Note for Sensors And Transducers - ST by Suman Kumar Acharya

Topics ( 4 )

Text from page-1

EE3.02/A04 Instrumentation 1. Transducers and Sensors 1. Transducers and Sensors By the end of this section you should be able to: • • • • Discuss the definitions/specifications by which sensors are characterised. Describe common methods for converting a physical parameter into an electrical quantity and give examples of transducers, including those for measurement of temperature, strain, motion, position and light. Explain how to make sensitive measurements using a Wheatstone bridge, including balancing and offset compensation. Describe systems for measuring motion, temperature, strain and light intensity. 1.1. Definitions In this course we will be studying Electrical Measurements, and we will necessarily interplay between techniques and hardware used to sense the quantity we wish to measure, techniques and hardware used to process the signal generated by the sensors and also algorithms to interpret the final result. We will be, therefore be dealing with transducers, sensors and actuators. Transducers: Devices used to transform one kind of energy to another. When a transducer converts a measurable quantity (sound pressure level, optical intensity, magnetic field, etc) to an electrical voltage or an electrical current we call it a sensor. We will see a few examples of sensors shortly. When the transducer converts an electrical signal into another form of energy, such as sound (which, incidentally, is a pressure field), light, mechanical movement, it is called an actuator. Actuators are important in instrumentation. They allow the use of feedback at the source of the measurement. However we will pay little attention to them in this course. The study of using actuators and feedback belongs to a course in Control theory. A sensor can be considered in its bare form, or bundled with some electronics (amplifiers, decoders, filters, and even computers). We will use the word instrument to refer to a sensor together with some of its associated electronics. The distinction between a sensor and an instrument is extremely vague, as it is increasingly common to manufacture integrated sensors. What follows is equally applicable to sensors and/or instruments. The discussion is also applicable to circuits, such as amplifiers, filters, mixers and receivers. Signal processing circuits are, in a sense, instruments. It is not very important that both input and output signals are, for example, voltages. 1.2. The linear model of a Sensor There is a fair amount of jargon associated with sensors, used to describe the usefulness or quality of a piece of hardware. Sensor specification terms are often used in an erroneous or misleading way, especially in the advertising literature of equipment manufacturers; they tend to manipulate definitions in order to make their product appear better than it is. It is always a good idea to investigate the precise meaning of specifications, before accepting them. Below we attempt the definition of some important specifications from the engineering point of view. CP Imperial College, Autumn 2008 1-1

Text from page-2

EE3.02/A04 Instrumentation 1. Transducers and Sensors The following discussion refers to an implicit linear model for the sensor. A sensor is assumed to be linear so that its response y to a stimulus x is idealised to have the form: y ( x ) = Ax , 0 ≤ x ≤ x max , A>0 (1.1) Please note that we have defined the stimulus to be positive. This makes it easier to define quantities such as the threshold, and consequently makes it easier to understand that there may exist gaps in the response of an instrument! 1.2.1. Sensitivity The constant A in (1.1) is called the sensitivity or the transducer gain or, simply, the gain of the sensor. To simplify the discussion we also take the gain to be positive. The linear model satisfies the definition of linearity, as it should: y ( x + z ) = A( x + z ) = y ( x) + y ( z ) (1.2) Please note that the response of a sensor defined this way exhibits no time dependence. Such an idealised sensor has no memory and its output instantly tracks the input. In the more general case we may know the steady state transfer function of the sensor. We can define the sensitivity as the derivative of the output with respect to the input: S= ∂y . ∂x (1.3) This is a partial derivative. As we shall see below, the sensor will exhibit sensitivities to other ambient (e.g. temperature) or operating parameters (e.g. a supply voltage). It is essential to study the sensor with all other (usually unintended) stimuli held constant. Sensitivity is, in a few words, the ratio of electrical output to signal input (input transducer), or physical output to electrical input (output transducer). e.g., a temperature sensor may be quoted as 50 µV/K. and a loudspeahker as 90dBspl/W. However, the term sensitivity may also be used in its usual electronic sense, i.e. the %change of some property of a device (eg gain) as a result of a % change in some parameter, (eg the ambient temperature). For clarity, we will refer to this as the cross-sensitivity of x on y. The sensitivity is also called the Gain of the sensor or instrument. The term sensitivity is occasionally misused to refer to the minimum detectable signal, i.e. the sensor’s detectivity or threshold, which , incidentally, equals the noise floor of the sensor. 1.2.2. Threshold and detectivity No sensor will respond to arbitrarily small signals. Signals in the range between zero and the sensor threshold xmin will not cause the output of the sensor to change. The existence of a threshold is related to nonlinearity and noise. A stimulus which is too small for the output to exceed the noise floor is considered to be smaller than the threshold. Nonlinearity can play a role CP Imperial College, Autumn 2008 1-2

