Dear Dr. Makhdoon:

There is a small number of answer on the RG system, in which I had participate, some of them linked below..

https://www.researchgate.net/post/How_can_I_analyze_grain_and_grain_boundary_resistance_from_impedance_data

https://www.researchgate.net/post/why_there_is_a_term_complex_number_in_the_impedance

https://www.researchgate.net/post/How_can_I_analyze_the_impedance_spectrum_of_ceramic

https://www.researchgate.net/post/How_do_I_interpret_my_impedance_spectroscopy_results

https://www.researchgate.net/post/Is_a_new_type_of_dielectric_spectroscopy_for_liquids_possible2

https://www.researchgate.net/post/When_does_peak_position_of_e_coincide_with_the_loss_tangent_tand_peak

https://www.researchgate.net/post/I_m_looking_for_dielectric_properties_of_frozen_sugar_solutions_Can_anyone_give_me_some_references

https://www.researchgate.net/post/What_is_the_reason_behind_high_loss_modulus_in_a_dielectric_measurement

https://www.researchgate.net/post/What_is_the_purpose_of_using_frequency_window_1_kHz_to_1_MHz_while_measuring_the_dielectric_constant_of_any_material2

https://www.researchgate.net/post/Why_does_dielectric_constant_increases_or_decrease_with_temperature

https://www.researchgate.net/post/How_can_I_analyze_grain_and_grain_boundary_resistance_from_impedance_data

https://www.researchgate.net/post/What_is_the_physical_interpretation_of_ac_and_dc_conductivity_in_relation_to_the_dielectric_constant

In specific, with relation to question, it is possible to reach some insight if  further attention is payed to the origin of function-of-Impedance selected to analysis. One or other question has been ascribed to the representation of function Impedance, which can be transformed in another function as: Electric Modulus M, Admmitance Y and complex permittivity E, all function are complex numbers. Below in attachment  is the Transformation Table adapted of Macdonald, 1987.

 Table Impedance Function Transformation

A priori the transformation allows further “vision of properties since, some transformation involves the reciprocal of a previous function that change the magnitude of values. Furthermore, from basic mathematics, imaginary part of previous function is transformed in real part of new function giving new opportunity windows to analyses the electrical and dielectrical phenomena.

Seems that the parameter called of critical frequency is in fact the most frequent relaxation frequency that in ideal Z’ versus Z”” plot is defined at highest value of Z” (imaginary part of Impedance function). In a Impedance diagram the frequency of measurement is lowest at right side of diagram. In the Admittance plot showed, the line showing positive slope present highest frequencies values, 10^3 or 10^4 up to 10^6 Hz. In this way, long range conductivity (dc conductivity) based on shifting of carrier at long distances is view in lowest domain frequency, left side of plot.

Best regards

Marcos Nobre

Dr Makhdoom - a semicircular (Nyquist) plot in the complex plane is the classic frequency response of a single pole low pass filter. The critical frequency Fc on the Nyquist plot occurs at the peak of the semicircle, corresponding to a phase lag of 45 degrees. I don't see why you feel that it should be lower down. Check the references Dr Nobre has given, I'm sure you will find a more thorough explanation there! Best wishes with your research.