2.1. Model of the Transducer
In the conventional EMFM, a pair of sensing electrodes are directly immersed in the conductive fluid, which would result in the formation of electrode-electrolyte interfaces. Since the conductive fluid moves in a direction perpendicular to the magnetic field with some velocity, induced voltage would be generated between the electrodes. With the effect of the Lorentz force, charged ions in the solution are attracted in the electrode-electrolyte interface [
17,
18,
19]. This phenomenon is usually called the electrical double layer (EDL), the impedance of which could be described by a constant phase element (CPE) as follows:
where the exponent
n is between 0.5 and 1.0, with 0.5 corresponding to the case of a Warburg impedance, a charge diffusion dominated element, and 1.0 to a pure capacitance element.
Some contributions have been made on the model of the transducer [
5,
7]. An equivalent circuit that models the transducer full of fluid is illustrated in
Figure 1. The impedance (denoted “
”) between the electrode (
or
) and the ground comprises a parallel connection of a capacitor
representing the distributed capacitance between the electrode and the ground, a resistance
representing the solution resistance and a constant phase element (CPE)
, which describes the behavior of the EDL, wherein the latter are connected in series with each other. Then, the impedance
could be represented as:
Figure 1.
Equivalent circuit model of the transducer full of fluid.
Figure 1.
Equivalent circuit model of the transducer full of fluid.
Consider some values of
in the range of 0.25 up to 250 kΩ, which corresponds to the solution conductivity from 2000 down to 2 μS/cm,
,
n = 0.8,
= 100 pF representing a short signal line. According to Equation (
2), the Nyquist plot of the impedance
over a frequency range from 2 Hz–10 kHz (restricted from the limited practical resources) is shown in
Figure 2. In the low frequency range, the impedance
appears as a line of a certain slope, mainly dependent on the EDL impedance
. Yet, in the intermediate frequency range,
is mostly determined by the solution resistance
, especially as the reactance of
is close to zero. In the relatively high frequency range,
, contributed mainly by the capacitance
and the solution resistance
, appears as a semicircle at the condition of
large enough.
Figure 2.
Nyquist plot of the impedance for known parameters.
Figure 2.
Nyquist plot of the impedance for known parameters.
Thus, the impedance contains the properties of the EMFM. By using the electrical impedance spectroscopy technique, the related valuable parameters could be extracted from the impedance spectra of the transducer.
2.2. Fluid-Velocity Signal Loss under Different Modes of Electrical Excitation
Without electrical stimulus, the electrode measurement loop of the EMFM is illustrated schematically in
Figure 3a. It consists of flow-induced voltage
, electrode impedance
and an INA, whose input impedance is
and amplification coefficient is
. Deduced from Ohm’s law, the differential voltage of the INA’s input signal is given by:
Figure 3.
Principal diagram of the electrode measurement loop of the electromagnetic flow meter (EMFM). (a) No electrical excitation; (b) parallel electrical excitation; mode (c) serial electrical excitation mode.
Figure 3.
Principal diagram of the electrode measurement loop of the electromagnetic flow meter (EMFM). (a) No electrical excitation; (b) parallel electrical excitation; mode (c) serial electrical excitation mode.
To acquire the impedance spectrum of
, there must be another known stimulus fed into the transducer via the sensing electrodes. As shown in
Figure 3b, a common method is to import the electrical excitation in parallel connection mode, wherein the electrical excitation module, consisting of an excitation voltage source and a sample resistance
, is in parallel connection with the input of the INA via an analog switch
. Supposing the internal impedance of the voltage source
is zero, the impedance of the excitation module could be written as
, with
denoting the switch impedance. Since magnetic excitation alternates with electrical excitation in the time domain in this research,
=0 in the period of magnetic excitation. Then, the differential voltage of the INA’s input signal
is:
From Equation (
3) and Equation (
4), the fluid-velocity signal loss percentage in the parallel electrical excitation mode could be calculated as:
Generally,
is of the order of magnitude of
, and that of
would not be larger than
, even at the condition of serious electrode adhesion. Moreover,
is equivalent to
during magnetic excitation, due to the fact that
under off-state is of the order of magnitude of
to
, much larger than
. Thus, Equation (
5) becomes:
Consider the impedance
in cases such as low medium conductivity or serious electrode contamination. The related fluid-velocity signal loss percentages are summarized in
Table 1. Supposing the value of
=0.1 MΩ corresponding to the medium conductivity of 10 μS/cm,
is calculated for some values of
in the range from 0.1–10 MΩ at the conditions of
= 10 MΩ, 30 MΩ, 100 MΩ. If
= 10 MΩ,
= 0.99% at
= 100 kΩ;
= 9.08% at
= 1 MΩ;
= 49.75% at
= 10 MΩ. If
= 100 MΩ,
= 0.1% at
= 100 kΩ;
= 0.99% at
= 1 MΩ;
= 9% at
= 10 MΩ. In addition, to obtain a wide measurement range, several sample resistors with resistances of different orders of magnitude are required to be connected in parallel to match the measured impedance
, making the fluid signal loss more serious.
