Math and Measurement
for Biomed Techs
© D. J. McMahon 2014
rev 140923
With few exceptions, medical
parameters 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
=
10
= mega
x 1,000,000,000
=
109
= giga
6
Scientific Notation
& Standard Prefixes
Number
101
Prefix
deka-
102
Symbol
da
Number
10-1
Prefix
Symbol
deci-
d
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 & Constants
Name
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
S
Siemens
(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
S.I. Units that are especially important
for Biomed Technology:
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
Pressure as a column of mercury:
Important Pressure Conversions for Biomeds
Most useful factors in red
PSI
KiloPascal
cm of H2O
mm of Hg
atm
millibar
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
0.0147
0.1
1.02
0.75
0.0009
1
1 PSI =
1 millibar =
cf handout
Atmospheric Pressure at Increasing Elevations
Elevation
in of Hg
mm of Hg
35
0
29.92
760
30
500
29.38
746
1000
28.86
733
3000
26.82
681
4000
25.84
656
5430
24.40
620
10
14411
19.80
502
5
29035
8.90
225
0
25
20
15
0
2000
4000
6000
8000
10000
12000
14000
16000
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.
The 68–95–99.7 Rule:
• Standard deviation:

• 1 standard deviation:
X    68.27%
• 2 standard deviations:
X  2  98.45%
• 3 standard deviations:
X  3  99.73%
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:
Vout
dbv = 20 log -------Vin
dB – power:
Pout
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 Accuracy
Which is more important in measurements?
Precise and
accurate
Imprecise but
accurate
Precise but
inaccurate
Imprecise and
inaccurate
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.
-8
So if a solution has a hydrogen concentration of 1 x 10 , then
its pH is 8.
pH range and values of common substances
Poiseuille's Law :
( pwah-zwee )
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)
π r4 ΔP
Q = -----------8ηL
:
π r ΔP
Q = -----------8ηL
4
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.
Poiseuille's Law :
Example from pulmonary physiology:
“Apparent power” vs
“Real Power”
or
VA vs Watts
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)
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.”