ES2292308A1 - Method for measuring electrical conductivity of liquid through conductivity cell, involves applying electrical stimulation signal through two electrodes and measuring same electrode voltage or electrical current produced in liquid - Google Patents
Method for measuring electrical conductivity of liquid through conductivity cell, involves applying electrical stimulation signal through two electrodes and measuring same electrode voltage or electrical current produced in liquid Download PDFInfo
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- 239000007788 liquid Substances 0.000 title claims abstract description 47
- 238000000034 method Methods 0.000 title claims abstract description 14
- 230000000638 stimulation Effects 0.000 title abstract 3
- 230000000737 periodic effect Effects 0.000 claims abstract description 3
- 238000005259 measurement Methods 0.000 claims description 18
- 230000005284 excitation Effects 0.000 claims description 7
- 239000000523 sample Substances 0.000 claims description 2
- 230000000694 effects Effects 0.000 abstract description 6
- 239000003990 capacitor Substances 0.000 abstract 1
- 238000001566 impedance spectroscopy Methods 0.000 description 4
- BASFCYQUMIYNBI-UHFFFAOYSA-N platinum Chemical compound [Pt] BASFCYQUMIYNBI-UHFFFAOYSA-N 0.000 description 4
- 238000006243 chemical reaction Methods 0.000 description 2
- 239000003792 electrolyte Substances 0.000 description 2
- 230000014509 gene expression Effects 0.000 description 2
- 150000002500 ions Chemical class 0.000 description 2
- 239000000203 mixture Substances 0.000 description 2
- 229910052697 platinum Inorganic materials 0.000 description 2
- OKTJSMMVPCPJKN-UHFFFAOYSA-N Carbon Chemical compound [C] OKTJSMMVPCPJKN-UHFFFAOYSA-N 0.000 description 1
- RTAQQCXQSZGOHL-UHFFFAOYSA-N Titanium Chemical compound [Ti] RTAQQCXQSZGOHL-UHFFFAOYSA-N 0.000 description 1
- 230000007797 corrosion Effects 0.000 description 1
- 238000005260 corrosion Methods 0.000 description 1
- 230000003111 delayed effect Effects 0.000 description 1
- 238000001514 detection method Methods 0.000 description 1
- 230000006866 deterioration Effects 0.000 description 1
- 238000006073 displacement reaction Methods 0.000 description 1
- 238000005868 electrolysis reaction Methods 0.000 description 1
- 229910002804 graphite Inorganic materials 0.000 description 1
- 239000010439 graphite Substances 0.000 description 1
- 239000000463 material Substances 0.000 description 1
- 238000000691 measurement method Methods 0.000 description 1
- 230000010349 pulsation Effects 0.000 description 1
- 238000004092 self-diagnosis Methods 0.000 description 1
- 229910001220 stainless steel Inorganic materials 0.000 description 1
- 239000010935 stainless steel Substances 0.000 description 1
- 229910052719 titanium Inorganic materials 0.000 description 1
- 239000010936 titanium Substances 0.000 description 1
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
- G01N27/00—Investigating or analysing materials by the use of electric, electrochemical, or magnetic means
- G01N27/02—Investigating or analysing materials by the use of electric, electrochemical, or magnetic means by investigating impedance
- G01N27/04—Investigating or analysing materials by the use of electric, electrochemical, or magnetic means by investigating impedance by investigating resistance
- G01N27/06—Investigating or analysing materials by the use of electric, electrochemical, or magnetic means by investigating impedance by investigating resistance of a liquid
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R27/00—Arrangements for measuring resistance, reactance, impedance, or electric characteristics derived therefrom
- G01R27/02—Measuring real or complex resistance, reactance, impedance, or other two-pole characteristics derived therefrom, e.g. time constant
- G01R27/22—Measuring resistance of fluids
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Abstract
Description
Procedimiento para medir la conductividad eléctrica de líquidos.Procedure to measure conductivity liquid electric.
Instrumentación de medida y control.Measurement and control instrumentation.
La conductividad de un líquido es un parámetro muy interesante que aporta información sobre la presencia de iones en el seno de dicho líquido. Para medir la conductividad basta aplicar la ley de Ohm: si se inyecta una señal (tensión o corriente) en el líquido, se obtiene una respuesta (corriente o caída de tensión) en el líquido que depende de cuál sea su conductividad. Pero mientras las corrientes eléctricas en los circuitos electrónicos están formadas por electrones, las corrientes eléctricas en un líquido se deben al desplazamiento de los iones que hay en él. Para convertir las corrientes electrónicas en corrientes iónicas se emplean electrodos conductores inmersos en el líquido. En los electrodos se produce una reacción química que realiza dicha conversión de un tipo de corrientes a otro.The conductivity of a liquid is a parameter very interesting that provides information about the presence of ions within said liquid. To measure conductivity, just apply Ohm's law: if a signal is injected (voltage or current) in the liquid, you get a response (current or voltage drop) in the liquid that depends on what your conductivity. But while the electric currents in the electronic circuits are made up of electrons, the electrical currents in a liquid are due to the displacement of the ions in it. To convert electronic currents in ionic currents conductive electrodes immersed in the liquid A chemical reaction occurs in the electrodes that performs said conversion from one type of stream to another.
