The Nature of Dielectric Materials
Most solid materials are classified as insulators because they offer very
large resistance to the flow of electric current. Metals are classified as
conductors because their outer electrons are not tightly bound, but in most
materials even the outermost electrons are so tightly bound that there is
essentially zero electron flow through them with ordinary voltages. Some
materials are particularly good insulators and can be characterized by
their high resistivities:
Resistivity (Ω-m)
Glass
1012
Mica
9 x 1013
Quartz (fused) 5 x 1016
Resistivity (Ω-m)
Copper 1.7 x 10-8
The familiar parallel plate capacitor equation with free space as
an insulator under vacuum is given by:
Q0 ε o A
Co =
=
V
L
Plates connected to a constant voltage
supply V.
Qo is the charge on the plates.
Co is the capacitance of the parallel
plate capacitor in free space (vacuum).
The electric field E is defined as the gradient
of the potential then:
V
εo absolute permittivity or the permittivity of
a vacuum, the value of 8.85×10-12 F/m
A is the area
L is the separation between the plates
C is the capacitance (charge storage ability per unit
voltage)
E =
L
Q0
V
DO =
= εo = εoE
A
L
DO - is dielectric displacement = dipole moment per unit volume = charge per
unit area, C/m2
If there is a material medium
between the plates, then the
capacitance (C) increases by
a factor εr, where εr is called
the dielectric constant or
relative permittivity
Under vacuum
Q0 ε o A
Co =
=
V
L
C
Q
ε
εr = = =
ε o C o Qo
With a material medium
Q εA
C= =
V
L
The increase in stored capacity is due to the
polarization of the dielectric material by the applied
field
Qo
Co =
V
+Qo
Co
Q
C=
V
Dielectric
–Qo
+Q
C
–Q
i (t)
E
E
V
V
V
(a)
(b)
(c)
Fig. 7.1: (a) Parallel plate capacitor with free space between plates.
(b) As a slab of insulating material is inserted between the plates,
there is an external current flow indicating that more charge is stored
on the plates. (c) The capacitance has been increased due to the
insertion of a medium between the plates.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
A dielectric is any polarizable material. All materials are
polarizable, so all materials are dielectrics, even air.
The increase in capacitance is
due to the polarization of the
medium in which positives and
negative charges are displaced
with respect to their equilibrium
position.
Polarization can be viewed as:
Charge per unit area or Dipole per
unit volume
DO = ε O E
D = DO + P
D = εE = ε r ε O E
Where P is the polarization (total dipole moment/volume)
P = E ε O (ε r − 1) = E ε O χ e
χe is the electrical susceptibility
Energy Stored in a Capacitor
The quantity of energy stored in a capacitor is given by the equation
Q 2 C ×V 2
1
Energy = Q × V =
=
2
2C
2
EnergyMax
C=
εA
L
(
C = 220 μF
Vmax = 400 V
)
2
C × VMax
220 × 10−6 F (400V )
=
=
= 17.6 joules
2
2
V =EL
2
Energy =
Electric _ Energy _ Density =
ε × E2
2
εr × εO × E2
2
The larger the dielectric constant the higher the
energy that can be stored.
AL
Dielectric strength
Desired Properties of a dielectric medium:
ability to increase capacitance
insulating behavior or low conductivity so that the charges are not simply
conducted from one plate of the capacitor to the other through the dielectric.
Many dielectrics can sustain very high internal electric fields before electrical
breakdown. The maximum electric field that can be sustained is called the
dielectric strength of the material.
The voltage across the dielectric can not be increased without limit. Very
high electric fields (>108 V/m) can excite electrons to the conduction band
and accelerate them to such high energies that they can, in turn, free other
electrons, in an avalanche process (or electrical discharge). The field
necessary to start the avalanche process is called dielectric strength or
breakdown strength. This is the maximum electric field to which a dielectric
material can be subjected without breaking down or discharging
εr
Example: We want to make a simple parallel plate capacitor that can store
4×10-5 C at a potential of 1000 V. The separation between the plates is to be
0.2 mm. Calculate the area of the plate required if the dielectric is (a) vacuum,
(b) polyethylene, (c) water, and (d) BaTiO3. The relative permittivity for
polyethylene, water and BaTiO3 are 2.26, 78.3, and 3,000 respectively.
