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Transcript
```Temperature Coefficient of Resistance
Temperature dependent resistivity of metals and its behavior as a feedback control.
Edward Oshaben
Cleveland State University College of Engineering
Feb. 18, 2009
INTRODUCTION
Since the early evolution of man, the art and science of feedback control design has been
a practice of imaginative minds. Without the incorporation of feedback loops into those systems
that both directly and indirectly affect our daily lives, the world as we know it would be out-ofcontrol in a scientific sense. Not all great innovations in feedback control, however, were
designed by man. The natural world, left untouched from human design, exhibits some of the
most efficient use of feedback control. All things in the natural world have a desire to maintain
stability. In nature and more generally in physics, most systems incorporate some form of
feedback control to maintain stability when subjected to external stresses. The increase in the
resistance of a metallic conductor due to increase in its temperature is an example of such natural
feedback mechanism.
In this paper, I derive a mathematical model for the behavior of a metallic conductor’s
resistance, as well as an explanation of the mechanism which affects its dynamics, and a
discussion of perhaps why quantum mechanics has designed it to behave as such. The
temperature dependent behavior of the resistivity of metals is not an idea designed by man, but
rather is a natural phenomenon discovered by man. If, in a fictional sense, one were to assume
that this phenomenon was an actual idea born from human innovation, one could also search for
an explanation of what would drive human innovation to create such an idea. The purpose of this
paper is to provide such a hypothetical explanation.
PHYSICAL EXPLANATION
Before discussing the “idea” behind this feedback model, it is necessary to provide a
physical explanation of the mechanisms that cause temperature-dependent resistivity. When an
electrical circuit of a known resistance, R, is subjected to a voltage, V, a current, I, flows in the
circuit. The value of I is defined by Ohm’s Law as
I=V/R (1).
The passage of current through a resistance dissipates power, P, in the form of heat. The value of
P is defined as
ܲ = ‫ܫ‬ଶܴ =
௏మ
ோ
(2).
From thermodynamics, we know that the energy, Q, required to raise the temperature of a solid
mass is defined as
ܳ = ݉ܿ∆ܶ (3),
where m is the mass of the object (in kg), c is specific heat of the material (in J/kg-K), and ∆ܶ is
the change in temperature of the object (in Kelvin). Calculus methods define Q as the integral of
the power over time,
ܳ = ‫( ݐ݀ ܲ ׬‬4).
From equations 2 through 4, we can see how current passing through a resistance dissipates
power (2), the power transfers energy over time (4), and the transferred energy raises the
temperature of the conducting mass (3).
Now we shall look, at an atomic level, at what happens inside the conductor that 1)
causes resistance and 2) how temperature affects the resistance. Metallic conductors consist of a
periodic crystal lattice of ions with an outer band of valence electrons which are free to move
about the lattice. These outer electrons are the charge carriers for current conduction. In a real
conductor, there exist internal electromagnetic forces which impede the flow of the valence
electrons and change the direction of their travel, and this impedance is quantified as the
resistivity of the metal. The phenomenon of electrons being impeded by these internal forces is
known as electron scattering [1]. Electron scattering results from a breakdown of the perfect
periodic potential of the rigid lattice. The main sources of this breakdown are 1) the displacement
of the ions from their fixed positions due to thermal oscillations, 2) the presence of impurities, 3)
imperfections in the crystals such as dislocations, grain boundaries and stacking faults, 4)
electron-electron interactions, and 5) magnetic scattering if the ions possess magnetic moments
which are not ordered [1]. Other mechanisms exist, but are more specifically related to the
particular metal.
Temperature dependence is caused by the first mechanism listed: the displacement of
ions due to thermal oscillations. As the lattice rises in temperature, the amplitude of the thermal
oscillations increases. In quantum mechanics, this is explained by an increase in the number of
phonons- packets of vibrational energy- emitted by the crystal lattice. These phonons interfere
with the electrons by diverting their directional paths away from flowing along the applied
electric field, and thus lower their drift velocities in the direction of the applied electric field. The
measureable effect is a decrease in current for a given applied voltage, and hence an increase in
resistance.
MATHEMATICAL MODEL
Let us assume a piece of wire with initial resistance ܴ଴ for a given initial temperature ܶ଴ .
This piece of wire is surrounded by an infinite heat sink that is also at ܶ଴ . Now let us assume that
a voltage V is applied across the ends of the piece of wire. The equation for resistance R as a
function of temperature T is
ܴ − ܴ଴ = ܴ଴ ߙ(ܶ − ܶ଴ ) (5),
where ߙ is the temperature coefficient of resistance (in ‫ି ܭ‬ଵ ), which is a property of the material.