El paso de la corriente eléctrica a través de la interfaz entre el electrodo y el líquido experimenta una dificultad que depende de la composición y área del electrodo, de la composición (y por ende de la conductividad) del líquido, de la intensidad de dicha corriente, de su frecuencia, y de otros parámetros (temperatura del líquido, rugosidad de los electrodos, etc.). Esta dificultad al paso de la corriente se modela mediante una impedancia eléctrica para cada electrodo, de modo que al inyectar una señal en el líquido, la respuesta obtenida depende no sólo de la conductividad del líquido sino también de los electrodos utilizados. Por consiguiente, si se aplica tensión y se mide corriente con los mismos electrodos, ésta puede que dependa más de los electrodos que de la conductividad del líquido que se desea medir. Si se inyecta corriente, la caída de potencial entre los electrodos con los que se ha inyectado la corriente puede que se deba en su mayor parte a los electrodos y muy poco al líquido.The passage of electric current through the interface between the electrode and the liquid experiences a difficulty which depends on the composition and area of the electrode, the composition (and therefore of the conductivity) of the liquid, of the intensity of said current, its frequency, and others parameters (liquid temperature, electrode roughness, etc.). This difficulty in passing the current is modeled by an electrical impedance for each electrode, so that at inject a signal into the liquid, the response obtained depends not only of the conductivity of the liquid but also of the electrodes used Therefore, if voltage is applied and measured current with the same electrodes, this may depend more on the electrodes that of the desired conductivity of the liquid to size. If current is injected, the potential drop between electrodes with which the current has been injected may be due mostly to the electrodes and very little to the liquid.
Este grave efecto de la impedancia de los electrodos ha motivado la búsqueda de diferentes estrategias para obviarla. Una estrategia es utilizar dos electrodos para inyectar una corriente y dos electrodos diferentes para medir la caída de tensión en el líquido. Si el circuito de medida de tensión tiene una impedancia de entrada mucho mayor que la impedancia de los dos electrodos de medida de tensión, por estos dos electrodos apenas circulará corriente eléctrica alguna, de modo que la tensión detectada entre ellos será la producida en el seno del líquido por la corriente inyectada. El inconveniente de esta estrategia es que la sonda de medida debe incorporar cuatro electrodos en vez de dos, con lo que resulta más cara, a la vez que aumenta la complejidad de los circuitos electrónicos que miden la caída de tensión entre el segundo par de electrodos.This serious effect of the impedance of electrodes has motivated the search for different strategies to ignore it. One strategy is to use two electrodes to inject a current and two different electrodes to measure the drop of tension in the liquid. If the voltage measurement circuit has an input impedance much greater than the impedance of the two voltage measuring electrodes, for these two electrodes just some electric current will circulate, so that the voltage detected among them will be produced within the liquid by the injected current. The drawback of this strategy is that The measuring probe must incorporate four electrodes instead of two, which is more expensive, while increasing the complexity of the electronic circuits that measure the voltage drop between the Second pair of electrodes.
Si se desea emplear sólo dos electrodos, cabe pensar en una estrategia basada en medir no sólo la impedancia total entre dichos electrodos, sino también la impedancia de cada uno de los elementos del circuito eléctrico formado por los dos electrodos y el líquido que haya entre ellos. Para ello hay que modelar dichas impedancias y concebir un método que permita identificarlas por separado. Un modelo convencionalmente aceptado es el de la figura 1: R_{s} es la resistencia que ofrece el líquido al paso de la corriente eléctrica; Z_{CPE} es la impedancia que modela a los dos electrodos; y C_{eq} es la capacidad equivalente entre los electrodos en ausencia de líquido, e incluye la capacidad del cable que conecta los electrodos a los circuitos electrónicos de excitación y detección. Aunque hay dos electrodos, su efecto se modela mediante una única impedancia porque interesa su efecto global: si se trata de dos electrodos iguales, sus impedancias serán iguales y se pueden sumar; y si un electrodo tiene un área mucho mayor que el otro, su impedancia será mucho menor, de forma que predominará la impedancia del electrodo más pequeño.If it is desired to use only two electrodes, a strategy based on measuring not only the total impedance between said electrodes, but also the impedance of each of the elements of the electrical circuit formed by the two electrodes and the liquid between them . To do this, these impedances must be modeled and a method designed to identify them separately. A conventionally accepted model is that of Figure 1: R s is the resistance offered by the liquid to the passage of electric current; Z CPE is the impedance that models the two electrodes; and C eq is the equivalent capacity between the electrodes in the absence of liquid, and includes the capacity of the cable that connects the electrodes to the electronic excitation and detection circuits. Although there are two electrodes, their effect is modeled by a single impedance because their global effect is of interest: if they are two equal electrodes, their impedances will be equal and can be added; and if one electrode has a much larger area than the other, its impedance will be much lower, so that the impedance of the smaller electrode will predominate.
