⫺48-dB and ⫺51-dB attenuation, respectively. The simulated and measured results are in good agreement. 5. CONCLUSION We have presented a design for a narrowband CPW BPF at 2.4 GHz. The folded open-stub-loaded structure reveals an improved slow-wave factor. In addition, this structure can be applied to the circuit load, not only to reduce the circuit size, but also to achieve a sharper TB and better Q of the filter. The filter design has proved to be an efficient one, and its attractive applications in MIC/MMIC are foreseeable. REFERENCES 1. D.F. Williams and S.E. Schwarz, Design and performance of coplanar waveguide bandpass filters, IEEE Trans Microwave Theory Tech MTT-31 (1983), 558 –566. 2. J.-T. Kuo and E. Shih, Wideband bandpass filter design with three-line microstrip structures, IEEE MTT-S Int Microwave Symp Dig 3 (2001), 1593–1596. 3. A. Gorur, C. Karpuz, and M. Alkan, Characteristics of periodically loaded CPW structures, IEEE Microwave Guided Wave Lett 8 (1998), 278 –280. 4. S.-S. Liao, H.-K. Chen, Y.-C. Chang, and K.-T. Li, New coplanarwaveguide folded ␭/4 open-stub structure, IEEE ICMMT Dig, Beijing, China (2002), 966 –969. 5. S.-S. Liao, P.-T. Sun, H.-K. Chen, and X.-Y. Liao, Compact-size coplanar waveguide bandpass filter, IEEE Microwave Wireless Compon Lett 13 (2003), 241–243. 6. N.I. Dib, G.E. Ponchak, and L.P.B. Katehi, A theoretical and experimental study of coplanar waveguide shunt stubs, IEEE Trans Microwave Theory Tech MTT-41 (1993), 38 – 44. structure must be accounted for. Many authors have theoretically and experimentally investigated the mutual impedance of antenna systems that consist of various skewed dipole [5, 6] configurations. For these arrangements, it is possible to achieve accurate results using simple expressions derived from a two-port network representation of the coupling between the antenna pairs [7]. However, this approach cannot be employed when the two dipoles are fed at the same feed point [8]. In this paper, we present a new equivalent circuit model that is used to calculate the mutual resistance and reactance between two collocated half-wave dipoles, which are orientated at angles between 0° and 90° and excited at a common feed point. The validity of the model is confirmed by the good agreement demonstrated between these results and the values computed when the dipoles are excited individually [7]. The scattering parameters used in both models have been measured using balanced dipoles that were designed to resonate at 960 MHz. At frequencies away from resonance, the experimental scattering parameters are modified by the loading effect of the quarter-wavelength folded baluns. However, this is shown to be insignificant over a bandwidth of almost 10% and, therefore, in this range it is possible to accurately determine the mutual impedance values using the new equivalentcircuit technique. EQUIVALENT CIRCUIT MODEL To determine the mutual impedance between two dipole antennas, consider the following equations for the equivalent two-port network representation: 冉冊 (1) 冉冊 (2) © 2004 Wiley Periodicals, Inc. Z 1 ⫽ Z 11 ⫹ Z 12 MUTUAL IMPEDANCE BETWEEN TWO DIPOLE ANTENNAS FED BY SINGLE SOURCE Z 2 ⫽ Z 22 ⫹ Z 12 M. Amin and R. Cahill High-Frequency Electronics Laboratories School of Electrical and Electronic Engineering Queen’s University Belfast Ashby Building, Stranmillis Road Belfast BT9 5AH, N. Ireland, UK ABSTRACT: We present a new circuit-model approach which can be used to compute the mutual impedance between two dipoles fed at the same feed point. The validity of the method is confirmed by comparison with mutual impedance values obtained when the dipoles are individually excited and orientated at angles between 0° and 90°. