Measurement of the phase of a complex impedance by thresholding

Abstract

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

Description

Title of the invention: Measurement of the phase of a complex impedance by thresholding

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

[0002] The concept of complex impedance generalizes that of resistance for sinusoidal signals at a given frequency f. In the case of an electric dipole, the complex impedance Z is defined by Z - y 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 a circuit with N ports (the dipole corresponding to the case N=l) we can define an impedance ^. .= Ü I • In other words, the impedance Zÿ is the (complex) ratio between the l J phasor representing the voltage across port "i" and the phasor representing the current entering (or leaving, depending on the convention adopted) port "j" when the current entering all other ports is zero. The various Z^ terms form the impedance matrix of the multiport element. In the following, the impedance term and the symbol "Z" will be used to designate both the impedance of a dipole and a Zÿ term of 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 module and phase: Z = where IZI is the ratio between the effective values ​​of the voltage and current and q> their phase shift.

[0004] Generally speaking, the impedance varies with the frequency of the electrical signals considered. To characterize an electrical element it is therefore necessary to measure its impedance(s) in a more or less extensive 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 from the prior art make it possible to measure the phase of a complex impedance.

[0007] (Angrisani 2001) discloses a measuring method in which a resistor of known value is connected in series to 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 particularly, 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 having filtered these two signals using predictive filters with finite impulse response to limit the impact of noise on the detection of zero; - The phase shift of the two signals is calculated from their internal product and their mean square value.

[0008] (Schrôder 2004) also determines the phase of a complex impedance by measuring the phase shift between two voltage signals. This phase shift measurement can be carried out 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 measurement of the phase of the impedance of an electrical element in a particularly simple manner, implementing a device of low complexity, 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) applying to said electrical element an excitation signal oscillating at a known frequency f; b) acquiring a first analog signal, variable over time, representative of a voltage at the terminals of the electrical element; c) acquiring a second analog signal, variable over time, representative of a current through the electrical element; d) determining a first digital value representative of an amplitude of said first analog signal and a second digital value representative of an amplitude of said second analog signal; e) performing hysteresis thresholding of said first and second analog signals; f) determining a third digital value representative of a time difference between an instant of crossing of a threshold by said first analog signal and an instant of crossing of said or another threshold by said second signal analog; and g) determining an estimate of said phase of the complex impedance of the electrical element as a function of said first, second and third digital values, as well as a fourth digital value representative of the frequency f of the excitation signal.

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

[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 the 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 comprise a time-digital conversion operation of a time offset between the first and second rising edge, or between the first and second falling edge.

[0014] - Step g) may comprise: gl) determining a first approximation of said phase from said third digital value representative of a time shift and said fourth digital value representative of the frequency f of the excitation signal; g2) determining a phase correction term as a function of the first and second numerical values; g2) determining said estimate 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 particularly, said fourth numerical value can be determined by calculating the product between said third digital value representative of a time shift and said fourth digital value representative of 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 representative of a said threshold.

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

[0018] Another object of the invention is an apparatus for measuring the phase of a complex impedance of an electrical element comprising: - a first analog-digital converter configured to receive as input a first analog signal varying over time, representative of a voltage between two terminals of the electrical element, and convert it into a first digital signal; - a second analog-digital converter configured to receive a second analog signal varying over time, representative of a current at through the electrical element, and convert it into a second digital signal; - a first Schmitt trigger for generating a first square wave signal by thresholding said first analog signal; - a second Schmitt trigger for generating 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 digital value representative of an amplitude of said first digital signal, of a second digital value representative of an amplitude of said second digital signal, of a third digital value representative of a time shift between the first square wave signal and the second square wave signal and of a fourth digital value representative of the frequency J of the excitation signal.

[0019] According to particular embodiments of such an apparatus:

[0020] Said digital circuit may comprise: - a time-to-digital converter for determining said third digital value; - a calculation module for determining a first approximation of said phase from said third digital value and a fourth digital value representative of a frequency f of said first and said second analog signal; - a lookup table for determining a phase correction term as a function of the first and second numerical values; and - an adder module for determining 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 apparatus may also comprise a third digital converter - analog to generate a fifth digital value representative of a threshold voltage common to said first and second Schmitt triggers, said correspondence table being configured to determine said phase correction term as a function of the first, second and fifth digital values.

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

[0023] Other characteristics, details and advantages of the invention will emerge on reading the description given with reference to the appended drawings given by way of example and 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 apparatus of [Fig.l];

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

[0027] [Fig.4] and [Fig.5] graphs illustrating the dependence of said phase measurement error as a function of the threshold voltage for different amplitude values ​​of the signal representative of the current through the electrical element and the same amplitude value of the signal representative of the voltage at 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 allowing, in principle, to measure the phase shift between two sinusoidal analog signals uv and ub. If these two signals are representative, respectively, of the voltage at the terminals of an electrical element and of the current flowing through it, this phase shift measurement allows to determine the phase of the complex impedance of the element.