Solution: 900, 400, 11.5 and 0.3 cm2
Dielectric Behavior
Mechanisms:dipole formation/orientation
electronic (induced) polarization: Applied electric field displaces
negative electron “clouds” with respect to positive nucleus.
Ionic materials (induced) polarization: Applied electric field displaces
cations and anions in opposite directions
molecular (orientation) polarization: Some materials possess
permanent electric dipoles (e.g. H2O). In absence of electric field,
dipoles are randomly oriented. Applying electric field aligns these
dipoles, causing net (large) dipole moment.
Ptotal = Pe + Pi + Po
Types or Mechanism of Polarization
Electronic Polarization: It may be induced (to some degree) in all atoms.
Displacement of the center of the negative electron cloud off the nucleus
(only present when there is an electric field)
- - - + - -
No field
- + -- -- -
Electric
Field
E
C x O
Electron cloud
Atomic
nucleus
(a) A neutral atom in E = 0.
Center of negative
charge
pinduced
(b) Induced dipole moment in a field
Fig. 7.3: The origin of electronic polarization.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
The magnitude of the electric dipole moment is P = q x d, where d is the
distance between dipoles
Ionic Polarization - Only occurs in ionic materials
An applied field displaces cations in one direction and anions in another:
-
+
-
+
-
-
+
-
+
-
+
+
-
+
-
+
-
+
-
+
-
+
No electric field
+
-
+
-
+
+
-
-
+
-
+
+
+
-
+
Electric Field
E
-
+
+
+
+
Molecular polarization, occurs
in all insulating molecules;
oils, polymers, H2O…
p+
p–
(a)
x
Cl–
Na+
p'+
p'–
(b)
E
Fig. 7.8: (a) A NaCl chain in the NaCl crystal without an applied
field. Average or net dipole moment per ion is zero. (b) In the
presence of an applied field the ions become slightly displaced
which leads to a net average dipole moment per ion.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Orientation Polarization
Only in materials which possess permanent dipole moments
example
δ+
H
No field
H
O
Electric Field
δ-
+Q
τ
Cl°
po = aQ
H+
po
F
–Q
(a)
pav = 0
(c)
pav ≠ 0
(b)
θ
F=QE
E
E
(d)
Fig. 7.9: (a) A HCl molecule possesses a permanent dipole moment, po
(b) In the absence of a field, thermal agitation of the molecules results
in zero net average dipole moment per molecule. (c) A dipole such as
HCl placed in a field experiences a torque which tries to rotate it to
align po with the field E. (d) In the presence of an applied field the
dipoles try to rotate to align with the field against thermal agitation.
There is now a net average dipole moment per molecule along the
field.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Total Polarization: P = Pe + Pi + Po
(electronic + ionic + orientation)
Capacitors
Parallel plate capacitor. Apply a voltage; charge
Q accumulates on the plates
Place a material between the plates - Q increases e.g.
H 2O
Polarization Vector, P
Electric Dipole Moment (- to +)
-q
d
P
P = qd
P=Zqd
+q
The process of dipole alignment is called Polarization
Surface charge density or Dielectric Displacement:
D (C/m2) ∝ Ε (electric field V/m)
D = εΕ D = dielectric displacement
In the presence of an electric
field, a force will tend to orient
the electric dipole with the
applied field
Example (Electronic Polarization)
Suppose that the average displacement of electrons relative to the nucleus in a copper
atom is 1x10-8 Angstroms when an electric field is imposed on a copper plate.
Calculate the electronic polarization. Data: Copper (Z=29 and lattice parameter =
3.6151 Angstroms)
Solution
( 4atoms / cell )(29electrons / atom )
30
3
Z=
=
2
.
