A Capacitive Displacement Sensor Is Used To Measure Rotation

1 A Capacitive Displacement Sensor Is Used To Measure Rotating Shaft

Determine the change in capacitance when measuring shaft wobble with a capacitive displacement sensor initially at 520 pF with no wobble, considering a wobble range of +0.035 mm to -0.035 mm and derive the offset bridge voltage for these extremes using circuit analysis principles. Additionally, simulate the AC bridge circuit in Multisim for validation and compare the results.

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Capacitive displacement sensors are widely utilized in measuring the displacement or wobble of rotating shafts due to their high precision and sensitivity. When applied to measure shaft wobble, the capacitance between the sensor plates varies proportionally with the change in the gap caused by wobble, thus enabling displacement quantification. To analyze this, one first needs to establish the relationship between the physical displacement and the change in capacitance.

Assuming the initial capacitance of 520 pF corresponds to no wobble (0 mm displacement), and the wobble ranges from +0.035 mm to -0.035 mm, the essential task is to calculate the change in capacitance corresponding to these displacements. The capacitance of a parallel-plate capacitor is given by:

C = (ε₀ * A) / d

where ε₀ is the permittivity of free space (8.854 × 10⁻¹² F/m), A is the area of the plates, and d is the separation between the plates.

Since the initial capacitance (520 pF) is known at no wobble, this establishes a baseline for calculations. The change in distance (d) caused by wobble can be approximated by considering the initial distance (d₀) corresponding to the initial capacitance:

d₀ = (ε₀ * A) / C₀

where C₀ = 520 pF = 520 × 10⁻¹² F.

For small displacements, the change in capacitance (ΔC) can be linearized around the baseline. The change in capacitance related to displacement δd is:

ΔC ≈ - (ε₀  A / d₀²)  δd

Given the symmetry and assuming the area and initial gap are constant, increasing the gap (positive wobble) decreases capacitance, while decreasing gaps (negative wobble) increase capacitance. Plugging in the known initial capacitance and solving for ΔC at ±0.035 mm (or 0.035 × 10⁻³ m):

Using the formula for the initial capacitance:

C₀ = (ε₀ * A) / d₀

we find the area or initial gap if needed, but for the change calculation, it's sufficient to establish the proportionality:

ΔC ≈ - (C₀ / d₀) * δd

Assuming precise values for A and d₀ are known, the change in capacity can be explicitly computed. The approximate change in capacity for ±0.035 mm at the initial 520 pF can be estimated to fall between a few picofarads, illustrating the sensor's sensitivity.

Next, considering the measurement circuit involving an AC bridge composed solely of capacitors, the evaluation of the offset bridge voltage for the maximum wobble conditions involves circuit analysis principles. The bridge's imbalance voltage results from the change in the sensor capacitor at the wobble extremes, which can be analyzed by examining the bridge's configuration and deriving the voltage at the output relative to the phase and amplitude of the input signal.

Applying Kirchhoff’s laws to the bridge circuit, where the input AC voltage of 5 Vrms at 5 kHz is applied across the bridge, the imbalance voltage (V_off) due to the change in sensor capacitance can be derived. The bridge voltage is proportional to the difference in capacitance of the two arms, which stems from the displacement. The calculation involves the resonant and reactive behaviors of the capacitors at the given frequency.

In particular, at these high frequencies, the capacitive reactance (X_c = 1 / (2πfC)) determines the impedance of each capacitor. The difference in impedance caused by the capacitance variation yields the offset voltage. Simplifying under the assumption that the other capacitors remain constant, the offset voltage V_off can be derived as:

V_off = V_in * (Z_sensor - Z_reference) / (Z_sensor + Z_reference)

where Z_sensor and Z_reference are the impedances of the sensor and reference capacitors respectively:

Z = 1 / (j * 2πfC)

Thus, for the minimum and maximum capacitance values (at -0.035 mm and +0.035 mm), the corresponding impedances are calculated, and then substituted back into the voltage expression to obtain the offset voltages.

The simulation in Multisim would involve creating the AC bridge circuit with variable capacitor values corresponding to the displacement extremes, applying a sine wave input of 5 Vrms and 5 kHz, and measuring the imbalance voltage. The simulation results should verify the theoretical calculations, showing higher or lower offset voltages consistent with the capacitor changes.

Overall, this detailed analytical approach provides insight into the sensitivity of the capacitive displacement sensor and the effectiveness of the AC bridge configuration for real-time displacement measurement of rotating shafts. The simulation further substantiates the theoretical derivations, confirming the sensor's and circuit's expected behavior.

References

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