[0030] The device of [Fig.l] comprises two Schmitt flip-flops BS1, BS2 having two threshold voltages VLh>0 V and VHl<0 V. The output of a Schmitt flip-flop becomes high when the signal at its input exceeds VLh, then remains high as long as said signal falls below VHl. For VLh0 V and VHl0 V, the Schmitt flip-flop becomes a simple zero comparator. If such a comparator were used, electronic noise would induce multiple and random switchings when the input signal crosses zero; for this reason, it is generally preferred 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 triggers convert the input sinusoidal signals uv and u, into two square wave signals and (Jf respectively. A time-to-frequency converter The digital TDC (time-to-digital converter) receives these signals as input and provides at its output a digital value a T which constitutes an estimate of the time shift AT between the rising edges (or, equivalently, the falling edges) of these signals. As can be seen in [Fig.2], this time shift is in turn proportional to the phase shift q> between the two analog input signals uv and ub. Also, the digital output of the TDC converter constitutes (to within a multiplicative factor, equal to the frequency f of the input signals) an estimate of this phase shift.

[0032] A disadvantage of using Schmitt triggers instead of simple zero comparators is that the time at which the VLH threshold (or VHl if we are interested in falling edges) is crossed does not depend only on the phase of the input signals, but also on their amplitude. Also, a difference in the amplitude of the

[0033] uv and u signals, will distort the measurement of their phase shift. This is illustrated in the case of [Fig.3], where the uv and u signals are perfectly in phase with each other, but not same amplitude. We can see that the most intense signal, uv, crosses the V LH 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 of the phase shift between these two signals, which are in reality in phase with each other. More precisely, we find = A (p = f A Tlh, / being the frequency of the input signals.

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

[0035] The phase estimation error A (p can be calculated by analyzing the signals IjH around TLH time: [00361 VLH = A cos(a>Tw)^Tli1 = -L cosa (¾2)

[0037] Vlh = Bcos(MT lh+ AT LH Y)~ AT LH = ^o^^

[0038] where A and B are the (real) amplitudes of the sinusoidal analog signals uv and u,, and co the pulsation of the signals, co = 2æ / .

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

[0040] ArLH = .qœr,(^).cf^

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

[0042] a (-¾2 j - Lri \ ^ / \ /

[0043] [Fig.4] is a graph of the phase error A (p[H) as a function of the threshold voltage VLH for different values ​​of amplitude B ranging from 1.5V to 2.85V, while amplitude A is considered fixed at 3V. In the case where the signal-to-noise ratio (SNR) of the input signals is not too high - for example SNR > 25 dB - we can take VLH low in front of A and B, for example VLH <0.2V. In these conditions, A <p peut être approché par une fonction linéaire, comme illustré par la [Fig.5].

[0044] Indeed, by developing the expression of A (p obtained above in Taylor series to the first order around Vlh — $ we find

[0045] 4^ = ^(¾2).^¾¾2)^.¾2)¾^ = (3 A )

[0046] An idea underlying the invention is that the error A tp^ can be pre-calculated as a function of the amplitudes A and B and the threshold voltage VLh, quantities which can be easily measured using analog-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 an apparatus 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 GS generator applies a sinusoidal excitation signal sext(t), of possibly variable frequency f, to the terminals of the EL element. The signal sext(t) can be a current or voltage signal. The GS generator also provides its sort with a digital value representative of the frequency / . The GS generator can, for example, be controlled so that / scans, continuously or discretely, a spectral band of interest.

[0050] The device of [Fig.6] receives on a first input port the first analog signal uv(t) representative of the voltage across the terminals of the EL element and on a second input port the second analog signal u^t) representative of the current flowing through the latter. For example, the signal uv(t) can be directly the voltage across the terminals of the EL element and u^t) a voltage across a resistor connected in series to EL. A first analog-digital converter ADC1 samples and converts the analog signal uv into a digital signal Uv. Similarly, a second analog-digital converter ADC2 samples converts the analog signal u, 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 supply as output square wave signals Üv, Ür. 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 increasing 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 mêmes signaux. Les seuils VHl et VLH sont fixés par des valeurs de tension fournies en entrée aux bascules de Schmitt BS1, BS2. La tension VLH, en outre, est convertie au format numérique par un troisième convertisseur analogique - numérique ADC3.

[0052] The digital signals Uv, Ub VTh and f, as well as the square wave signals (J y, Ü are supplied 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 comprises a time-digital converter TDC which receives as input the square wave signals Üy, Ûj and - as explained above with reference to [Fig.l] - generates a digital value A T representative of an offset time between their rising edges. A multiplier module MM calculates the product of a T by the frequency / to determine a first estimate <p du déphasage q>between the signals uv(t) and u^t), and therefore of the phase of the complex impedance of the EL element.

