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# The Engineer's Notebook

The Engineer's Notebook is a shared blog for entries that don't fit into a specific CR4 blog. Topics may range from grammar to physics and could be research or or an individual's thoughts - like you'd jot down in a well-used notebook.

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### System Definitions

Posted December 26, 2006 4:49 PM by amichelen

Today I would like to present some definitions (besides being and engineers I teach engineering) that many times either are not well remembered or we never learned them in details.

Systems and devices in general must conform to a set of definitions and standards to describe their behavior and characteristics. This is necessary in order for professionals to effectively interchange ideas. In this offer I will present some important definitions and standards that may be useful to all engineers.

Error

This is one of the most important quantities when determining the accuracy of a measurement. In this context it is defined as the difference between the actual value of a variable (quantity) and its measured value. In control systems it is defined as the difference between the actual value and the desired value of a variable. Figure 1 shows a calibration curve of a sensor where the input variable is X and output variable is Y. The figure shows the actual value of a measurement and the measured value. Note that the error represents a range of values around the actual value and is normally represented as a percentage.

Figure 1: Error Determination

Transfer Function

A general system can be represented by blocks that describe different interconnected parts. Sensors are followed by transducers, for instance. Each one of these blocks is a graphical representation of the mathematical function of the relationship between the output and the input of the block. In technical terms such a block is called a Transfer Function.

Figure 2: General Transfer Function

Figure 2 shows a typical transfer function of a general device. The transfer function T is a function of the input, the output, and -- in the case of a dynamic transfer function -- it is also a function of time, t. T describes the relationship between the output and the input of the block. In general, for linear systems, this relationship is simply the ratio of the output over the input as it is indicated in the figure.

Transfer functions are composed of two parts (in some systems one of these component does not exist): the dynamic part and the static part. The dynamic transfer function is the relationship between the output and the input when the input changes with time. The static transfer function is the relationship between output and input when the input is not a function of time. Static transfer function can always be represented by simple fraction of output over input. Dynamic transfer functions are normally represented by differential equation.

Example 1: Static Transfer Function

Figure 3 is an example of a static transfer function. This transfer function represents the relationship between the distance traveled by a car and the amount of gasoline in the tank. As it can be seen there is a linear relationship between the amount of gasoline in the tank and the distance traveled. Therefore the transfer function is static.

Figure 3: Example 1

Accuracy

The accuracy of a device is the combined maximum overall error (non--linearity, Hysteresis, repeatability, etc.) that is expected when measuring a variable. This term is normally expressed as the inaccuracy or uncertainty of the measurement. It can be expressed in several ways. The most common are as follows:

· Measured variable:The inaccuracy is a percentage of any value measured.

· Full-scale percentage (FS): The inaccuracy or uncertainty of any measure is expressed as a percentage of the full scale reading of the instrument.

· Span percentage: The uncertainty is expressed as percentage of the range of measuring capability of the instrument.

· Percentage of actual reading: In this form the inaccuracy is a percentage of the value of the actual reading.

An example follows.

Example 2:

A voltmeter with a reading span of 5 -- 20 volts is used to make a measurement. The reading results in a value of 8 V. Determine the error and the possible range of voltages for the reading if the accuracy is (a) ±2% of any measurement, (b) ±0.5% FS, (c) ± 0.6% of span, and (d) ± 0.8 of reading.

Solution:

1) Error = ± (0.02)(8 V) = ± 0.16 V. The possible range of voltages for this reading is: 7.84 - 8.16 V.

2) Error = ± 0.005)(20 V) = ± 0.1 V. The possible range of voltages for the reading is: 7.9 - 8.1 V.

3) Error = ± 0.006)(20 - 5) V = ± 0.09 V. The possible range of voltages for the reading is: 7.91 - 8.09 V.

4) Error = ± 0.008)(8 V) = ± 0.064 V. The possible range of voltages for the reading is: 7.936 - 8.064 V.

Accuracy for Digital Signals

For digital systems the most important source of error is in the inaccuracy of the digital representation of analog signals. Based on this the accuracy is defined as the percentage deviation of the analog signal per bit of the digital signal. For example, suppose an analog-to-digital converter (ADC) has a resolution of 0.525 volts per bit, and an accuracy of ± 2%. This implies that in order to set an output bit an input (analog) voltage change of 0.525 ± (0.525)(0.01) = 0.525 ± 0.005 V (0.52 - 0.53 V) should be applied.

Sensitivity

Sensitivity is defined as the amount of change in the output of a device for a given change in the input. A device is highly sensitive if it exhibits a big change in the output for a small change in the input signal. Normally, sensitivity is equivalent to the value of the transfer function of an instrument or sensor. For example, if an LVDT outputs 5 V for every 2 mm of motion, then its sensitivity and its transfer function are both equal a to 2.5 V/mm.

