Dielectric Characteristics Analysis of Aluminum Electrolytic Capacitors Based on Linear Response Function
Authors: Mukaiyama Daisaku , Yamamoto Masayoshi 
Published: February 2020

Introduction

The electrical characteristics of aluminum electrolytic capacitors are usually measured in frequency domain. The measured data of capacitance and dissipation or equivalent series resistance (ESR) has been treated individually for each frequency, and the “LCR” model has been developed by utilizing the measurement data in frequency domain (1, 2). Therefore, these models don’t show any relation between capacitance and dissipation which should follow the Kramers-Kronig relation. In this paper, we discuss the dielectric characteristics of aluminum electrolytic capacitors based on linear response theory. Furthermore, a useful formula had been derived to analyze the dielectric properties of aluminum oxide layer in aluminum electrolytic capacitors.

Experiments

The specifications of the capacitors under test are as follow: “400V / 280μF / D25mm X L40mm”. These capacitors were fabricated with the following raw materials. Anode foil: 120HB15S-580V (HEC), Cathode foil: 50CK (JCC), Spacer Paper: PE4-30 (NKK), Electrolyte: resistivity 600Ωcm at 30oC (an organic acid ammonium salt in a solvent mainly containing ethylene glycol). The impedance parameters, |Z|, Capacitance, ESR, tanδ (dissipation factor ) were measured for 3 pieces of the capacitors under tests, in a frequency range from 20Hz to 1MHz, with respect to temperatures in the range from -40oC to 105oC.

Results and Discussion

The electric current of the dielectric absorption in aluminum electrolytic capacitors has a time dependency of ∝ t-n (0 < n ≈< 1) (3). That means the linear response function of the dielectric of aluminum electrolytic capacitors is also dependent on t-n. This has been proved through mathematical derivation, and the equations will be presented in the final manuscript. In this discussion, it is considered that the aftereffect (εr0 >> 1) should be much larger than the instantaneous response (εr∞), the mathematical term (ε0E) shown in eq. [1] can be neglected, therefore, the electric flux density is equal to the dielectric polarization and the eq. [2], [3], [4] and [5] can be obtained. Then, the expressions of the real and imaginary parts of relative permittivity εr, and the tanδ were derived as shown in eq. [7], [8] and [9] based on the linear response function φ0(t) = A / tn (0 < n ≈< 1). Furthermore, to consider the linear response function of aluminum electrolytic capacitors, the parameter τ∞ was proposed. Where the parameter τ∞ is defined as time interval until the dielectric relaxation is saturated. The parameter of the linear response function is expressed by A = (1-n) / τ∞
1-n as shown in eq. [5]. These formulae based on the linear response function are consistent with the impedance measurement data. (see Fig.1 and Fig.2). These figures can be obtained by calculating only two key parameters n(θ) and τ∞(θ). Where, θ is temperature in degrees Celsius. That means these two parameters can explain the properties of εr ′ and εr″ of the dielectric of aluminum oxide layer in aluminum electrolytic capacitors against frequency and temperature while fulfilling the Kramers – Kroning relation. Furthermore, the formula of this linear response function expressed by n(θ) and τ∞(θ) can explain the dielectric absorption phenomenon such as the recovering voltage of aluminum electrolytic capacitors. The voltage recovery speed of aluminum electrolytic capacitors is dependent on the temperature and it follows “Double relaxation time for every 10degC reduction” which is expressed by the eq. [14]. If the relaxation time of the voltage recovery phenomenon is proportional to the relaxation time τ∞ in eq. [4] and [5], it leads the capacitance temperature coefficient : 6.91x10-4 per 1oC as shown in the calculation results of eq. [15]. And this should refer to the result of 7.529x10-4 per 1oC in this paper result eq. [10].

Conclusion

The linear response function contains the whole information of the dielectric, and it can explain the temperature and frequency characteristics of complex permittivity of aluminum oxide of aluminum electrolytic capacitors by only two key parameter n(θ) and τ∞(θ). They satisfy the relation of capacitance and dissipation which is expected by the Kramers – Kroning relation. Furthermore, this theory can be applicable for the relaxation phenomenon such as the recovery voltage of aluminum electrolytic capacitors.

Reference:

R. H. Broadbent, Electrochem. Technol., 6, 163 (1963)

S. G. Parler, IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 929–935, 2003.

J. M. Albella et al., J. Appl. Electrochem. 14 (1984) 9-14

D. A. McLean, J. Electrochem. Soc. 108, 48 (1961)

Figure 1