46
×
10
electrons
/
m
(3.6151×10 −10 )3
Where Z is the number of electrons (electronic polarization) per unit
volume
P = Z ×q×d
P = (2.46 × 10 electrons / m )(1.6 × 10
30
P = 3.94 × 10− 7 C / m 2
3
−19
−8
o
C / electron)(10 A)(10−10 m / A)
Example (Ionic Polarization)
Calculate the increase in separation of Cs+1 and Cl-1 in a CsCl crystal
when an ionic polarization of 4x10-8C.m-2 is achieved by the application of
an electric field. Data: lattice parameter a=0.402nm, ionic radii 0.165nm
for Cs+1 and 0.181nm for Cl-1.
Solution
Use the equation
P = Z ×q ×d
Where Z is the number of charges per unit volume i.e. (dipoles per cell) x (charges per
cell) per unit volume
(
1dipole _ per _ cell )(1 _ ch arg e _ per _ dipole )
Z=
(0.402 ×10 ) m
−9 3
3
_ per _ cell
Z = 1.54 ×10 28 ch arg es.m −3
P
4 ×10 −8 C .m −2
d=
=
Z × q 1.54 ×10 28 ch arg es.m −3 × 1.6 ×10 −19 C / ch arg e
(
d = 1.62 × 10 −17 m = 1.62 × 10 −8 nm
) (
)
Frequency Dependence of the Dielectric Constant
++++++
-------Alternating Current. (Applied voltage
+ + + + +
+ + + +
+
-
-
-
--------
Electric Field
-
-
-
-
++++++
-
As frequency increases, dielectric
constant decreases as orientation
and ionic components go to zero.
-
Sometime dipoles can’t keep up with
changing electric field:
-
or electric field changes direction
with time)
Dipoles try to reorient with field. (This
requires time)
Relaxation Frequency = 1/time to
reorient
Electric Field
Interfacial and
space charge
Orientational,
Dipolar
εr'
Ionic
Electronic
εr''
εr' = 1
ƒ
10–2
1
102
104
106
Radio
108
1010
1012
Infrared
1014
1016
Ultraviolet light
Fig. 7.14: The frequency dependence of the real and imaginary parts
of the dielectric constant in the presence of interfacial, orientational,
ionic and electronic polarization mechanisms.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Define the permittivity or dielectric constant of a material by:
H2O is a polar liquid; εr ~ 80
Typical ionic solids; εr ~ 10
Air; εr ~ 1
BaTiO3 :-
Below 120°C, BaTiO3 is ferroelectric with aligned dipoles.
Residual dipole disorder gives εr~200-1000
At ~127°C, tetragonal → cubic phase transition.
Dipoles randomise and εr increases to ~5,000-10,000
Q
εr =
Q vac
Electrolyte
Al2O3
Anode
Al foils
Cathode
Al
Al
Al case
(a)
(b)
Fig. 7.31: Al electrolytic capacitor.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
The anode of an Al electrolytic capacitor is an
aluminum foil of extreme purity. The effective
surface area of this foil is greatly enlarged (by a
factor of up to 200) by electrochemical etching in
order to achieve the maximum possible
capacitance values.
The type of etch pattern and the degree of etching
is matched to the respective requirements by
applying specific etching processes. Etched foils
enable very compact Al electrolytic capacitor
dimensions to be achieved and are the form used
almost exclusively nowadays. The electrical
characteristics of Al electrolytic capacitors
The dielectric layer of an Al electrolytic capacitor is created by anodic oxidation (forming)
to generate an aluminum oxide layer on the foil. The layer thickness increases in
proportion to the forming voltage at a rate of approximately 1,2 nm/V. Even for capacitors
for very high voltages, layer thicknesses of less than 1 μm are attained, thus enabling very
small electrode spacings. This is one reason for the high volumetric efficiency achieved
(e.g. in comparison to the minimum thickness of a paper dielectric, 6 to 8 μm).
Example
A 2mm thick porcelain dielectric is used in a 60 Hz circuit. Calculate the voltage required
to produce a polarization of 5x10-7 C.m-2. Use Table.
