above results for the maximum gain of arbitrary omnidirectional, or directional, antennas apply to any polarization. But one degree of freedom is required to generate CP (either Type 1 or Type 2), and one degree of freedom is required to obtain directionality (TM and TE modes). Achieving both requires two degrees of freedom (two TM sources and two TE sources). Limiting the number and type of sources limits the type of polarization and the maximum gain that may be obtained.

Maximum Gain and Minimum Q for Various Combinations of Elementary Sources SourcesPolarizationPatternMax GainMin Q J z (or J x, or J y )LP-1Omni1.5Q0Q0 M z (or M x, or M y )LP-1Omni1.5Q0Q0 J z ± M z LP-2Omni1.5Q 0 /2 J z ± M y LP-2Directional3Q 0 /2 J z ± j M z CP-2Omni1.5Q 0 /2 J z ± j M y LP-2Bi-directional1.5Q 0 /2 J x ± j J y CP-1Bi-directional1.5Q0Q0 M x ± j M y CP-1Bi-directional1.5Q0Q0 J x ± j J y and M x ± j M y CP-2Directional3Q 0 /2

Cross-Polarization of Maximum Gain Arbitrary Antennas. Co-pol and cross-pol far-fields of an optimum gain arbitrary CP antenna: The co-pol and cross-pol far-fields of an optimum gain arbitrary LP antenna are: Using Ludwig’s Third Definition for cross-pol: (cross-pol field is identically zero) Cross-pol field is identically zero, if Ludwig’s definition is used. This makes sense in terms of optimality, but also gives validation to Ludwig’s definition. Results from “Polarization of maximum gain antennas”, IEEE APS, July 2007.

Overall Conclusions It is possible to achieve reduced Q for electrically small antennas by employing both TM and TE modes, regardless of polarization. It is possible to achieve maximum gain for an arbitrary (non- supergain) antenna, with a directional pattern, for any polarization, by using at least two modes of each type, to give at least two degrees of freedom. An optimal arbitrary maximum gain antenna will have no cross- polarization in the far-field.

Transmit vs. Receive Antennas All of the above results apply only to transmitting antennas. We (Kwon and Pozar) have recently developed corresponding optimal results for arbitrary receiving antennas, and this work has been submitted for publication.

Thank You !