Table 1.
Fluid-velocity signal loss in the parallel electrical excitation mode.
Table 1.
Fluid-velocity signal loss in the parallel electrical excitation mode.
| |
---|
| | |
---|
0.1 | 0.99% | 0.33% | 0.1% |
1 | 9.08% | 3.22% | 0.99% |
10 | 49.75% | 24.81% | 9% |
The electrode measurement loop of the EMFM in the serial electrical excitation mode proposed in this paper, is illustrated schematically in
Figure 3c. The electrical excitation module, consisting of a current source
(and
) and a resistance
, serving as a current-to-voltage converter (
=3.6 kΩ), is in series between the sensing electrode and the input of the INA. Moreover, there is a pair of sample capacitances
connected to the inputs of the INA. During magnetic excitation, the excitation source with zero current (
and
) is identical to an ideal open circuit. On the basis of transient circuit analysis, the differential voltage of the INA’s input signal
is:
where
. Having
, so Equation (
7) becomes:
By Equation (
8), provided that the time interval between the magnetic excitation starting and the fluid-velocity signal sampling is larger than
, the fluid-velocity signal loss percentage would be smaller than the value of
.
Table 2 illustrates the related time intervals (
) to some values of
in the range from 0.1–10 MΩ at the conditions of
= 1 nF, 10 nF, 100 nF. For most applications, such as medium conductivity measurement, electrode adhesion evaluation and empty pipe detection, the capacitance
would be selected with a value equal or smaller than 10 nF. If
= 10 nF,
= 50 ms at
= 1 MΩ;
= 500 ms at
= 10 MΩ. The above time intervals are feasible for the reason that the magnetic excitation cycle of the EMFM is usually 16 or 32 multiples of the power frequency cycle (20 ms in a 50-Hz system). Nevertheless, for special applications, such as the property detection of the electrode adhesion, to obtain the parameters of the EDL, the capacitance of
is suggested to be larger than 10 nF. At this time, even when the electrode is seriously contaminated (not fully insulated), the signal loss still could be small.
Table 2.
Suitable time intervals in the serial electrical excitation mode.
Table 2.
Suitable time intervals in the serial electrical excitation mode.
| (ms) |
---|
= 1 nF | = 10 nF | = 100 nF |
---|
0.1 | 0.52 | 5.2 | 52 |
1 | 5 | 50 | 500 |
10 | 50 | 500 | 5000 |
Therefore, if the condition
(t denotes the time interval) was fulfilled, the conventional flow-rate measurement equation [
20] would be also available in the serial electrical excitation mode, given by:
where
B is the magnetic flux density,
D is the diameter of the tube in meters and
ν is the flow velocity.
2.3. EIS Measurement Principle in Serial Electrical Excitation Mode
The schematic diagram of a dual-excitation EMFM based on the PV cell is shown in
Figure 4. It consists of a pair of sensing electrodes (
and
), a pair of specially-designed PV converters, a pair of sample capacitances
and an INA. Each PV converter consists of a PV cell and a resistor
(a current-to-voltage converter) in parallel connection. During electrical excitation,
= 0. The PV cell (
or
), activated by an adjacent LED(
or
), which is driven by a controllable AC current (
or
), generates the photovoltaic current (
or
) based on the photovoltaic effect. The current signal for electrical excitation could be expressed mathematically as:
Figure 4.
Schematic diagram of the dual-excitation EMFM based on a PV cell.
Figure 4.
Schematic diagram of the dual-excitation EMFM based on a PV cell.
With the resistor
converting current to voltage, the terminal voltages of the two PV cells are
and
, respectively. Then, the voltage signal for electrical excitation becomes:
Thus, the differential voltage of the INA’s input signal could be calculated as:
where
is the impedance of the sample capacitance
. Since the signal
generated for electrical excitation is a periodic odd signal, according to the Fourier series theory,
could be represented as an infinite summation of sinusoidal harmonics.
where
and
are the magnitude and phase of the
i-th harmonic, respectively.
At the same time, the resulting signal also could be expressed as an infinite summation of sinusoidal harmonics.
where
and
are the magnitude and phase of the
i-th harmonic, respectively. Therefore, substituting Equation (
13) and Equation (
14) into Equation (
12), solving for
yields:
where
is the impedance
at the frequency
,
. Then, from Equation (
15), the amplitude and phase at the frequency
could be written as: