Math and Measurement for Biomed Techs. D. J. McMahon rev cewood

Transcription

1 Math and Measurement for Biomed Techs D. J. McMahon rev cewood

2 Standard Prefixes Number Prefix Symbol Number Prefix Symbol 10 1 deka- da 10-1 deci- d 10 2 hecto- h 10-2 centi- c 10 3 kilo- k 10-3 milli- m 10 6 mega- M 10-6 micro- µ (mu) 10 9 giga- G 10-9 nano- n tera- T pico- p peta- P femto- f exa- E atto- a Most frequently used: 1/100 = 10-2 = centi x 1,000 = 10 3 = kilo 1/1 000 = 10-3 = milli x 1,000,000 = 10 6 = mega 1/ = 10-6 = micro x 1,000,000,000 = 10 9 = giga 1/ = 10-9 = nano

3 Scientific Notation Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. In scientific notation all numbers are written in the form m 10 n (m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any real number. In normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. (1 m < 10) Thus 350 is written as This form allows easy comparison of numbers, as the exponent n gives the number's order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as ). The 10 and exponent are often omitted when the exponent is 0. much more:

4 Dimensional Analysis (the Unit Factor Method) Dimensional Analysis is a problem-solving method that uses the fact that any number can be multiplied by one without changing its value. Unit factors are made from any two terms that describe the same or equivalent value. examples: Because times 1 is equal to divided by 1, you can invert these unit factors as you wish. Your goal is to arrange all your unwanted units so they cancel out. Solve: much more:

5 With few exceptions, medical parameters are always expressed in the metric (SI) system. SI: (Système International d'unités)

6 Physical Units & Constants Name Symbol Unit Unit Name Charge Q C Coulomb Voltage V V Volt Resistance R Ohm Conductance G S Siemens (mho) Capacitance C F Farad Inductance L H Henry Mag. Induction B T Tesla Frequency F Hz Hertz Power P W Watt Energy E J Joule

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8 SI Units that are especially important for Biomed Technology: Mass: Length: Time: kilogram (kg) meter (m) second (s) Force: newton (N) (kg. m / s 2 ) Energy: joule (J) (N. m) = (watt. sec) Power: watt (W) (N. m / s) Pressure: pascal (Pa) (N/m 2 ) and Torr (.133 kp) = (mm of Hg column) Light: candela (cd) (luminous intensity) lumen (lm) (cd. sr) (light flux) lux (lx) (lumens / m 2 ) (illuminance) Magnetic Flux: Tesla (T)

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10 Pressure as a column of mercury barometer manometer

11 Manometers, two styles; Pressure measured as a column of mercury: barometer manometer absolute pressure closed end (contains vacuum) measures atmospheric pressure 760 mmhg normal (at sea level) higher atmospheric pressure = higher number (pressure in outer space vacuum would be 0 mmhg) gauge pressure open end (contains ambient air pressure) measures applied pressure 0 mmhg is unpressurized state higher applied pressure = higher number

12 Pressure Conversions (most useful conversions shown in bold) PSI KiloPascal cm of H 2 O mm of Hg atmosphere millibar 1 PSI = KiloPascal = cm of H 2 O = mm of Hg = atmosphere = millibar = remember: 1 atmosphere = 14.7 psi = 760 mmhg = 1033 cmh 2 O = 101 kpa Hg: 760 mm = inches H 2 O: 1033 cm = 33.9 feet