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 42: 187–189, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20247 Key words: antennas; mutual impedance; coplanar skewed dipoles; crossed dipoles INTRODUCTION The mutual coupling [1] between closely spaced radiating elements can affect the beam shape, radiated power, phase, and input VSWR of an antenna. For example, to accurately model an array of quadrifilar helix antennas [2, 3], the mutual coupling between the helical and crossed-dipole radial sections [4] of the radiating I1 , I2 where Z 1 and Z 2 are the driving-point impedances, Z 11 and Z 22 are the self-impedances of dipoles #1 and #2, respectively, and Z 12 is the mutual impedance. Therefore, the input impedance to antenna 1, when the second dipole is passively loaded with an impedance Z p , is given by the following expression [7]: Z 1 ⫽ Z 11 ⫺ Received 20 January 2004 I2 , I1 冉 2 Z 12 . Z p ⫺ Z 22 冊 (3) The mutual resistance and reactance can therefore be determined knowing Z 1 , Z 11 , Z 22 , and Z p . However, this approach cannot be used to determine the mutual impedance if the two antennas are fed by single source, because, as shown in Figure 1, it is not possible to isolate one of the dipoles from the load Z p . A crossed dipole fed by a single source can be represented by a shunt circuit with a voltage source at the feed terminal and currents I 1 and I 2 in the two arms. The current ratio in the equivalent parallel circuit can be expressed as the ratio of the driving-point impedances of the individual antennas, given by I1 Z2 ⫽ . I2 Z1 (4) By substituting the values of Z 1 and Z 2 from Eqs. (1) and (2), we obtain MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 3, August 5 2004 187 2 Z 12 ⫺ 2Z inZ 12 ⫹ Z in共Z 11 ⫹ Z 22兲 ⫺ Z 11Z 22 ⫽ 0. (8) The solution yields the mutual impedance between the dipoles when Z in , Z 11 , and Z 22 are known. This equivalent-circuit approach does not require the input port of one of the antennas to be passively terminated, and therefore Eq. (8) can be used to solve the coupled-dipole problem when these are driven in parallel by connecting the terminals together at the same feed point. RESULTS In order to validate the new circuit model, the mutual impedance values of a crossed dipole, with one arm rotated between 0° and 90° and fed by a single source, were compared to those obtained when the elements are individually excited. The measured scattering parameters were used to determine the coupling resistance and reactance in Eqs. (8) and (3), respectively. Figure 1 shows the dimensions of the parallel-fed crossed dipole, which was designed to resonate at 960 MHz at ␪ ⫽ 90°. This antenna was constructed using 2.1-mm-diameter semi-rigid cable, with an SMA connector at the input to feed the orthogonal conductors. The same arm dimensions were used to construct the two individual dipoles, and both structures employed a folded balun consisting of a ⬃␭/4 (75 mm) conductor, which was soldered to one of the dipole arms and the cable sheath, as shown in Figure 1. The input impedance of the dipole is modified by the loading effect of the balun, except at resonance, where it presents an open circuit. At other frequencies, the measurements are a parallel combination of the antenna and the balun impedance and, therefore, for the dipole Z a alone this, can be expressed as Figure 1 Parallel-fed crossed dipole I 1 共Z 22 ⫺ Z 12兲 ⫽ . I 2 共Z 11 ⫺ Z 12兲 (5) By substituting Eq. (5) into Eqs. (1) and (2), we then obtain expressions for the driving-point impedances of the two dipoles: Z1 ⫽ 2 Z 11Z 22 ⫺ Z 12 , Z 22 ⫺ Z 12 (6) Z2 ⫽ 2 Z 11Z 22 ⫺ Z 12 . Z 11 ⫺ Z 12 (7) Since these are fed in parallel, the input impedance Z in of the antenna is simply Z in ⫽ Z 1 Z 2 /(Z 1 ⫹ Z 2 ). By replacing Z 1 and Z 2 with Eqs. (6) and (7), respectively, the following quadratic equation is obtained: 188 Figure 2 Effect of balun on the input impedance of a half-wave dipole: (a) input impedance, real part; (b) input impedance, imaginary part. – – impedance (magnitude) of balun Z b ; — impedance with balun Z ab ; - - - impedance without balun Z a ;
