Method of measuring impedance using Gaussian white noise excitation

Abstract

A method of impedance measurement of a device under test (DUT) is disclosed based on a random excitation signal, the method comprising the steps of generating the random excitation signal, applying the generated random excitation signal to the DUT through two points of a data acquisition board (DAQ) and re-structuring the converted random excitation signal through a plurality of iterative calibration loops, wherein spectral phase of the random excitation signal is derived from a discrete uniform distribution and its time domain amplitude is controllable. The random excitation signal is a structured Gaussian White Noise (GWN) signal or sequence, which is generated based on the user-defined input parameters such as white noise power level, frequency range between the minimum and maximum frequencies (F.sub.min and F.sub.max), and frequency step (F.sub.step).

Claims

1. A method of impedance measurement of a device under test (DUT) based on a random excitation signal, the method comprising the steps of: generating the random excitation signal; re-structuring the random excitation signal through a plurality of iterative calibration loops, wherein a spectral phase of the random excitation signal is derived from a discrete uniform distribution, leading to a controllable time domain amplitude; applying the re-structured random excitation signal to the DUT through two points of a data acquisition board (DAQ) for impedance measurement.

2. The method of claim 1, wherein the random excitation signal is a structured Gaussian White Noise (GWN) signal or sequence.

3. The method of claim 2, wherein the GWN signal or sequence is generated based on user-defined input parameters comprising at least one of: white noise power level, frequency range between the minimum and maximum frequencies (F.sub.min and F.sub.max), and frequency step (F.sub.step).

4. The method of claim 3, wherein the GWN signal or sequence comprises N selected sampling frequency points such that N=(F.sub.max−F.sub.min)/F.sub.step.

5. The method of claim 2, wherein the GWN signal has a flat power spectral magnitude over a wide bandwidth while having a random phase.

6. The method of claim 2, wherein the re-structured GWN signal is iteratively calibrated.

7. The method of claim 1, wherein the discrete uniform distribution is obtained from a physically-true random number generator.

8. The method of claim 1, wherein the random excitation signal is converted to a voltage or current value using the DAQ, for potentio-static or galvano-static measurements respectively.

9. The method of claim 1, wherein the generated random excitation signal is in time domain.

10. The method of claim 9, wherein the random excitation signal is converted from the time domain to frequency domain prior to re-structuring the converted random excitation signal.

11. The method of claim 10, wherein the random excitation signal is converted from the time domain to frequency domain using Fast Fourier Transform (FFT).

12. The method of claim 1, wherein the plurality of iterative calibration loops maintains a random phase of the random excitation signal and re-adjusts a magnitude of the random excitation signal, thereby ensuring a uniform flat power spectrum over a test frequency range set by a user.

13. The method of claim 12, wherein the test frequency range is 2 mHz to 200 kHz.

14. The method of claim 1, wherein the plurality of iterative calibration loops results in increasing amplitude of weakened frequencies due to software and hardware signal processing.

15. The method of claim 1, wherein the DUT is a precision standard cell.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The subject matter that is regarded as the invention is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The foregoing and other aspects, features, and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings in which:

(2) FIG. 1A is a flowchart depicting the series of steps followed in order to generate the fully conditioned and calibrated signal X.sub.c(t) of flat spectral magnitude and random phase starting with a GWN sequence X.sub.i(t) which is generated based on the user-defined input parameters: Power, F.sub.min, F.sub.max and F.sub.step.

(3) FIG. 1B depicts typical time domain and frequency domain representations of samples of the voltage signals V.sub.i(t), V.sub.w(t), V.sub.u(t) and V.sub.s(t), which correspond respectively to the signals Xi(t), X.sub.w(t), X.sub.u(t) and X.sub.s(t) in FIG. 1A.

(4) FIG. 1C is a flowchart depicting the steps followed for impedance measurement using the re-structured random signal excitation V.sub.c(t), which corresponds to the signal X.sub.c(t) in FIG. 1A.

