Principles of pressure transducers, resonance, damping and frequency response
Article Outline
- Abstract
- Pressure
- Pressure measurement
- General transducer principles
- Transducer development
- Resonance, damping and frequency response
- References
- Copyright
Abstract
Blood pressure is a determinant of blood flow, and is the sum of hydrostatic and dynamic pressures. Intravascular pressures can be measured directly using intravascular pressure sensors, or with external transducers connected by a fluid column. Early pressure transducers consisted of wire strain gauges, but these have been superseded by semiconductor devices, which have become increasingly mass-produced and miniaturized, using production techniques common in microelectronics. Performance of pressure-monitoring systems is affected by physical factors including resonance and damping. This article examines the physical principles that underlie transducer design and function, and the sources of error and inaccuracy.
Keywords: Damping, frequency, hydrostatic, kinetic, micromachining, piezoresistive, pressure, resonance, strain gauge, transducer
After reading this article, you should be able to:
Pressure
Blood pressure is a determinant of blood flow – that is, flow in a vascular bed is linearly related to the pressure gradient across it. Pressure (P) is the magnitude of a force (F) applied to a specific area (A). The SI unit, the Pascal (Pa), is defined as Newtons per square metre (N/m2).
Static liquid in a container exerts ‘hydrostatic pressure’ (Ps), which increases with increasing depth from the surface. At the bottom of the container, the relationship between force and area is:
Moving liquids exert ‘dynamic pressure’ (Pd) parallel to the direction of flow due to the kinetic energy of the liquid. The pressure exerted is related to liquid density and flow velocity:
Bernoulli's equation expresses the total pressure exerted by the liquid (P0) as P0 = Ps + Pd.1 The human vascular system is neither a cylinder of fluid, nor a single horizontal tube filled with flowing blood, but it is clear that blood pressure in different parts of the vascular tree varies, according to the height above the lowest point (static pressure), and the velocity of blood flow at the point of measurement (dynamic pressure).2
Pressure measurement
Gases expand to fill the volume they occupy, and equilibrate rapidly within a mechanical gauge. Non-compressible fluids, including blood, do not expand like gases, and the viscosity of the fluid precludes the rapid entry into and exit from simple gauges. With the rapid cyclical variation of blood pressure, incomplete equilibration of pressures with a mechanical gauge would impair accurate measurement.
To directly measure blood pressure, therefore, a system must consist of, either,
General transducer principles
Pressure sensors are transducers – energy converters – which convert the kinetic and potential energy within a fluid to electrical energy, proportional to the magnitude of the pressure within it.
The generic design of pressure transducers quantifies the deflection of a square or circular diaphragm – placed between measurement and reference chambers – across which there is a pressure difference. The deflection of the diaphragm in sensing systems is frequently less than the thickness of the diaphragm itself, and is linearly related to the applied pressure. Pressure is either measured relative to a vacuum (absolute pressure), or relative to ambient pressure (gauge pressure). Pressure transducers in clinical use measure gauge pressure, that is, relative to atmospheric pressure (Patm) (Figure 1).
Figure 1
Cross-section of a transducer assembly. The sensor element is shown in red, and magnified schematically below. The fluid channel is shown in blue.
The deflection of the diaphragm is measured using a form of strain gauge (Box 1). The original resistance-wire strain gauge, invented in 1936, measured increased electrical resistance of a wire, bonded to a diaphragm, in response to stretching or ‘strain’.3 Resistivity (ρ) (Ωm) is an innate property of a material, which defines how strongly it opposes conduction of electrical current.4 A wire bonded to the diaphragm of a pressure transducer, of length l, radius r and resistivity ρ, has resistance (R),
Stress (s) (Pa or N/m2) is force per unit area (i.e. pressure or ‘load’). Strain (e) (%) is change in length relative to original length. If the relationship between stress and strain is linear, the material is said to obey Hooke's Law, and s/e is a constant called Young's Modulus (E) (Pa or N/m2) which describes the ‘stiffness’ or ‘elasticity’ of the material.
