Transcript Physics: Principles and Applications

Lecture 3
• Demos: capacitor separation with distance
• Introduce dielectrics (foam,glass) and see
the voltage decrease as the capacitance
increases.
Electrical Sensors
Employ electrical principles to detect
phenomena.
• May use changes in one or more of:
• Electric charges, fields and potential
• Capacitance
• Magnetism and inductance
Some elementary electrical
sensors
Thermocouple
Thermistor
Variable Capacitor
Review of Electrostatics
• In order to understand how we can best
design electrical sensors, we need to
understand the physics behind their
operation.
• The essential physical property measured by
electrical sensors is the electric field.
Electric Charges, Fields and
Potential
Basics: Unlike sign charges attract, like sign charges repel
Coulombs’ Law: a force acts between two point
charges, according to:
 Q1Q2 rˆ
F
4 0 r 2
The electric field is the force per unit charge:

 F
E
Q1
How do we calculate
the electric field?
Electric Field and Gauss’s Law
We calculate the electric field using Gauss’s Law.
It states that:
  Q
 E  ds 
S
Seems very abstract,
but is really useful
0
Point or Spherical charge
What is the field around a point charge (e.g. an electron)?
The electric field is everywhere perpendicular to a spherical
surface centred on the charge.
Electric field vectors
 
Q
2
So  E  ds  E  4r 
0
S
Gaussian surface
We recover Coulombs Law!
Q
E
2
4r
The same is true for any distribution of charge which is
spherically symmetric (e.g. a biased metal sphere).
Line of Charge
For a very long line of charge (eg a
wire), the cylindrical surface has electric
field perpendicular to a cylindrical
surface.
 
Q
So  E  ds  E  2rL 
0
S

E

2 0 rL 2 0 r
Q
Where  = linear charge density
(Coulombs/meter)
Plane of Charge
For a very large flat plane of charge
the electric field is perpendicular to a
box enclosing a segment of the sheet
So
 
Q
 E  ds  E  A 
S
Q

E

A 0  0
0
Where  = Charge/Unit area on the surface
Electric Dipole
• An electric dipole is two equal
and opposite charges Q separated
by a distance d.
• The electric field a long way
from the pair is
1 Qd
1 p
E

3
4 0 r
4 0 r 3
• p = Q d is the Electric Dipole moment
• p is a measure of the strength of the
field generated by the dipole.
Electrocardiogram
• Works by measuring changes in electric
field as heart pumps
• Heart can be modeled as a rotating dipole
• Electrodes are placed at
several positions on the
body and the change in
voltage measured with
time
Electrocardiogram
• Interior of Heart muscle cells negatively charged at rest
• Called “polarisation”
• K+ ions leak out, leaving interior –ve
• Depolarisation occurs just proir to contraction:
Na+ ions enter cells
Occurs in waves across the heart
Re-polarisation restores –ve charge in interior
---
++++
Polarisation
++
++
-----
Depolarisation
Electrocardiogram
• Leads are arranged in pairs
• Monitor average current flow at specific time in a portion
of the heart
• 1 mV signal produces 10 mm deflection of recording pen
• 1 mm per second paper feed rate
-
A
A
+
C
+
B
-
+
B
C
Electric Potential
The ECG measures
differences in the
electric potential V:


E  V
The Electric Potential is the Potential ability to do
work.
Alternatively: Work = Q  V
Where V =
V1  V2
For uniform electric fields:
V
| E |  V | E | d
d
Electric fields on conductors.
• Conductors in static electric fields are at uniform
electric potential.
• This includes wires, car bodies, etc.
• The electric field inside a solid conductor is zero.
Dielectric Materials
• Many molecules and crystals have a non-zero Electric
dipole moment.
• When placed in an external electric field these align with
external field.
• The effect is to reduce the
strength of the electric field within
the material.
• To incorporate this, we define a
new vector Field, the electric

displacement,
D
Electric Displacement

of dielectric materials. Then the electric
D is independent



field E is related to D by:
1  1 
E
r , 0 , 
 r 0
D

D
Are the relative permittivity, the
permittivity of free space and the
absolute permittivity of the material.
As shown in the diagram, there is
torque applied to each molecule. This
results in energy being stored in the
material, U. This energy is stored in
every molecule of the dielectric:
 
U  p E
Capacitance.
Remember that the electric field near a plane
of charge is:
Q

E

A 0  0
In the presence
of a dielectric:

E

V
d
E  V 
d

Q
dQ
Since   , V 
A
A
So the Potential difference is proportional
to the stored charge.
Cylindrical Capacitor
Can make a capacitor out of 2 cylindrical conductors
2L
C
b
ln( )
a
Sensing using capacitance.
So the charge Q = CV
Where C = Capacitance, V = Potential difference.
For a parallel plate capacitor:
Easily Measured
Area of plate
A
C
d
Properties of Material
Distance between plates
We can sense change in A, ε, or d and
measure the change in capacitance.
Measurement of Capacitance
Capacitors have a complex resistance
V (t )
1

i (t )
jC
We measure capacitance by probing with an AC
signal.
Directly measure current i(t) with known V(t) and
frequency ω.
For extreme accuracy, we can measure resonant
frequency with LC circuit.
Example: water level sensor
Measures the capacitance
between insulated conductors
in a water bath
Water has very different
dielectric properties to air (a
large )
As the bath fills the effective
permittivity seen increases, and the
capacitance changes according to:
Ch 
2

H  h(1   )
b
ln( )
a
Example: The rubbery Ruler
Invented by Physicists here to measure fruit growth.
http://www.ph.unimelb.edu.au/inventions/rubberyruler/
Spiral of conductor embedded in a flexible “rubbery” compound
As the sensor expands, the
A
distance between the plates
C
increases causing capacitance to
d
decrease.
The rubbery ruler
Spiral of conductor embedded in a
flexible “rubbery” compound
Invented by Physicists here to measure fruit growth.
http://www.ph.unimelb.edu.au/inventions/rubberyruler/
As the sensor expands, the
distance between the plates
increases causing capacitance to
decrease.
A
C
d
Lecture 4
• Piezoelectric demo (stove lighter and
voltmeter)
Piezoelectric sensors
Mechanical stress on some crystal lattices results in a
potential difference across the solid.
This is an extremely useful effect. Reversible too!
• For quartz, stress in x-direction results in a potential
difference in the y-direction.
• This can be used as a traffic weighing and counting
sensor!
• A piezoelectric sensor can be thought of as a capacitor,
with the piezoelectric material acting as the dielectric.
The dielectric acts a generator of electric charge resulting
in a potential V across the capacitor.
• The process is reversible. An electric field induces a
strain in the material. Thus a very small voltage can be
applied, resulting in a tiny change in the size of the
crystal.
Characterisation of Piezoelectrics
We quantify the piezoelectric effect using a vector of Polarisation.
 


P  Pxx  Pyy  Pzz
Pxx  d11 xx  d12 yy  d13 zz
Pyy  d 21 xx  d 22 yy  d 23 zz
Pzz  d31 xx  d32 yy  d 33 zz
Where dmn are coefficients, i.e.
numbers that translate applied
force to generated charge and
are a characteristic of the
piezoelectric material.
Units are Coulomb/Newton.
Characterisation of Piezoelectrics
Piezo crystals are transducers;
They convert mechanical to electrical energy.
The conversion
efficiency is given by
the coupling
coeffient:
K mn 
2
d mn
Y
 0 mn
Where Y is Young’s Modulus = Stress/strain

F Force
Y  ,   stress  
dl
A Area
l
The charge generated is proportional to the applied force
The charge generated in
the X-direction from an
applied stress in y
Qx  d12 Fy
Using our Q = CV, we get a generated voltage
Qx d12 Fy
V

C
C
 r 0 A
The capacitance is:
C
l
So the Voltage is
V
d12lFy
 0 r A
Area of
electrodes
Thickness of
crystal
Some piezoelectrics
Numerical Example.
What is the sensitivity of 1 mm thick, BaTiO3 sensor with an
electrode area of 1 square cm?
2
d12lF
781103 F
7
.
8

10
F
V

=
 r  0 A 1700 8.8 10121104
1.5 1012
So
V
 5.2  1010
F
Volts/Newton
This is a big number because the effective capacitance is
so small. In the real world the voltage is smaller.
C=
1.5 1012
 1.5nF
3
110
Very Small!
Atomic Scale Microscopy
Use Piezoelectric crystals as
transducers to do atomic scale
microscopy
Piezoresistive Sensors
The stress on a material is
F Ydl
 