(5) FIG. 2A shows a standard electrical calibration cell on which the measurements were carried out (DUT). The voltage V.sub.m(t) is measured between nodes A and B, and the current I.sub.m(t) is calculated as the voltage measured between nodes B and C divided by the 100-Ohm resistance.

(6) FIG. 2B depicts time domain and frequency domain components of the initial voltage signal V.sub.i(t) compared to the measured voltage signal V.sub.m(t) within different frequency ranges between F.sub.min and F.sub.max and with different frequency steps, F.sub.step. The entire frequency range between F.sub.min and F.sub.max has been segmented into four intervals in order to help setting a limit on the maximum permissible voltage amplitude of V.sub.m(t).

(7) FIG. 2C depicts the measured voltage and current corresponding to the same frequency ranges as in FIG. 2B.

(8) FIG. 2D shows the complete signals V.sub.i and V.sub.m both in time and frequency domains, previously depicted in FIG. 2B over four different intervals.

(9) FIG. 2E shows the complete measured voltage and current signals both in time and frequency domains, as previously depicted in FIG. 2C.

(10) FIGS. 3A-C is a comparison of the measured impedance (magnitude, phase, and Nyquist plot) using the proposed method and that measured using a Biologic VSP-300 electrochemical station in the form of 3A—magnitude of impedance vs. frequency, 3B—impedance phase angle vs frequency, and 3C—Nyquist plot of real vs. imaginary parts of impedance. FIG. 3D represents the relative deviation of measured impedance magnitudes between the proposed method and that measured using a Biologic VSP-300 electrochemical station.

DETAILED DESCRIPTION OF THE INVENTION

(11) The aspects of a method for fast impedance measurement, according to the present invention will be described in conjunction with FIGS. 1-3. In the Detailed Description, reference is made to the accompanying figures, which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. It is to be understood that other embodiments may be utilized and logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims.

(12) In accordance with the present invention, a structured Gaussian White Noise (GWN) signal to be used for impedance measurement is proposed. By definition, a time series (w.sub.t: t=1, 2, . . . , n) in which the elements w.sub.i are independent and identically distributed following a standard normal distribution with zero mean and finite variance σ.sup.2 (i.e. (w.sub.t˜N(0, σ.sup.2)), and with no serial correlation (i.e. Cor (w.sub.i, w.sub.j)≠0, ∀ i≠j) is known as a GWN. In the frequency domain, the GWN has a constant power spectral density over a limited bandwidth. Therefore, post-processing, conditioning and calibration procedures are applied on the initial GWN signal in order to engineer a final multi-value (i.e. not binary) signal of fixed power, flat magnitude, and random phase over a wide bandwidth. Experimental verification of the technique is conducted using a precise standard impedance cell over the frequency range 2 mHz-200 kHz, and its accuracy is compared to that of a research-grade Bio-logic VSP-300 electrochemical station. The results show a maximum of 2.2% deviation between the two measurements with a total speedup of 6.8 times using the proposed method in accordance with the present invention.

(13) Considering the algorithm used in accordance with the present invention, and unlike the multi-sine based random phase signal, a GWN sequence is generated to cover the target frequency range with a predefined frequency step. The amplitude of the signal is then adaptively increased to strengthen its power at selected frequency points. Subsequently, the proposed signal is calibrated to mitigate software and hardware sampling errors. FIG. 1A shows a sequential flowchart of the series of steps followed to engineer and produce the proposed signal X.sub.c(t). A GWN signal is first generated using Matlab's wgn function with the user-defined parameters: white noise power level, frequency range between the minimum and maximum frequencies, F.sub.min and F.sub.max, and the frequency step, F.sub.step. Thus, the signal contains N points such that N=(F.sub.max−F.sub.min)/F.sub.step. Next, this GWN signal X.sub.i(t) undergoes a signal conditioning procedure. Windowing (Blackman window) is applied onto the signal in the time domain (giving X.sub.w(t)) which is a common practice in order to avoid spectral leakage. Fast-Fourier-Transform (FFT) is then used to convert the latter from time to frequency domain in order to adjust the amplitudes at the target frequencies, i.e. those between F.sub.min and F.sub.max, while keeping the random phases untouched. The goal is to maintain a relatively flat power spectrum which can be achieved using the formula:

(14) $A_1={\mathrm{CF}}_s\frac{1}{N}\underset{k=1}{\overset{N}{.Math.}}{a}_k$

(15) CF.sub.s is a software calibration factor multiplied by the average amplitude of the signal (where a.sub.k is the amplitude of the individual frequency components at the target frequencies, and A.sub.1 is the obtained flat amplitude). CF.sub.s is dependent on the user-defined specifications of the random signal, particularly its assigned power level. For the present invention, CF.sub.s was assigned a value between 2 and 3. In accordance with the present invention, FIG. 1A is a flowchart depicting the series of steps followed in order to generate the fully conditioned and calibrated signal X.sub.c(t) of flat spectral magnitude and random phase starting with a GWN sequence X.sub.i(t) which is generated based on the user-defined input parameters power, F.sub.min, F.sub.max and F.sub.step. FIG. 1B depicts time domain and frequency domain representations of samples of the signals V.sub.i(t), V.sub.w(t), V.sub.u(t) and V.sub.s(t). FIG. 1C is a flowchart depicting the steps followed for impedance measurement using the engineered signal V.sub.c(t).

(16) The newly calculated amplitude A.sub.1 is assigned to all frequencies within the target frequency range resulting in the signal X.sub.u(t) (as shown in FIG. 1A). However, this amplitude adjustment procedure might affect the windowing performed earlier, as well as generating values that are impractical for a Digital-to-Analog Converter (DAC) with a given resolution (which will transform the signal X.sub.c(t) to either a voltage or a current excitation). For example, if the DAC has a maximum resolution of 10 mV between two consecutive points, a smaller step value has to be rounded up or down. Thus, values between 1 mV and 10 mV have to be rounded. Therefore, as shown in the flowchart, inverse FFT is applied to X.sub.u(t), and windowing and rounding are applied again. The signal is then re-converted to the frequency domain and the amplitudes of the target frequencies are re-adjusted. Frequencies with reduced amplitudes are re-enforced with amplitude above the average so that when windowing and rounding are applied again, they remain high enough. This iterative procedure is repeated several times until the number of weakened amplitudes is minimized.

(17) The formula used to further increase the weakened frequencies at each iteration is:
A.sub.2=CF.sub.h×A.sub.1

(18) CF.sub.h is a hardware calibration factor multiplied by the previously calculated A.sub.1. The new amplitude A.sub.2 is assigned only to the affected or weakened frequencies. Upon exiting the calibration loop, a signal X.sub.s(t) is obtained. This signal is then transformed either to a voltage or a current signal using a DAC depending on the required EIS measurement mode, and subsequently applied on the device under test (DUT). However, prior to that, the DAC output signal is re-sampled and compared with the originally generated X.sub.s(t) to make sure there is no drop in any amplitude due to the DAC limited accuracy. If this happens, weakened frequencies are re-enforced. This process is referred to as the hardware calibration which results in the fully calibrated signal X.sub.c(t) suitable for applying to the DUT. FIG. 1B shows the time domain (upper plot) and frequency domain components (middle and lower plots) of an example of the proposed signal (set to be a voltage excitation in this work) and the subsequent transformations made upon it with F.sub.min and F.sub.max being 1 Hz and 50 Hz, respectively, and F.sub.step=0.25 Hz.