impedance from NEC simulation MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 3, August 5 2004 extracting the experimental scattering parameters from the simple antenna shown in Figure 1, we therefore obtain accurate results when the new equivalent-circuit model is used to calculate the coupling levels. CONCLUSION We have shown that the two-port coupling-network approach cannot be used to obtain the mutual impedance of a crossed dipole which is fed by a single source. The limitation imposed on obtaining the coupling coefficient has been overcome by a new equivalent-circuit model which only requires the impedance values of the individual dipoles and the driving-point impedance of the parallel-fed antenna. The model was validated against the measured and simulated results and was shown to give accurate mutual impedance values when the rotation angle between the two dipoles was varied. This circuit model provides a useful tool for investigating the electromagnetic coupling effects that are inherent in closely coupled structures, such as parallel-fed quadrifilar helix antennas. ACKNOWLEDGMENT M. Amin is supported by a scholarship from the Ministry of Science & Technology, Pakistan. REFERENCES Figure 3 Mutual impedance between crossed dipoles vs. angle: (a) input impedance, real part; (b) input impedance, imaginary part. — NEC simulation; E dipoles fed individually; { crossed dipoles fed by a single source Za ⫽ Z abZ b , 共Z b ⫺ Z ab兲 (9) where Z ab and Z b are the frequency-dependent impedance values of the antenna with the balun connected and of the balun, respectively. Figure 2 shows the measured impedance values of Z ab and Z b over the frequency range 850 –1200 MHz. These values have been used in Eq. (9) to calculate the impedance of the radiating structure Z a , and are plotted in Figure 2 in order to demonstrate the close agreement with the results generated by the numerical electromagnetic code (NEC) [9]. The loading effect of the folded balun is insignificant (compare Z a with Z ab ) over the range 926 –1003 MHz, for which the VSWR is ⱕ 2. Therefore, the mutual-coupling values can be accurately determined over this bandwidth using Eq. (8) [as well as Eq. (3)]. The S 11 response for a single dipole was measured at 960 MHz to determine the value of Z 11 (and Z 22 ) in Eqs. (3) and (8). Similarly, Z 1 was measured at the input port with the dipole separated by 2 mm from an identical antenna, which was terminated in a load impedance Z p of 50⍀. The experimental data was obtained when the orientation angle between the arms was varied from 0° (parallel) to 90° (perpendicular). Figure 3 shows the mutual resistance and reactance versus the angle calculated from Eq. (3). The experimental values are in good agreement with the results obtained using the input impedance predictions from the NEC model. The driving point impedance Z in of the parallel-fed crossed dipole shown in Figure 1 was measured over the same range of angles and the mutual impedance was obtained from Eq. 8. Figure 3 presents the coupled resistance and reactance plots, which are in good agreement with both the NEC predictions and the experimental results for the individually excited dipoles. By 1. J.D. Kraus, Antennas, 2nd ed., McGraw Hill, New York, 1988. 2. B. Desplanches, J.C. Louvigne, A. Sharaiha, and C. Terret, Analysis of the mutual coupling in finite arrays of printed quadrifilar helical arrays, Microwave Opt Technol Lett 28 (2001), 34 – 40. 3. R. Cahill, I. Cartmell, G. Van Dooren, K. Clibbon, and C. Sillence, Performance of shaped beam quadrifilar antennas on the METOP spacecraft, Proc IEE Microwave Antennas Propagat 145 (1998), 19 –24. 4. V.F. Fusco, R. Cahill, and R.L. Li, Quadrifilar loop antenna, IEEE Trans Antennas Propagat AP-51 (2003), 115–120. 5. J.H. Richmond, Coupled linear antennas with skew orientation, IEEE Trans Antennas Propagat AP-18 (1970), 694 – 696. 6. J.H. Richmond and N.H. Geary, Mutual impedance of nonplanar-skew sinusoidal dipoles, IEEE Trans Antennas Propagat AP-23 (1975), 412– 414. 7. R.C. Hansen, Microwave scanning antennas, Academic Press Inc., New York, 1966. 8. B.Y. Toh, R. Cahill, and V.F. Fusco, Understanding and measuring circular polarisation, IEEE Trans Educ 46 (2003), 313–319. 9. NEC-Win Professional V1.1a, Nittany Scientific Inc., Riverton, 1997. © 2004 Wiley Periodicals, Inc. MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 3, August 5 2004 189