In transducers of the capacitive cum conductive type, invariably non linear conductance contributes to an erroneous measurement of the Capacitance part. For instance, while the measurement of moisturized transformer oil’s dielectric properties are measured with a transformer ratio arm bridge, the separation of the capacitive component from the non-linear ionic conductance appears to mask the results considerably. So, a study was made of the composite signal component due to the complex impedance R + jX as an analytic function. Due to the effects dependent on impurities and stray causes, the two components of this impedance function, the conductance and the capacitance, do not fit into an exact analytic function. Mainly, the effects of non-linearity in conductance exhibited by the impure parts are to be eliminated. Since the capacitance part is mostly affected only by additive stray effects which can be taken into account (by a dummy measurement with no material between electrodes), the conductance part cannot be easily removed of the impure, interfering components.

In the case of ionic conductivity, such as the moisture with salt absorbed in the oil specimen, the same is having a non-linear ohmic relation and the effect is to introduce a wrong component into the separated signal for the capacitance component from the bridge cell.

So, in order to resolve this problem, the fact that the Hilbert transform is the link between the two components of a linear analytic signal [1], has been used and found to give the corrective procedure. Because the signal from the capacitive component has no non-linearity (except under high electric fields, for the case of the transformer oil bridge measurement), the Hilbert transform of the bridge signal for an excitation burst to the bridge provides the linear part of the conductance component, such as
$$R_{lin} +jX$$
And by simple arithmetic, the non linearity due to the conductance can be eliminated. Transducers based on conductance or capacitance measurements for physical quantities are quite many and well known. Many physical phenomena involve these two properties. In optical measurements, substances absorb light and refraction also takes place. The net property is a complex refractive index, a complex function of frequency, given by
$$\nu (\omega )=\sigma +j\epsilon \text{ (1)}$$ Where σ denotes the conductance and + j the dielectric constant. If such a complex function is analytic, it satisfies the Kramers-Kronig relations [2] given by
$$X_1(\omega )=\frac{1}{\pi }P{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d{\omega }^1}\frac{X_2({\omega }^1)}{{\omega }^1-\omega }\text{ (2a)}$$ And $$X_2(\omega )=-\frac{1}{\pi }P{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d{\omega }^1}\frac{X_2({\omega }^1)}{{\omega }^1-\omega }\text{ (2b)}$$ $$Q(\omega )=\frac{-j\omega 1}{\left|\omega \right|}=-j\mathrm{sgn}\omega \text{ (3a)}$$ $$q_{{}_n}=\{{}_{\frac{-2}{\pi n}fornodd}^{0forneven}\text{ (3b)}$$

In transducers of the capacitive cum conductive type, invariably non linear conductance contributes to an erroneous measurement of the capacitance part. As an example, in the measurement of transformer oil’s dielectric properties using a transformer ratio arm bridge, we get the two components, namely the capacitance and the conductance. These components are separated from the signal output voltage from the bridge using phase sensitive detection. Here, two detecting circuits are involved; one gives the in-phase component, the other gives the imaginary or Quadrature component, as χ(ω) = χ₁(ω) + jχ₂(ω). Here the χ denotes the impedance of the measuring cell.
If the signals are phase sensitively detected, there arise the two components -- in-phase or the real part and the out of phase or quadrature part.
If V_d is the detected output signal voltage, then,

Direct or in-phase component

Quadrature Component

When the properties are measured for a pure substance, the two components will give the conductance and capacitance respectively. However, this separation of the capacitive component from the conductance is subject to errors whenever there are interactive disturbances, causing one component to interfere with the other. One reason is because of the nonlinear nature of ionic conductance and another is dead time. Nonlinearity in the conductance component can arise when the current to voltage relationship is not linear due to the property of ionic conduction [4]. So, when we measure a capacitance change it appears on the conductance and vice versa, with the result that one appears to mask the other. So, we cannot infer that the quadrature component is only due to capacitance.

At a particular frequency, the capacitance C gives rise to a reactance X at a frequency ω given by X = 1/ωC, while the conductivity gives rise to a resistance R. So, a study was made of the composite signal component due to the complex impedance R + jX as an analytic function. Due to the effects dependent on impurities and stray causes, the two components of this impedance function, the conductance and the capacitance, may not fit into an exact analytic function. Mainly, the effects of nonlinearity in conductance exhibited by the impure parts are to be eliminated. The capacitance part is not so much affected as does the conductance, except by additive stray effects which are correctible (by shielding and using guard electrodes and making dummy measurement with no material between electrodes). But the absorption (or conductance) component of a substance cannot be easily separated from the impure, interfering components.

