A P P E N D I X
C
Basic Electronics
S.N. Whittlesey and J. Hamill
T
his appendix gives a brief overview of the
elementary electronic circuit concepts that
are relevant to the collection of human movement
data. The topics discussed include basic electronic
components, Ohm’s law, circuit diagrams, and the
functions of several common lab instruments, such
as amplifiers and electrogoniometers. Students
interested in further detail or sample problems on
any particular topic should refer to textbooks on
electronics or linear circuits (such as Schaum’s Outline [O’Malley 1992] or Winter and Patla 1997). The
focus here is on simple, steady state circuit concepts
and how they apply to common measurements of
human movement.
Electronics notation and symbols are standardized
across different fields. Here, we use the notations
and symbols given in this table
Symbol
A circuit diagram is a formal means of representing
an electric circuit. We use these diagrams in this
appendix to illustrate different examples. Circuit
diagrams have many conventions, the most common
of which are that
• components are represented by standard icons
with their sizes noted,
• wires are represented by straight lines for zero
resistance,
• wires are drawn only in north-south-east-west
directions,
• a connection of two wires is indicated by a solid
dot,
• one wire passing over another is indicated by a
short loop, and
Basic SI Electrical Units
Quantity
CIRCUIT DIAGRAMS
SI unit SI abbreviation
Current
I
ampere
A
Voltage
V
volt
V
Resistance
R
ohm
⍀
Capacitance
C
farad
F
Power
P
watt
W
• interface points are indicated by open dots and
labeled.
Circuit diagrams can, of course, become very complicated. The conventions just listed are displayed in
figure C.1. This diagram shows the symbols for various components: a 9-volt (V) battery and its ground,
a 100-ohm (⍀) resistor, a 10 ⍀ variable resistor, and
a 1-microfarad (␮F) capacitor. A discussion on these
electrical components and several principles of electricity follows.
245
246
Appendix C ________________________________________________________________________
10 Ω
9V
1 µF
100 Ω
V
º Figure C.1 Circuit diagram of a 9 V battery powering
two resistors and a capacitor. The lower side of the battery is grounded. The voltage (V) is the quantity that we
measure. The 10 ⍀ resistor has a variable resistance with
a maximum of 10 ⍀.
ELECTRIC CHARGE, CURRENT,
AND VOLTAGE
Electric charge can be either positive or negative,
depending on whether we are dealing with protons
or electrons. Electricity is the flow of electrons
through some medium, whether through a wire in
a house or lightning through the air. The basic SI
unit of electric charge is the coulomb (C). It represents about 6.25 ⫻ 1018 electrons. The rate of flow of
electricity, or current, has units of amperes, or amps
(A); 1 A is a flow rate of 1 C/s. As practical examples,
consider that a handheld calculator requires a few
microamperes (mA) to operate, a D-cell battery supplies about 100 mA, a car battery offers a maximum
of about 2 A, and a typical house circuit provides
20 A. Current flow occurs when there is a difference between the electrical potential energy at two
sites. This potential difference is called a voltage.
One volt is defined as 1 joule ( J) of energy per C of
charge. A D-cell battery offers 1.25 V, a car battery
offers 12 V, and house electricity averages 110 V. A
human electromyography (EMG), in contrast, is on
the order of ␮V.
A point of zero voltage is called a ground. This
is never an absolute quantity, but rather a defined
reference point in a circuit. Thus, two circuits can
have their own grounding references, but there may
be a potential difference between the grounds of the
two circuits. For example, in a small battery-powered
circuit such as a clock or flashlight, ground is typically defined as the negative terminal of the battery
powering the circuit. In house applications, ground
is defined as the potential of the surrounding soil.
This is accomplished by connecting the circuit to a
metal rod driven into the earth. This house ground
is different from the ground in any battery-powered
circuit unless a connection is made between them.
As another example, jump-starting a car is dangerous because potential differences can exist between
two cars; even though the battery in each car is 12 V,
their tires insulate them from the road (which is
the ground). In human movement, we often see
these principles applied in EMG recording because
different voltage potentials can exist over the skin
surface of the body depending on what muscles
are active. We often record EMG with a separate
grounding plate on a bony landmark away from the
musculature.
