Phase measurement of a complex impedance by thresholding

Abstract

A method for measuring the phase of the complex impedance of an electrical element (EL) comprising: applying an excitation signal (sex) to said electrical element (EL) at a known frequency; acquiring a first (uV) and a second (uI) analog signal representing a voltage and a current, respectively; converting the first and second analog signals to digital format; performing thresholding with hysteresis on the first and second analog signals; measuring a time lag between the instant of the first analog signal crossing a threshold and the instant of the second analog signal crossing a threshold; and determining an estimate of the phase of the complex impedance of the electrical element as a function of the amplitudes of the first and second analog signals converted to digital format, the time lag, and the frequency of the excitation signal. Apparatus for implementing this method. Figure for the abstract: Fig. 6

Description

Title of the invention: Phase measurement of a complex impedance by thresholding

[0001] The invention lies in the field of electronic instrumentation. It relates to more specifically on a method and apparatus for measuring impedance complex of an electrical element.

[0002] The notion of complex impedance generalizes that of resistance for signals sinusoidal at a given frequency f. In the case of an electrical dipole, the impedance The complex number Z is defined by Z - j- where U is the phasor (complex number) representing the amplitude and phase of the voltage across the dipole and I the phasor representing the amplitude and phase of the current flowing through it. More generally, in the case of an N-port circuit (the dipole corresponding to the case N=1) we can define an impedance 7 1 . In other words, the impedance Zÿ- is the (complex) ratio between the lJ~ The phasor represents the voltage across port "i" and the phasor represents the current entering (or leaving, depending on the convention adopted) port "j" when the current entering all other ports is zero. The various Zy terms form the impedance matrix of the multiport element. In the following, the term impedance and the symbol "Z" will be used to refer to both the impedance of a two-terminal device and a Zy term in the impedance matrix of a multiport.

[0003] Being a complex number, the impedance Z can be decomposed into a complex part and an imaginary part - Z=R+jX, where "j" here designates the imaginary unit - or into magnitude and phase: Z = \Z\e^, where IZI is the ratio between the effective values ​​of the voltage and current and q> their phase shift.

[0004] In general, impedance varies with the frequency of the electrical signals considered. To characterize an electrical element, it is therefore necessary to measure its impedance(s) over a more or less wide frequency band. We therefore write Z(f), IZ(f)l and <p(f) pour désigner, respectivement, une impédance complexe, son module et sa phase en fonction de la fréquence f.

[0005] Several techniques have been developed to measure the phase of an impedance, <p(f), en fonction de la fréquence.

[0006] Several methods known in the prior art allow the phase of a complex impedance to be measured.

[0007] (Angrisani 2001) discloses a measurement method in which a resistor of known value is connected in series with the element to be characterized and a sinusoidal excitation signal is applied to said element through this resistor. The impedance of The element to be characterized can be determined from the measurement of the voltage u(t) of the excitation signal and that, v(t), of a node located between the known resistance and the element to be characterized. More specifically, two methods are proposed to determine the phase of said impedance: - Either the zero crossings of the signals u(t) and v(t) are detected, after filtering these two signals using predictive filters with finite impulse response to limit the impact of noise on zero detection; - Either the phase shift of the two signals is calculated from their inner product and their mean squared value.

[0008] (Schröder 2004) also determines the phase of a complex impedance by measuring the phase shift between two voltage signals. This phase shift measurement can be performed by detecting the zero crossings of said signals, or by analytical calculation from amplitude and phase parameters of said signals, determined by interpolation.

[0009] The solutions proposed by (Schrôder 2004) and (Angrisani 2001) require the performance of fairly complex calculations and, in the case of (Angrisani 2001), the implementation of a specific power sensor.

[0010] The invention aims to overcome, at least in part, the aforementioned drawbacks of the prior art. More specifically, it aims to enable a particularly simple measurement of the phase impedance of an electrical element, using a low-complexity device, which may in particular be based on a microcontroller or an FPGA.

[0011] An object of the invention is a method for measuring the phase of the complex impedance of an electrical element comprising the following steps: a) apply to said electrical element an excitation signal oscillating at a known frequency f; b) acquire a first analog signal, variable over time, representative of a voltage across the terminals of the electrical element; c) acquire a second analog signal, variable over time, representative of a current through the electrical element; d) determine a first numerical value representing an amplitude of said first analog signal and a second numerical value representing an amplitude of said second analog signal; e) perform a thresholding with hysteresis of said first and second analog signal; f) determine a third numerical value representing a time lag between the instant a threshold is crossed by said first analog signal and an instant of crossing said or another threshold by said second analog signal; and (g) determine an estimate of said phase of the complex impedance of the electrical element as a function of said first, second and third numerical values, as well as a fourth numerical value representative of the frequency f of the excitation signal.