Differentially, we can write (5) as
ܴ݀ = ܴ଴ ߙ ݀ܶ (6).
Similarly, (3) can be expressed as an infinitesimally small amount of energy dQ added will result
in a differential change in temperature as
݀ܳ = ݉ܿ ݀ܶ (7).
We can substitute (7) into (6) for dT and get
ܴ݀ = ܴ଴ ߙ
ௗொ
௠௖
(8).
Using heat transfer analysis, dQ can be defined as
݀ܳ = ܳሶ௛௘௔௧ ݀‫ ݐ‬− ܳሶ௖௢௢௟ ݀‫( ݐ‬9),
where ܳሶ௛௘௔௧ is the power dissipated P from (2), and ܳሶ௖௢௢௟ is the rate of heat transfer from the
wire to the infinite heat sink. The equation for surface heat transfer is ܳሶ௖௢௢௟ = ℎ‫ܣ‬൫ܶ௦௨௥௙௔௖௘ −
ܶ௦௜௡௞ ), where h is the heat transfer coefficient (another material property) (in ܹൗ݉ଶ ‫ ) ܭ‬and A is
the surface area of the wire (in ݉ଶ ). We can assume that ܶ௦௨௥௙௔௖௘ is the temperature of the wire
T and that ܶ௦௜௡௞ is the temperature of the infinite heat sink ܶ଴ (which remains constant over time)
so therefore
ܳሶ௖௢௢௟ = ℎ‫ ܶ(ܣ‬− ܶ଴ ) (10).
Substituting (9) into (8) we get
ܴ݀ = ܴ଴ ߙ
(ܳሶ௛௘௔௧ − ܳሶ௖௢௢௟ ) (11).
ௗ௧
௠௖
Substituting (2) and (10) into (11) we get
ܴ݀ = ܴ଴ ߙ
ௗ௧
௏మ
( − ℎ‫ ܶ(ܣ‬− ܶ଴ )) (12).
௠௖ ோ
The (ܶ − ܶ଴ ) term can be replaced from (5) into (12) and gives
ܴ݀ = ܴ଴ ߙ
ௗ௧
௠௖
(
௏మ
ோ
− ℎ‫ܣ‬
ோିோబ
ோబ ఈ
) (13).
Finally, we can re-write (13) in differential form as
ௗோ
ௗ௧
=
ோబ ఈ௏ మ
௠௖
௛஺
ܴିଵ − ௠௖ ܴ +
௛஺ோబ
௠௖
(14).
This is the mathematical model for the resistance of a wire with an initial resistance of ܴ଴ , that is
subjected to an applied and voltage V. Through qualitative analysis of the model in (14), one can
௛஺
identify the second term − ௠௖ ܴ as the driving force (due to heat transfer) for negative feedback.
The first term
ோబ ఈ௏ మ
௠௖
ܴିଵ exhibits the effect of the forcing function V, which increases the
resistance due to temperature-dependence quantified by ߙ, while at the same time, R is fed back
into that same term and dampens the rate of increase as R increases. It is the dampening effect of
ܴିଵ that embodies the internal feedback phenomena discussed in this paper.
We can generically solve for the rate of change of temperature dT/dt as a function of resistance
R=f(t). By combining equations (7), (9), (2), and (10) we get
ௗ்
ௗ௧
௏మ
௛஺
= ௠௖ோ − ௠௖ (ܶ − ܶ଴ ) (15).
We know that in steady-state, the temperature is constant, and thus dT/dt=0. We can solve for the
steady-state temperature ܶௌௌ by setting equation (15) equal to zero and solving for T. In doing so,
we get
௏మ
ܶௌௌ = ௛஺ோ + ܶ଴ (16).
Now we can incorporate the effect of the temperature feedback loop by substituting (5) into (16)
for R. This will result in a quadratic equation in terms of Tss that takes the form
ଶ
+ ℎ‫ܴܣ‬଴ (1 − 2ߙܶ଴ )ܶௌௌ + ℎ‫ܴܣ‬଴ (ߙܶ଴ଶ − ܶ଴ ) − ܸ ଶ = 0 (17).
ℎ‫ܴܣ‬଴ ߙܶௌௌ
Using the quadratic formula, the solution for Tss is
ܶௌௌ =
ିଵ±ටଵା
ଶఈ
రഀೇమ
೓ಲೃబ
+ ܶ଴ (18).