Para describir Z_{CPE} se han propuesto a su vez distintos modelos (ver por ejemplo el artículo de L. A. Geddes, Historical evolution of circuit models for the electrode-electrolyte interface, publicado en Annals of Biomedical Engineering vol. 25, págs. 1-14, 1997); algunos modelos son tan sencillos como una simple capacidad, otros son un poco más complejos pues incluyen además una resistencia en paralelo con dicha capacidad. En cualquier caso, la impedancia resultante tiene un parte real y una parte imaginaria, y para identificar cada uno de los elementos que intervienen en dichas partes, incluida la resistencia del líquido, basta hacer tantas mediciones independientes como parámetros distintos se desee identificar. Un método de medida común es inyectar, simultánea o sucesivamente, estímulos sinusoidales de distintas frecuencias y medir el efecto del estímulo a la frecuencia respectiva. Con un par de frecuencias, por ejemplo, se pueden identificar dos elementos distintos (ver por ejemplo la patente US 6 369 579 B1). En este caso, las dos frecuencias de medida deben estar suficientemente separadas para que los errores inevitables en todo proceso de medición no impidan obtener dos ecuaciones realmente independientes. Si se desea identificar más elementos, hay que inyectar más señales sinusoidales de frecuencias diferentes. Cuando se inyectan varias señales sinusoidales, se habla de espectroscopia de impedancias (ver por ejemplo el libro de J. Ross Macdonald, Impedance spectroscopy, Wiley, 1988). También se pueden inyectar señales que no sean sinusoidales, pero las señales sinusoidales tienen la ventaja de que llevan a sistemas de ecuaciones bastante simples a partir de los cuales es relativamente fácil obtener las expresiones analíticas de los elementos que se desea identificar. Ahora bien, se pueden emplear también estímulos con formas de onda que den respuestas más fáciles de medir que las correspondientes a los estímulos sinusoidales, si se es capaz de resolver el sistema de ecuaciones resultante. Con estímulos de dos niveles (señal cuadrada), por ejemplo, se han podido identificar dos componentes (ver por ejemplo la patente US 5 334 940), e incluso tres componentes (tal como se describe por ejemplo en el artículo de J. Lario-García y R. Pallàs-Areny, Measurement of three independent components in impedance sensors using a single square wave, publicado en Sensors and Actuators A, Vol. 11, págs. 164-170, 2004). También se ha propuesto la utilización de estímulos con forma de onda triangular (J. Wu and J. P. Stark, A low-cost aproach for measuring electrical conductivity and relative permitivity of liquids by triangular waveform voltage at low frequencies, publicado en Measurement Science and Technology, vol. 16, págs. 1234-1240, 2005), pero hay que realizar muchas medidas analógicas y el tiempo de cálculo es elevado. Otro método se basa en medir la caída de tensión entre los electrodos al cabo de unos microsegundos después de aplicar un escalón de corriente (B. Carkhuff y R. Cain, Corrosion sensors for concrete bridges, publicado en IEEE Instrumentation and Measurement Magazine, vol. 6, núm. 2, págs. 19-24, 2003). Este método, sin embargo, sólo determina la conductividad del líquido, no la impedancia de los electrodos, de forma que no obtienen esta información que puede ser importante para el autodiagnóstico, pues una impedancia de electrodo excesivamente alta puede ser un indicio de su deterioro o ensuciamiento.Different models have been proposed to describe Z_ {CPE} (see, for example, the LA Geddes article, Historical evolution of circuit models for the electrode-electrolyte interface , published in Annals of Biomedical Engineering vol. 25, pp. 1- 14, 1997); some models are as simple as a simple capacity, others are a bit more complex because they also include a resistance in parallel with that capacity. In any case, the resulting impedance has a real part and an imaginary part, and to identify each of the elements involved in these parts, including the resistance of the liquid, it is enough to make as many independent measurements as different parameters to identify. A common measurement method is to inject, simultaneously or successively, sinusoidal stimuli of different frequencies and measure the effect of the stimulus at the respective frequency. With a pair of frequencies, for example, two distinct elements can be identified (see, for example, US 6 369 579 B1). In this case, the two measurement frequencies must be sufficiently separated so that the inevitable errors in any measurement process do not prevent obtaining two really independent equations. If you want to identify more elements, you have to inject more sinusoidal signals of different frequencies. When several sinusoidal signals are injected, we speak of impedance spectroscopy (see for example J. Ross Macdonald's book, Impedance spectroscopy , Wiley, 1988). It is also possible to inject signals that are not sinusoidal, but sinusoidal signals have the advantage that they lead to fairly simple systems of equations from which it is relatively easy to obtain the analytical expressions of the elements to be identified. However, stimuli with waveforms that give answers that are easier to measure than those corresponding to