[0012] According to particular embodiments of such a process:

[0013] - During step e), the thresholding of the first analog signal can generate a first square wave signal comprising a first rising edge and a first falling edge and thresholding of the second analog signal generates a second square wave signal comprising a second rising edge and a second falling edge, and step f) may include a time-to-digital conversion operation of a time offset between the first and second rising edges, or between the first and second falling edges.

[0014] - Step g) may include: gl) the determination of a first approximation of said phase from said third numerical value representing a time shift and said fourth numerical value representing the frequency f of the excitation signal; g2) the determination of a phase correction term as a function of the first and second numerical values; g3) the determination of said estimation of the phase of the complex impedance of the electrical element by calculating the sum of said first approximation and said phase correction term.

[0015] - More specifically, said fourth numerical value can be determined by calculation of the product between said third numerical value representing a time shift and said fourth numerical value representing the frequency f of the excitation signal.

[0016] - The determination of said phase correction term can also be carried out based on a fifth numerical value representing said threshold.

[0017] - The determination of a phase correction term can be carried out by means of from a lookup table.

[0018] Another object of the invention is a device for measuring the phase of a complex impedance of an electrical element comprising: - a first analog-to-digital converter configured to receive as input a first analog signal that varies over time, representing a voltage between two terminals of the electrical element, and to convert it into a first digital signal; - a second analog-to-digital converter configured to receive a second time-varying analog signal, representative of a current through the electrical element, and convert it into a second digital signal; - a first Schmitt flip-flop to generate a first square wave signal by thresholding said first analog signal; - a second Schmitt flip-flop to generate a second square wave signal by thresholding said second analog signal; And - a digital circuit configured to determine an estimate of said phase of the complex impedance of the electrical element as a function of a first numerical value representing an amplitude of said first digital signal, a second numerical value representing an amplitude of said second digital signal, a third numerical value representing a time offset between the first square wave signal and the second square wave signal and a fourth numerical value representing the frequency / of the excitation signal.

[0019] According to particular embodiments of such a device:

[0020] Said digital circuit may include: - a time-to-digital converter to determine said third numerical value; - a calculation module to determine a first approximation of said phase from said third numerical value and a fourth numerical value representing a frequency f of said first and second analog signal; - a lookup table to determine a phase correction term based on the first and second numerical values; and - an adding module to determine said estimate of said phase of the complex impedance of the electrical element by calculating the sum of said first approximation and said phase correction term.

[0021] - The device may also include a third digital converter - analog to generate a fifth digital value representative of a threshold voltage common to said first and second Schmitt flip-flops, said lookup table being configured to determine said phase correction term as a function of the first, second and fifth digital value.

[0022] - Said digital circuit can also be configured to receive as input said fifth numerical value.

[0023] Other features, details and advantages of the invention will become apparent from the description given with reference to the accompanying drawings provided by way of example, which represent, respectively:

[0024] [Fig. 1], the functional diagram of a measuring device not falling within the scope of the invention;

[0025] [Fig.2], an illustration of the operating principle of the device in [Fig.1];

[0026] [Fig.3], an illustration of the phase measurement error induced by a difference amplitude of input signals;

[0027] [Fig.4] and [Fig.5] graphs illustrating the dependence of said phase measurement error on the threshold voltage for different amplitude values ​​of the signal representing the current through the electrical element and the same amplitude value of the signal representing the voltage across its terminals; and

[0028] [Fig.6], the functional diagram of a measuring device according to an embodiment of the invention.

[0029] Fig. 1 is the functional diagram of a hypothetical measuring device which, in principle, allows the phase shift between two sinusoidal analog signals uv and ub to be measured. If these two signals are representative, respectively, of the voltage across an electrical element and the current through it, this phase shift measurement allows the phase of the complex impedance of the element to be determined.

[0030] The device in [Fig. 1] comprises two Schmitt flip-flops BS1, BS2 with two threshold voltages VLh > 0 V and VHl < 0 V. The output of a Schmitt flip-flop goes high when the signal at its input exceeds VLH, and then remains high as long as said signal falls below VHl. For VLh 0 V and VHl 0 V, the Schmitt flip-flop becomes a simple zero comparator. If such a comparator were used, electronic noise would induce multiple and random switching during a zero crossing of the input signal; for this reason, it is generally preferable to use Schmitt flip-flops—also called hysteresis comparators—with a hysteresis AV = VLH - VHl of the same order of magnitude as the peak amplitude of the noise affecting the input signal.