The extent to which the wire elongates under pressure is determined by a constant: Young's modulus (E). This is the ratio of tensile stress in the wire (T = F/πr2) to the relative elongation (strain) of the wire (ε = Δl/l), and is expressed in units of pressure.5
The extent to which the thickness of the wire contracts under stretch is determined by the Poisson ratio (υ), such that Δr/r = −υ (Δl/l).5 Most materials have a Poisson ratio of around 0.3.4 So,
Metals are homogeneous materials that are electrically isotropic, that is, their electrical properties are equally distributed within the material, and the values of those properties are identical irrespective of the direction in which they are measured. Therefore, in wire strain gauges, resistance change is essentially independent of material resistivity, which changes negligibly with deformation.
Gauge factor (G) is equivalent to the relative change in resistance (ΔR/R), and is used to describe the performance of a transducer. Resistance is the easiest electrical property to measure accurately at low cost, but with small changes in resistance – in the region of 0.01–0.1% of base resistance – gauge factors for these sensors are low.2, 6
The change in resistance of strain gauge resistors may be accentuated by arranging them in a ‘Wheatstone bridge’ configuration. The ratio of voltage change to supply voltage (ΔV/Vs) is linearly related to ΔR/R, and is typically adjusted to have an output of 5 μ V/Vs/mmHg. The use of double-bonded diaphragms exaggerates the output resulting from diaphragm deflection, by causing stretching of resistors in one plane at the same time as compression in another (Figure 2).
Figure 2
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.
Transducer development
Pressure sensing developments over the last six decades have focused on increasing accuracy, miniaturization and reducing cost of production.
Piezoresistive sensors employ the characteristic of crystal semiconductors (e.g. silicon, germanium) of varying resistivity (ρ) in response to applied stress, due to the inhomogeneous, crystalline structure of semiconductors, which are electrically anisotropic. Strain alters the crystalline structure of semiconductors, which affects the resistivity of the material. As a result, the gauge factor of silicon piezoresistors strain gauges is one to two orders of magnitude greater than those of metal alloys.5
Early models of semiconductor strain gauges in the 1950s were bonded to metal diaphragms, but these were superseded by silicon diaphragms, with piezoresistors etched into the semiconductor material (Figure 2, Figure 3).
Figure 3
Performance of a measuring system can be described in the following terms.2, 10 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.
The advantages of piezoresistive sensors are that silicon has three times the tensile strength of steel wire, and they are less susceptible to the hysteresis and drift. They are, however, susceptible to temperature errors, namely temperature coefficients of offset (TCO) and temperature coefficients of resistance (TCR). Compensation circuitry is needed, using ‘unstrained’ resistors, affected by the same temperatures as the strain gauge.2, 5
Capacitative sensors are based on parallel plate capacitors, where a metal or metal-coated diaphragm functions as one plate of the capacitor. Capacitance refers to the ‘storage’ of charge by polarization of two adjacent metal plates. Deflection of the diaphragm under pressure reduces the gap between the plates, increasing capacitance (C),
The advantages of capacitative sensors are that they are the most precise sensors, with low susceptibility to temperature error. The capacitance change may be linear with pressure, but sensors may be susceptible to ‘stray capacitance’, which occurs when electromagnetic fields produce capacitance effects between adjacent circuits.2, 6
Capacitance sensors are larger than piezoelectric sensors, affecting frequency response, and are more expensive to produce, a consideration in mass-produced systems.
Micromachining of microelectromechanical systems (MEMS) consists of the etching of resistive or capacitative elements into single crystal or polycrystalline silicon diaphragms. The miniaturization of sensors has tracked the evolution of microelectronics, as the manufacture of integrated circuits employs the same techniques.
Since the 1980s, increasingly precise micromachining has resulted in diaphragm dimensions decreasing from around 1 cm to 0.02 cm. Temperature compensation circuits, Wheatstone bridges and signal conditioning circuitry are also etched onto the silicon wafer, enabling inclusion of the entire sensor electronics ‘on-chip’, facilitating error-reduction and compensation. Capacity to produce hundreds of thousands of sensors on a silicon wafer has also enabled the development of low-cost, disposable blood pressure transducers.5
Resonance, damping and frequency response
Fluid motion in the fluid column between the blood vessel, intravascular cannula, connecting catheter and transducer is most closely represented by a mass suspended on a spring, referred to as an harmonic oscillator.
When it is ‘bounced’, it oscillates briefly, and then comes to rest. If ‘driven’ by an external displacing force, like a sine wave generator or the arterial pressure waveform, it oscillates at the frequency of that force, and at an amplitude proportional to it.
The mass-spring system and the fluid column are both modelled as second-order dynamic systems with one degree of freedom. The variable of interest (x) is the amplitude of motion in both systems. The mechanical behaviour of a second-order system, is determined by three factors:
In a hypothetical, frictionless mass-spring system, a displacement force would cause it to oscillate indefinitely at the system's natural frequency (ω0), which is determined by the stiffness of the spring (k) and the mass of the system (m):
In reality, all second-order systems, including the fluid column, are subject to frictional forces or damping factors, which means that oscillations occur at the damped natural frequency (ωd). The performance of the catheter–fluid column–transducer system can therefore be described in terms of natural frequency and the degree of damping.
Resonance is the tendency of system to oscillate with greater amplitude at the natural frequency than at other frequencies. When the external driving frequency of the system is the heart rate (or a harmonic frequency of the blood pressure waveform), the resulting resonance in a pressure monitoring system causes overshoot at both peak and trough values, overestimating systolic and slightly underestimating diastolic pressure.8
Damping is the reduction of the amplitude of oscillations in an oscillating system, caused by energy loss. In the fluid column, the extent of damping is determined by inertance (viscous friction within the fluid column), impedance (changes in the calibre of the fluid path), and compliance in the tubing (elastic catheter material or air bubbles).9 Using Poiseuille's Law to calculate flow in the fluid column, the damping coefficient (ζ) is calculated as follows:
where η = fluid viscosity. The relationship between natural frequency and damping coefficient is shown in Figure 4. An excellent article by Kleinman shows the derivation of both equations.8
Figure 4
(a) An external driving force causes a second-order system to oscillate. Amplitude increases as driving frequency (ωA) 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
From the above equations, we can see that the setup of a blood pressure-monitoring system may introduce error by the following mechanisms:
Frequency response, also known as dynamic bandwidth or dynamic range, describes the range of oscillation frequencies over which pressure can be accurately represented by a monitoring system. For heart rates in the range 30–180 beats per minute, the fundamental frequency of pulsatile pressures transmitted from a blood vessel will therefore be 0.5–3 Hz.
Fourier analysis of the arterial pressure waveform reveals 8–10 significant harmonics (component waves which are multiples of the fundamental frequency) with frequencies of up to 30 Hz. To avoid resonance at the high heart rates, the frequency response of the monitoring system should therefore exceed 30 Hz. This is determined by the upper limit of the ‘flat range’ of the frequency–amplitude curve, which is approximately two- thirds of the natural frequency of the system (Figure 4a).
References
- Romero D. Bernoulli's law. http://scienceworld.wolfram.com/physics/BernoullisLaw.html (accessed 24 May 2011).
- Bicking R. Fundamentals of pressure sensor technology. http://www.sensorsmag.com/sensors/pressure/fundamentals-pressure-sensor-technology-846 (accessed 29 Mar 2011).
- Ruge AC. Strain responsive apparatus. United States Patent 2322319, 22/06/1943.
- . Analysis and design principles of MEMS devices. 1st edn. Amsterdam: Elsevier; 2005;
- . Micromachined pressure sensors: review and recent developments. Smart Mater Struct. 1997;6:530–539
- Williams CD. Introduction to sensors. http://newton.ex.ac.uk/teaching/CDHW/Sensors (accessed 29 Mar 2011).
- . Direct blood pressure measurement–dynamic response requirements. Anesthesiology. 1981;54:227–236
- . Understanding natural frequency and damping and how they relate to the measurement of blood pressure. J Clin Monit. 1989;5:137–147
- . Direct measurement of blood pressure by liquid-filled catheter manometer systems. Eindhoven University of Technology; 1981;
- Interpreting specifications of pressure sensor accuracy. http://www.sensorsone.co.uk/news/54/Interpreting-specifications-of-pressure-sensor-accuracy.html (accessed 21 May 2011).
PII: S1472-0299(11)00238-4
doi:10.1016/j.mpaic.2011.10.010
© 2011 Elsevier Ltd. All rights reserved.