sinusoidal stimuli can also be used, if the resulting system of equations is capable of solving. With two-level stimuli (square signal), for example, two components have been identified (see for example US Patent 5 334 940), and even three components (as described for example in the article by J. Lario- García and R. Pallàs-Areny, Measurement of three independent components in impedance sensors using a single square wave , published in Sensors and Actuators A, Vol. 11, pp. 164-170, 2004). The use of triangular waveform stimuli has also been proposed (J. Wu and JP Stark, A low-cost aproach for measuring electrical conductivity and relative permitivity of liquids by triangular waveform voltage at low frequencies , published in Measurement Science and Technology, vol. 16, pp. 1234-1240, 2005), but many analog measurements have to be made and the calculation time is high. Another method is based on measuring the voltage drop between the electrodes after a few seconds after applying a current step (B. Carkhuff and R. Cain, Corrosion sensors for concrete bridges , published in IEEE Instrumentation and Measurement Magazine , vol. 6, No. 2, pp. 19-24, 2003). This method, however, only determines the conductivity of the liquid, not the impedance of the electrodes, so that they do not obtain this information that may be important for the self-diagnosis, since an excessively high electrode impedance may be an indication of its deterioration or soiling.
El inconveniente de los modelos basados en elementos simples (resistencias y capacidades) es que su validez depende mucho del material de los electrodos y de la conductividad del líquido. En concreto, dejan de ser válidos cuando la resistencia del líquido es del orden de magnitud de la impedancia de los electrodos, y cuando el comportamiento de los electrodos no se puede modelar con pocos elementos simples de valor constante en todo el margen de frecuencias de las señales inyectadas para la medición y para todo el margen de conductividades de interés. Cuando Z_{CPE} no se puede describir con ninguna combinación de componentes electrónicos elementales (resistencias y capacidades), se emplea lo que se ha convenido en denominar un elemento de fase constante (ver por ejemplo el libro de S. Grimnes y O. G. Martinsen, Bioimpedance and bioelectricity basics, Academic Press 2000). En este caso, si se considera que la impedancia de C_{eq} es mucho más alta que la del conjunto electrodos-líquido, la impedancia entre los terminales del circuito de la figura 1 tiene la siguiente expresión matemáticaThe drawback of models based on simple elements (resistances and capacities) is that their validity depends greatly on the material of the electrodes and the conductivity of the liquid. In particular, they cease to be valid when the resistance of the liquid is of the order of magnitude of the electrode impedance, and when the behavior of the electrodes cannot be modeled with few simple elements of constant value over the entire frequency range of the signals injected for measurement and for the entire range of conductivities of interest. When Z CPE cannot be described with any combination of elementary electronic components (resistors and capacities), what has been agreed to be called a constant phase element is used (see for example the book by S. Grimnes and OG Martinsen , Bioimpedance and bioelectricity basics , Academic Press 2000). In this case, if the impedance of C eq is considered to be much higher than that of the electrode-liquid assembly, the impedance between the terminals of the circuit of Figure 1 has the following mathematical expression
donde \omega es la frecuencia angular (pulsación) del estímulo con el que se mide la impedancia, R_{s} es la resistencia del líquido entre los dos electrodos, A y \alpha son los parámetros que definen la impedancia de los electrodos (\alpha vale entre 0 y 1). Así pues, para obtener R_{s} hay que descontar de la impedancia total la debida a los electrodos, y esto implica determinar los parámetros A y \alpha. Dado que estos parámetros no son constantes sino que cambian en función precisamente de la conductividad del líquido y de la densidad de corriente, no se puede emplear una corrección fija para cada tipo de electrodos concreto. Una opción es identificarlos mediante espectroscopia de impedancias, pero esto exige inyectar estímulos sinusoidales (ver patente US 6 369 579 B1 o J. Ross Macdonald, Impedance Spectroscopy, Wiley, 1988).where? is the angular frequency (pulsation) of the stimulus with which the impedance is measured, R s is the resistance of the liquid between the two electrodes, A and? are the parameters that define the impedance of the electrodes ( α is valid between 0 and 1). Thus, to obtain R s it is necessary to deduct from the total impedance due to the electrodes, and this implies determining the parameters A and α. Since these parameters are not constant but change depending on precisely the conductivity of the liquid and the current density, a fixed correction cannot be used for each specific type of electrode. One option is to identify them by impedance spectroscopy, but this requires injecting sinusoidal stimuli (see US Patent 6 369 579 B1 or J. Ross Macdonald, Impedance Spectroscopy , Wiley, 1988).