[0031] Schmitt flip-flops convert the sinusoidal input signals uv and u into two square wave signals Ûv and Ü, respectively. A time-to-digital converter (TDC) receives these signals as input and provides at its output a digital value AT that constitutes an estimate of the time shift AT between the rising (or, equivalently, falling) edges of these signals. As can be seen in [Fig. 2], this time shift is in turn proportional to the phase shift θ between the two analog input signals uv and ub. Thus, the digital output of the TDC constitutes (up to a multiplicative factor equal to the frequency f of the input signals) an estimate of this phase shift.

[0032] One drawback of using Schmitt flip-flops instead of simple zero-separating comparators is that the instant of crossing the VLH threshold (or VHl if one (is interested in falling edges) does not depend solely on the phase of the input signals, but also on their amplitude. Therefore, a difference in amplitude of the

[0033] signals uv and u, will distort the measurement of their phase shift. This is illustrated in the case of [Fig. 3], where the signals uv and u are perfectly in phase with each other, but not of the same amplitude. It can be seen that the more intense signal, uv, crosses the VLH alone at a time t=TLH, while the less intense signal, ub, crosses it later, at a time t=TLH+ ATLH. The time shift ATLH induced by the difference in amplitude of uv and u leads to a non-zero estimate p of the phase shift between these two signals, which are actually in phase with each other. More precisely, we find = A (p^ = f A Tijj- / being the frequency of the input signals.

[0034] The situation does not change if the phase shift cp between the two input signals is indeed non-zero: the difference between their amplitudes introduces an estimation error A of said phase shift.

[0035] The phase estimation error Am can be calculated by analyzing the LH signals around the TLH time:

[0036] VLH = ACOS( üfTLH )^Tlh = ^CO^ ( ) G)

[0037] Vui = B cos(œ(TLH + ^LH)) A TLH = ^(cos-l(-jr- ) ) -TLH^

[0038] where A and B are the (real) amplitudes of the sinusoidal analog signals uv and u,, and co the angular frequency of the signals, w = 2^- / .

[0039] By removing TLH from the two preceding equations we obtain:

[0040] A Tlh = - Xco^y—) ) )

[0041] The phase error A (p) is therefore given by:

[0042] AI ( LH 1 i / ^LH ) (4) A (pLH = cos1 ( -g- ) - cos- ' \ v 7

[0043] [Fig.4] is a phase error graph A <pLff en fonction de la tension de The VLH threshold is set for different values ​​of amplitude B ranging from 1.5V to 2.85V, while amplitude A is considered fixed at 3V. If the signal-to-noise ratio (SNR) of the input signals is not too high—for example, SNR > 25 dB—VLH can be taken to be low compared to A and B, for example, VLH < 0.2V. Under these conditions, A <pLH peut être approché par une fonction linéaire, comme illustré par la [Fig.5].

[0044] Indeed, by expanding the expression for A (pLH obtained above as a first-order Taylor series around V lu ~ 0, we find

[0045] A 1 (YjAL ] 1 1 Y-LLL 1 l 77 YlB. ) = A — J - — J 4 - - — J 4 ô - ” / = “ - “ ( 1 2 ) y (5) \ ABJ v LH

[0046] One idea underlying the invention is that the error A (pLfJ) can be pre-calculated as a function of the amplitudes A and B and the threshold voltage VLH, quantities that can be easily measured using analog-to-digital converters (VLh can also be considered known by system design). This makes it possible to compensate for said error to obtain an improved estimate of the phase shift between the input signals:

[0047] A^ / ®

[0048] Fig. 6 illustrates the functional diagram of a device according to an embodiment of the invention, implementing this principle.

[0049] In [Fig. 6], the reference EL represents an electrical element (more particularly, a dipole, comprising two terminals forming a single port) whose complex impedance phase is to be determined. A generator GS applies a sinusoidal excitation signal sext(t), of optionally variable frequency f, to the terminals of the element EL. The signal sext(t) can be a current or voltage signal. The generator GS also provides its source with a numerical value representing the frequency f. The generator GS can, for example, be driven such that f sweeps, continuously or discretely, a spectral band of interest.