Voltage and current are related (as is discussed
later in this appendix), and this is often a source of
confusion. The basic principles, stated previously,
must be remembered: Current is the flow of electrons
and voltage is a potential energy difference that can
cause electron flow. If current is flowing between
two sites, then there must be a voltage difference
between them. However, there can be a voltage
difference without current flowing; in that case,
there is no complete circuit for the current to flow
through. For example, there is a voltage difference
between the terminals of a wall outlet, regardless of
whether an appliance is connected to it. Current
only flows between the terminals when an appliance
is connected to them and turned on. An extreme
example is that birds can land on a high-voltage overhead power line without being harmed. The same
principle applies to electrical line workers: As long
as workers are highly insulated from the ground, it
is possible for them to touch the wire with their bare
hands. When contact is made, a person is thousands
of volts higher than the ground, but because virtually no current can flow through the insulation, the
worker is unharmed. However, when a power line is
broken in a storm and one end falls to the ground,
touching the wire can be fatal because making contact with the wire connects a circuit to the ground.
Circuits are often difficult to conceptualize
because they cannot be visualized directly. A measurement instrument, such as a voltmeter, oscilloscope,
or computer, must be used to establish the state of a
circuit. This is an abstract task, and it can be helpful
to use the flow of a fluid through a pipe system as
an analogy. Electric current (amperage) is analogous
to the rate of fluid flow through the pipe (i.e., liters
per second). Voltage is analogous to the pressure in
the pipe system. Thus, if water is flowing through a
hose, there must be a pressure difference between
the ends of the hose; however, we can have a closed,
_________________________________________________________________________ Appendix C
pressurized container with no water leaking out of it.
Flow implies that a potential energy difference exists.
The fact that a potential energy difference exists,
however, does not imply that something is flowing.
Other fluid examples will be offered throughout this
appendix to illustrate key points.
We most often think of voltage as the strength
of a power supply. However, it is also an important
quantity that we measure. In biophysical systems, we
almost always measure a voltage, not a current. This
is primarily a matter of ease of use and the relative
durability of voltmeters as compared to ammeters.
When we speak of a biophysical signal, we are referring to a time-varying voltage produced by a human
subject or some device attached to it.
Some resistors have variable resistances. A
common type of variable resistor is the potentiometer, often called a pot. Some potentiometers can
be adjusted by turning them (a rotary pot), whereas
others slide linearly. Volume controls on radios can
take both forms, as can dimmer switches for indoor
lighting.
Most circuits include multiple resistances. Thus,
it is important to understand how resistors act when
connected together. The two basic manners of connecting are in a series and parallel. In a series connection, there is one path. One resistor follows the
other, and all current flowing through one resistor
must also flow through the other (figure C.2). The
total resistance of two resistors in series is equal to
the sum of the resistances, that is,
RESISTORS
Electrical resistivity is a fundamental material property: As electrons pass through a material, energy
is dissipated as heat. Resistance is a measure of this
effect in a specific object. Resistance is measured in
units of ohms (⍀), and thus resistivity has units of
ohms per meter (⍀/m). In other words, the resistance of an object is a function of the resistivity of
its material as well as the object’s dimensions. In
particular, resistance is directly proportional to the
length of the material. Returning to fluid flow, resistivity is analogous to the friction that exists between
a fluid and the pipe that it flows through; resistance
is analogous to the total frictional force of the pipe
system. The total resistance of a pipe depends on its
frictional characteristics as well as its length. Electrical resistivities of materials vary over many orders of
magnitude. For example, copper wire has a resistivity of about 10– 4 ⍀/m; human skin, 20 to 50 k⍀/m;
semiconductors such as silicon are around 105 ⍀/m;
and wood, about 1013 ⍀/m.
247
R = R1 + R 2
R1
º Figure C.2
(C.1)
R2
Circuit diagram of two resistors in series.
If more resistors are added to the series, the total
resistance is equal to the sum of each resistance:
R = R1 + R 2 + R 3 + . . . + R n
(C.2)
In a parallel connection, there is branching (figure
C.3). The total current flowing through the system
is divided between two or more resistors. The total
resistance R of two or more resistors in parallel is
given by
1
1
1
1
=
+
+. ..+
R R1 R2
Rn
EXAMPLE C.1 ______________________
(C.3)
R1
Estimate the resistance of 1 cm of copper wire using
the resistivity just given, 10– 4 ⍀/m.
See answer C.1 on page 280.