A
l
Strain = dl/l
A cylinder stretched by a Force F keeps constant volume
but l increases and A decreases.
A
l
l
l 2
Resitance R  A  vol
Sensitivity of the sensor is
dR 2 l

dl
vol
Longer wires give more sensitivity
Characterizing Piezoresistors
Normalised resistance is a linear function of strain:
Where e is the strain, and
Se
dR
 See
R
is the gauge factor or sensitivity of the strain.
Metals
Semiconductors
2  Se  6
40  Se  200
Semiconductor strain gauges are 10 to 100 times more
sensitive, but are also more temperature dependent.
Usually have to compensate with other types of sensors.
Piezoresistive Heat Sensors.
Resistive Temperature Detectors: on demand “RTD”s
RTD’s used at Belle
Thin platinum wire deposited on a substrate.
Other piezoresistive issues
• Artificial piezoelectric sensors are made by poling; apply a
voltage across material as it is heated above the Curie point
(at which internal domians realign).
• The effect is to align natural dipoles in the crystal. This
makes the crystal a Piezoelectric.
• PVDF is of moderate sensitivity but very resistant to
depolarization when subject to high AC fields.
• PVDF is 100 times more resistant to electric field than the
ceramic PZT [Pd(Ze,Ti)O3] and useful for strains 10 times
larger.
Example: acceleration Sensor.
• Piezoelectric cable with an inner copper core.
• The piezoelectric acts as an insulator, clad by an outer metal
sheath and flexible plastic and rubber coating.
• Other configurations exist: see
www.pcb.com/techsupport/tech_accel.aspx
Inner copper core
Piezoelectric
Outer metal sheath or braid
Plan view of cable
Remember that F=ma , so if the sensor
mass is known, then the force measured
can be converted into an acceleration.
Applications for piezoelectric
accelerometers
• Vibration monitor in compressor blades in turboshaft aircraft.
• Detection of insects in silos
• Automobile traffic analysis (buried in highway):
traffic counting and weighing.
• Force and pressure sensors (say, monitoring jolts to
packages).
• Tactile films: thin silicone rubber film (40 m)
sandwiched between two thin PVDF films.
If tactile sandwich is compressed, the mechanical coupling in the PVDF/rubber/PVDF
sandwich changes, the measured AC signal changes, and the demodulation voltage
changes
Lecture 5
Pyroelectric Effect.
Generation of electric change by a crystalline material when
subjected to a heat flow.
Closely related to
Piezoelectricity.
BaTiO3, PZT and
PVDF all exhibit
Pyroelectric effects
Primary Pyroelectricity.
Temperature changes shortens or elongates individual dipoles.
This affects randomness of dipole orientations due to thermal
agitation.
Secondary Pyroelectricity
Quantitative Pyroelectricity.
Pyroelectric crystals are transducers: they convert thermal to
electrical energy.
The Dipole moment of the bulk
pyroelectric is:
M=Ah
Where  is the dipole moment per unit volume, A is the
sensor area and h is the thickness
From standard dielectrics, charge on electrodes, Q =  A
The dipole moment, , varies with temperature.
dPS
PQ 
dT
Is the pyroelectric charge coefficient, and Ps is
the “spontaneous polarisation”
The generated charge is Q = PQ A T
dE
Pv =
is the pyroelectric voltage coefficient and E is the
dTField.
electric
The generated voltage is QV = Pv h T (h is the thickness)
The relation between charge and voltage
coefficients follows directly from Q = CV
PQ
PV

dPS
  r 0
dE
Seebeck and Peltier Effects.
Seebeck effect: Thermally induced electric currents in circuits
of dissimilar material.
Peltier effect: absorption of heat when an electric current
cross a junction two dissimilar materials
The dissimilar materials can be different species, or the the
same species in different strain states.
The Peltier effect can be
thought of as the reverse of
the Seebeck effect
Seebeck effect
Free electrons act as a gas. If a metal rod is hot at one end
and cold at the other, electrons flow from hot to cold.
So a temperature gradient leads to a voltage gradient:
dV
dT

dx
dx
Where  is the absolute
Seebeck coefficient of the
material.
When two materials with different  coefficients are
joined in a loop, then there is a mis-match between the
temperature-induced voltage drops.
The differential Seebeck coefficient is:
AB = A - B
Thermocouples
dVAB   AB dT
The net voltage at the junction is
So the differential Seebeck
coefficient is also
 AB
dV AB

dT
This is the basis of the thermocouple sensor
Thermocouples are not necessarily linear in response.
E.g. the T – type thermocouple has characteristics
V  a0  a1T  a2T 2
Where the a’s are material properties:
V  0.0543 4.094102 T  2.874105 T 2
The sensitivity is the differential Seebeck coefficient
 AB
dV AB

 a1  2a2T  4.094 10  2  5.748 10 5 T
dT
Independent of geometry, manufacture etc. Only a function of
materials and temperature.
Seebeck effect is a transducer which converts thermal to
electrical energy.
Can be used as solid state thermal to electrical energy
converter (i.e. engine)as well as an accurate temperature
sensor.
Seebeck engines are currently not very efficient but are
much more reliable than heat engines. They are used by
NASA for nuclear powered deep-space probes.
Peltier Effect.
If electric current is passed through a dissimilar material
junction, then the heat may be generated or absorbed.
The change in heat dQ =p I dt
(where p is the Peltier constant (unit of voltage))
Can be used to produce heat or cold as required.
Eg. Cooling high performance Microprocessors.
Lecture 6
Magnetism
The density of a magnetic
field (number of magnetic
field lines passing through a
given surface) is the
magnetic flux:
 
 B   B  dS
Units of flux are Webers.
Tesla/m2
Photos of flux gate
magnetometers, used for
sensing magnetic fields
down to a few microtesla,
which is about the size of
the earth’s magnetic field.
Sources of Magnetism
A solenoid produces lines of flux as shown (in blue).
Note that the magnetic field lines are continuous with no
source or sink
Inside the solenoid the magnetic flux density is:
B  nI
Where n = number of turns of wire.
= permeability of the core material.
I = current through the core.
Active solenoids have many uses in sensor technologies.
Solenoids make inductive sensors which can be used to detect
motion, displacement, position, and magnetic quantities.
There are permanent magnets (ferromagnets) too; these are
very useful for small compact sensors…..
Magnetic fields increase
inside a permanent
magnet.
The magnetic field inside
a magnetic material is
usually denoted H.

 
B  0 ( H  M )
Magnetisation (M) is the average
magnetic moment of the magnet. It is a
measure of how much all the domains
are pointing in the same direction.
Residual
inductance in
Gauss – how
strong the
magnet is. Also
called
remanence or
retentivity
B
Characteristics of
permanent magnets
H
Coercive force in
Oersteds -Resistance to
demagnetization
We can also plot magnetisation instead of flux
density to get a similar hysteresis curve.
Maximum Energy Product (MEP),
(B x H) in gauss-oersteds times 106.
The overall figure of merit for a
magnet.
Temperature coefficient %/°C,
how much the magnetic field
decreases with temperature.
Some other
figures of merit
for permanent
magnets- these
are commonly
listed in data
tables
Some common permanent magnets.
Typical Magnetic and Physical Properties of Rare Earth
Magnet Materials
Density
Magnetic
Material
Maximu
m
Energy
Product
BH
(max)
Resid
ual
Induct
ion
Br
Coerciv
e
Force
Hc
Intrinsi
c
Coerci
ve
Force
Hci
Curie
Temp.
Normal* Maximum Operating Temp.
lbs/in
g/cm
MGO
Gaus
s
Oersted
s
Oerste
ds
F°
C°
F°
C°
SmCo 18
0.296
8.2
18.0
8700
8000
20000
482
250
1382
750
SmCo 20
0.296
8.2
20.0
9000
8500
15000
482
250
1382
750
SmCo 24
0.304
8.4
24.0
10200
9200
18000
572
300
1517
825
SmCo 26
0.304
8.4
26.0
10500
9000
11000
572
300
1517
825
Neodymi
um
27
0.267
7.4
27.0
10800
9300
11000
176
80
536
280
Neodymi
um 27H
0.267
7.4
27.0
10800
9800
17000
212
100
572
300
0.267
7.4
30.0
11000
10000
18000
176
80
536
280
0.267
7.4
30.0
11000
10500
17000
212
100
572
300
0.267
7.4
35.0
12300
10500
12000
176
80
536
280
Neodymi
um
30
Neodymi
um
30H
Neodymi
um
35
Some rare earth magnetsnotice how the small
spheres are strong enough
magnets to support the
weight of the heavy tools.
These structures
were created by
the action of rare
earth magnets on
a suspension of
magnetic particles
(a ferrofluid).
A movie of ferrofluid reacting to a magnetic field from
a rare earth magnet.
Hard disk reading heads
use permanent magnets.
Note that the hysteresis curves
for magnetisation (J or M) and
flux density (B) are slightly
different.
The maximum energy product is
the maximum energy that can be
obtained from the magnet. In
practice, it is the ‘strength’ of a
permanent magnet.
Magnetic Induction
Time varying fluxes induce electromotive force (emf, i.e. a
voltage difference) in the circuit enclosing the flux Φ=BS
d B
emf  
dt
The sign of the voltage
is such as to make a
current flow whose
magnetic field would
oppose the change in
the flux.
Induced currents also happen for solid
conductors- they are called eddy
currents
Small current loops are set up in the
material to create a magnetic field that
opposes the applied field.
We can add a second
solenoid to intercept the
flux from the first
Assuming the same cross section
area and no flux leakage, a voltage
is induced in the second coil:
d B
V  N
dt
N= number of turns in the
solenoid coil
Assuming B is constant over
area A gives a more useful
relation :
d ( BA)
V  N
dt
This second coil is called the pickup circuit. We get a signal
in this circuit if the magnitude of the magnetic field (B)
changes or if the area of the circuit (A) changes.
We get an induced voltage if we:
• Move the source of the magnetic field (magnet, coil etc.)
• Vary the current in the coil or wire which produces the
magnetic field
• Change the orientation of the magnetic field in the source
• Change the geometry of the pickup circuit, (eg. stretching or
squeezing)
Self Induction.
The magnetic field generated by a
coil also induces an emf in itself.
This voltage is given by:
d ( n B )
V 
dt
Note that the voltage is only
induced for a changing flux.
The number in parenthesis is called the flux
linkage, and is proportional to the current in
the coil.
n B  Li
The constant of proportionality is labeled
the inductance, L.
d ( n B )
di
V
 L
dt
dt
Most induction sensors
measure the change in L; e.g.
as a result of motion.
L
We can therefore define the inductance
V
di
dt
Induction notes.
The defining
equation is:
V
L
di
dt
Induced voltage is proportional to current change
Voltage is zero for DC (inductors look like short circuit to DC)
Voltage increases linearly with rate of change of coil current
Voltage polarity different for increased and decreased current in
same direction
Induced Voltage in direction which acts to oppose change in
current
Calculating inductance
Inductance can be calculated from geometry
For a closely packed coil it is
If n is the number of turns per unit
length, the number of flux
linkages in a length l is
Plugging in the expression B
for a solenoid gives:
n B
L
i
N B  (nl)  ( BA)
N B
2
L
  0 n lA
i
Note that lA is the volume of the solenoid, so keeping n
constant and changing the geometry changes L
Inductors and complex resistance
In an electronic circuit,
inductance can be represented as
complex resistance, like
capacitance.
V
 jL
i
i(t) is a sinusoidal current having a frequency =2f
Two coils brought near each
other one coil induces an emf in
the other
V2   M 21
di1
dt
Where M21 is the coefficient of mutual inductance between
the coils.
Mutual inductance.
For a coil placed
around a long cylinder:
For a coil placed around a torus,
mutual inductance is
M  0R nN
2
 0 N1 N 2 h b
M
ln( )
2
a
Example: Motion Sensor.
Pickup coil with N turns, moves into the gap of a permanent magnet
Flux enclosed by the loop is:
The induced
voltage is:
b  Blx
d B
d
dx
V 
  N ( BLx )  nBl
 nBlv
dt
dt
dt
Cross-section of a magnetic position sensor
The Hall Effect.
When an electron moves
through a magnetic field it
experiences a sideways
force:
F  qvB
q is electron charge
v is the electron velocity
B is the magnetic field
This gives rise to an potential
difference across an appropriate
sensor.
Qualitative Hall effect
The direction of the current and magnetic fields is vital in
determining size of the potential difference.
The deflecting force
shifts the electrons in
the diagram to the right
side.
This deflection produces the transverse Hall potential VH
Quantitative hall effect
At fixed temperature, VH= h I B sin()
I is the current, B is the magnetic field,  is the angle between the
magnetic field and the Hall plate, h is the Hall coefficient.
h depends on the
properties of the material
and is given by:
1
h
Ncq
• N is the number of free charges per unit volume
• c is the speed of light
• q is the charge on the carrier (+ve if holes).
Control current flows through
the control terminals
Ro is the differential
output resistance
Output is measured
across the
differential output
terminals
Ri is the control
resistance
Hall effect sensors are almost always Semiconductor devices.
Parameters of a Typical sensor.
Control Current
Control Resistance, Ri
Control Resistance, Ri vs Temperature
Differential Output Resistance, Ro
Output offset Voltage
Sensitivity
Sensitivity vs Temperature
Overall Sensitivity
Maximum B
Note the significant temperature
sensitivity.
Also note need to use a constant
current source for control.
3 mA
2.2 k Ohms
0.8%/C
4.4 K Ohms
5.0 mV (at B=0 Gauss)
60 micro-Volts/Gauss
0.1%/C
20 V/Ohm-kGauss
Unlimited
Piezoresistance of silicon
should be remembered; makes
semiconductor sensors very
sensitive to shocks.
End of Electrical sensors
Summary:
• Magnetism essentials
• Permanent Magnets
• Inductance
• Hall effect