We only keep the solution
ܶௌௌ =
ିଵାටଵା
ଶఈ
రഀೇమ
೓ಲೃబ
+ ܶ଴ (19)
because the other solution ܶௌௌ =
ିଵିටଵା
ଶఈ
రഀೇమ
೓ಲೃబ
+ ܶ଴ corresponds to time ‫ݐ < ݐ‬଴ which this model
does not apply to.
ANALYSIS
When solving for the steady-state temperature of the wire, from (19) one can show that
ܶௌௌ =
ିଵାටଵା
ଶఈ
రഀೇమ
೓ಲೃబ
+ ܶ଴. For purposes of comparison, if we were to open the temperature
feedback loop, and assume R remained constant at a value of ܴ଴ , from (16) we would find the
௏మ
steady-state temperature to be ܶௌௌሖ = ௛஺ோ + ܶ଴ . Comparing the two results, we can observe that
బ
by feeding the temperature back into the resistance equation through a gain ofܴ଴ ߙ, the steady௏మ
state temperature is lowered by a factor of ∆ܶ௦௦ = ܶௌௌሖ − ܶௌௌ = ௛஺ோ − (
బ
రഀೇమ
೓ಲೃబ
ିଵାටଵା
ଶఈ
). Of course,
∆ܶ௦௦ must be a positive number in order for the feedback loop to indeed lower the steady-state
temperature. Setting ∆ܶ௦௦ > 0 and simplifying, it can be shown that this condition is satisfied
ଶఈ௏ మ
when ௛஺ோ > 0. This condition is true for allߙ > 0, and ߙ is indeed greater than zero for all
బ
metallic conductors (ߙ is not necessarily greater than zero for superconductors or semiconductors because their resistance is determined by different mechanisms not discussed in this
paper). This result proves the effectiveness of incorporating the feedback loop.
DISCUSSION
The primary objective of this paper is to uncover the “idea” behind the feedback control
that is the temperature coefficient of resistivity. If this behavior were actually a control designed
by man, what would have driven him to dream up such an idea? The temperature of the
conductor presents a problem in that once it reaches a critical point, the atomic bonds of the
conductor will break. In any system that dissipates energy in the form of heat, cooling must exist
for the temperature to be limited to less than a critical value. This cooling usually comes in the
form of forced convection or radiation into a cooling medium. The heat energy is transferred to
an external system. The unique thing about the temperature feedback loop discussed in this paper
is that the same mechanism that causes the temperature rise (thermal vibrations of the crystal
lattice) also works to govern the power dissipation. In actuality, the work energy that engineers
think of as being wasted as heat due to resistive power dissipation is not completely being
“wasted”. Some of that heat energy actually gets transferred back to the carrier electrons and
negatively impacts their kinetic energy via electron scattering. If the ions of the crystal lattice
could not transfer any energy back to the electrons, then they would have a higher vibrational
energy (temperature) and the interatomic bonds would be broken at a lower level of resistive
power dissipation. In essence, efficiency is improved via electron scattering. The underlying idea
is that the heat energy can be used as a negative feedback to maintain stability within the lattice
structure. When resources are tight, and there is nowhere to transfer the heat energy to, it can be
transferred back to the same working fluid (carrier electrons) to work against the same
mechanism that created the heat.
CONCLUSION
We derived a mathematical model for the resistance
ௗோ
ௗ௧
=
ோబ ఈ௏ మ
௠௖
ܴିଵ −
௛஺
௠௖
ܴ+
௛஺ோబ
௠௖
, for a
given mass of conductor with an initial resistance at an initial temperature, having a given
coefficient of resistivity, a given specific heat capacity, and a given surface area, subjected to a
constant voltage, and cooled through surface heat transfer into an infinite heat sink also at the
initial conductor temperature. We showed the decrease in steady-state temperature due to the
௏మ
incorporation of negative feedback to be ∆ܶ௦௦ = ௛஺ோ − (
బ
ିଵାටଵା
ଶఈ
రഀೇమ
೓ಲೃబ
), and that this decrease is
always positive for ߙ > 0. Most importantly, we identified the idea as taking otherwise
destructive heat energy and using it to dampen the same mechanism which is creating it.
REFERENCES
[1] R.D. BARNARD, THERMOELECTRICITY IN METALS AND ALLOYS. London. Taylor &
Francis LTD. 1972.
```
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