A novel formula for the quality factor calculation for the multiphase microwave dielectric ceramic mixtures

Determining the Quality Factor of Dielectric Ceramic Mixtures with Dielectric Constants in the Microwave Frequency Range

“…The complex dielectric constant of a ceramic is given by the classical harmonic oscillator model 1,5,6,25,26,29 where ε ∞ is a constant permittivity of the material, n represents the number of Lorentz oscillators, ω is the working frequency at GHz, i 2  = −1; while ω pj , ω oj , and γ j are the plasma frequency, natural resonant frequency, and damping factor of the j -th Lorentz oscillator, respectively. If one separates the real ( ε r ) and imaginary ( ε i ) parts of equation (3) as 25,26,29 then the real and imaginary parts of the complex dielectric constant are given by 29 The dielectric loss (tan δ ) of a ceramic is defined as 1,5,6,29 Since ω oj / ω is ~10 3 , equation (7) can be rewritten as 1,5,6,25,26,29 …”

“…If one separates the real ( ε r ) and imaginary ( ε i ) parts of equation (3) as 25,26,29 then the real and imaginary parts of the complex dielectric constant are given by 29 The dielectric loss (tan δ ) of a ceramic is defined as 1,5,6,29 Since ω oj / ω is ~10 3 , equation (7) can be rewritten as 1,5,6,25,26,29 …”

“…In this case, the real part of the dielectric constant can be predicted by the revised Maxwell-Wagner equation 7,8,25,26 where l is the l -th ceramic, V l is its volume molar ratio, ε rl is its dielectric constant, and m means the number of ceramics mixed. A k value can be obtained through fitting the experimental data with equation (9).…”

“…Otherwise, the mixed material will be a single phase solid solution 4,5,17–24 . In previous work, the solid solution was treated as a two-phase composite 16,17,20–24 , where the dielectric constant can be derived from the Maxwell-Wagner equation 7–9,25,26 where V represents the volume molar ratio of the second material, ε r is the dielectric constant of the composite, ε r 1 and ε r 2 are the dielectric constants of the two ceramics, respectively. By using a parallel or serial capacitor model, the parameter k in equation (1) corresponds to +1 and −1, individually 7 .…”

“…From the definition of Q (it will be given in equation (8) in the next section) at the microwave frequency range, the quality factor of the mixture is related to both the dielectric constant and the resonant frequency 1,5,6,25,26,29 . Equation (2) only considers the quality factor of the two initial materials and is a simple superposition relationship.…”