INSTRUMENT SYSTEM MODELS AND CALIBRATION

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MODELS OF INSTRUMENT SYSTEMS :A mathematical model relates the input and output of a system or sub-system. In other words it is a formula relating the input and output. The instrument is usually drawn as a block with the input and output shown. The mathematical model is written inside the block. The general symbol for signals is q but specific symbols may be used. The suffix i denotes the input and o the output.
When the input and output is a simple ratio, the model is just a number representing the ratio of output to input. It is often denoted by G, especially if it is a gain. In such case $G = \frac{\theta_o}{\theta_i}$. If the input and output have different units, then G has units also.

Some sensors have non linear equations and we cannot represent the relationship with a simple ratio so must use the full equation. For example a differential pressure flow meter has an equation Flow rate = $C{{(\Delta p)}^{\frac{1}{2}}}$.
Where C is a constant and $\Delta p$ is the differential pressure.

MODELS FOR COMPLETE SYSTEMS : A complete instrument system is made up from several sub-systems connected in series. The best way to deduce the input or output of a complete system is a step by step analysis of the information passing through. Consider the case of a D.P. flow meter. The meter converts flow rate into differential pressure. The d.p. is then converted into current and the current is indicated on a meter.

You have just seen how to work out problems involving instrument systems with different subsystems connected in series. The following is true for all types of systems.
In many cases each block may have a model that can be written as a ratio of output to input $G= \frac{\theta_o}{\theta_i}$. (This is not always true). In such cases we can easily work out the model for the complete system as follows. Consider three systems with model equations G1, G2 and G3 connected in series.

Now consider that if the three make up a single system the overall transfer function is $G_overall = \frac{\theta_o}{\theta_i}$
If we multiply G1 x G2 x G3 we have $(\frac{\theta_1}{\theta_i})(\frac{\theta_2}{\theta_1})(\frac{\theta_o}{\theta_2}) = (\frac{\theta_o}{\theta_i}) = G_overall$.
From this we conclude that the model for systems in series is obtained by multiplying the individual equations (ratios) together. Before doing this, make sure that the units are compatible.

INSTRUMENT ERRORS: Any given instrument is prone to errors either due to aging or due to manufacturing tolerances. Here are some of the common terms used when describing the performance of an instrument.
RANGE: The range of an instrument is usually regarded as the difference between the maximum and minimum reading. For example a thermometer that has a scale from 20 to 100⁰C has a range of 80⁰C. This is also called the FULL SCALE DEFLECTION (f.s.d.).
ACCURACY :The accuracy of an instrument is often stated as a % of the range or full scale deflection. For example a pressure gauge with a range 0 to 500 kPa and an accuracy of plus or minus 2% f.s.d. could have an error of plus or minus 10 kPa. When the gauge is indicating 10 kPa the correct reading could be anywhere between 0 and 20 kPa and the actual error in the reading could be 100%. When the gauge indicates 500 kPa the error could be 2% of the indicated reading.
REPEATABILITY :If an accurate signal is applied and removed repeatedly to the system and it is found that the indicated reading is different each time, the instrument has poor repeatability. This is often cau sed by friction or some other erratic fault in the system.
STABILITY : Instability is most likely to occur in instruments involving electronic processing with a high degree of amplification. A common cause of this is adverse environment factors such as temperature and vibration. For example, a rise in temperature may cause a transistor to increase the flow of current which in turn makes it hotter and so the effect grows and the displayed reading DRIFTS. In extreme cases the displayed value may jump about. This, for example, may be caused by a poor electrical connection affected by vibration.
TIME LAG ERROR : In any instrument system, it must take time for a change in the input to show up on the indicated output. This time may be very small or very large depending upon the system. This is known as the response time of the system. If the indicated output is incorrect because it has not yet responded to the change, then we have time lag error.
A good example of time lag error is an ordinary glass thermometer. If you plunge it into hot water, it will take some time before the mercury reaches the correct level. If you read the thermometer before it settled down, then you would have time lag error. A thermocouple can respond much more quickly than a glass thermometer but even this may be too slow for some applications.
When a signal changes a lot and quite quickly, (speedometer for example), the person reading the dial would have great difficulty determining the correct value as the dial may be still going up when in reality the signal is going down again.
RELIABILITY :Most forms of equipment have a predicted life span. The more reliable it is, the less chance it has of going wrong during its expected life span. The reliability is hence a probability ranging from zero (it will definitely fail) to 1.0 (it will definitely not fail).
DRIFT : This occurs when the input to the system is constant but the output tends to change slowly. For example when switched on, the system may drift due to the temperature change as it warms up.

INSTRUMENT CALIBRATION : Most instruments contain a facility for making two adjustments. These are

• The RANGE adjustment.
• The ZERO adjustment.

In order to calibrate t he instrument an accurate gauge is required. This is likely to be a SECONDARY STANDARD. Instruments calibrated as a secondary standard have themselves been calibrated against a PRIMARY STANDARD.
PROCEDURE : An input representing the minimum gauge setting should be applied. The output should be adjusted to be correct. Next the maximum signal is applied. The range is then adjusted to give the required output. This should be repeated until the gauge is correct at the minimum and maximum values. CALIBRATION ERRORS:
RANGE AND ZERO ERROR: After obtaining correct zero and range for the instrument, a calibration graph should be produced. This involves plotting the indicated reading against the correct reading from the standard gauge. This should be done in about ten steps with increasing signals and then with reducing signals. Several forms of error could show up. If the zero or range is still incorrect the error will appear as shown.

HYSTERESIS and NON LINEAR ERRORS : Hysteresis is produced when the displayed values are too small for increasing signals and too large for decreasing signals. This is commonly caused in mechanical instruments by loose gears and linkages and friction. It occurs widely with things involving magnetisation and demagnetisation.
The calibration may be correct at the maximum and minimum values of the range but the graph joining them may not be a straight line (when it ought to be). This is a non linear error. The instrument may have some adjustments for this and it may be possible to make it correct at mid range as shown.