Kramers–Kronig Test Applied to Impedance Measurements of Electrical Circuits

Introduction 

In this post, the Kramers-Kronig relations are used to verify the results of an impedance measurement experiment performed by the Zurich Instruments MFIA Impedance Analyzer or MFLI Lock-in Amplifier equipped with the MF-IA option. The Kramers-Kronig relations state that the real part of the impedance can be obtained from its imaginary part and vice versa, if either of them is available for all frequencies. In some situations, measuring one of the real or imaginary components of impedance is more straightforward and so the Kramers-Kronig relations can be used to obtain the other one. The sample tested here is a compound RC circuit with the series resistor \(R_s=2.2\ \text{k}\Omega\), the parallel resistor \(R_p=6.8\ \text{k}\Omega\), and the capacitor \(C=1\ \mu\text{F}\) as shown in Fig. 1.

Fig. 1. A compound RC circuit including a capacitor, a parallel resistor and a series resistor.

Theoretical Background  

Considering the impedance of a circuit as a voltage response of a system to its current excitation, i.e. \(Z=\frac{V}{I}=R+jX\), we can use systems theory to obtain a useful relation between the real and imaginary parts of impedance, i.e. the resistance \(R\) and the reactance \(X\). The fact that a physical system is causal implies that its system function is analytical which means that its real part can be obtained from its imaginary part and vice versa. The bridge between the real and imaginary parts of an analytical function which decays with \(1/\omega\) or faster as \(\omega\rightarrow\infty\), is the Hilbert transform [1]. Moreover, the impedance measurement can be confined to only positive frequencies because the response of a physical system is real-valued which results in a Hermitian symmetry of the system function. Combining the Hilbert transform (causality) and Hermitian symmetry (real-valued response), we obtain the following Kramers-Kronig relations for the real and imaginary parts of impedance [2].

\[X = \frac{2\omega}{\pi}\int_0^\infty{\frac{R(u)}{u^2-\omega^2}du}\\ R = R_\infty + \frac{2}{\pi}\int_0^\infty{\frac{uX(u)}{\omega^2-u^2}du}\]

where \(R_\infty\) is the resistance of the circuit at infinite frequency, e.g. the series resistance in Fig. 1 which is \(2.2\ \text{k}\Omega\). It is worth mentioning that there is a singularity at \(u=\omega\) in the above integrals; therefore, the Cauchy principal value of the integrals must be obtained to calculate the above expressions [3]. Generally speaking, the Kramers-Kronig relations are used to analyze the dielectric constant or susceptibility of materials [4]; however in certain conditions, it is possible to use it for impedance of electrical circuits.

Experimental Verification 

Using the Zurich Instruments impedance analyzer, we measure the impedance of the circuit in Fig. 1 for a frequency range from \(10\ \text{mHz}\) to \(80\ \text{kHz}\). Fig. 2 depicts the measured values of the resistance and reactance of the sample.

Fig. 2. Impedance spectroscopy; the measured real and imaginary components of the circuit impedance versus frequency.

According to the Kramers-Kronig relations, we can obtain the resistive part of impedance from its reactive part and vice versa. Since the relations include a singularity point at the desired frequency \(u=\omega\) and thus, the Cauchy principal value of the integrals must be calculated, one should be careful in how to extract one component from the other. In the example of Fig. 1, calculating the real part of impedance from its imaginary part leads to a more stable and less noisy results while the imaginary part obtained from the real part will be noisier. The reason is that the imaginary part is close to zero at some frequencies while the corresponding real part is high and a small perturbation in the real part results in a large fluctuation in the imaginary part through calculating the integral.

Fig. 3. Resistance of the hybrid RC circuit obtained by measurement (solid red curve) and calculated by the Kramers-Kronig relation (dashed blue curve).

Fig. 3 compares the resistance of the hybrid RC circuit obtained by two methods: 1) direct measurement by the MFIA Impedance Analyzer (solid red curve) and 2) calculation from the measured reactance using the Kramers-Kronig relation (dashed blue curve). Therefore, it is fair to say that the figure shows both direct and indirect measurements. All the Kramers-Kronig calculations are done by MATLAB. The programming code for calculating the Kramers-Kronig expressions and plotting the impedance is available here as well as the measured values in this CSV file. As it is evident from Fig. 3, the results obtained from the Kramers-Kronig equations are in good agreement with the measurement. Fig. 4 shows the reactance of the circuit derived by the same two methods. As it is clear from the figure, there are some discrepancies between the direct measurement and the calculation of reactance. Especially, in very low frequencies the results obtained from the Kramers-Kronig are noisy because the corresponding resistance is large so that a small perturbation causes a large fluctuation in reactance.

Fig. 4. Reactance of the hybrid RC circuit obtained by measurement (solid red curve) and calculated by the Kramers-Kronig relation (dashed blue curve).

Conclusion

The Kramers-Kronig relations allow us to obtain the real (imaginary) part of impedance from its imaginary (real) part. They can be used to verify the results of an impedance measurement or to calculate one component from the other in case it is more straightforward to perform the measurement only for one component. There are some serious restrictions in applying Kramers-Kronig relations. First of all, one needs to have real or imaginary part of impedance for the whole frequency range, theoretically from DC to infinity, in order to obtain the other component. Another difficulty is the calculation of Cauchy principal value due to the singularities in the integral. This makes the results highly sensitive to any kind of noise and fluctuation. The ability of MFIA Impedance Analyzer to perform measurements at very low frequencies (mHz range) with high precision (low noise) overcomes the challenges in applying Kramers-Kronig testing to electrical circuits.

References

  1. Wikipedia: “Hilbert transform.”
  2. Wikipedia: “Kramers–Kronig relations.”
  3. Wikipedia: “Cauchy principal value.”
  4. M. Alem, “Impact of modulation instability on distributed optical fiber sensors,” PhD Thesis, EPFL, Lausanne, 2016. DOI: 10.5075/epfl-thesis-7007