Impedance fitting

When measuring components with a network analyzer, often is useful to fit a simple circuit model to the measured impedance.
The method described in [1] and [2] can be used to easily compute the equivalent circuit model parameters.
Below are some common equivalent circuits used, which are the same usually available on the Agilent Impedance Analyzers, and their recommended usage according to the Agilent User's Manual. The circuits impedance formula is also shown, both in its usual form and also in the form needed to apply the method described in the referenced articles.

Fitting procedure overview

Given a generic impedance function written in polynomial form Generic impedance function expression the following error function can be defined fitting error function where F_i are the measured impedance function values at the frequencies s_i.
By differentiating the error function eta with respect to all the unknown coefficients a_i and b_j and then setting all these derivatives to zero, a system of equations is obtained which allows to determine the unknown coefficients.
The matrix system has the canonical form matrix equation where X is a (p x p) matrix and a and y are vectors of length p, where p=n+m.

The matrix system can be written using sub-matrices as Block matrix where the sub-matrices and sub-vectors are defined as X_A X_B X_C A_A A_B Y_A Y_B

The matrix equation can then be solved by a inv(X) y In practice two problems arise: one is that X is a complex matrix, y is also a complex vector while we expect a to be a real vector. The other problem is that the condition number of X is often very large, which can cause large errors when computing the coefficients.

Circuit A

Circuit A

Typical usage : inductor with high core losses

Impedance : Impedance function to be used for fitting :


Circuit B

Circuit B

Typical usage : inductor or resistor

Impedance (already in a format suitable for fitting) :


Circuit C

Circuit C

Typical usage : high-value resistor

Impedance (already in a format suitable for fitting) :


Circuit D

Circuit D

Typical usage : capacitor

Impedance : Impedance function to be used for fitting :


Circuit E

Circuit E

Typical usage : resonator

Impedance : Impedance function to be used for fitting :


References:

[1] L. Vestling and J. Ankarcrona "A General Small-Signal Series Impedance Extraction Technique," IEEE Microwave and Wireless Components Letters, vol. 12, no. 7, pp. 249-251, July 2002.
[2] L. Vestling "Design and Modeling of High-Frequency LDMOS Transistors," Ph.D. Thesis, ISBN 91-554-5210-8, Uppsala University, Feb.2002.