Hysteresis

Some instruments (mechanical sensors, for instance) exhibit the peculiarity that a different reading results for a specific input, depending on weather the input value is approached from higher or lower values. This phenomenon is called hysteresis (derived from the Greek word that means deficiency), and it is related to the history of the instrument. Figure 4 shows a typical graph of hysteresis for an instrument. Notice that for the same input we get two different outputs depending on weather we approach the input from curve A (increasing) values or curve B (decreasing) values.

Figure 4: Hysteresis

Resolution

The resolution of a measurement is defined as the minimum measurable input value. For instance, if the slider of a potentiometer moves in such a way that for every turn of the winding the resistance changes by 2 Ω , then the potentiometer cannot provide a resistance smaller than 2 Ω .

Linearity

The output of sensors and other devices is a function of the input values. It is very important that for every different output value there is a unique input value that is related to this output. Therefore, a simple linear relationship between the output and the input is desirable to ensure that for each input value there is only one output value associated with it.

A linear relationship between two variables can be expressed algebraically by the equation of a straight line. Figure 5 represents a general sensor whose input is X and its output is Y.

Figure 5: General sensor transfer function

The mathematical relationship between these two variables is given by

Y = mX + b (1)

where m is the slope of the line and b is its intercept with the vertical axis. Figure 6 shows the graph of Eq. (1).

Figure 6: Linear plot (a calibration curve)

The following example illustrates the general procedure to determine the linear relationship between the output and the input.

Example 3:

A thermistor (temperature sensor) changes linearly from 200 Ω to 500 Ω as the temperature changes from 20° to 120°. Determine the linear equation relating resistance (output) to temperature (input).

Solution:

Using equation (1) we set up two equations as follows:

200 Ω = (20°)m + b

500 Ω = (120°)m + b

Solving these two equations for the two unknowns - m and b - we get

m = 3 Ω / °

b = 140 Ω

Then the resulting equation is

R = 3m + b

(I will treat dynamic transfer functions in a coming blog.)

Bibliography:

"Process Control Instrumentations Technology", Curtis D. Johnson. Prentice Hall. ISBN:0-13-938200-3

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Participant

Join Date: Jan 2007
Posts: 2
#1

### Re: System Definitions

01/05/2007 8:13 PM

Engineers that are not utterly familiar with these trivial definitions are not engineers! Maybe some non-technical folk might be interested, but somehow I doubt it. So the purpose is what?

Power-User

Join Date: Mar 2005
Posts: 105
#2

### Re: System Definitions

01/06/2007 10:17 AM

My dear trgregory,

What you say is not completely true. There are many engineers that are not utterly familiar with all these terms. Most engineers, however, have seen these definitions at least when they were studying engineering, but as time passes and if they do not need these definitions in their daily engineering work, certainly they forget.

So the purpose of this entry is very simple: to share and to remember with my fellow engineers (I suppose you are an engineer also. If so what type if I may ask?) these important terms and definitions. No harm implied.

I have a couple of questions for you: what is the purpose of your comment that does not make any contribution to the blog intellectual discussion? Why don't you add another, non-trivial, definition like Gain Flatness, P1dB power, Output Intercept Point known as IP3, VSWR, PSRR, and so forth?

Abe

Participant

Join Date: Jan 2007
Posts: 2
#3

### Re: System Definitions

01/08/2007 8:16 PM

Dear Abe Michelen,

I am an electrical engineer. I never intimated that what you posted was harmful in any way, it just seems like a waste of time.

"Most engineers...but as time passes and if they do not need these definitions in their daily engineering work..."

Then why would they be interested in being reminded of them?

"So the purpose of this entry is very simple: to share and to remember with my fellow engineers.." I guess this is a nostalgic trip down memory lane.

"what is the purpose of your comment that does not make any contribution to the blog intellectual discussion?"

IMHO posting a load of simplistic definitions, which are found in elementary text books, isn't contributing to any intellectual discussion!

trgregory

Guru

Join Date: Mar 2007
Location: South-east corner of Spain 50 48 49.24N 2 28 27.70W
Posts: 1595
#4

### Re: System Definitions

05/12/2007 6:22 AM

I´m glad to see these old nuggets of information! and yes I had forgotten them! It´s nice to be reminded about things because the more you are reminded, the more you remember. wouldn´t you feel stupid if you placed an order for some machining work for say 100000 parts and when you got them back they were wrong because you didn´t know the basics of SPC! Whoops, sorry gov, but I forgot to tell them the tolerance ranges!!

Keep on rockin Abe

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