Solution
P = E ε O (ε r − 1) = (6 − 1) × 8.85 × 10
V = 22.6volts
−12
V
−7
×
=
5
×
10
2 × 10 − 3 m
CRYSTAL SYMMETRY:
Crystal structures can be divided into 32 classes, or point groups,
according to the number of rotational axes and reflection planes they
exhibit that leave the crystal structure unchanged.
Twenty of the 32 crystal classes are piezoelectric. All 20 piezoelectric
classes lack a center of symmetry.
Any material develops a dielectric polarization when an electric field is
applied, but a substance which has such a natural charge separation even
in the absence of a field is called a polar material.
Whether or not a material is polar is determined solely by its crystal
structure. Only 10 of the 32 point groups are polar.
Under normal circumstances, even polar materials do not display a net
dipole moment. As a consequence there are no electric dipole equivalents
of bar magnets because the intrinsic dipole moment is neutralized by "free"
electric charge that builds up on the surface by internal conduction or from
the ambient atmosphere.
Polar crystals only reveal their nature when perturbed in some fashion that
momentarily upsets the balance with the compensating surface charge.
The possibility of inorganic crystals being polar (pyroelectric or piezoelectric) is
strictly a function of their point group symmetry.
Polar Materials
Solid with a natural charge separation even in the absence of a field
Crystals comprising cations and anions can be classified into four types,
according to their polar behavior:
• Piezoelectric materials: There is coupling between electrical and mechanical
energies. For example, an applied stress results in the generation of
polarization.
• Pyroelectric materials: Pyroelectricity refers to the change in polarization by
changes to the structure from thermal effects. A material with a temperature
dependent polarization. This requires a unique polar axis.
• Ferroelectrics: A subgroup of pyroelectric materials in which the spontaneous
polarization can be reoriented between “equilibrium” states by applying an
electric field.
All ferroelectrics are both pyroelectric and piezoelectric.
The possibility of inorganic crystals being polar (pyroelectric or piezoelectric) is
strictly a function of their structure (point group symmetry)
•Ferroelectrics: Ferroelectric materials possess a natural electric
polarization. A subgroup of pyroelectric materials in which the spontaneous
polarization can be reoriented between “equilibrium” states by applying an
electric field. All ferroelectrics are both pyroelectric and piezoelectric. Not all
piezoelectric materials are pyroelectric.
Ferroelectrics are materials which possess an electric polarization in the
absence of an externally applied electric field such that the polarization can
be reversed if the electric field is reversed. Normally materials are very nearly
electrically neutral on the macroscopic level. However, the positive and
negative charges which make up the material are not necessarily distributed
in a symmetric manner. If the sum of charge times distance for all elements
of the basic cell does not equal zero the cell will have an electric dipole
moment which is a vector quantity. The dipole moment per unit volume is
defined as the dielectric polarization.
Piezoelectric materials: Piezoelectricity refers to a materials property that the
polarization (or electric field) of the material can be changed by mechanical
perturbation of the structure. There is coupling between electrical and
mechanical energies. For example, an applied stress results in the generation
of polarization.
PIEZOELECTRIC EFFECT:
The piezoelectric effect is a linear, reversible electromechanical interaction
occurring in materials possessing the proper symmetry properties. The direct
piezoelectric effect is the production of an electric polarization by a strain;
the converse piezoelectric effect is the production of a stress by an electric
field. Piezoelectric materials have wide applications as transducers transferring mechanical motion into electricity or electricity into mechanical
motion. One of the most wide spread examples is a quartz resonator. The
quartz resonator converts the electrical potential energy of a battery into a
steady beat that becomes the oscillator (counter) of a watch.
Pyroelectricity : It is a property of dielectric materials, which show a
temperature-dependent, macroscopic (permanent or spontaneous)
polarization P, i.e. they generate surface charges as a result of a
temperature change ΔT(t). These charges can either be detected directly or
as a pyroelectric current I(t).
PYROELECTRICITY: Spontaneous polarization is temperature dependent, so
a good perturbation probe is a change in temperature which induces a flow of
charge to and from the surfaces. This is the pyroelectric effect. All polar
crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred
to as the pyroelectric classes.
