Analysis and evaluation of differential inductive transducers for transforming physical parameters into usable output frequency signal

This paper presents a novel hybrid oscillating inductive sensor circuit converting deflection oscillations into useful signal and thus being used to support an oscillatory circuit mechanism and management of their resulting oscillation in a real-time implementation. This circuit does not only detects and optimizes the deflections produced in an oscillation in instrumentation electronics, but also provide an improved sensory with high accuracy, sensitivity, responsiveness and operating range. The design therefore help in transforming the deflection deviations in the in and out movement of the core into useful output content. A simulation and derivation of the oscillating circuit in see-saw convulsion bar sensing system has been implemented. Simulation and derivations shows how the oscillating circuits of the bar are converted into frequency, duty cycle, current, voltages for further wireless applications.


INTRODUCTION
Sensors have proven important in converting immeasurable quantities such as wind velocity, vibrations, oscillation, turning effects of forces, deviations, illness etc into qualitative and quantitative electrical signal suitable for real time implementation by an intelligent oscillation harvesting mechanism.For an accurate measurement of such oscillation parameter, an efficient oscillation detector is essential (Figure 1).This is capable of converting a see-saw convulsion bar into frequency response, duty cycle response, voltage response.The sensor output should be such that the oscillation/vibration, their motion, damping etc is determined.Based on the sensing inductive principle used, the designed circuit can be categorized into a vibration sensor and pressure sensor applied even in height location such as; sagging and oscillations of electrical transmission line, turning moment in railroad, signal upgrade from degradation in GSM mask (Ezzat and Cheng, 2011;Edgarcio et al., 1988;Grover and Deller, 1999;Hameed et al., 2012).
The sensor output should be such that electromechanical deviations in oscillation circuit and their effects servomechanism are determined.The vibration sensor can be categorized according to the sensing element such as resistive, capacitive, inductive and linear variable differential transducer (Mohammed et al., 2012 Mohan et al., 2008).Piezo-resistive sensors have good linearity and acceptable sensitivity, but suffer from the problem of inaccuracies (Mohan et al., 2009;Ravindra, 2006).Capacitive pressure/oscillation sensors exhibit features of higher sensitivity and lower temperature hysteresis, but they are usually nonlinear (Ravindra, 2006;Saxena and Sahu, 1994;Slamwomir, 2007).Conventional LVDT-based pressure/oscillation sensors possess good linearity, highest sensitivity and lowest temperature hysteresis, but such devices have got bulky physical structures (Saxena and Sahu, 1994;Slamwomir, 2007;Texas Instruments, "LM555 Data sheet", 2003;Udaya and Duleepa, 2011).This paper presents a sensor capable of harnessing oscillations in circuits into pulse able signal.Each aspect of this finding has been put into different sections.

STRUCTURAL MODEL AND DESIGN
The structure of the proposed oscillating circuit is shown schematically in Figure 2. The oscillation of the actual component of the sensor varying differentially, maximizing the circuit variations and deviations into a derivable and pulse able signal is shown in Figure 3.As shown in Figures 2, 3 and 4, a vertical core of a varying height of 4 to 36 mm is embedded into an open stator made up of magnetic coils thereby providing the inductive change in relation to circuit vibrations.The oscillation of the circuit provides the force or pressure needed to bring about the inductance of the coil as the core is displaced in and out of the coils mounted in a stator of a servomechanism.This displacement is proportional to the force or pressure and the inductance of the core.The timer circuit when connected to the coils in Figures 1 and 2 will give rise to Figures 3 and 4.This RL3R circuit gives us a square wave with a frequency depending on the inductance value of the coil as shown in Figure 5.

MATHEMATICAL MODEL OF THE CIRCUIT
A differential transducer is one that simultaneously senses two separate sources and provides an output proportional to the difference between the sensing.Considering the idea in Figure 4, the inductances of the two coils change in a differential manner when the overhanging bar is moving in a seesaw manner.At position 'x', the inductance of the right hand side of the coil is given by Equation (1), while that of the left hand side is given by Equation (2), giving the total equation as follows: Details of the derivation of equation 1 and 2 are shown in the APPENDIX A-1 to A-6.A linear relationship is shown in Figure 6 demonstrating the in and out oscillation of the device.

Frequency output for inductive change delta L
The inductance of left inductor is given by L 1 and can be expressed as: Whereas the inductance of second inductor is given by L 2 The frequency of RL3R-timer circuit for single inductor is given by: f=1.58 The frequency of the first inductor is: The frequency of the second inductor is: The frequency of total inductive change is: (5) Substituting the value of L 1 in Equation ( 5) we get: (6) Now solving for the other case of ∆f 2, we obtain; Substituting the value of L 2 in Equation ( 7), (10) The plot for frequencies of the in and out movement of the core in the coil is given thus.

