Approximation and Correction of Measuring Transducer’s Characteristics

A very frequent case is that the characteristics of measuring transducers are non-linear and, apart from that, the output signal value may depend on several input values or unwanted influencing factors such as temperature, humidity etc. The need to adjust the form of information about the measured value to human perception possibilities, or to simplify the further processing of measurement information generally requires such adjusting of the slotted line element characteristics so that the resultant characteristics of the slotted line is linear, with assumed error, in the adopted coordinate system. (Bucci et al., 2000; Grzybowski @ Wagner, 1971; Iglesias @ Iglesias, 1988; Patranabis et al., 1988; Patranabis @ Ghosh, 1989)


Introduction
A very frequent case is that the characteristics of measuring transducers are non-linear and, apart from that, the output signal value may depend on several input values or unwanted influencing factors such as temperature, humidity etc. The need to adjust the form of information about the measured value to human perception possibilities, or to simplify the further processing of measurement information generally requires such adjusting of the slotted line element characteristics so that the resultant characteristics of the slotted line is linear, with assumed error, in the adopted coordinate system. (Bucci et al., 2000;Grzybowski @ Wagner, 1971;Iglesias @ Iglesias, 1988;Patranabis et al., 1988;Patranabis @ Ghosh, 1989) When the measuring transducer's characteristics is very non-linear, for example in the case of certain transducers for light intensity measurement, a problem with selecting a relevant A/D transducer may occur. All methods of correcting measuring transducer's characteristics cannot eliminate the error related to insufficient A/D transducer resolution. That issue is illustrated by figure 1. is sufficient to process the voltage signal U' x to an appropriate digital code N' 2 in the area in which the inclination of measuring transducer's characteristics is high. On the other hand, in the area of small inclination of the measuring transducer's characteristics, the changes in output voltage U" x of the transducer are too small to be properly processed by the A/D converter to digital code N" 2 . That problem may be partially bypassed using an A/D converter of high resolution (line 3). However, a better solution would be to adjust the processing characteristics of an A/D converter to the measuring transducer's characteristics (curve 4). The A/D converter characteristics should be described by a function reverse to the function describing the measuring transducer's characteristics and then the following linear relation would be obtained N=c x.
That chapter discusses issues related to the approximation of measuring transducer's characteristics and their correction in an analog to digital (A/D) converters The thesis will consider only static characteristics of measuring transducers.

Transducers with successive compensation
The block diagram of a transducer with weight compensation is presented in figure 2.1.

S/H U x
Digital-to-analog converter It consists of a sample-and-hold circuit, a comparator, a digital-to-analog converter, an approximating register and a control circuit. The transducer's operating principle can be found in readily available literature.
In transducers with weight compensation, the only available element, through which the shape of a transducer processing characteristics may be influenced is the reference voltage U REF . For that reason, the change of processing characteristics can be achieved by generating a non-linear reference voltage in relation to the digital output signal N: where: g(·) -measuring transducer's characteristics approximating function, a -proportionality coefficient.
The circuit of adjusting transducer processing characteristics with weight compensation by means of reference voltage generation method has been presented in figure 2.2.
www.intechopen.com Converter circuit consists of a comparator, a control circuit, a digital register and a reference voltage generator containing two coding circuits CODI and CODII, three digital-to-analog converters D/AI, D/AII and D/AIII and an  adder. D/A III converter is required to have an input for external reference voltage U 0III , thanks to which the multiplication function may be performed: where: k III -processing coefficient of D/A converter III. U III -output voltage of C/A converter III N" -digital signal value controlling D/A converter III, dependent on bits B i-1 ,  B 0 That allows for an adequate change of the u value of voltage spike to one bit, and hence the change of inclination of the generated voltage characteristics. The measuring transducer's characteristics should be approximated by a range-linear function with the assumed error, figure 2.3 a), while the range points A i , A j , … should be selected so that their values meet directly, or with the proportionality coefficient, the values N i , N j , … of the A/C transducer output signal. More significant bits (B n , B i ) of the approximating register, through the coding circuit KODI control the D/A converter I. The CODI circuit, which may be ROM memory, contains a board converting the signal value from the register to such a signal value controlling the D/A converter, as to ensure proper processing characteristics, figure 2.3 b).
The transducer converting process is conducted in the same way as for a linear transducer, figure 2.4, i.e. bits are issued one after another, starting from the most significant. When voltage from the generator exceeds the value of the voltage processed recently, the preset bit is withdrawn and another bit is incorporated. Testing all the more significant bits B n ,  B i means that a specified approximation range has been selected, in which the processed input signal is located (status of bits B n ,  B i provides the values N i , N j ). Appropriate inclination of the characteristics required for a given approximation range is received by changing the reference voltage of D/A converter III. The value of that voltage is obtained from D/A where:  I -coefficient value received from cell of memory COD I with address N'. k i -coefficient values of transducer processing.  II -coefficient value received from cell of memory COD II of address N'. N ' -more significant part of number N of control circuit (bits B n ,  B i ). N " -less significant part of number N of control circuit (bits B i-1 ,  B 0 ).
After summing up voltages from converters D/A in the adder, U ERF voltage is obtained, defined by the relation: The manner of adjusting A/D converter processing characteristics with weight compensation described above requires that the linearized measuring transducer's characteristics is approximated by a range-linear function. Therefore, designing the circuit should be started by determining the approximating function with the assumed error. However, in order not to complicate the shaping circuit, the lengths of approximation ranges should be equal.
The additional errors, which may be introduced by the shaping circuit, are errors mainly from the D/A converter I, especially the linearity error, and the adder. The errors of other transducers have a smaller impact since the values of voltages generated by them are smaller than the voltage from D/A converter I.
The above-mentioned shaping circuit, in a simplified configuration, was completed in a universal temperature meter cooperating with transducers Pt-100, Pt-1000, Ni-100, PtRh-Pt, NiCr-NiAl, Fe-CuNi. The simplified scheme of the meter which allows explanation of its Because the assumptions require that the absolute error of temperature measurement, submitted by the meter, to be within 1 °C for platinum transducers, 2 °C for nickel transducer and 5 °C for thermocouples, the characteristics of those transducers could be approximated by four sections. Thus, in the place of D/A converter I and the circuit COD I, a voltage generator, containing dividers R1 -R7 and half of analog selector MAX384, has been used. The DAC08 circuit has been used as a C/A III converter. That is a digital-toanalog converter with current output whose inclination of characteristics can be selected by changing the current input I IN . The current can be changed through selecting, in the second part of the analog selector, one of resistors R8 -R11. For each measuring transducer, a separate module containing the selector along with dividers of voltages and current resistors was intended. Module selection was done by specifying the signal for input EN.
Converter circuit operates like an ordinary converter with successive approximation, with addition that in the first four timings (bits B 9 and B 8 ), the approximation range is determined, where the U x voltage is located. The voltage from the generator, supplied to K comparator's reversing input, has the beginning of the approximation range value. The value of DAC08 transducer input current is also determined for the selected approximation range. In the other processing timings, the less significant bits are activated (B 7 -B 0 ) and, through voltage drop on resistor R caused by DAC08 transducer current output, the exact definition of U x voltage value takes place. The status of bits B 9 -B 0 directly provides the value of measured temperature.
The conducted meter measurements indicated that it meets the parameters adopted in the assumptions concerning ambient temperatures 5 -35 °C. Processing time was 2.5 ms.
Short converter processing time with successive approximation was used in the meter to measure carbon monoxide concentration. The TGS 2444 measuring transducer manufactured by Figaro USA Inc was used in the meter. In that transducer, the value of carbon monoxide concentration is processed into resistance. The measuring range is within 30 to 1000 ppm. Transducer characteristics has been presented in figure 2.6. It presents relative changes of output resistance, related to nominal resistance. Nominal resistance is given for concentration 100 ppm and ranges from 13.3 k to 133 k. In applied transducer the nominal resistance was 15.2 k.
www.intechopen.com Transducer characteristics, presented in figure 2.6, shows a great change of inclination. For 30 ppm, the inclination is -1,52 k/ppm and for 1000 ppm --1,26 /ppm (TGS 2611). With such high non-linearity of the measuring transducer's characteristics, using an A/C transducer with influenced processing characteristics is beneficial.
Because the meter was expected to be a portable device with battery power supply, a microcontroller MSP430F435 equipped with an LCD field direct control module was used in it. There is an A/C transducer with weight compensation with resolution of 12 bits and processing frequency of up to 200 khz within the internal structure of the micro-controller.
Calculations proved that the resolution of that transducer is too small to receive a proper resolution of measurement for high gas concentrations. Therefore, a circuit capable of adjusting transducer processing characteristics needed to be used.
The value of the digital A/D converter output signal is specified by the relation: The reference voltage U REF is accessible from the outside, which enables manipulation of transducer processing characteristics. The simplified scheme of the meter has been presented in figure 2.7. The device consists of a microcontroller MSP430F435 manufactured by Texas Instruments, an LCD field controlled directly from the microcontroller, a double D/A converter, an RS measuring transducer, highly stable resistors R1, R2 and R3, a heating circuit (elements T2 and R4), a circuit of switching on the measurement resistor (elements T1 and R5) and elements related to power supply and keyboard not shown in the figure.
The measuring transducer requires relevant control, which entail cyclic switching on the Rh heater and in the appropriate phase, switching on the RS measurement resistor, figure 2.8.
Proper operation of a measuring transducer requires that the heating is switched on for 14 ms and, after 981 ms from turning the heating off, the measurement resistor's power supply is switched on. The measurement resistor activation time is to last 5 ms and the measurement should be taken during that period.
The RS measuring transducer is connected with resistor R1 in a divider circuit, which enables changing the measuring transducer's resistance to a voltage corresponding to it. Output voltage from the divider, after strengthening x6 in the amplifier, is supplied to A/D converter input. VCC voltage value was adopted as 3.3V (voltage supplying the microcontroller). The value of resistor R1 has been selected to obtain the broadest range of voltage changes U x . Calculations, which have been omitted here, provided the value of 10 k, which provides the range of voltage changes from 0.47 V to 2 V. The chart of relation between gas concentration s and U x voltage value has been presented in figure 2.9.
The function presented in figure 2.9 is approximated by a range-linear function. With approximation error not exceeding 0.1%, 8 approximating sections were sufficient for approximation. However, the approximating function does not need to be continuous on range endpoints. The values of range points and coefficients k i of section inclination in particular ranges have been presented in figure 2.10 Inclination coefficients k i are determined by the relation:  The measurement takes place in two phases. In the first phase, input An1 is active and the initial voltage measurement U x takes place on the divider output. In that phase, the reference voltage for an A/C transducer is the micro-controller internal voltage. That measurement provides information in which range voltage U x is located. Then, for a given range, voltage U oi is generated from D/A converter output A for the amplifier. The value of that voltage is selected so that for the value U x = U Pi , the voltage on amplifier output is equal to zero. At the same time, from the transducer output B, a reference voltage U REF is generated for A/D converter. The value of that voltage should allow to obtain a satisfactory measurement resolution in a given range.
The voltage on output B of D/A converter is set proportionally to inclination coefficient value k i . Such proceeding stems from the fact that the measurement value in i-th range is specified by the relation: www.intechopen.com The value of voltage difference U x = U Pi , after processing in A/D converter, can be specified using the formulation (7): where: k -amplifier gain N xi -value of processing result in i-th range. Thus: If we assume that a linear relation has to be met: then the reference voltage should meet the condition: Meeting the aforementioned condition may be difficult owing to a limited range of changes in reference voltage. In that particular case, in accordance with catalogue data, the minimum value of that voltage cannot be smaller than 1.4 V, and the maximum may not be greater than VCC -0.2 V. For that reason, it has been assumed that the reference voltage value should ensure the required measurement resolution: The final measurement is calculated using the formulation (11). The shaping circuit introduces two kinds of errors. The first is connected with the approximation of the measuring transducer's characteristics by a range-linear function. It stems from the possible finite values of coefficients k i and threshold voltages U Pi , what is caused by their values being generated by converter D/A. The fact that the approximating sections do not have to be continuous in range points, facilitates approximation.
The second type of errors stems from the accuracy of generating the U REF voltage and errors of A/D converter. D/A converters are 10-bit converters with errors: linearity -1 LSB, offset -1 mV, characteristics inclination -0.5 LSB. Those errors will have the greatest impact at endpoints of the range, where there is greatest inclination of measuring transducer's characteristics. Taking into account the above values, it has been calculated from formulation (11) that the maximum absolute error s should not exceed the value of 8 ppm.
The conducted simulation research, in which the measuring transducer was replaced with a resistance decade, indicated that the relative measuring error d, (referred to the endpoint of the measuring range) in the whole measuring range did not exceed the value of 0.7%, figure 2.11.

Dual slope converters
Dual slope analog-to-digital converters with are very popular owing to their simple structure, high reliability, high resolution, high processing accuracy and were one of the first to be integrated. They are mainly intended for measuring constant voltages and most often cooperate with different kinds of measuring transducers.
The block diagram of a double integral converter has been presented in figure 3.1 and the waveforms in particular points of the transducer circuit in figure 3.2. The converting process takes place in two phases. In the first phase, which lasts a given time T 0 , the processed voltage U x is connected to the integrator's input through key k 1 . Then, after time T 0 , voltage U x is disconnected and reference voltage with polarization opposite to the www.intechopen.com polarization of voltage U x is connected to the integrator's input. The level of voltage U K at the comparator output indicates the polarity of the processed voltage U x . Reference voltage integration lasts until the moment when voltage U I at the integrator's output returns to the initial state which will result in activation of comparator K. Time T of reference voltage integration determines the result of processing.
Assuming the ideal integrator and comparator operation, voltage U I at the integrator's output is defined by the relation: ττ  (15) where: = RC -integrator time constant If voltage U x is constant over processing, then relation (15) gives the following: Voltage U REF also has a constant value in a transducer with linear processing and then: Double integral converters typically have a resolution limited to about 13 -14 bits. It stems from the fact that, with an increasing transducer resolution, the comparator must differentiate between increasingly smaller voltages at the integrator's output. The consequence is the extension of the comparator's response time and its unstable operation. In addition, the presence of noise, interference, hysteresis of the comparator and its insensibility threshold results in the fact that minimum voltage distinguished by the comparator is limited. Increasing transducer resolution, at the same time of voltage integration, would require increasing the clock generator's signal frequency. The abovementioned factors would lead to transducer's unstable operation.
The need to increase transducer resolution to 16 -18 bits resulted in preparing a modified double integral transducer circuit whose scheme has been presented in figure 3.3 and the waveforms of signals in selected points of the circuit -in figure 3.4. Fig. 3 Fig. 3.4. Waveforms of signals in selected points of transducer circuit.
In comparison to the basic circuit of double integral converter, the circuit presented in figure  3.3 has been expanded with two input buffers, two additional comparators and elements allowing cyclical zeroing of the transducer circuit. The control circuit has also been changed.
The modified transducer circuit operates in the following manner: first, zeroing of the set integrator -comparator K1 takes place. That takes place when keys k4, k5 and k6 are closed.
Other keys are open. Over that period, the integrator and the comparator K1 are enclosed by the feedback loop and the voltage on the integrator's reversing input reaches the value of that amplifier's offset voltage. Because the inputs of buffering amplifiers are also dense to the mass, the difference of offset voltages of input amplifier and the integrator amplifier is depositing on capacitor C z . That voltage is remembered on the capacitor at all times throughout the process of converting. As a result, the impact of all amplifiers' offset voltage is disposed.
During integration of the processed input voltage U x which lasts for T 0 , when voltage U I at integrator's output exceeds the value of voltage U p , which is the reference voltage for the comparator. Then the comparator activates, which, in turn, causes such action of the control circuit US that, synchronically with impulse f n of the clock generator switching on takes place -respectively to polarization of the processed voltage -of key k2 or k3 supplying the reference voltage with polarization opposite to the polarization of the processed voltage to the integrator's input. It is summed up together with the processed voltage. Under the influence of the reference voltage, the voltage at the integrator's output reduces its value. If the value of that voltage falls below voltage U p of the comparator, a change of the comparator's condition takes place and, as a consequence, the control circuit will turn off the reference voltage key again synchronically with the clocking impulse. If the value of the processed voltage is high, that process may be repeated several times. For that converter to operate properly, the absolute value of reference voltage should be greater or at least equal to the absolute value of processed voltage. After the period T 0 , the reference voltage is switched on, as in a regular double integral converter.
Because the voltage at the integrator's output returns to the initial state after the period of processing, that proves that the charge supplied from the processed U x voltage source and the reference U REF voltage source are of equal value. Therefore, an appropriate equality can be written: If we assume that R1=R2 and the processed voltage value is constant during integration, the following is obtained: where: T -total time of reference U REF voltage integration.
The processing method has changed in the aforementioned transducer, but function (19) describing the relations occurring in it, is the same as function (16) for dual slope converter.
From the point of view of adjusting processing characteristics of A/D double integral converter, the relation (16) is important, which indicates the process of reference U REF voltage integration. That means that using a generator generating an appropriate waveform of reference voltage will enable obtaining a desired A/D converter characteristics, (Janiczek, 1993).
In the transducer circuit presented in figure 3.5, the set of integrator, comparator, time constant and RC keys remains unchanged. On the other hand, the reference voltage generator -correction voltage, consists of an adder containing an operational amplifier, a D/A converter and EPROM memory. The meter block along with the control block US and the generator block f n , form a circuit generating signals controlling keys k 1 and k 2 . In addition, more significant bits of the meter block are at the same time control signals for address inputs of the EPROM memory. When the second phase of double integral converter operation starts -the phase of reference voltage integration U o , through closed key k 2 , reference voltage U o is applied to the integrator's input. The voltage is the sum of (with appropriate coefficients specified by resistors R2, R3 and R4), constant voltage U REF and voltage generated by C/A transducer, controlled by signals from the EPROM memory. Signals from the meter, controlling the EPROM memory, in selected time moments t 0 , t 1 , ... t n result in sending relevant codes controlling the C/A transducer from the memory, which Stepped reference voltage, integrated in the second phase of double integral converter operation, gives, as a result, the voltage at the integrator's output described by a rangelinear function. The voltage waveform at the integrator's output during processing has been presented in figure 3.7. Time T 0 is the time of the processed voltage integration.  where: U oi -reference voltage in the t i -t i+1 time interval t -moment of completion of reference voltage integration phase. After integration, the relation is obtained: Since time slices t i -t i+1 are the same throughout the entire reference voltage generating time and amount to t, and the integration time t of reference voltage ends in moment t n , the formulation (21) can be presented in the following form: where: N -value of input signal processing result. Taking into consideration that the reference voltage is equal: the formulation (24) takes the form: Because, as it seems from the above-mentioned formulations, the function describing the output signal on the integrator in the phase of reference voltage integration is a range-linear function, the measuring transducer's characteristics should also be approximated by a range-linear function: where:  i -coefficients of section inclination in i-th range x i -range points Formulations (24) and (27) lead to believe that to obtain a linear relation: the condition must be fulfilled: where: x -range length.
Range-linear function inclination coefficients obtained from the generator of function containing the C/A transducer, have finite values with discretization corresponding to D/A www.intechopen.com converter's output signal resolution. That fact should be considered when calculating the function approximating the measuring transducer's characteristics. In addition, the number of values that the approximating function coefficients can have is equal to the number of values generated by the D/A converter. In addition, due to simplicity of the shaping circuit's structure, the approximation ranges should have the same length, and, if possible, the number of ranges should be the power of number 2.
The range of changes in coefficient values of approximating sections' inclination is determined by the following relation: From formulations (17) and (23), it seems that accuracy of double integral converter depends on the reference voltage stability. Therefore, also in the transducer with adjusted characteristics, the processing accuracy will depend on voltage generation accuracy U o , with the assumption that the approximation error of the measuring transducer's characteristics by a range-linear function is negligibly small.
The formulation (30) shows that maintaining the assumed coefficient  accuracy depends on the stability of resistors R2, R3, R4 and voltage U REF as well as on the accuracy of generating voltage U g by D/A converter. The currently available parameters of resistors and sources of reference voltages are such that their impact on the accuracy of coefficients  may be omitted. Thus, the main source of errors will be D/A converter's errors.
The most important D/A converter's errors which should be factored in when determining the generated reference voltage accuracy include: characteristics linearity error, characteristics inclination change error and zero drift. D/A converter characteristics zero drift reduction as well as other analog blocks of slotted line can be performed by means of cyclical zeroing of the slotted line.
The method of adjusting transducer processing characteristics described above is one of the most effective, especially for converters of high non-linearity. An example may be the methane concentration meter with a measuring transducer TGS 22611 E00 manufactured by Figaro USA Inc. (TGS 2611). That is a resistance transducer and it measures methane concentration ranging from 300 ppm to 10000 ppm. Its characteristics, determining the relation of output resistance Rs to reference resistance Ro depending on methane concentration has been presented in figure 3.8. Resistance Ro, defined at concentration 5000 ppm, depending on a specific transducer may vary from 0.68 k to 6.8 k. The transducer used in the described meter, had 1.2 k. Thus, transducer output resistance changes were ranging from 0.9 k to 3.6 k.
The currently available integrated A/D converters do not provide the feature of adjusting their processing characteristics. That is why analog elements that can be found in microcontroller internal structures were used in the structure. The MSP430FG4618 microcontroller manufactured by Texas Instruments was used in the meter for measuring methane concentration. Its structure contains three operational amplifiers, a 12-bit D/A converter, resistors and a set of analog switches. The circuit of dual slope converter, containing those elements, has been presented in figure 3.9.
Rs/Ro Fig. 3.8. Characteristics of transducer for measuring methane concentration.
The integrator with amplifier OA0 and comparator OA2 is enclosed by the zeroing circuit with a capacitor C z storing offset voltage. Voltage V z produces artificial mass, in relation to which the input voltage and the reference voltage received from D/A transducer is determined. The capacitor C p provides an appropriate shift of the output signal level from the D/A converter. Resistors inside the micro-controller have been used as resistor R of the integrator. The measuring transducer is included in the divider circuit, which simplifies the measuring circuit. Since voltage VCC, supplying the divider, is, at the same time, the reference voltage for the D/A converter. Changes in that voltage do not affect the accuracy of measurement. The transducer control circuit has been based on the structure of module Timer A and an appropriate program.
The simplified scheme of the meter circuit has been presented in figure 3.10. It is clear, that almost the entire transducer structure is located inside the micro-controller. Only capacitors and divider resistors are located outside. The diagram has omitted the heater activation circuit and the meter's keyboard.
For selected concentration values, voltage values on bridge diagonal have been calculated and for points obtained in such way, the second-degree approximating polynomial U(s) has been calculated. Its waveform have been specified in figure 3.11.  Testing of errors produced by the shaping circuit took place in such way that the transducer output resistance, for selected concentration values, was simulated by a resistance decade and the difference between the theoretical concentration value and the value indicated by the meter was defined. The value of relative error d, calculated with respect to the maximum range value, has been shown in figure 3.12. As it stems from the error chart, its maximum value does not exceed 0.6% and is present at the beginning of the range.

Transducer with impulse feedback
Transducer with impulse feedback is also classified into integrating transducers. Its great advantage is that it is a very simple circuit, but it processes signals of just one polarity, although that limitation generally is not disturbing in the case of cooperation with measuring transducers. Especially since it is very well suited for cooperation with measuring resistance transducers. Due to that simplicity and easy control, they are used in measuring devices cooperating with different measuring transducers. Block diagram of a transducer with impulse feedback has been presented in figure 4.1, and voltage waveforms in selected points of the transducer in figure 4.2. The transducer circuit consists of an integrator with operational amplifier, capacitor C and resistors R1, R2, comparator K, control block CB, key k and reference voltage source U o . The output signal of frequency f being the function of the processed input voltage U x is obtained at the comparator's output. If numeric values are to be obtained from the transducer, then the meter calculates output signal impulses for a given time T 0 .
The circuit operates in the following manner: processed voltage U x is supplied to the integrator, through resistor R1, at all times. As a result, voltage U I at integrator's output continues to increase until it reaches the reference level U p . At that time, the comparator is activated and, as a consequence, control block CB is activated which, for time t p , closes key k supplying the integrator with reference voltage with polarization opposite to the polarization of input voltage U x through resistor R2. That results in change of voltage www.intechopen.com waveform at the integrator's output. After time t p , key k is turned off and the processing cycle starts over. In the first phase of transducer operation, voltage increase at the integrator's output can be described by the relation: In the second phase of transducer operation, in which reference voltage is added to the integrator, voltage increase at the integrator's output can be described by the following relation: In the steady state, voltage increases at the integrator's output are equal: Thus, summing up the integrals from formulations (31) and (32)  On the other hand, when the reference voltage U 0 is a variable voltage over the processing period, but it maintains constant value over period t p , figure 4.3, then that condition can be described by relation (39)  Formulation (40) corresponds to formulation (30) for double integral converter and, as a result, shows that the characteristics of a transducer with impulse feedback can be adjusted in a similar way as it has been presented for a double integral converter. Also, all presented considerations are also valid here with one reservation concerning time t p .
In the transducer circuit presented in figure 4.1, time t p is generated by a monostable circuit which does not provide too high accuracy. It may be remedied by introducing modifications in the transducer circuit, presented in figure 4.4. (Janiczek, 1992)  The modification of transducer operation involves generation of time t p -it begins not when the comparator is activated, but synchronically with the first impulse of generator f n , which comes when the comparator is activated. That is illustrated in figure 4.5. Fig. 4.5. Voltage waveforms in modified transducer circuit.
As a result, time slice t p may be generated by division of frequency f n in the divider. That results in a certain deviation of transducer output signal frequency, but after averaging over time interval t 0 , that does not affect the processing accuracy. Also, time interval T 0 may be obtained in the same manner. Thus, times t p and T 0 can be identified as: Thanks to such solution, the processing accuracy of a transducer depends only on the constancy of relation R1/R2 and accuracy of generating voltage U o . In addition, there is a possibility to adjust transducer characteristics in the above-mentioned circuit by changing duration time t p of the feedback impulse.
Assuming, in formulation (37), that both the values of processed voltage U x and reference voltage U o are constant, the following is obtained: After considering the relations (43) and transformation, the following formulation is obtained: from which it can be concluded that the characteristics of a transducer with impulse feedback can be adjusted by changing the degree of meter division specifying impulse duration t p .
Assuming, as previously, that values q i are constant over p long ranges, the following relation is obtained: which shows that in such circuit of a transducer with impulse feedback, measuring transducers characteristics approximated by a range-linear function may be adjusted in a very wide range. Moreover, in that transducer, a very high processing resolution and accuracy can be obtained, which practically depends on the offset voltage constancy of integrator's amplifier, the constancy of R1/R2 resistors' relation and the constancy of reference voltage source U o .
The above-mentioned method of adjusting processing characteristics of transducer with impulse feedback has been applied in the meter for measuring carbon monoxide concentration. A transducer TGS 203 manufactured by Figaro Engineering Inc. (Japan) whose simplified scheme has been presented in figure 4.6. was used in the meter. (Janiczek, 2009) The A/D converter circuit consists of an integrator, comparator, keys k1 -k5, voltage divider R2, R3 and R4 and reference resistor R1. Rs is the measuring transducer's resistance. Particular phases of transducer operation are controlled by the micro-controller manufactured by Texas Instruments -MSP430F415. Additionally, to improve transducer resistance to changes in temperature and supply voltage, the phase of zeroing the integrator's amplifiers and comparator has been introduced. The zeroing voltage is stored on capacitor C z . In order for the circuit to be supplied with one voltage, a divider of voltage R2/R3, R4 providing the reference voltage U z for the integrator has been introduced. That voltage is equal approximately to half the supply voltage. As a result, voltages applied to resistors R1 and Rs, through keys k1 and k2, have opposite polarizations in relation to voltage U z . The reference voltage U p for the comparator is obtained from divider R2, R3/R4. The circuit is zeroed over the T z time, figure 4.7. Keys k1 and k2, are open and keys k3, k4 and k5 are closed. The integrator's and comparator's amplifiers are enclosed by the feedback loop, and capacitor C z is charged up to the integrator's amplifier offset voltage. After zeroing the circuit, keys k4 are opened and key k1 is closed for the entire duration of processing and the transducer o p e r a t e s i n r e g u l a r m o d e o f t r a n s d u c e r w i t h i m p u l s e feedback. Comparator's output impulses are counted in one of the meter's micro-controllers and a different meter unit generates time slices T 2 of feedback impulse duration. Using the basic relation (38) describing transducer operation and taking account of relation (4.4) describing the measuring transducer's characteristics, the following is obtained: where: N -number of impulses counted in the meter, T 0 -processing duration, U z -artificial mass voltage. For proper charge balance over the T 2 period, it has been assumed that resistor R1=150  and U z =VCC/2, hence the formulation (45) takes the form: In order to obtain a linear relation between the number of counted impulses N and concentration s, the following condition must be fulfilled: T 2 (N) = c·s 0.95 The measuring transducer's characteristics has been approximated with ten sections, which ensured approximation error not greater than 0.3%. Using relation (45), coefficient values determining times T 2 have been calculated.
Conducted tests have indicated that a stable resolution of so constructed A/D converter was reached at the level of 14 bits. The impact of temperature changes in the range 0 -30 o C did not exceed 0.02%. The main factor of that error was changes of R1 resistor values and change of R2/R3, R4 divider's pitch. Change of supply voltage VCC within +/-10% was not noticed. Those parameters significantly exceed the requirements necessary to measure carbon monoxide concentration but they show the capabilities of the measuring circuit.
Conducted meter characteristics tests, in which the measuring transducer was simulated with a variable resistor, indicated that the relative measuring error d (with regard to the range endpoint) did not exceed the value 0.3%, figure 4.9.

Conclusion
This chapter presents a method for digital adjustment of analog-to-digital converter transfer function. If the measuring transducer's characteristics is non-linear, it is necessary to use suitable correction methods. When the measuring transducer's characteristics is very nonlinear, for example in the case of certain transducers for light intensity measurement, a problem with selecting a relevant A/D transducer may occur.
A converter with digital adjustments is a useful tool for correcting non-linearity of transducers, especially those with high linearity, in case of which the numerical correction would lead to the lost of resolution. The conducted experiments proved the suitability of the proposed new solution of converter. Such designed analog-to-digital converter achieves a stable resolution and small error of the approximation function. Measurement is a multidisciplinary experimental science. Measurement systems synergistically blend science, engineering and statistical methods to provide fundamental data for research, design and development, control of processes and operations, and facilitate safe and economic performance of systems. In recent years, measuring techniques have expanded rapidly and gained maturity, through extensive research activities and hardware advancements. With individual chapters authored by eminent professionals in their respective topics, Applied Measurement Systems attempts to provide a comprehensive presentation and in-depth guidance on some of the key applied and advanced topics in measurements for scientists, engineers and educators.