13 Portland State University: The Big Barometer In the atrium of the Engineering Building stands the world s tallest barometer. At 14.2 m tall (46'8"), the PSU Barometer is made of repurposed 2" i.d. pyrex glass pipe and has a 14 gallon reservoir. The thick-walled glass pipes were salvaged during a renovation of Science Building 2. They had previously been used for over 35 years as chemistry lab drain pipes. The PSU barometer uses vacuum pump oil instead of water as the working fluid. Vacuum pump oil has several advantages in this application such as extremely low vapor pressure and the lowest available fluid density. Low vapor pressure is important for maintaining a vacuum over the fluid column. The low density of the vacuum oil allows the PSU Barometer s fluid column to reach a nominal height of 12.4m and +/- excursions of 0.4m. A water barometer is limited to a height of 10m due to the density of water. The construction of the barometer received front page coverage in The Oregonian. In addition to breaking a world record, the architect of the barometer, Tom Bennett was named the 2013 Outstanding Supporter of Research by the Sigma Xi Columbia Willamette chapter. The PSU Barometer is primarily as a teaching tool for the Civil and Environmental Engineering fluidrelated lab courses. It is available for other PSU Departments and for science and engineering education outreach programs. If you would like to contact Maseeh College about using the PSU Barometer in science education curriculum or to schedule a tour of the college and see it in person please barometer@cecs.pdx.edu vacuum pump oil: 12.4 m = 40.7 feet

14 Atmospheric Pressure at Increasing Elevations Elevation in of Hg mm of Hg (Denver) (Mt Rainier) (Mt Everest) elevation vs inches of Hg

15 Significant Figures > In any measurement, we can t claim more accuracy than physical reality allows. > Always ask: - Plus or minus how much? - In digital displays, is there last digit bobble? > The number of significant figures is NOT improved by multiplying errors. > Three digits (sometimes four) is usually the best you can do!

16 Measurement Errors - Static Error Misreading displays or limitations of equipment > parallax reading of an analog meter > interpolation of the scale on an analog meter > last-digit bobble on a digital meter Dynamic Error > errors caused by changing values during measurement Instrument Insertion Errors > loading of the device under test by the tester

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18 Mean, Median, and Mode Mean: Sum of all values divided by the number of quantities. Usually called the average. Median: It is the middle value in the data set, that is, the value where exactly half of the values are above it and half below it. Mode: The most frequently occurring value in a set of data. Example: Given these measurements (after you rank them in order): mean value = 950 (the sum) / 9 = median value = 106 mode = 107

19 Mean, Median, and Mode

20 Standard Deviation (σ) > Widely used measure of variability or dispersion of data. > Standard deviation serves as a measure of how far the samples of data are spread out. > A large standard deviation indicates that the data points are far from the mean; a small standard deviation indicates that they are clustered closely around the mean.

21 The Rule: Standard deviation: σ (sigma) 1 standard deviation: X 68.27% 2 standard deviations: X % 3 standard deviations: X %

22 The mean is the same, but. = 1 = 2 = 3

23 Precision vs Accuracy Precision: the closeness of many measurement points to each other Accuracy: the closeness of many measurement points to a reference think of accurate as actual or Remember P.A.R.T. : Precision and Accuracy mean Repeatability and Trueness

24 Precision vs Accuracy Which is more important in measurements? Precise and accurate Imprecise but accurate Precise but inaccurate Imprecise and inaccurate

25 Root Mean Square RMS In electronics, used to express AC current or voltage as its equivalent DC current or voltage. V RMS = V peak V peak = V RMS accurate for a sine wave only So, 110 VAC (rms) = peak voltage of volts

26 RMS voltage is the equivalent heating voltage of AC :

27 sine wave A sine wave is a curve with this shape: Signals in the electromagnetic spectrum (heat, radio, light, x-rays, etc) have a speed of 3x10 8 meters/second. With this standard speed, frequency and wavelength are reciprocal to each other. 300 MHz (3x10 8 cycles per second) signals have a wavelength (λ) of 1 meter. The sine wave has a pattern that repeats. The length of this repeating piece of the sine wave is called the wavelength (λ). The wavelength can be found by measuring the length or distance between one peak of a sine wave and the next peak. All periodic waves can be made by adding up sine waves of different frequencies and amplitudes. This is called Fourier Analysis, sometimes calculated by the Fast Fourier Transform (FFT).