The property of pyroelectricity is the measured change in net polarization (a
vector) proportional to a change in temperature. The total pyroelectric
coefficient measured at constant stress is the sum of the pyroelectric
coefficients at constant strain (primary pyroelectric effect) and the piezoelectric
contribution from thermal expansion (secondary pyroelectric effect).
Pyroelectric materials can be used as infrared and millimeter wavelength
detectors.
Piezoelectricity
Piezoelectricity or pressure electricity: Unusual phenomena in which polarization
is induced and an electric field is established across a sample when it is
mechanically stressed.
Similarly, the same crystal also exhibits mechanical strain when it experiences
an electric field.
Force
P
P=0
(a)
(b)
V
The direction of mechanical deformation (extension or compression)
depends on the direction of the applied field, or the polarity of the voltage.
Only crystals with a special crystal structure can exhibit piezoelectricity that
which has no center of symmetry.
V
V
(c)
(d)
Fig. 7.35: The piezoelectric effect. (a) A piezoelectric crystal
with no applied stress or field. (b) The crystal is strained by an
applied force which induces polarization in the crystal and
generates surface charges. (c) An applied field causes the crystal
to become strained. In this case the field compresses the crystal.
(d) The strain changes direction when the field is reversed, and
now the crystal is extended. The dashed rectangle is the original
sample size in (a).
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Consider the cubic unit cell. When unstressed, center of mass (c.m. of the –
ve charges at the corner of unit cell coincides with +ve charge at center
therefore, no net polarization occurs and P=0.
Under stress, unit cell becomes strained. However, c.m. of the –ve charges
still coincides with +ve charge and net polarization is still 0. P=0 for strained
crystal. This is generally true for crystals with center of symmetry.
Force
If we draw a vector from O
(position of an arbitrary
point charge) to any charge,
then the reverse vector will
point to the same type of
charge: we call O, or any
other point charge, a center
of symmetry
P=0
O
(a)
P=0
(b)
Fig. 7.36: A cubic unit cell has a center of symmetry. (a)
In the absence of an applied force the centers of mass for
positive and negative ions coincide. (b) This situation does
not change when the crystal is strained by an applied force.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
If we draw a vector from O to
any charge, then the reverse
vector will point to an opposite
charge. The unit cell is said to
be non-centrosymmetric.
When unstressed, c.m. of –ve
charges coincides with c.m. of
the +ve charges, both at O,
therefore, no net polarization
occurs and P=0.
Under stress, the +ve charge at
A and –ve charge at B both
become displaced inwards to A’
and B’ respectively. The two
c.m.’s become shifted and
there is now a net polarization
P for the strained crystal.
A
y
A'
x
P
P=0
O
B'
B
(a)
(b)
A''
P=0
P
B''
(c)
Fig. 7.37: A hexagonal unit cell has no center of symmetry. (a) In
the absence of an applied force the centers of mass for positive
and negative ions coincide. (b) Under an applied force along y the
centers of mass for positive and negative ions are shifted which
results in a net dipole moment P along y. (c) When the force is
along a different direction, along x, there may not be a resulting net
dipole moment in that direction though there may be a net P along
a different direction (y).
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
A
y
A'
The direction of induced
x
polarization depends on the
P
P=0
O
direction of applied stress.
In the above case, P
B'
B
appears in the same
direction as applied stress
(a)
(b)
A''
along y.
If the stress is applied along
P=0
x, A and B are displaced
P
outwards to A’’ and B’’
respectively, resulting in
B''
shift of c.m.’s away from
each other in y direction
therefore P appears along y
direction.
Generally, an applied stress in one direction can give rise to induced
polarization in other crystal directions and reversing the stress reverses the
polarization. Crystals with no center of symmetry exhibit piezoelectricity.
Piezoelectric and Ferroelectric Materials
When mechanical pressure is applied to a piezoelectric
material, the crystalline structure produces a voltage
proportional to the pressure.
Conversely, when a piezoelectric material is subjected to
an electric field, the structure changes in shape, producing
dimensional changes in the material.
