IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. zyxw 37, NO. zyxw 4, APRIL 1989 zyx 506 Communications Design of a Matching Network for an HF Antenna Using the Real Frequency Method OMAR M. RAMAHI, MEMBER, IEEE, AND zyxwvutsrqp RAJ MITTRA, FELLOW, IEEE Abstract-The real frequency method (RFM) is used to design a matching network for an electrically short loaded dipole. The RFM is demonstrated to be superior than other analytical and numerical techniques in the sense that it yields the maximum flat transducer power gain possible, and that is does not require any analytical modeling of the load impedance to be matched. For this reason, the RFM is found to be well suited for matching distributed systems such as antennas. I. INTRODUCTION Altshuler zyxwvutsrqpon [ 11 showed that the dipole antenna can be considered as an open-ended transmission line, and, thus, one could excite a traveling wave along the antenna by placing a resistance equal to the “characteristic resistance” of the antenna at a suitable location along the antenna. Based on these findings, Halpem and Mittra [2] employed the Numerical Electromagnetic Code (NEC) to obtain significant improvement in the matching bandwidth of a dipole antenna; however, they found that the lumped loading approach enables one to improve the bandwidth over only a portion of the HF band. Wheeler [3] and Chu zyxwvutsrqpo [4] have shown that the matching bandwidth of small antennas is subject to fundamental limitations that are related to the effective volume of the antenna. Consequently, for an antenna with size limitations, it is unlikely that any further improvement of significant proportions in the antenna bandwidth can be achieved over that obtained by Halpem and Mittra without employing a matching network at the feed. The most common methods for broad-band matching, which are applicable to the matching problem under consideration, are the gain- bandwidth theories of Fano [6] and Youla [7]. These theories require the approximation of the antenna impedance data by an analytic function realized as a driving point impedance. An analytic form of the transfer function (load-equalizer system) is assumed which is then adjusted to meet he gain-bandwidth restrictions. These restrictions take the form of a set of complex algebraic and integral equations which must be satisfied simultaneously. Because the solution of these equations presents great numerical difficulties, and because of the complexity involved in the approximation of the load by a driving- point function, the gain-bandwidth approaches of Fano and Youla are not easy to implement in practice. Furthermore, it is important to realize that the gain-bandwidth theories yield optimum matching only when the power gain characteristic is restricted to a prescribed type, e.g., Chebyshev, elliptic, etc. Therefore, they fail to guarantee an optimum design. In this work we use the real frequency method (RFM) developed by Carlin [5] to design a matching network for an electrically short Manuscript received July 20, 1987; revised March 18, 1988. The authors are with the Electromagnetics Communication Laboratory, IEEE Log Number 8825320. University of Illinois, Urbana, IL 61801. loaded dipole antenna. The RFM is numerical, and its most important feature is its capability to work with either experimental or numeri- cally simulated impedance data, and it does not call for any analytical description of the impedance function that is being matched. In the next section we present a summary of the RFM theoretical formulation. 11. REALFREQUENCY METHOD The broad-band matching problem considered in this study is the design of a lossless two-port network that couples a load to a resistive generator, as shown in Fig. 1, such that the power delivered to the load is maximized over the frequency band of interest. The problem is readily formulated by dealing with the transducer power gain (TPG), which is defined as the ratio of the power delivered to the load to the power available at the generator. The TPG is expressed as follows: For an ideal match, the error function 11 - TPGI is minimized over the frequency band of interest. It is important to emphasize that the resulting design will be optimum in the sense that the transfer function of the matching network approximates the maximum flat transducer power gain possible for the prescribed bandwidth and the prescribed load. For complete realizability of the equalizer, both the resistance R4(w) and reactance zyxwv X,(o) must be known. The relation between the real and imaginary parts was derived in [8], and it takes the form of the following integral equations: dA + R(w) zyx A Jo dh . I (3) Without loss of generality, the impedance function is assumed to be of the minimum-reactance type. Consequently, it is only necessary to determine the resistance function R,(w) that will provide optimum matching, since the reactance function X,(W) is directly derived from it. A key step in the real frequency method is the approximation of the resistance function R4(u) by a number of straight-line segments with break points at 0 < w1 < w2 < zyxw w3 < . . * < U,,. The choice of the break points is made such that they span the frequency band of interest. In order to express the reactance function in terms of the resistance function, the latter has to be specified over the entire frequency spectrum; this can be accomplished by setting R,(w) = 0 beyond the nth break point. The resistance function R4(u) is now expressed as a linear combination of the individual total excursions of each of the straight- line segments. We have h’ Rq(u)=rO+ ak(w)rk (4) k= 1 0018-926X/89/0400-0506$01 .00 zyxwvu 0 1989 IEEE