Impedance Spectroscopy
533
3.2 Complex Nonlinear Least Squares (CNLS) Fitting
Graphical analysis of impedance data is beneficial in obtaining only rough idea about
the parameters of the system. Fitting method such as CNLS applied to impedance data
is a good technique for characterization of material electrode system. Macdonald and
Gaber were the first to apply the CNLS method to IS. In this method, the measured
data is fitted to a known mathematical model which includes the response of an
equivalent circuit. CNLS technique not only gives values of parameters but also
yields estimated standard deviations. The values of standard deviation play a pivotal
role in deciding the crucial parameters that affect the model. The weighted sum of
squares of real and imaginary residuals (difference between value of data at given
frequency and the value deduced by model at the same frequency) is minimized by
a software program to produce CNLS fits [55, 56]. The reciprocal of the calculated
error variance for value of real data and its corresponding imaginary value are the
used weights. The subjective part of least squares fitting is weighting. Weighting
is of paramount importance since it has substantial effects on the output of fitting.
Variance models which are simplified are used to get values to be utilized in fitting.
Example of variance model is unity weighting which considers all weights to be equal
to unity. Another model is proportional weighting (PW). In PW, the error variance of
each data is equal to its square value. PW is helpful since an impedance data covers
a wide range. But this model gives biased estimates of parameter value.
CNLS technique fits all data simultaneously and complex models containing 10 or
even more unknown parameters can be fitted. This method of fitting data ensures that
the Kronig-Kramer (KK) equations are satisfied. So, carrying out lengthy integrations
to check the validity of KK is deemed unnecessary. However, CNLS fitting is not
advantageous in the absence of any proper mathematical model or unavailability of
equivalent circuit.
4 Applications of IS
4.1 Aqueous Electrochemistry
Oomen and Sluyters were the first to utilized the complex plane analysis in the
field of aqueous electrochemistry in 1960 [57]. Their report contains the explanation
regarding the capability of IS in determining the kinetic parameters of an aqueous
electrochemical system [57]. The authors studied the impedance response of an electrolyte at room temperature using a hanging mercury drop cell. The potential and
current equation is solved for an electrochemical reaction under controlled conditions. The rate of transfer of charge is kept very slow as compared to the diffusive
process. The contribution of diffusion to impedance is obtained by solving the diffusion equation by applying proper boundary conditions under small ac stimulus. In
all experiments performed during that decade similar to one conducted by Oomen
533
3.2 Complex Nonlinear Least Squares (CNLS) Fitting
Graphical analysis of impedance data is beneficial in obtaining only rough idea about
the parameters of the system. Fitting method such as CNLS applied to impedance data
is a good technique for characterization of material electrode system. Macdonald and
Gaber were the first to apply the CNLS method to IS. In this method, the measured
data is fitted to a known mathematical model which includes the response of an
equivalent circuit. CNLS technique not only gives values of parameters but also
yields estimated standard deviations. The values of standard deviation play a pivotal
role in deciding the crucial parameters that affect the model. The weighted sum of
squares of real and imaginary residuals (difference between value of data at given
frequency and the value deduced by model at the same frequency) is minimized by
a software program to produce CNLS fits [55, 56]. The reciprocal of the calculated
error variance for value of real data and its corresponding imaginary value are the
used weights. The subjective part of least squares fitting is weighting. Weighting
is of paramount importance since it has substantial effects on the output of fitting.
Variance models which are simplified are used to get values to be utilized in fitting.
Example of variance model is unity weighting which considers all weights to be equal
to unity. Another model is proportional weighting (PW). In PW, the error variance of
each data is equal to its square value. PW is helpful since an impedance data covers
a wide range. But this model gives biased estimates of parameter value.
CNLS technique fits all data simultaneously and complex models containing 10 or
even more unknown parameters can be fitted. This method of fitting data ensures that
the Kronig-Kramer (KK) equations are satisfied. So, carrying out lengthy integrations
to check the validity of KK is deemed unnecessary. However, CNLS fitting is not
advantageous in the absence of any proper mathematical model or unavailability of
equivalent circuit.
4 Applications of IS
4.1 Aqueous Electrochemistry
Oomen and Sluyters were the first to utilized the complex plane analysis in the
field of aqueous electrochemistry in 1960 [57]. Their report contains the explanation
regarding the capability of IS in determining the kinetic parameters of an aqueous
electrochemical system [57]. The authors studied the impedance response of an electrolyte at room temperature using a hanging mercury drop cell. The potential and
current equation is solved for an electrochemical reaction under controlled conditions. The rate of transfer of charge is kept very slow as compared to the diffusive
process. The contribution of diffusion to impedance is obtained by solving the diffusion equation by applying proper boundary conditions under small ac stimulus. In
all experiments performed during that decade similar to one conducted by Oomen