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

[0055] The digital values ​​A, B and VLH (the latter, from the third analog-to-digital converter ADC3) make it possible to calculate the correction term - A ç by applying equation (4), its linear approximation (5) or a polynomial approximation. The calculation is typically carried out using a three-input LUT lookup table.

[0056] Finally, an adder module MA calculates a corrected estimate (p' by applying the correction term - A® to the first estimate ~ LH

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

[0058] - The digital circuit CN can be implemented by means of a microprocessor programmed in a timely manner (in which case the different “blocks” and “modules” of [Fig.6] are realized in software), an FPGA or an ASIC.

[0059] - The LUT correspondence table can be produced by means of a device of dedicated memory, or 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 an FPGA or ASIC implementation) or by a calculation routine (in the case of a software implementation), instead of a lookup table, particularly if the linear approximation of equation (5) is used.

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

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

[0063] - The value of the threshold VLh (or VHl if we are interested in the falling edges) can be predefined and stored in a 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 drifts to be taken into account.

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

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

[0066] (Angrisani 2001): L. Angrisani, L. Ferrigno, Reducing the uncertainty in real-time impédance 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 impédance spectroscopy, Measurement Science and Technology, Volume 15, Issue 7, 2004, Pages 1271 - 1278

Claims

Claims

1. 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; b) acquiring a first analog signal (uv), variable over time, representative of a voltage across the terminals of the electrical element; c) acquiring a second analog signal (uO, variable over time, representative of a current through the electrical element; d) determining a first digital value (A) representative of an amplitude of said first analog signal (uv) and a second digital 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 signals;f) determining a third digital value (a T) representative of a time difference between an instant of crossing of a threshold by said first analog signal and an instant of crossing of said or another threshold by said second analog signal; and g) determining an estimate (; <p ') de ladite phase l’impédance complexe l’élément électrique en fonction desdites première, deuxième et troisième valeurs numériques, ainsi que d’une quatrième valeur numérique représentative la fréquence f du signal d’excitation. [revendication 2] procédé selon revendication 1 dans lequel, lors l’étape e), le seuillage premier analogique génère un créneau (ûv) comprenant front montant descendant     comprenant descendant, lequel f) comprend une opération conversion temps - d’un décalage temporel (atlh) entre montant, ou descendant. 3] l’une des revendications précédentes g) : gl) détermination première approximation (^) ( v) à partir a t) d’excitation ; g2) terme correction (-aq>LH) as a function of the first and second numerical values; g2) determining said estimation ÇO of the phase of the complex impedance of the electrical element by calculating the sum of said first approximation and said phase correction term.

4. Method according to claim 3 wherein said fourth digital value is determined by calculating the product between said third digital value (AT) representative of a time shift and said fourth digital value representative of the frequency f of the excitation signal.

5. Method according to one of claims 3 or 4 in which the determination of said phase correction term (-Aq>LH) is also carried out as a function of a fifth digital value (VLh) representative of a said threshold.

6. Method according to one of claims 3 to 5 in which the determination of a phase correction term (-Aq>LH) is carried out by means of a look-up table (LUT).

7. Apparatus for measuring the phase of a complex impedance of an electrical element (EL) comprising: - a first analog-digital converter (ADC1) configured to receive as input a first analog signal (uv), variable over time, representative of a voltage between two terminals of the electrical element, and convert it into a first digital signal (Uv); - a second analog-digital converter (ADC2) configured to receive a second analog signal (u^, variable over time, representative of a current through the electrical element, and convert it into a second digital signal (U^; - a first Schmitt flip-flop (BS1) for generating a first square wave signal (Ûy) by thresholding said first analog signal; - a second Schmitt flip-flop (BS2) for generating a second square wave signal (Üp by thresholding said second analog signal;and - a digital circuit (CN) configured to determine an estimate (^') of said phase of the complex impedance of the electrical element in; function of a first digital value (A) representative of an amplitude of said first digital signal (Uv), of a second digital value (B) representative of an amplitude of said second digital signal (UO, of a third digital value (AT) representative of a time shift between the first square wave signal (Üy) and the second square wave signal (Üy) and of a fourth digital value representative of the frequency / of the excitation signal.

8. Apparatus according to claim 7 wherein said digital circuit comprises: - a time-to-digital converter (TDC) for determining said third digital value (AT); - a calculation module for determining a first approximation (^) of said phase (q>) from said third digital value (AT) and a fourth digital value representative of a frequency f of said first and said second analog signal; - a correspondence table for determining a phase correction term (-Aq>LH) as a function of the first and second digital values; and - an adder module for determining said estimation ((p') of said phase of the complex impedance of the electrical element by calculating the sum of said first approximation and said phase correction term.

9. Apparatus according to claim 8 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 triggers, said look-up table (LUT) being configured to determine said phase correction term (-Aq>LH) as a function of the first, second and fifth digital values.

10. Apparatus according to one of claims 8 and 9 wherein said digital circuit (CN) is also configured to receive as input said fifth digital value (VLH).

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