Principles of pressure transducers, resonance, damping and frequency response

Cross-section of a transducer assembly. The sensor element is shown in red, and magnified schematically below. The fluid channel is shown in blue.

Four resistors are etched into the silicon diaphragm, which is supported by a silicon frame and connected to compensation circuitry by aluminium connectors. A cavity below the diaphragm is etched into the opposite surface.

Performance of a measuring system can be described in the following terms. Accuracy describes how closely the measured value reflects the actual value (i.e. the absolute difference between the actual and measured values). Linearity (a) describes the closeness of fit of output values to a straight line of evenly spaced increasing pressures. The line of best fit is usually plotted using the method of least squares. Precision or repeatability (b) describes the degree to which repeated measurements under identical conditions will produce the same measured value (i.e. the measure of spread of the output values). Hysteresis (c) describes the inherent tendency of a transducer to give different outputs when consecutively applied pressures are either increasing or decreasing. The mechanisms are complex and involve internal friction and dynamic lag between the input and output. Gain errors (d) occur when the transducer is operated outside of a specific temperature range. Electrical resistance, conductivity and diaphragm stiffness are all affected by variations in temperature. Null offset errors (e) describe a non-zero output when no pressure is applied. Zero drift describes a change in output over time when no pressure is applied.

(a) An external driving force causes a second-order system to oscillate. Amplitude increases as driving frequency (ω_A) approaches natural frequency (ω₀). With increased damping, there is a decrease in the frequency at which maximum amplitude occurs (maximum line). As damping coefficient (ζ) tends towards 1, the resultant curve has a wider flat range than in the undamped system. The upper limit of the flat range of an optimally damped system is shown at X. (b) The damping coefficient modifies the behaviour of a second-order system, when a displacement force is applied: • The undamped system (ζ = 0) oscillates at its natural frequency (entirely hypothetical). • The underdamped system (0 < ζ < 1) oscillates with lower frequency and amplitudes than the undamped system. • The critically damped system (ζ = 1) does not oscillate, but rapidly returns to baseline, with no overshoots. • The over-damped system (ζ > 1) shows delayed return to baseline, with no overshoot. Optimal damping (ζ) has been quoted between 0.6 and 0.7, such that overshoot is limited after a displacement force is applied, and oscillations decay rapidly. (c) In the cannula–catheter–transducer system, the operation and release of the fast flush device produce a square pressure wave followed by a small number of oscillations at the system's natural frequency. The ratio of adjacent oscillation amplitudes, A₁ and A₂, can be used to calculate damping coefficient as follows,⁸