La presente invención consiste en un procedimiento para determinar la conductividad eléctrica de líquidos mediante dos electrodos inmersos en el líquido que inyectan una señal de excitación periódica cuya amplitud tiene sólo dos niveles: l_{0} y -l_{0} (señal cuadrada, figura 3a) o tres niveles distintos: l_{0}, -l_{0} e l_{1}, (pulsos, figura 3b) o tiene forma triangular (figura 3c). La respuesta del líquido al estímulo se mide con los mismos electrodos de la forma siguiente: a partir de un instante de referencia de la señal de excitación, se mide la amplitud de la respuesta en un número n de puntos de la forma de onda resultante (figura 2); la ubicación temporal de dichos n puntos se elige de antemano de forma arbitraria pero conocida. A partir de la relación matemática que da la respuesta del conjunto liquido-electrodos en función de la señal de excitación y la impedancia del sistema, modelada según la figura 1 y con C_{eq} supuesta muy pequeña, sustituyendo los valores de las amplitudes de la respuesta en los n instantes elegidos se forma un sistema de n ecuaciones que se puede resolver para determinar, como máximo, tantos parámetros como puntos de medida se haya elegido.The present invention consists of a method for determining the electrical conductivity of liquids by means of two electrodes immersed in the liquid that inject a periodic excitation signal whose amplitude has only two levels: l 0 and - l 0 (square signal figure 3a) or three levels: l _ {0}, - l _ {0} and {1} l (pulses, 3b) or is triangular (figure 3c). The response of the liquid to the stimulus is measured with the same electrodes as follows: from a reference instant of the excitation signal, the amplitude of the response is measured in a number n of points of the resulting waveform ( figure 2); the temporary location of said n points is chosen in advance in an arbitrary but known way. From the mathematical relationship given by the response of the liquid-electrode assembly as a function of the excitation signal and the impedance of the system, modeled according to Figure 1 and with assumed very small C eq, replacing the values of the amplitudes From the answer in the n chosen moments a system of n equations is formed that can be solved to determine, as a maximum, as many parameters as measurement points have been chosen.
Con este procedimiento se consigue: (a) emplear sólo dos electrodos; (b) emplear una señal de excitación más fácil de generar que un conjunto de señales sinusoidales de distinta frecuencia; (c) determinar no sólo la conductividad del líquido, sino también la impedancia de los electrodos, incluso si ésta es grande. Esto permite utilizar, por ejemplo, electrodos de titanio o de acero inoxidable, que son mucho más robustos que los electrodos de platino, platino ennegrecido, o grafito, comunes en aplicaciones de laboratorio.With this procedure it is possible to: (a) use only two electrodes; (b) use an easier excitation signal of generating a set of different sinusoidal signals frequency; (c) determine not only the conductivity of the liquid, but also the impedance of the electrodes, even if this is big. This allows, for example, titanium electrodes or stainless steel, which are much more robust than electrodes of platinum, blackened platinum, or graphite, common in applications from laboratory.
La figura 2 muestra una forma de realizar el método. En este caso la señal de estímulo es una corriente (A1) y se detecta la caída de tensión V_{x} en la impedancia Z(s) (1) que modela el conjunto formado por los electrodos (A2) y electrolito (A3). s es el operador de Laplace y vale j\omega, donde j es el número imaginario que vale j = \sqrt{-1} y \omega es la frecuencia angular (pulsación) de la señal aplicada.Figure 2 shows a way of performing the method. In this case the stimulus signal is a current (A1) and the voltage drop V x is detected at the impedance Z ( s ) (1) that models the set formed by the electrodes (A2) and electrolyte (A3 ). s is the Laplace operator and is worth j \ omega, where j is the imaginary number that is worth j = \ sqrt {-1} and \ omega is the angular frequency (pulse) of the applied signal.
La tensión medida en bornes de la impedancia Z(s) (1) de la figura 2 cuando se aplica una fuente de corriente cuadrada con transformada de Laplace l(s) esThe voltage measured at terminals of the impedance Z ( s ) (1) of Figure 2 when a square current source with Laplace transform l ( s ) is applied is
Un método para obtener la transformada inversa de Laplace de V(s) es descomponer la señal cuadrada de amplitud l_{0} y período 2T como la suma de señales escalón retrasadas un tiempo T la una respecto de la otra. Para un sistema lineal, la tensión en bornes de la impedancia eléctrica v(t) se puede obtener como la suma de la tensión producida por cada escalón de corriente a la entrada.One method of obtaining the inverse Laplace transform of V ( s ) is to decompose the square signal of amplitude l0 and period 2 T as the sum of stepped signals delayed one time T from one another. For a linear system, the voltage at terminals of the electrical impedance v ( t ) can be obtained as the sum of the voltage produced by each current step at the input.
La transformada de Laplace de un escalón de corriente esThe Laplace transform of a step of current is
La tensión en bornes de la impedancia de la red de la figura 2 cuando se aplica un escalón de corriente esThe terminal voltage of the network impedance of figure 2 when a current step is applied is
Para obtener la tensión en bornes de la impedancia Z(s) en el dominio del tiempo se calcula la transformada inversa de Laplace de (4).To obtain the voltage at terminals of the impedance Z ( s ) in the time domain, the inverse Laplace transform of (4) is calculated.
La transformada inversa del primer término del (5) es la de un escalón de tensión,The inverse transform of the first term of (5) is that of a voltage step,
Utilizando las tablas de la transformada inversa de Laplace se obtiene la transformada inversa del segundo término de (5).Using the tables of the inverse transform from Laplace you get the inverse transform of the second term of (5).
Identificando el término de la izquierda de (7) con el segundo término de (5) se tieneIdentifying the term on the left of (7) with the second term of (5) you have
Por lo tanto, (5) es igual aTherefore, (5) is equal to
Una señal cuadrada de corriente se puede descomponer en suma de escalones u(t-MT),A square current signal can be decomposed in sum of steps u ( t-MT ),
donde u(t) es el escalón unidad, l_{0} es la corriente de pico de la señal cuadrada y T es el semineríndn de la señal cuadrada.where u (t) is the unit step, l _ {0} it is the peak current of the square signal and T is the semineríndn of the square signal.
Debido al carácter capacitivo del elemento de fase constante Z_{CPE}, cuando \alpha > 0, éste se comporta como un integrador cuando se aplica una fuente de corriente (dependencia de t^{\alpha} en (8)). Al medir conductividad es necesario que la tensión media aplicada a una célula de conductividad sea cero para evitar efectos de electrólisis. Para que se cumpla esta condición, la señal cuadrada de corriente aplicada ha de tener la amplitud del primer escalón u(t) = 0,5, y la amplitud del segundo escalón u(t-T) = -1,5.Due to the capacitive nature of the constant phase element Z CPE, when?> 0, it behaves like an integrator when a current source is applied (dependence on t ? In (8)). When measuring conductivity it is necessary that the average voltage applied to a conductivity cell be zero to avoid electrolysis effects. For this condition to be fulfilled, the square signal of applied current must have the amplitude of the first step u ( t ) = 0.5, and the amplitude of the second step u ( tT ) = -1.5.
La tensión en bornes de la impedancia Z(s) (1) producida por la corriente cuadrada (9) en el instante t esThe terminal voltage of the impedance Z ( s ) (1) produced by the square current (9) at time t is
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Para cualquier instante de tiempo dado, la tensión en bornes de Z(s) será el resultado de la suma de tensiones producidas por los escalones de corrientes previos. Para un tiempo t tal que 2MT \leq t < (2M + 1)T, siendo M un número entero y positivo, se puede considerar que t = 2MT + m\DeltaT, donde \DeltaT = T/N es el incremento del instante de medida. N es el número de medidas que se debe realizar más uno, y puede ser cualquier número positivo mayor que uno y 0 < m < N. La tensión en el instante t es entoncesFor any given time, the voltage at terminals of Z ( s ) will be the result of the sum of voltages produced by the previous current steps. For a time t such that 2 MT \ leq t <(2M + 1) T, M being a positive integer, can be considered to t = 2 MT + m \ DeltaT, where \? T = T / N is the increase of the instant of measurement. N is the number of measurements that must be taken plus one, and it can be any positive number greater than one and 0 < m < N . The tension in the instant t is then
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Los tres primeros términos de (11) son debidos a
los tres primeros escalones, de amplitudes +0,5l_{0},
-1,5l_{0} y +2l_{0}. El segundo término es la suma
de las tensiones producidas por todos los escalones de corriente
con amplitud -2l_{0} (en 3T, 5T...). El
tercer término es la suma de las tensiones producidas por todos los
escalones de corriente de amplitud +2l_{0} (en 4T,
6T...). En régimen permanente (M >> 1), cada
término de la ecuación (11) se puede escribir
comoThe first three terms of (11) are due to the first three steps, of amplitudes +0.5 l 0, -1.5 l 0 and +2 l 0. The second term is the sum of the voltages produced by all current steps with amplitude -2 l 0 (in 3 T , 5 T ...). The third term is the sum of the voltages produced by all levels of current amplitude +2 l _ {0} (in 4 T, 6 T ...). In permanent regime ( M >> 1), each term of equation (11) can be written
how
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Sustituyendo (13) (14) y (15) en (11) se tieneSubstituting (13) (14) and (15) into (11), have
Simplificando la ecuación (16) se obtiene la tensión en bornes de la impedancia eléctrica Z(s) cuando se aplica una señal cuadrada de corriente en régimen permanente,Simplifying equation (16), the voltage at terminals of the electrical impedance Z ( s ) is obtained when a square current signal is applied in permanent mode,
donde \Psi_{m}(\alpha, M, N) es la suma de las contribuciones de los escalones para t = 2MT + m\DeltaT, M es el número de periodos transcurridos desde que se aplicó la señalwhere \ Psi_ {m} (?, M , N ) is the sum of the contributions of the steps for t = 2 MT + m \ DeltaT , M is the number of periods elapsed since the signal was applied
yY
siendo N el número de medidas de tensión que se realizan más uno (e.g. N = 4 si se miden tres tensiones), m\DeltaT es el instante de medida dentro del semiperiodo T, y m es un entero entre 1 y N. \Psi_{m}(\alpha,M,N) es un sumatorio que, cuando M es suficientemente grande, que es el caso general, depende exclusivamente de \alpha y N, y que describe las contribuciones que todos los escalones de corriente anteriores aportan a las tensiones medidas (18).where N is the number of voltage measurements that are made plus one (eg N = 4 if three voltages are measured), m \ DeltaT is the instant of measurement within the half-period T , and m is an integer between 1 and N. \ Psi_ {m} (?, M, N) is a summation that, when M is large enough, which is the general case, depends exclusively on? and N , and that describes the contributions that all previous current steps contribute to the measured tensions (18).
A partir de las tres medidas de tensión V_{1}, V_{2}, y V_{3}, se calcula el parámetro \phi(N)From the three voltage measurements V 1, V 2, and V 3, the parameter ph ( N ) is calculated
y a partir de esta relación se obtiene a mediante cualquier programa comercial de matemáticas que permita hacer ajuste de curvas (Matlab®, Mapple®, Mathematica®, Excel®, Mathcad® ...). Por ejemplo, si se han medido las tensiones V_{1}, V_{2}, y V_{3} en los instantes T_{1}=T/4, T_{2}=2T/4 y T_{3}=3T/4, tenemos N = 4, yand from this relationship is obtained through any commercial math program that allows curves to be adjusted (Matlab®, Mapple®, Mathematica®, Excel®, Mathcad® ...). For example, if the voltages V 1, V 2, and V 3 have been measured at times T 1 = T / 4, T 2 = 2 T / 4 and T 3 = 3 T / 4, we have N = 4, and
La constante A se obtiene mediante una combinación lineal de cualquier par de tensiones medidas V_{m}, por ejemplo, cuando m =1 y m = 2,The constant A is obtained by a linear combination of any pair of measured voltages V m, for example, when m = 1 and m = 2,
A partir de A y \alpha se puede determinar la resistencia del líquido R_{s} tomando por ejemplo m = 1 en la ecuación de V_{m} (la tensión medida V_{1}),The resistance of the liquid R s can be determined from A and α by taking for example m = 1 in the equation of V m (the measured voltage V 1),
Por ultimo, a partir de R_{s} y de la constante de célula K se obtiene la conductividad del líquido,Finally, the conductivity of the liquid is obtained from R s and the cell constant K ,
La constante de célula K es un factor geométrico que se determina midiendo la conductancia de un líquido patrón de conductividad conocida.The K cell constant is a geometric factor that is determined by measuring the conductance of a standard liquid of known conductivity.
Claims (3)
- --
- al conjunto electrodos-líquido se aplica un generador de señal de excitación que es una fuente de corriente,to the electrode-liquid assembly a generator is applied of excitation signal that is a current source,
- --
- la señal eléctrica de excitación es periódica y es una señal cuadrada, tiene dos niveles de amplitud distintos, o es una señal de pulsos, tiene tres niveles de amplitud distintos, dondethe electrical excitation signal is periodic and is a square signal, it has two different amplitude levels, or it’s a pulse signal, It has three different amplitude levels, where
- --
- la respuesta medida es la caída de tensión en bornes de los electrodos de la célula de conductividad, y dondethe measured response is the voltage drop at electrode terminals of the conductivity cell, and where
- --
- estas tensiones se miden en tres o más instantes de tiempo predeterminados,these tensions are measured in three or more instants of time default
- --
- a partir del valor de la constante de célula K, que es conocida, se determina la conductividad eléctrica del líquido mediante el cálculo la resistencia del líquido R_{s}, y del cálculo los parámetros A y \alpha del elemento de fase constante CPE, que modela la impedancia de los electrodos.from the value of the cell constant K , which is known, the electrical conductivity of the liquid is determined by calculating the resistance of the liquid R_, and the calculation of the parameters A and α of the constant phase element CPE, which models the impedance of the electrodes.