Text from page-3

EE3.02/A04 Instrumentation 1. Transducers and Sensors as well. Consider an enhancement mode MOSFET as a voltage sensor (MOSFETs are used as very high impedance voltage or charge probes in high end “active” oscilloscope probes). Clearly such an instrument cannot respond to voltages smaller than the MOSFET threshold voltage. A sensor will also fail to respond to stimuli which are arbitrarily large. A sensor will necessarily have a range or a full scale xmax . The full range of a sensor can be limited by compression or by clipping. (Note that clipping is an extreme example of compression!) Since both compression and clipping are manifestations of nonlinearity we conclude that all sensors are non-linear. 1.2.3. Zero offset A real sensor will deviate from the idealised linear model. The smallest improvement we can make to the description of an assumed linear sensor is the addition of a constant zero offset as follows: y ( x ) = b0 + Ax (1.4) This is not a linear form, despite the fact that it is described by a first order polynomial. This is called an affine relation. The constant b 0 is called the zero offset of the sensor. The zero offset can be defined in two ways: The sensor reading when the input is zero, or the value of the stimulus required to make the output zero. The zero offset is simple to correct. By subtracting b 0 from y we recover a linear description of the sensor: y′ ( x ) = y ( x ) − b0 = Ax (1.5) 1.3. Non-linearity While still retaining the time independence assumption we can introduce non-linearity in the model of the sensor: y′ ( x ) = Ax + b2 x 2 + b3 x 3 + " = Ax + g ( x ) (1.6) The function g ( x ) describes how much the sensor response deviates from its linearised description. There are several ways to describe linearity or nonlinearity , each one of them described by a different term. The terms linearity and nonlinearity are conjugate, are used interchangeably, and often a value of linearity is quoted as non-linearity, and vice-versa. Nonlinearity is usually measured in relative units, either as a percentage of the maximum full scale reading of the sensor or the instrument, or locally as a percentage of a reading. Ideally we wish the nonlinearity to vanish, so it must be proportional to g ( x ) We can define the absolute non-linearity locally, at x, as: g ( x) , x ∈ [ xmin , xmax ] Ax It is more common to compare the maximum of g ( x ) to the range of the sensor: δ y ( x) = CP Imperial College, Autumn 2008 (1.7) 1-3

Text from page-4

EE3.02/A04 Instrumentation δy = max g ( x ) A ( xmax − xmin ) 1. Transducers and Sensors = max y ( x ) − b0 − Ax A ( xmax − xmin ) , x ∈ [ xmin , xmax ] (1.8) Although this is the correct description of nonlinearity, sometimes the definition is given as: max g ( x ) δy = (1.9) y ( xmax ) This form may be used in the presence of large zero offset in order to make the nonlinearity appear smaller than it is! Nonlinearity results not only in a discrepancy between what the instrument reads and what the linear model describes. It also leads to a gain error, usually referred to as the differential nonlinearity. The differential nonlinearity may be defined as the discrepancy, due to the non-linear character of the sensor’s transfer function, of the sensor gain from its modelled gain. The differential nonlinearity at a value x of the stimulus is simply: δ A ( x) = g′ ( x) (1.10) A We may, of course, be interested in the maximum value of δ A over the sensor range: δA = max g ′ ( x ) (1.11) A 1.4. Memory effects 1.4.1. Linear sensors –Laplace transforms and Convolution The models we have developed so far are not entirely adequate, especially when we are concerned with very fast measurements. In this case we must account for the possibility that the sensor can internally store energy. Its internal energy content can modify the sensor’s behaviour. As a result the output of a sensor depends on previous measurements the sensor made, or, equivalently, the sensor exhibits memory. The time dependence of the response of a linear sensor is well known. A sensor can still be linear if its response is described by a linear differential equation: N ∑ An n=0 ∂n y K ∂k x = B ∑ k ∂t n k =0 ∂t k (1.12) We can take the Laplace transform of this description to conclude that: N ⎛ K ⎞ y ( s, X ) = ⎜ ∑ Bk s k ∑ An s n ⎟ x = H ( s ) X ( s ) (1.13) n=0 ⎝ k =0 ⎠ so that, in Laplace transform space, the sensor response is still linear in the stimulus x. The response of a sensor with a transfer function H ( s ) at time t is the convolution integral between the history of the stimulus x and the inverse Laplace transform h ( t ) of H ( s ) : CP Imperial College, Autumn 2008 1-4