In the case of ionic conductivity, such as the moisture with salt absorbed in the oil specimen, the same is having a nonlinear ohmic relation (Figure2) and its effect is to introduce a wrong component into the separated signal for the capacitance component from the bridge cell.

So, in order to resolve this problem, the fact that the Hilbert transform is the link between the two components of a linear analytic signal, has been used and found to give a fairly corrective procedure. The capacitive component has no non-linearity (except under very high electric fields, for the case of the transformer oil bridge measurement). The resistive component can be linked to it theoretically, by its Hilbert transform. If this is represented as R_lin, then, under ideal conditions, the bridge signal provides the linear part of the conductance component together with the capacitive component, as
$$R_{lin}+jX$$ $$V_{real=} \Sigma V_1[n]V_2{\left[n\right]}_{} \dots \left(8\right)$$
$$V_{quad}= \Sigma V_1\left[n+K\right]V\dots \left(9\right)$$ $$V_{H.Q}=òH\left\{V_d\right\}Sinwtdt \dots (10)$$

The samples of signals, the carrier frequency voltage, the detected bridge output and its Hilbert transform, for use in digital phase detection will look like (Figure 5).

The Hilbert transform phase detector extracts the phase error by using trigonometric computations. The Digital Carrier Oscillator used in this kind of phase detector produces two signals and in-phase signal I = cos (ω₀t) and a quadrature signal Q = sin ( ω_ot).

If the input signal is a digital signal of the form

Then by trigonometric operations,

Where ‘u’ represents the Hilbert transform of the signal u
Since the analytic counterpart of the signal has to be determined for a discrete signal, the use of a Discrete Hilbert transform by Digital signal processing [4] done using Cizek’s algorithm [5]. This evaluates the discrete Hilbert Transform of a sequence x(v) as the first summation being from v = 0 to N-1 and the second from k = 1 to (N−1)/2 if N is odd and (N/2 -1) if N is even, where N is the number of samples. This can also be done through a matrix format for easy program in

The results of the measurements, when repeated with the bridge gave repeatable results among several sets of prepared samples from substances (Figure 6). The capacitance changes with addition of moisture content to the substance were also indicative of moisture content correctly.

The experiments used a Transformer Ratio Arm (TRA) bridge with a digitization bridge output and signal processing. In identifying the substances among their groups, such as raw rice and boiled rice for example, the curve of capacitance change provided a clear indicator. With the Hilbert transform based detection, the curves were same with all sample preparations. Out of these, two results of practical importance are given below.

1. In rice varieties, the experiments were done in flat large packed cells with measured addition of pure water (Figure 7).

2. The measurements were able to reveal, by their corrected components of conductance/capacitances, how the expensive “Ponni” variety had better quality in comparison to fast yield plantation varieties of rice like IR-20, “Kuruvai R” etc. The figure 9 shows the range of measured output with moisture

for the varieties. The Y axis shows the change in percent value of ΔX_c/X_c. It is to be noted that the higher priced rice has a larger range of variation. These high quality varieties are harvested with plantation duration of six months (“Ponni”) while the “Kuruvai” type is a short duration (about 3 months) yielding variety. So, this range of variation in the curve is an indicator of the quality, the large range of variation occurring for the high priced rice variety. The Boiled rice is a type which is prepared by boiling the rice with husk and drying, with subsequent husk removal. Since the boiling causes the range of measured capacitance component to be more, the curve shows it accordingly, after the non-linearity correction using the equation 911]. The curve shows a different trend for this substance. We do not want to guess the physical reason for this. A calibration curve was prepared using substance cells with varying amounts of water added by weight and homogenized. The results were tabulated and programmed into the microcomputer. The system is for developing a direct reading instrument for quality assessment of grain samples.

The results proved to be linearly related because of the relationship of the causal signal of conductance and capacitance in a specimen with the help of the Hilbert Transform based Quadrature phase sensitive detection, performed by digital signal processing. The method is likely to be applicable for many other physical measurements using the principle of conductance or capacitance changes even in presence of nonlinearity precisely. Thus, we have developed a measurement scheme for tackling nonlinearity in transducers for quite many physical measurements, for which two examples have been provided.

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