Parent Category: 2021 HFE

*By Dr. John Howard and Steve Jalil*

An extension of Tresselt’s(1) 90-degree hybrid theory introduces an improvement of bandwidth compared to the present theory. Using four weighting methods of Gibb’s phenomena reduction, low coupling ripples are achieved for a very wide bandwidth.

Introduction: Tresselt has provided an equi-ripple, multi-octave 90-degree hybrid design capable of developing 90-degree hybrids exceeding 18:1 bandwidth with increased ripple. Application of four weighting correction methods to the Gibb’s phenomena extends the bandwidth of the 90-degree hybrids up to 120:1 with reduced ripple.

**Theoretical and Computational Results:**

The conventional frequency response of a 90° hybrid coupler can be given by equation 1 below.

Equation 1 exhibits the Gibbs phenomenon. Applying appropriate weighting functions in order to improve the Gibbs phenomenon results also in diminishing all other response ripples. One may seek to have all ripple values equal (2) but as long as the Gibbs ripple is below some desired value there is no reason to introduce further computational complications.

A 3dB 90-degree hybrid coupler is unrealizable due to the very high coupling impedance required at the center section of the 90-degree hybrid. Therefore, two -8.343dB symmetrical couplers are connected in tandem to achieve the -3dB response required.

Tresselt shows that the inverse Fourier Transform of equation 1 is related to the even mode impedance of the hybrid coupler by

Where p(x) is the reflection coefficient distribution along the length of the hybrid coupler and u is defined in reference 1. Using the method provided by Tresselt the impedance variation from the center to the end of the 90-degree hybrid coupler is given in figure 1 through 4.

Fig. 1 • Even Mode Impedances along the length of the -8.343 hybrid. Due to symmetry, only half the hybrid is shown. Weighting function A.

Fig. 2 • Even Mode Impedances along the length of the -8.343 hybrid. Due to symmetry, only half the hybrid is shown. Weighting function B.

Fig. 3 • Even Mode Impedances along the length of the -8.343 hybrid. Due to symmetry, only half the hybrid is shown. Weighting function C.

Fig. 4 • Even Mode Impedances along the length of the -8.343 hybrid. Due to symmetry, only half the hybrid is shown. Weighting function D.

Where u is related to the hybrid coupler bandwidth.

Table 1 shows the bandwidth and ripple level for each one of the four weighting functions applied to reduce the Gibbs phenomenon in this paper.

Weighting Function |
Bandwidth |
Gibbs’ Ripple Level (dB) |

None |
223:1 |
1.43 |

A |
121:1 |
.52 |

B |
83:1 |
.2 |

C |
63:1 |
.08 |

D |
51:1 |
.04 |

Conclusion: An extension of Tresselt’s 90-degree hybrid theory is presented. The reduction of coupling ripple vs bandwidth has been calculated. It is shown that with a prescribed Gibbs phenomenon level, bandwidth can be substantially increased.

**About the Authors**

Dr. John Howard and Steve Jalil, Electromagnetic Technologies Industries, Incorporated, 50 Intervale Road, Boonton, NJ 07005, USA).

**References**

1. Tresselt, C.P.: “The Design and Construction of Broadband, High Directivity, 90-Degree Couplers Using Nonuniform Line Techniques”, IEEE Trans. On Microwave Theory and Techniques, pp. 647–656, June 29, 1966.

2. E. G. Cristal and L. Young, “Theory and tables of optimum symmetrical TEM-mode coupled-transmission line directional couplers,” IEEE Trans. on Microwave Theory and Techniques, vol. MTT-13, pp. 545–558, September 1965.

3. Howard, J., and Lin, W.C.: ‘Comments on ‘Design and analysis of a decade bandwidth 180° hybrid coupler’’, IET Microw. Antennas Propag., 2014, Vol. 8, Iss. 12, pp. 999–1003.

### January 2023

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