Frequency change as a function of displacement
From the previous equation, the frequency change as a function of displacement is given thus as: That of the second coil is; (12) The plot for frequencies of the in and out movement of the core in the coil is given as; these change in frequencies from Equations ( 11) and ( 12), gives a symmetrical behaviour with respect to the displacement produced by the oscillatory motion of the core.This aforementioned behaviour gave rise the plot in Figure 7

Frequency as a function of oscillations
Generally, vibrations in movement can be described or expressed by the following equations.In the case the seesaw bar shown in Figures 2 through 4, we assume that the seesaw bar in the springs has a movement which is a simple harmonic motion according to the following equation; (13) The vibrations found in electronic circuit and in machines can be converted into useful frequencies as shown in Equations 14 and 15. ( To check the effect of the sea saw bar frequency, Equations ( 13), ( 14) and ( 15) are used.Assuming a coil of length of 20 mm, number of turns = 100, cross sectional area A=pi(r*r) where r=1 mm, an iron core with relative permeability 4728, and R= 1k ohm, then by changing the value of fs, the plots for the output frequency as a measure of x is obtained.Figure 8 shows the effect of the circuit as it oscillates, converting the deviations into these useful frequency values when the harmonics from the see-saw in the convulsion are given as 10, 100, 1000 and 10,000 Hz respectively.

SIMULATION RESULT AND ANALYSIS
A simulation of the designed mathematical model and its necessary derivations for frequency, frequency hysteresis, voltage and the current responses are carried out.The results obtained confirm the mathematical derivation plots with the simulation.Figure 9 result of simulation of both circuits; the right and the left one using the same assumption that was used in the calculations.The green line represents the frequency output of the left side and the red one is for the circuit in the right hand side as a measure of the core displacement.Figure 9(b) shows the output frequency when the sea saw frequency fs= 100 Hz, for three time periods, in every period, the displacement value change from 0 to and from to 0. Figure 10(a) to (d) shows the result of simulation when the sea saw bar frequency is change.When the frequency of sea saw bar is 10 Hz, x changes from 0 to , and time t, is changed from 0 to 50 ms, 0 to 100 ms, 0 to 1000 ms and 0 to 10000 ms according to Equation ( 15).The results obtained confirm those of the derivations shown in this work.A slight difference in simulation analytical deduction is due to the environmental effect on the circuit.The table for the different frequencies giving rise to frequency hysteresis is given in Table 1.This table gives rise to Figure 6, showing the frequencies of the see-saw behavior when the core is differentially in and out of its coil as the circuit oscillate.
A linear relationship exist between the inductance of the first coil, which increases when the core is moving in, while the core for the second coil is moving out, with decrease in inductance.The inductance change is calculated and plotted in Figure 6.These inductance change leads to a frequency change and hysteresis for the output of the timers as seen from derivations and simulation results.This shows that the frequency is decreasing when the core is going in and increases when the core goes out.At equilibrium state when the sea saw bar is horizontal, the displacement x is equal to half of the coil length, and the value of the inductance for both coils is the same.This is clearly shown in Figure 6 at the point when x= 10 mm, the inductance L=4.67 mH for the two coils.The result of the inductance equality at this point is reflected to the frequency value as Figure 7.
The main difference between derivations and simulation result is seen when the coil is almost fully out.From derivation point, when the coil is fully out x=0, the frequency is larger than 800 Mega Hz, which is very large compared to other frequency values which ranges from 0.18 to 3.4 Mega Hz.On the other hand, simulation result shows that maximum frequency occurs when the core is fully out and this is at 3 MHz as shown Figure 7. Although, the result from derivations and simulation has shown that the sea saw bar frequency has no effect on the output frequency values, the frequency counter calculate the value which is equal to the time period of the timer output for every reading.So, if the sea saw frequency fs exceeds a certain value, the frequency counter will give wrong frequency output.This value is derived as shown in Equation ( 16).converted into frequency output in the processing part of the sensor using a timer circuit.Simulation of the proposed system shows a similar result with that of theoretical derivation.The sea saw bar frequency effect is checked and the limitation for this value.This was found in Equation ( 16).Finally, this novel design provides a platform for electro-mechanical and servomechanism system deviation in form of oscillation and vibration to be harvested into useful electrical signal by the help of designed differential inductive sensor using both coils transducers as shown in Figure 3.The result obtained appears to be at minimum value of 0.001% of FSO which serve as improvement and introduction frequency hysteresis results in both derivations and in simulations.

Figure 1 .
Figure 1.(a) A balance see-saw of overhanging bar, (b) Differential manner of an overhanging seesaw bar.

Figure 2 .
Figure 2. Varying differential cores in a see-saw.

Figure 4 .
Figure 4. Signal produced from circuit oscillation and vibrations.

Figure 6 .
Figure 6.A plot of the inductance variation with the displacement.
(a) showing the reciprocal behaviour of the in and out movement, harnessing the deviations in terms of oscillation in servomechanism and electro-mechanical devices.The frequency change with the displacement is given in Figure 7(b).

Figure 7 .
Figure 7. (a): A plot of Frequency change in symmetrical variation with respect to core displacement, (b): A plot of Frequency with respect to core displacement.

Figure 8 .
Figure 8. Plot of the Output Response for oscillating circuit for varying values of Fs.

FrequencyFigure 9 .
Figure 9. (a): A simulation of Frequency change in symmetrical variation with respect to core displacement, (b): Simulated Output for frequency with three time periods.

Conclusion
Differential sensing system architecture is proposed as shown in Figures1 to 4. This consists of two coils with varying cores in which a change in the core position will result in inductance change.These changes can be

Table 1 .
Simulation details of frequency hysteresis.
Figure10 (a-d): Simulation values for different harmonics in the see-saw.