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|---|---|---|---|
| ES200502219A ES2292308A1 (en) | 2005-09-08 | 2005-09-08 | Method for measuring electrical conductivity of liquid through conductivity cell, involves applying electrical stimulation signal through two electrodes and measuring same electrode voltage or electrical current produced in liquid |
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|---|---|---|---|
| ES200502219A ES2292308A1 (en) | 2005-09-08 | 2005-09-08 | Method for measuring electrical conductivity of liquid through conductivity cell, involves applying electrical stimulation signal through two electrodes and measuring same electrode voltage or electrical current produced in liquid |
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| ES2292308A1 true ES2292308A1 (en) | 2008-03-01 |
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| Application Number | Title | Priority Date | Filing Date |
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| ES200502219A Pending ES2292308A1 (en) | 2005-09-08 | 2005-09-08 | Method for measuring electrical conductivity of liquid through conductivity cell, involves applying electrical stimulation signal through two electrodes and measuring same electrode voltage or electrical current produced in liquid |
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| ES2343720A1 (en) * | 2008-12-12 | 2010-08-06 | Universitat Politecnica De Catalunya | Procedure to measure a modeled electrical impedance with a specific network of three components. (Machine-translation by Google Translate, not legally binding) |
| WO2018076325A1 (en) * | 2016-10-31 | 2018-05-03 | City University Of Hong Kong | Method and apparatus for use in electric circuit |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4119909A (en) * | 1977-03-31 | 1978-10-10 | Radian Corporation | Pulsed DC transient conductivity measurement system |
| US4516077A (en) * | 1983-06-14 | 1985-05-07 | The United States Of America As Represented By The Secretary Of The Navy | Apparatus for and a method of measuring the intrinsic time constant of liquids |
| DE3517772A1 (en) * | 1985-05-17 | 1986-11-20 | Dr. A. Kuntze GmbH & Co KG, 4000 Düsseldorf | Method for measuring the electrolytic conductivity of fluids |
| ES2041776T3 (en) * | 1987-03-30 | 1993-12-01 | Yokogawa Europe B.V. | METHOD AND DEVICE TO MEASURE THE CONDUCTIVITY OF A LIQUID, WITH WHICH THE INFLUENCE OF POLARIZATION IS COUNTERRESTED |
| ES2219047T3 (en) * | 1998-05-13 | 2004-11-16 | Johnsondiversey, Inc. | METHOD FOR MEASURING CONDUCTIVITY INCLUDING THE PROBE POLLUTION COMPENSATION. |
-
2005
- 2005-09-08 ES ES200502219A patent/ES2292308A1/en active Pending
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4119909A (en) * | 1977-03-31 | 1978-10-10 | Radian Corporation | Pulsed DC transient conductivity measurement system |
| US4516077A (en) * | 1983-06-14 | 1985-05-07 | The United States Of America As Represented By The Secretary Of The Navy | Apparatus for and a method of measuring the intrinsic time constant of liquids |
| DE3517772A1 (en) * | 1985-05-17 | 1986-11-20 | Dr. A. Kuntze GmbH & Co KG, 4000 Düsseldorf | Method for measuring the electrolytic conductivity of fluids |
| ES2041776T3 (en) * | 1987-03-30 | 1993-12-01 | Yokogawa Europe B.V. | METHOD AND DEVICE TO MEASURE THE CONDUCTIVITY OF A LIQUID, WITH WHICH THE INFLUENCE OF POLARIZATION IS COUNTERRESTED |
| ES2219047T3 (en) * | 1998-05-13 | 2004-11-16 | Johnsondiversey, Inc. | METHOD FOR MEASURING CONDUCTIVITY INCLUDING THE PROBE POLLUTION COMPENSATION. |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| ES2343720A1 (en) * | 2008-12-12 | 2010-08-06 | Universitat Politecnica De Catalunya | Procedure to measure a modeled electrical impedance with a specific network of three components. (Machine-translation by Google Translate, not legally binding) |
| WO2018076325A1 (en) * | 2016-10-31 | 2018-05-03 | City University Of Hong Kong | Method and apparatus for use in electric circuit |
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