[0050] The device in [Fig. 6] receives, at a first input port, the first analog signal uv(t) representing the voltage across element EL, and at a second input port, the second analog signal u(t) representing the current flowing through it. For example, the signal uv(t) can be directly the voltage across element EL and u(t) a voltage across a resistor connected in series with EL. A first analog-to-digital converter ADC1 samples and converts the analog signal uv into a digital signal Uv. Similarly, a second analog-to-digital converter ADC2 samples and converts the analog signal u(t) into a digital signal Ub

[0051] The signals uv(t) and u(t) are also supplied as input to respective Schmitt flip-flops BS1, BS2, which provide as output square wave signals Ûv, Uj. As explained above, with reference to [Fig. 2] and [Fig. 3], the rising edges of these square wave signals correspond to the crossing of a threshold VLh in the ascending direction by uv(t) and U(t) respectively, while their falling edges correspond to the crossing of a threshold VHl <Vlh dans le sens décroissant par ces same signals. The thresholds VH1 and VLH are set by voltage values ​​supplied as input to the Schmitt flip-flops BS1, BS2. The voltage VLh, in addition, is converted to digital format by a third analog-to-digital converter ADC3.

[0052] The digital signals Uv, Ub VTH and f, as well as the square wave signals Ûy, Üp are provided as input to a digital circuit CN which uses them to calculate an estimate of the phase shift between the signals uv(t) and u^t) - that is to say of the phase of the complex impedance of the electrical element EL.

[0053] The digital circuit CN includes a time-to-digital converter (TDC) which receives as input the square wave signals Ûv, Ûj and – as explained above with reference to [Fig. 1] – generates a digital value aT representing a time shift between their rising edges. A multiplier module MM calculates the product of AT by the frequency f to determine a first estimate of the phase shift θ between the signals uv(t) and u^t), and therefore of the phase of the complex impedance of the element EL.

[0054] The digital circuit CN also includes two logic blocks MX1, MX2 configured to extract the values ​​A and B from the digital signals Uv and U! representative of the amplitude of the analog signals uv(t) and u^t). For example, the MX1, MX2 blocks can determine local extrema of said signals and average their values, or interpolate them with perfect sinusoidal functions.

[0055] The numerical values ​​A, B and VLH (the latter, from the third converter) analog-to-digital ADC3) allows the calculation of the correction term - A ® by application of equation (4), its linear approximation (5) or a “LH Polynomial approximation. The calculation is typically performed using a table three-input LUT matching.

[0056] Finally, an MA adder module calculates a corrected estimate by applying the term correction ^LH at the first estimate

[0057] The invention has been described with reference to a particular embodiment, but variations are possible. For example:

[0058] - The digital CNC circuit can be implemented using a microprocessor programmed appropriately (in which case the various "blocks" and "modules" of [Fig.6] are implemented in software), an FPGA or an ASIC.

[0059] - The LUT lookup table can be created using a device dedicated memory, or to constitute a region of a microprocessor's memory.

[0060] - The determination of the correction term - A can be carried out by a circuit arithmetic (in the case of implementation by FPGA or ASIC) or by a routine of calculation (in the case of a software implementation), instead of a lookup table, particularly if the linear approximation of equation (5) is used.

[0061] - The MX1 and MX2 blocks can be omitted if a peak detector is provided upstream. of each of the analog-to-digital converters ADC1, ADC2

[0062] - Instead of the time shift between the rising edges of the Uv and Ub signals, it is It is possible to take into account falling fronts. In this case, the VHl threshold should be used for calculating the correction term.

[0063] - The value of the VLH threshold (or VHl if we are interested in descending fronts) can be predefined and stored in memory, instead of being acquired and converted to digital format. The use of an analog-to-digital converter ADC3, as in the embodiment of [Fig. 3], is, however, advantageous because it allows for the consideration of drift.

[0064] - Similarly, the frequency / can be a predefined value stored in memory, instead of being supplied by the FS generator. Conversely, it can also be determined from the analog signals uv(t) and u^t).

[0065] - Several implementations known to those skilled in the art are possible for the Time-to-digital converter (TDC): a simple high-frequency counter, a counter with a delay line, a dual-delay-line circuit, etc. References

[0066] (Angrisani 2001): L. Angrisani, L. Ferrigno, Reducing the uncertainty in real-time impedance measurements, Measurement, Volume 30, Issue 4, 2001, Pages 307-315,

[0067] (Schrôder 2004): Jens Schrôder and Steffen Doemer and Thomas Schneider and Peter Hauptmann, Analogue and digital sensor interfaces for impedance spectroscopy, Measurement Science and Technology, Volume 15, Issue 7, 2004, Pages 1271 - 1278

Claims

Demands

1. A method for measuring the phase of the complex impedance of an electrical element (EL) comprising the following steps: a) applying to said electrical element (EL) an excitation signal (sex) oscillating at a known frequency f; b) acquiring a first analog signal (uv), varying over time, representative of a voltage across the terminals of the electrical element; c) acquiring a second analog signal (u^), varying over time, representative of a current through the electrical element; d) determining a first numerical value (A) representative of an amplitude of said first analog signal (uv) and a second numerical value (B) representative of an amplitude of said second analog signal (u;); e) performing a thresholding with hysteresis of said first and said second analog signal;(f) determine a third numerical value (AT) representing a time lag between the time a threshold is crossed by said first analog signal and the time said or another threshold is crossed by said second analog signal; and (g) determine an estimate ( <p') de ladite phase de l’impédance complexe de l’élément électrique en fonction desdites première, deuxième et troisième valeurs numériques, ainsi que d’une quatrième valeur numérique représentative de la fréquence f du signal d’excitation ; dans lequel l’étape g) comprend : gl) la détermination d’une première approximation (^) de ladite phase (^) à partir de ladite troisième valeur numérique ( A F) représentative d’un décalage temporel et de ladite quatrième valeur numérique représentative de la fréquence f du signal d’excitation ; g2) la détermination d’un terme de correction de phase (-Aq> LH) depending on the first and second numerical values;g3) the determination of said estimate (^r) of the phase of the complex impedance of the electrical element by calculating the sum of said first approximation and said phase correction term.;

2. A method according to claim 1 wherein, in step e), the thresholding of the first analog signal generates a first signal in square (Ûv) comprising a first rising edge and a first falling edge and the thresholding of the second analog signal generates a second square signal (1¾ comprising a second rising edge and a second falling edge, and wherein step f) comprises a time-to-digital conversion operation of a time offset (ATLH) between the first and second rising edges, or between the first and second falling edges.

3. A method according to any one of the preceding claims wherein said fourth digital value is determined by calculating the product between said third digital value (AT) representing a time shift and said fourth digital value representing the frequency f of the excitation signal.

4. A method according to any one of the preceding claims wherein the determination of said phase correction term (-Aq>LH) is also carried out as a function of a fifth numerical value (VLH) representative of said threshold.

5. A method according to any one of the preceding claims wherein the determination of a phase correction term (-Aq>LH) is carried out by means of a lookup table (LUT).

6. A phase measuring apparatus for a complex impedance of an electrical element (EL) comprising: - a first analog-to-digital converter (ADC1) configured to receive as input a first time-varying analog signal (uV) representing a voltage between two terminals of the electrical element, and convert it into a first digital signal (UV); - a second analog-to-digital converter (ADC2) configured to receive a second time-varying analog signal (ul) representing a current through the electrical element, and convert it into a second digital signal (UI); - a first Schmitt flip-flop (BS1) for generating a first square wave signal (Üv) by thresholding said first analog signal; - a second Schmitt flip-flop (BS2) for generating a second square wave signal (1¾) by thresholding said second analog signal; and - a digital circuit (DC) configured to determine an estimate ($') of said phase of the complex impedance of the electrical element as a function of a first digital value (A) representing an amplitude of said first digital signal (UV), a second digital value (B) representing an amplitude of said second digital signal (UI), a third digital value (AT) representing a time offset between the first square wave signal (Ûy) and the second square wave signal (Üv) and a fourth digital value representing the frequency 1 of the excitation signal; in which said digital circuit comprises - a time-to-digital converter (TDC) for determining said third digital value (AT);- a calculation module to determine a first approximation ($) of said phase ( <p) à partir de ladite troisième valeur numérique ( A T) et d’une quatrième valeur numérique représentative d’une fréquence f desdits premier et dudit deuxième signal analogique ; - une table de correspondance pour déterminer un terme de correction de phase (-Aq> LH) as a function of the first and second numerical values; and - an adding module to determine said estimate ($') of said phase of the complex impedance of the electrical element by calculating the sum of said first approximation and said phase correction term.;

7. Apparatus according to claim 6 also comprising a third digital-to-analog converter (ADC3) for generating a fifth digital value (VLH) representative of a threshold voltage common to said first and second Schmitt flip-flops, said lookup table (LUT) being configured to determine said phase correction term (-Aq>LH) as a function of the first, second and fifth digital value.

8. Device according to any one of claims 6 or 7 in which said digital circuit (DC) is also configured to receive as input said fifth digital value (VLH).

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