A resistor is a device that resists electricity. Typical
resistor sizes vary from around 1 ⍀ to 1 M⍀. Knowledge of the resistances within a circuit is critical to
understanding its behavior. Indeed, we typically use
our knowledge of resistors to manipulate the flow of
current and perform the desired function. Also, in
human movement study, we often need to be aware
of the resistances in both our instruments and the
human body.
R2
º Figure C.3
Circuit diagram of two resistors in parallel.
For the case of two resistors in parallel, this
reduces to
R=
R1 R2
R1 + R2
(C.4)
248
Appendix C ________________________________________________________________________
EXAMPLE C.2 ______________________
a. What is the total resistance of two 10 ⍀ resistors
in series? In parallel?
See answer C.2a on page 280.
b. What is the total resistance of a 10 ⍀ resistor and
a 1 ⍀ resistor in series? In parallel?
See answer C.2b on page 280.
CAPACITORS
A capacitor is a device that stores electric charge; in
our analogy to fluid flow, a capacitor is equivalent
to a tank or a bucket that holds water. Its behavior is very different from that of a resistor and is
not discussed in detail here. The important thing
about capacitance is that it is a common physical
property that we often must account for. It typically
attenuates the voltage that we try to measure, and
its effects can be noticeable on certain data. For
example, high-speed devices such as telephones and
computer networks have very thin cables because
the capacitance of thicker cables would essentially
absorb the small amounts of electricity being sent
through them. This is analogous to the fact that
a garden hose holds water: Water does not come
out of the hose for a few seconds after the faucet is
turned on because the water must first fill the hose
to capacity. It is for this reason that some accelerometers have extremely thin cables. Similarly, EMG
electrodes are preamplified to provide a stronger
source of electricity that can overcome the capacitance of the wires.
Note that capacitance is not a bad factor, but
simply a factor that must be taken into account.
We in fact exploit the behavior of capacitors so that
radios can be tuned to different stations. Capacitors
can also be used to filter signals in the same way as the
digital filters introduced in chapter 2 and detailed in
chapter 11. Readers interested in relevant examples
may again refer to any linear-circuits text.
Along with capacitors, impedance is also important. Impedance, denoted Z, is a more general term
for all of the factors that limit electrical flow through
a circuit. Thus, impedance includes the net effects
of all resistors and capacitors in the circuit.
The symbol for a capacitor—two lines—represents the two plates that hold the electric charge.
Sometimes the plates are drawn parallel, but at other
times, the plate with the lower voltage is denoted
with a curved shape.
OHM’S LAW
Ohm’s law is perhaps the most fundamental law in
all of electronics. It states that the voltage across a
resistor equals the product of its resistance and the
current flowing through it:
(C.5)
V = IR
where V is the voltage across the resistor, I is the current through the resistor, and R is the magnitude of
the resistance. There are different ways to express
this law. If we increase the voltage in a circuit, the
current increases in proportion, and if we increase
the size of a resistor, the current decreases proportionately. When plotted, this function is a straight
line, as shown in figure C.4. This is a linear function.
It remains true regardless of the magnitude of the
current or how it changes over time. We express this
mathematically as
V(t) = I(t) R
(C.6)
Because of this linearity, resistor circuits are the most
straightforward to analyze, although they, too, can
get complicated.
V
Slope ⫽ R
I
º Figure C.4
Graphical representation of Ohm’s law.
Ohm’s law is analogous to fluid flow. The electrical resistance R corresponds to the resistance of the
piping, the current I corresponds to the volume rate
of fluid flow, and the voltage V corresponds to pressure. If we increase the voltage, more current flows,
in the same way that if we increase the pressure of
water in a pipe, more water flows. If we increase the
resistance in a circuit, less current flows—again, just
like water.
_________________________________________________________________________ Appendix C
EXAMPLE C.3 ______________________
a. What is the current flowing in this circuit?
10 Ω
A
20 V
where Z is the impedance, V is the voltage across the
circuit, and I is the current through it. If a circuit is
made up entirely of resistors, then the impedance is
equal to the resistance. However, for reasons beyond
the scope of our discussion, if the voltage varies with
time, we will observe the effects of the capacitance
of the circuit.
POWER LAWS
B
See answer C.3a on page 280.
b. In this circuit, what is the size of the resistor R?
9V
Sliding friction between two objects generates heat.