Examples of natural piezoelectric crystals:
Quartz (SiO2),
Rochelle Salt
Tourmaline.
Piezoelectric materials are anisotropic, that is their
mechanical, electrical and electromechanical properties
depend strongly on the crystal orientation.
Man-made piezoelectric ceramic example:
lead-zirconate-titanate (PZT),
lead-titanate (PbTiO2), lead-zirconate (PbZrO3),
and barium-titanate (BaTiO3).
Strictly speaking, these ceramics are not actually
piezoelectric but rather exhibit a polarized electrostrictive
effect.
PZT exhibits a cubic structure above a critical (Curie)
temperature.
During cooling (below the Curie temperature) the cubic
structure transform to a tetragonal or rhombohedral
structure.
Due to the non-centrosymmetry of this structure exhibits a
dipole moment.
Regions of the crystal with the dipoles having the same
direction are called “domains”.
In a PZT, the direction of polarization among neighboring domains is random,
so the ceramic element has no net polarization.
Net polarization is induced by exposing the PZT to a strong direct electric field,
to align all the individual domains towards one specific direction, the poling
direction. With this treatment the ceramic element increases in size in the
direction of the electric field. When the electric field is removed, most of the
dipoles are locked into a configuration near alignment. The element now shows
a permanent or net polarization.
Analogous to ferromagnetic materials,
piezoelectric materials exhibit a hysteresis
loop. An electric field is applied to the
piezoelectric until a maximum polarization is
achieved, then the electric field is removed
and the material exhibit a “remanent
polarization”. To eliminate the remanent
polarization, an reverse electric field is
induced, in opposite direction.
A compressive stress in the same
direction of poling, creates a voltage
of the same polarity as the poling
voltage. Voltage of the same polarity
as the poling voltage is produced by
a tensile stress perpendicular to the
direction of poling.
A tensile stress along the direction of poling produces a voltage with polarity
opposite to the poling voltage.
Conversely, if a voltage of the same polarity as the poling voltage is applied
to the piezoelectric, it will increase in size, while if the polarity of the applied
voltage is reverse it will shorten in size. Finally if a cyclic voltage is applied in
the direction of poling, the piezoelectric will change in dimensions cyclically
at the frequency of the applied voltage.
Piezoelectric effect basics
Apply mechanical stress -> Electric charge produced
Apply electric field -> Mechanical deformation produced
Dipole: each molecule has a polarization, one end is more negatively
charged and the other end is positively charged.
Monocrystal: the polar axes of all of the dipoles lie in one direction. -Symmetrical
Polycrystal: there are different regions within the material that have a
different polar axis. -- Asymmetrical
How to produce piezoelectric effect
a) Material without stress /
charge
b) Compress -> same polarity
c) Stretched -> opposite polarity
d) Opposite voltage -> expand
e) Same voltage -> compress
f) AC signal -> vibrate
Applications of piezoelectric
materials is based on
conversion of mechanical strain
into electricity (microphones,
strain gauges, sonar detectors,
audible alarms, ultrasonic
imaging, speakers)
Piezoelectric materials include
barium titanate BaTiO3, lead
titanate, lead zirconate PbZrO3,
quartz, ammonium dihydrogen
phosphate (NH4H2PO4).
Piezoelectric
Igniters
Hydrophones: A "Hydrophone"
is a device which will listen to,
or pick up, the acoustic energy
underwater. A hydrophone
converts acoustic energy into
electrical energy and is used in
passive underwater systems to
listen only.
Piezoelectric
Audiotone
Transducers
Liquid Atomization
Devices
Mechanical
vibrations
Piezoelectric
transducer
A
Oscillator
Elastic
waves in the
solid
B
Oscilloscope
Fig. 7.38: Piezoelectric transducers are widely used to generate
ultrasonic waves in solids and also to detect such mechanical
waves. The transducer on the left is excited from an ac source
and vibrates mechanically. These vibrations are coupled to the
solid and generate elastic waves. When the waves reach the
other end they mechanically vibrate the transducer on the right
which converts the vibrations to an electrical signal.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca