Math and Measurement for Biomed Techs

© D. J. McMahon 2014 rev 140923

With few exceptions, medicalparameters are always expressed

in the metric (SI) system.

Scientific Notation Most frequently used:

1/100 = 10-2 = centi

1/1 000 = 10-3 = milli

1/1 000 000 = 10-6 = micro

1/1 000 000 000 = 10-9 = nano

x 1,000 = 103 = kilo

x 1,000,000 = 106 = mega

x 1,000,000,000 = 109 = giga

Scientific Notation& Standard Prefixes

Number Prefix Symbol Number Prefix Symbol

101 deka- da 10-1 deci- d

102 hecto- h 10-2 centi- c

103 kilo- k 10-3 milli- m

106 mega- M 10-6 micro- u

109 giga- G 10-9 nano- n

1012 tera- T 10-12 pico- p

1015 peta- P 10-15 femto- f

1018 exa- E 10-18 atto- a

Physical Units & ConstantsName Symbol Unit Unit Name Dimension

Charge Q C Coulomb TI

Voltage V V Volt L2MT-3I-1

Resistance R Ohm L2MT-3I-2

Conductance G SSiemens

(mho)L-2M-1T3I2

Capacitance C F Farad L-2M-1T4I2

Inductance L H Henry L2MT-2I-1

Mag. Induction B T Tesla L2MT-2I-2

Frequency F Hz Hertz MT-2I-1

Power P W Watt T-1

Energy E J Joule L2MT-3

Force: Newton (Kg.m / s2)

Pressure: Pascal (N/m2) and Torr (mm of Hg column)

Light: Lumen (light flux) and Lux (lumens / m2)

Candela (Cd) (luminous intensity)

Power: Watt (N.m / s)

Energy: Joule (watt x sec)

Magnetic Flux: Tesla

S.I. Units that are especially important for Biomed Technology:

Pressure as a column of mercury:

PSI KiloPascal cm of H2O mm of Hg atm millibar

1 PSI = 1 6.89 70.3 51.7 0.068 68.9

1 KiloPascal = 0.145 1 10.19 7.5 0.0098 10

1 cm of H2O = 0.014 0.098 1 0.735 0.00096 0.98

1 mm of Hg = 0.019 0.133 1.36 1 0.0013 1.33

1 atmosphere = 14.7 101 1033 760 1 1013

1 millibar = 0.0147 0.1 1.02 0.75 0.0009 1

Important Pressure Conversions for Biomeds

Most useful factors in red

cf handout

Atmospheric Pressure at Increasing Elevations

0

5

10

15

20

25

30

35

0 2000 4000 6000 8000 10000 12000 14000 16000

Elevation in of Hg mm of Hg

0 29.92 760

500 29.38 746

1000 28.86 733

3000 26.82 681

4000 25.84 656

5430 24.40 620

14411 19.80 502

29035 8.90 225

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.

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”

Dynamic Error –

> errors caused by changing values during measurement

Instrument Insertion Errors –

> “loading” of the device under test by the tester

Mean, Median, and Mode

Standard Deviation

> 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.

• Standard deviation:

• 1 standard deviation:

• 2 standard deviations:

• 3 standard deviations:

68.27%X

2 98.45%X

3 99.73%X

The 68–95–99.7 Rule:

Root Mean Square

“RMS”In electronics, used to express AC current or voltage

as its equivalent DC current or voltage.

For a sine wave only,

VRMS = 0.707 × Vpeak

Vpeak = 1.414 × VRMS

RMS voltage is the equivalent “heating voltage” of AC :

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.

Decibels:

dB – voltage:

dbv = 20 log --------Vout

Vin

PoutdB – power: dbp = 10 log --------

Pin

Used mainly in amplifier comparisons

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

Precision vs Accuracy

Precision:

the closeness of many measurement points to each other

Accuracy:

the closeness of many measurement points to a reference

Remember “P.A.R.T.”:

“Precision and Accuracy mean Repeatibility and Trueness”

Precision vs AccuracyWhich is more important in measurements?

Imprecise butaccurate

Precise butinaccurate

Precise and accurate

Imprecise andinaccurate

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

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 1 to 14, taken from the exponent of the concentration of hydrogen ion.

So if a solution has a hydrogen concentration of 1 x 10 , thenits pH is 8.

+-

- 8

pH range and values of common substances

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)

> the tube’s length (L)

> the pressure difference along the tube (ΔP) :

Poiseuille's Law :

Q =π r4 ΔP

8 η L------------

( pwah-zwee )

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.

Q =π r4 ΔP

8 η L------------

Example from pulmonary physiology:

Poiseuille's Law :

Watts is “Real Power” -- the power (V x I) that does work.VA is “Apparent Power” -- the vector sum of real power (P) and reactive power (jQ).

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)

“Apparent power” vs “Real Power”or

VA vs Watts

Instrumentation Amplifier

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

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.

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.”