In a similar manner, electrical resistance generates
heat. This is simply a matter of energetics: If the electrical potential between sites is different and current
flows between them, the energy must be dissipated
in some manner, whether through a motor, light
bulb, or heating element. The power dissipated by
a resistor is given by
(C.8)
P = IV
R
2 mA
See answer C.3b on page 280.
c. Suppose a 110 V house outlet is wired to a 15 A
circuit breaker. What is the minimum resistance
that can be applied to this outlet?
See answer C.3c on page 280.
V
I
where P is the power dissipated by the resistor, I is the
current through it, and V is the voltage across it. That
is, the power dissipated by a resistor as heat is given
by the product of the current flowing though and
the voltage across it. Power, as in mechanical applications, has units of watts (W). With Ohm’s law, we can
also derive two other forms of the power law,
P = I 2 R and
In practice, we are not concerned with currents
flowing through loops, as these examples have illustrated. Instead, we often speak of the voltage drop
across a resistor, which has to do with the manner in
which you measure voltage. Since voltage is a potential difference between two points, a voltmeter measures the difference with two probes. For example,
in the circuit in example 2, if we placed one probe
before the 9 V battery and one probe after it, the
voltmeter would register a voltage gain of 9 V. If we
placed the probes across the resistor, the voltmeter
would measure a voltage drop of 9 V (i.e., it would
read –9 V). If you placed the leads across points A and
B, the voltmeter would register 0 V because there is
no resistance between these points. In measuring any
electrical device, there is a specific component across
which the changes in voltage are measured.
Earlier in this appendix, we discussed impedance.
When measuring impedance, the formula is analogous to Ohm’s law:
Z=
249
(C.7)
P=
V2
R
(C.9)
where R is the magnitude of the resistance. These
equations demonstrate that heating devices such
as ovens and hair dryers work by having low resistances. A heating element (a coil) is simply a resistor; as current flows through it, energy dissipates
as heat. Using the equation on the left, we see that
the power increases as the square of the current.
Therefore, a decrease in the resistance of the heating element causes a proportional increase in the
current.
EXAMPLE C.4 ______________________
a. What is the resistance of the heating coils of a
1,200 W toaster that runs on 110 V house circuitry?
See answer C.4a on page 280.
b. In an earlier example, we had a 110 V house outlet
on a 15 A circuit breaker. What is the maximum
wattage appliance you can plug into this outlet?
See answer C.4b on page 281.
250
Appendix C ________________________________________________________________________
MEASUREMENT OF
PHYSICAL SYSTEMS
Having discussed the basic behaviors of simple circuit
components, we now turn to the way we use these
components in the laboratory. We begin with a discussion of how we convert human movements into
electrical signals that our computers can measure.
rent. Suppose we have a variable resistor and connect
a voltage source across it as shown in figure C.5. The
standard nomenclature is to label the source voltage
Vin and the measured voltage Vout. For a simple circuit
like this, no matter how much the variable resistance
R V changes, Vout will always equal Vin, so this circuit is
useless for measuring changes in R V .
⫹
TRANSDUCERS
In the vast majority of cases in which we measure
physical quantities electrically, we measure changes
in voltage. This is a fundamental principle that
cannot be overemphasized. A 0 or a 1 in a computer
is represented by a voltage of 0 or 5 V, respectively.
When sound is transmitted through a wire to a
speaker, the changes in voltage are interpreted
as sound. When radio signals are transmitted to a
satellite, these, too, are registered by the voltages
they impart on the receiver. This is also true for the
measurement of EMG activity, force, and even the
reflections of body markers to a camera’s lens.
The process of converting a physical dimension
into a voltage is called transduction. A device that
performs this function is a transducer. Some of
the types of transducers are force, pressure, linear
displacement, rotary displacement, and acceleration transducers. The common principle in all of
these devices is that the quantity being measured
causes the resistance of the transducer to change.
For example, a force transducer (used in a force
platform) has tiny resistors that deform slightly when
force is applied. An electrogoniometer has a rotary
resistor that changes as it is rotated. When these resistances change, then, in accordance with Ohm’s law,
a constant current through a transducer causes the
voltage to change proportionately.
Vin
⫹
RV
Vout
⫺
⫺
º Figure C.5 Circuit diagram of a variable resistor connected to a voltage source.
In a modification of this circuit (figure C.6), a
resistor R is in series with the variable resistance.
