Feature 2: Dispersion Function


Dispersion is defined as the distortion due to the different group velocity versus frequency, especially for non-TEM propagation structure such as waveguide and microstripe or wideband application. The phrase, “Filter Dispersion” defined in SynMatrix incorporates both the dispersive effect as well as unwanted spurious which is close to the pass band. In microwave filter design, this function can simulate the asymmetrical “skirt effect” which may be caused by frequency dispersive effect or by unwanted spurious. This phenomenon widely exists in wide band filter design and waveguide filter design.

The dispersive function in SynMatrix can be categorized by two states:

  • Coupling schemes. Frequency dispersive impacts on Electrical coupling & Magnetic coupling
  • Spurious. Excited unwanted modes trigger spurious causing the asymmetrical slopes

The dispersive effect is the primary cause of asymmetrical slopes of the filter passband [1, 2]. In filter design, the coupling schemes can be classified as electrical coupling and magnetic coupling. As coupling can be impacted by the reactance slope due to the frequency dispersion, the passband may behave as asymmetrical slopes. SynMatrix offers positive and negative values to simulate the “asymmetrical” effect. It can assist computer-aided tuning especially for high dispersive structures.

For the BPF, one defintion of the coupling coefficient is defined as,

“ (valid for narrow bandwidth)

Where, “

Tyurnev, V. V., has studied the impact of the frequency dispersion on the coupling coefficient in tuned wideband filters with a fractional BW of 40% [3] and concludes that such formula is not valid for bandpass filter with wideband application due to the frequency dispersive effect [4] - such obtained coefficients from the above equation need to be optimized during the tuning process.


  • f0=1GHz, BW=0.05GHz, RL=25dB
  • Two designs with different coupling schemes will prove the different dispersive effect
    • Case1: Electrical Coupling
    • Case2: Magnetic Coupling

  • Dispersive Function results: Electrical Coupling

    Even for the narrow band application, it can still be seen that the frequency dispersive effect brings impactions to the overall performance. In general, positive value in dispersion function will simulate the filter structure which is dominated by electrical coupling(Capacitive coupling).

  • Dispersive Function results: Magnetic Coupling

    The positive value in dispersion function will simulate the filter structure with magnetic coupling (Inductive coupling) dominated. The dispersive value is dependent on the design parameter , such as fractional bandwidth, filter structure (waveguide or re-entrant), coupling realizations, etc..


Dispersion Value Coupling Scheme Asymmetrical Effec
Positive Value Electrical High side raising up;
lower side falling down
Negative Value Magnetic High side falling up;
lower side raising up

The above table summarizes the dispersion function in the practical application. The user may figure out their design requests as well as coupling schemes before applying this function. Please note that once the dispersion function has been applied, the overall performance needs to be fine-tuned in order to comply with design requests. The aim of this function is designed for assisting computer aided tuning work.

SynMatrix suggests that the user generate the coupling matrix in a normal way without applying dispersion function. More details will be addressed in “Computer Aided Tuning”.


  1. Belyaev, B. A. and V. V. Tyurnev, “Frequency-dependent coupling coefficients of microstrip resonators," Elektronnaya tekhnika. Ser. SVCh-Tekhnika, No. 4, 24-27, 1992 .
  2. Belyaev, B. A. and V. V. Tyurnev, “ Investigation of Frequency De-pendences of Coupling Coe±cients between Microstrip Resonators,” Krasnoyarsk, Kirensky Institute of Physics, Preprint No. 695F, 1991.
  3. Tyurnev, V. V., “Influence of the frequency dispersion of resonators' coupling coefficients on the accuracy of direct-synthesis formulas for microwave filters," J. of Communications Technology and Electronics, Vol. 54, No. 3, 298-301, 2009.
  4. Tyurnev, V. V., “COUPLING COEFFICIENTS OF RESONATORS IN MICROWAVE FILTER THEORY,” Progress In Electromagnetics Research B, Vol. 21, 47-67, 2010.