(19) The initial GWN voltage signal (V.sub.i(t)) is shown to be scattered randomly in the voltage-time series plot (see FIG. 1B) with close-to-zero mean value (0.8 mV) and a variance of σ.sup.2=60 mV.sup.2. Its corresponding phase was tested for randomness using Matlab's runstest function and found to be random with 95% confidence interval (see lower plot in FIG. 1B). The windowed signal Vw(t) is shown in the upper plot. Equation (1) is applied to modify the amplitude of V.sub.w(t), which results in the conditioned signal V.sub.u(t). Finally, the software calibrated signal V.sub.s(t) exhibits practically a frequency-independent amplitude with negligible deviation from the mean value (see middle plot in FIG. 1B). The spectral phases of these three signals are also shown in the lower plot of the figure. Finally, the flowchart in FIG. 1C shows the steps followed in order to measure the impedance of a DUT using V.sub.c(t). The measured voltage and current are referred to as V.sub.m(t) and I.sub.m(t) respectively in the figure. FFT is used to convert both signals to the frequency domain giving V.sub.m(s) and I.sub.m(s) (s=jω) on which windowing is applied. Finally, the spectral impedance for the DUT is computed from its definition Z(s)=Vm(s)/Im(s).

(20) Considering setup and data acquisition in accordance with the present invention, an experimental setup is used to test and verify the impedance measurement technique, consisting of a USB interfaced DAQ Analog Discovery 2 data acquisition board (Digilent) with a 100 MS/s sampling rate (the manufacturer's WaveForms software package was used to program the board), and a precisely calibrated standard linear, time-invariant impedance cell (from Bio-Logic). In accordance with the present invention, FIG. 2A shows a standard electrical calibration cell on which the measurements were carried out. FIG. 2B depicts time domain and frequency domain components of the initial voltage signal V.sub.i(t) compared to the measured voltage signal V.sub.m(t) within different frequency ranges and with different frequency steps and FIG. 2C depicts measured voltage and current corresponding to the same frequency ranges in FIG. 2B. FIG. 2D shows the complete signals V.sub.i and V.sub.m both in time and frequency domains, previously depicted in FIG. 2B as four intervals. FIG. 2E shows the complete measured current and voltage signals both in time and frequency domains, previously depicted in FIG. 2C.

(21) For clarity, the circuit diagram of the calibration impedance cell is shown in FIG. 2A. Following the procedure described previously in detail (see FIG. 1), the generated excitation signal is applied through the DAQ board between points A and C; i.e. V.sub.AC. Meanwhile the voltage signal at point B is read through the DAQ board and hence the current is calculated as V.sub.AB/100 ohm. As a result, the voltage and current needed for impedance calculation are both available. The measured impedance is compared to that obtained with a standard frequency sweep using single sine method on a state-of-the-art, research grade VSP-300 (from Bio-logic). The built-in FRA can cover the frequency range of 10 μHz up to 7 MHz with an accuracy of 1%/1° up to 3 MHz and 3% /3° to 7 MHz. The frequency range we covered in the testing of the invention was from 2 mHz to 200 kHz; i.e. 8 frequency decades, with maximum applied amplitude of +/−1.8 V.

(22) FIG. 2B-2E show a set of plots depicting the time domain and frequency domain components of the initial voltage signal V.sub.i(t), and the measured signals V.sub.m(t) and I.sub.m(t) using the setup in accordance with the present invention. In FIGS. 2B and 2C, each row represents a different frequency range (from F.sub.min to F.sub.max) covering a total of 2 mHz to 200 kHz, and different frequency steps (F.sub.step), while FIGS. 2D and 2E show the full combined range. The splitting of the frequency range into four segments is performed in order to minimize the effect of windowing. If the whole frequency range was to be covered using one signal only, the windowing procedure will affect the frequencies at the extremities, and hence increase the overall error. However, increasing the number of segments also implies increasing the measurement time. Table I summarizes the different applied experimental conditions along with the measurement time and relative deviation from the reference VSP-300 instrument results. In all experiments, the applied voltage was set within the limits of ±1.8V. The measurement time speedup factor (compared to the reference machine) is also shown in the Table.