We want to know the voltage Vout. To do this, we can
determine using Ohm’s law that the current, I, is
Vin
(C.10)
I=
R + RV
Because current flows through both resistors, we can
substitute it in Ohm’s law for R V and Vout:
Vin RV
(C.11)
Vout =
R + RV
This circuit is referred to as a voltage divider. Vout
for this circuit varies over an easily measurable
range when R and R V are of similar magnitudes.
The circuit is commonly used for a simple potentiometer.
EXAMPLE C.5 ______________________
Suppose we have a blood-pressure transducer
connected to a 10 mA current supply. As the pressure
changes from 80 to 120 mmHg, the transducer’s
resistance changes from 1,000 to 1,200 ⍀. What
will the voltage outputs be at these two pressures?
See answer C.5 on page 281.
VOLTAGE DIVIDERS
How would we measure a sensor with a variable
resistance? This is slightly more complicated than
the blood-pressure example, because most electrical
supplies have a constant voltage, not a constant cur-
⫹
Vin
⫹
RV
Vout
⫺
⫺
R
º Figure C.6 Circuit diagram of a resistor in series with a
variable resistance.
_________________________________________________________________________ Appendix C
EXAMPLE C.6 ______________________
LINEAR VARIABLE DIFFERENTIAL
TRANSDUCER
In the voltage divider, what is Vout for the following
cases?
The LVDT (figure C.8) is a common instrument that
measures a linear movement over a short range of
motion, typically less than 30 cm. Its main cylinder
contains a finely manufactured and calibrated linear
potentiometer. Therefore, its resistance changes linearly as it is moved. LVDTs can measure to within
fractions of a millimeter—a computer-controlled
milling machine, for instance, measures to within
2.5 ␮m. Common lab applications include treadmill
inclination adjustments, digital calipers, footwear
impact testers, and knee arthrometers (for measuring joint laxity or stiffness).
1. Vin = 15 V, RV = 100 ⍀, and R = 100 ⍀
2. Vin = 15V, RV = 110 ⍀, and R = 100 ⍀
3. Vin = 15V, RV = 100 ⍀, and R = 10 ⍀
4. Vin = 15V, RV = 110 ⍀, and R = 10 ⍀
See answer C.6 on page 281.
WHEATSTONE BRIDGES
Voltage dividers have two problems. In many sensors, the variability of resistance is small, often less
than 5%. Also, we often “zero” a sensor rather than
subtracting a constant voltage to establish zero for
the quantity we are measuring. These difficulties are
overcome with a Wheatstone bridge (figure C.7), a
circuit of two parallel voltage dividers. In its neutral
state, it has four equivalent resistances. When the
variable resistance changes, we can compare the
amount by which the variable resistance has changed
from its neutral state by measuring Vout.
R1
⫹
Vout
⫺
⫺
Vin
RV
⫹
251
⫺
R2
R3
º Figure C.8 Linear variable differential transducer as used
in a footwear impact tester.
º Figure C.7
Circuit diagram of a Wheatstone bridge.
EXAMPLE C.7 ______________________
In the Wheatstone bridge, what is the formula for Vout?
Let R1 = R2 = R3.
See answer C.7 on page 281.
COMMON LABORATORY
INSTRUMENTS
Many common laboratory instruments employ the
principles discussed in this appendix. These instruments include (1) the linear variable differential
transducer (LVDT), (2) the electrogoniometer, (3)
strain-gauge force transducers, and (4) amplifiers.
ELECTROGONIOMETER
An electrogoniometer, as its name suggests, measures
joint angles electronically. Its basic component is a
rotary potentiometer. Its internal structure is diagrammed in figure C.9. The end terminals (A and C)
are connected to the ends of the resistive material.
The middle terminal (B) is connected to a rotating
slider. As the knob of this slider is turned, the middle
contact moves across the resistive material. Because
resistance is a function of material length, we observe
the change in resistance. For example, if we have a
10 k⍀ potentiometer, the resistance from A to C will
measure 10 k⍀. As the rotating slider is moved from
A to C, we will measure a resistance across A and B
that changes from 0 to 10 k⍀, while the resistance
from B to C changes from 10 k⍀ to 0.
252
Appendix C ________________________________________________________________________
Rotating
slider
Resistive
material
A
C
B
º Figure C.9
º Figure C.11 Three types of strain gauges and a straingauged link for measuring axial loads.
Schematic of a rotary potentiometer.
Strain gauges are commonly used to measure
forces in human movement with such devices as
floor-mounted force plates, tension transducers,
pressure transducers, and even accelerometers. It
is also common in biomedical research to mount
gauges to orthoses and prostheses, as well as to
cadaver samples of bone, cartilage, and tendon.
AMPLIFIERS
An amplifier is a device that increases the voltage of
a signal. Figure C.12 shows how the idealized amplifier is designated in circuit diagrams.
º Figure C.10
linkages.
Vcc
Two electrogoniometers with a four-bar
⫹
Out
STRAIN-GAUGE FORCE
TRANSDUCERS
When force is applied to a material, it deforms. This
is called mechanical strain. Because resistance is a
function of material length, we observe a change in
resistance when a material is deformed. This is the
basic principle of a strain gauge. If we have a resistor
with a precisely known resistance glued to a deformable object, we can measure the object’s change in
resistance as it deforms. The gauges themselves are
usually much smaller in area than a postage stamp,
but equally as thin (see figure C.11). Once glued
to the surface of a structure, they bend with the
material without altering the structural properties.
Strain gauges are usually placed in a Wheatstone
bridge circuit.
⫺
Vdd
Figure C.12 Symbols for an operational amplifier. The
detailed form, on the right, labels the inputs and the power
supply for the amplifier Vcc and Vdd .
º
The most common type of amplifier is the operational amplifier. They are commonly installed on silicon chips. Unlike resistors and capacitors, op-amps are
active circuit elements and therefore require current
to power them. Op-amps have many different uses
and implementations. Two common connections
are the inverting and noninverting configurations
(figure C.13). The noninverting op-amp circuit
_________________________________________________________________________ Appendix C
Vin
⫹
Vout
⫺
RF
R
º Figure C.13
253
RF
Vin
R
⫺
⫹
Vout
Circuit diagrams for noninverting (left) and inverting (right) operational amplifiers.
increases the magnitude of the voltage it measures.
The inverting op-amp circuit increases the incoming
voltage and inverts it (i.e., takes the negative of it).
The performance of an amplifier is called the
gain. Gain is the ratio of the incoming voltage to
V
the amplified voltage, V . In op-amp circuits, gain
is controlled by altering the ratio of the resistors R F
and R . For noninverting configurations, the gain is
R
given by 1 + R , and for inverting configurations the
R
gain is - R . For a variable gain, a potentiometer may
be substituted for either R F or R .
Applications for amplifiers are too numerous to
mention. Their most common usage is for turning
weak electromagnetic waves into audible sound in
radios and cell phones. In human movement science,
we use them to measure tiny EMG, electrocardiographic, and electroencephalographic signals and
in force plates and other force transducers and in
accelerometers. They can also be used to construct
analog filters, integrators, and differentiators.
An important characteristic of op-amps and
other active circuits is their input impedance. This
is a measure of the sensitivity of the op-amp: High
input impedance means, in effect, that the op-amp
needs to draw very little current from the measured
quantity to function. This is very important in human
movement, because most biophysical signals are
out
in
extremely small. Ideally, the input impedance of
an EMG amplifier, for example, would be infinite.
Typically, amplifiers have input impedances of 1 M⍀,
but EMG or bioamplifiers have input impedances
of 10 M⍀ or more. EMG amplifiers require higher
impedances because skin can have resistances of
about 20 to 100 k⍀, and when unprepared, as high
as 2 M⍀ or more.
F
F
SUGGESTED READINGS
Bobrow, L.S. 1987. Elementary Linear Circuit Analysis. 2nd
ed. Oxford: Oxford University Press.
Cathey, J.J. 2002. Schaum’s Outline of Electronic Devices and
Circuits. 2nd ed. New York: McGraw-Hill.
Cobbold, R.S.C. 1974. Transducers for Biomedical Measurements: Principles and Applications. Toronto: John Wiley
& Sons.
Horowitz, P., and W. Hill. 1989. The Art of Electronics. 2nd
ed. Cambridge: Cambridge University Press.
Ohanian, H.C. 1994. Electric force and electric charge. In
Principles of Physics. 2nd ed. New York: Norton.
O’Malley, J. 1992. Schaum’s Outline of Basic Circuit Analysis.
2nd ed. New York: McGraw-Hill.
Winter, D.A., and A.E. Patla. 1997. Signal Processing and
Linear Systems for Movement Sciences. Waterloo, ON:
Waterloo Biomechanics.