28 Apparent power vs or VA vs True (or Real) Power Watts Watts is Real Power (P) - the power (V x A) that does work. VA is Apparent Power - the vector sum (S) of real power (P) and reactive power (Q). Reactive loads such as inductors and capacitors dissipate zero power, yet the fact that they drop voltage and draw current gives the deceptive impression that they actually do dissipate power. This phantom power is called reactive power. Apparent Power is used when sizing wiring and components. Real Power is what accomplishes useful work in the device. Apparent Power is always > Real Power if there is any reactive factor. Power Factor = W / VA ( 0.60 is typical)

29 Logarithmic Units: Decibels Bel was used in the telephone industry (named after Alexander Graham Bell). The Bel is usually too large for most applications, so it is rarely if ever used. Decibel (db) is one-tenth of a Bel. It is simply a means of logarithmically expressing the ratio between two signal levels. special cases: Used mainly in amplifier comparisons audio Volume Unit (VU): 0 VU = 1 mv at 1 KHz through 600 radio dbm: 0 dbm = 1 mv of RF through 50

30 Three General Categories of Measurement Direct measurement: Compare the parameter to some calibrated standard. Indirect measurement: Measure something other than the actual parameter that we want. Null measurement: Compare a calibrated source to an unknown value and adjust the unknown value until the difference between them is zero.

31 Measurement Standards: > International References: at the ISI > Primary Standards: at the NIST > Working Standards: NIST Traceable > Secondary Standards: on-site references > Gauges & Instruments: routine equipment

32 Resolution ( Definition ) The degree to which we can distinguish the individual elements of an output. eg: the lines in a video display test pattern, or the change of pitch in an audio signal

33 ph: The measure of acidity or alkalinity of any liquid Water always has a small amount of hydrogen (H + ) and hydroxide (OH - ) ions. ph is the numeric value from 0 to 14, taken from the negative of the exponent of the concentration of hydrogen ion. So if a solution has a hydrogen concentration of 1 x 10-8, then its ph is 8. ph = - log 10 [H + ] **smaller ph number is more acidic **

34 ph range and values of common substances

35 Poiseuille's Law : defines the flow (Q) of fluid passing a point along the tube in terms of: > the fluid's viscosity (η) > the tube's radius (r) ( pwah-zwee ) > the tube s length (L) > the pressure difference along the tube (ΔP) Q = π r4 ΔP η L

36 Q = π r4 ΔP η L In other words, The flow in a tube is directly proportional to the fourth power of the radius. This means that doubling the radius of the tube increases the fluid flow by a factor of 16.

37 Poiseuille's Law : Example from vascular physiology:

38 Poiseuille's Law : Example from pulmonary physiology:

39 Instrumentation Amplifier Advantages for physiological monitoring: > High Common Mode Rejection Ratio (CMRR): The capability of an instrument to reject a signal that is common to both input leads. [ CMRR = Differential Gain / Common Mode Gain ] > High input Z > Wide bandwidth > Low noise

40 Instrumentation Amplifier

41 Analog to digital conversion A process in which a continuously variable (analog) signal is changed into a multi-level signal without altering its essential content. The input is a voltage that varies among a theoretically infinite number of values (sine waves, speech, ECG, etc). The output has defined levels or states. The simplest digital signals are in binary values. Digital signals propagate more efficiently than analog signals, because digital impulses, which are well-defined and orderly, are easier for electronic circuits to distinguish from noise, which is chaotic.

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43 Quantization Error: Error resulting from trying to represent a continuous analog signal with discrete, stepped digital data. When the analog value being sampled falls between two digital steps. the analog value must be represented by the nearest digital value, resulting in a very slight error. The difference between the continuous analog waveform and the stair-stepped digital representation is quantization error. For a sine wave, quantization error will appear as extra harmonics in the signal. For music or program material, the signal is constantly changing and quantization error appears as wideband noise, cleverly referred to as quantization noise.

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