(23) TABLE-US-00001 TABLE I Time Time Relative F.sub.min/Hz F.sub.max/Hz F.sub.step/Hz duration/s speed up deviation 1 0.002 1 0.002 500 6.79x 1.87% 2 1 50 0.25 4 16.0x 2.12% 3 50 1000 4 0.250 8.80x 1.02% 4 1000 200000 50 0.020 37.8x 2.43% Overall (0.002 Hz-200 kHz) 504.27 6.81x 2.20%

(24) The overall speedup factor is clearly limited by the very low frequency segment between 2 mHz and 1 Hz. Nevertheless, a speedup of 6.81 times is achieved in this frequency range with higher speedup factors achieved at higher frequencies. Meanwhile, an average relative error of just 2.20% from the reference Bio-Logic machine is achieved over the entire frequency range indicating the excellent quality of the proposed measurement technique. FIG. 3A-D is a comparison of the measured impedance (magnitude, phase and Nyquist plot) using the proposed method and that of the VSP-300 station in the form of (a) magnitude of impedance vs. frequency, (b) impedance phase angle vs frequency, and (c) Nyquist plot of real vs. imaginary parts of impedance; (d) shows the average relative deviation of measured impedance between the two methods.

(25) In accordance with a preferred embodiment of the present invention, a fast impedance measurement system is proposed using a conditioned and iteratively calibrated GWN signal as an excitation. The signal is freely converted to a voltage or current for potentio-static or galvano-static measurements respectively using a suitable DAQ board. Using a standard impedance calibration cell, a speedup of 6.8 times is achieved compared to traditional single sine frequency sweep method over the frequency range 2 mHz to 200 kHz, with an error less than 2.2% compared to a reference electrochemical station. The complete setup is compact and inexpensive which is appealing for portable applications. Further improvements may be achieved by optimizing the intermediate conditioning steps performed on the initial signal.

(26) The proposed method is enhanced by designing the random excitation signal in such away that its spectral phase is derived from a discrete uniform distribution (obtained from a physically-true random number generator), but its time domain maximum amplitude remains controllable. As such impedances can be excited with adjustable amplitude signals (as low as 50 mV and up to 2V). Traditional impedance measurement techniques employed in commercial impedance analyzer devices do not depend on random signals. In previous works, a randomly generated signal (converted to a voltage or current) was directly applied to the unknown impedance in the time domain. The response (current or voltage) was then measured, converted to frequency domain via FFT and the corresponding impedance magnitude and phase are calculated. However, in the proposed technique, the generated random signal is not directly applied to the unknown impedance. This signal is first converted to the frequency domain and re-structured via iterative calibration loops. These calibration loops maintain the random phase of the signal but re-adjust the magnitude to ensure a uniform flat power spectrum over the target frequency range set by a user, while increasing the amplitude of the weakened frequencies due to software and hardware signal processing (e.g. windowing, rounding the voltage to values acceptable by the board, etc.).

(27) Also, as part of the frequency-domain iterative re-structuring of the random signal, it is converted back into the time domain where another re-structuring takes place to ensure that the signal satisfies the minimum resolution requirements (e.g. 10 mV resolution steps) and circumvent the impact of windowing. This re-structuring in turn might affect the power spectrum of some frequencies and therefore, the signal is re-calibrated again in the frequency domain, by increasing the amplitude of these weakened frequencies. This process of re-structuring and calibration alternating between the time-domain and the frequency-domain continues until the desired frequency-domain power spectrum for all target frequencies and the desired time-domain resolution are both satisfied (as depicted in FIG. 1). Owing to the above explained iterative procedure, the error in impedance measurement using the proposed technique is minimized when compared to standard non-random-based methods overcoming most practical limitations or random-based methods (as depicted in FIG. 2).

(28) Many changes, modifications, variations and other uses and applications of the subject invention will become apparent to those skilled in the art after considering this specification and the accompanying drawings, which disclose the preferred embodiments thereof. All such changes, modifications, variations and other uses and applications, which do not depart from the spirit and scope of the invention, are deemed to be covered by the invention, which is to be limited only by the claims which follow.