Applicability and power of complex nonlinear least squares for the analysis of impedance and admittance data

Abstract

An analysis is made of the ability of nonlinear complex least squares fitting to yield useful estimates of parameter values occurring in equivalent circuits. In addition, three-dimensional perspective plots are presented which show the full frequency response of either impedance or admittance data. Three different kinds of data are studied using these two methods. For a given set of four circuit elements (two resistors and two capacitors) artificially computer-generated impedance and admittance values are constructed for a Voigt circuit having two time constants. These exact results are then contaminated either by digit round-off or addition of zero mean, normally distributed random noise. A complex least squares analysis is then made on the contaminated data and the resulting estimates of the circuit parameters and their associated standard errors are compared to the exact starting values. For the case when one of the capacitors is vanishing small, the optimal range of measurement over the resulting semi-circle in the complex impdance plane is discussed in detail. For this case the accuracy of input data required to resolve the values of the two resistors when they are separated in size by two orders of magnitude is determined. The other extreme of two time constant which are not well separated is also studied. Again one obtains approximate estimates of both the range of frequencies and in the inherent accuracy of the data necessary to adequate resolve all four circuit parameters. The actual frequency dependent admittance of real ladder network of resistors and capacitors having three different time constant was measured on a Solartron type 1172 response analyzer. These data were then analyzed using the complex least squares procedure and the fitted circuit parameters were all in close agreement with their nominal experimental values. Finally, the admittance response of β-PbF 2 at 474 K was measured on the Solartron 1172. A complex least squares analysis of these data yields a good fit when an equivalent circuit employing a constant phase element is used. A three-dimensional perspective plot showing very clearly the agreement between fitted and measured values of the impedance is shown.