The Energetics Foundations Of Analyzing Magnets

References

Texts:

Laurendeau, Normand M. “Statistical Thermodynamics”. New York: Cambridge University Press, 2005

Hill, Terrell L. “An Introduction to Statistical Thermodynamics”. 1960. New York: Dover Publications Inc., 1986

Stone, Greg C. "Electrical Insulation for Rotating Machines". IEEE Press, 2004

Fleming, A. P. "Insulation and Design of Electrical Windings". New York: William Clowes and Sons, 1913

Carey, Van P. "Statistical Thermodynamics and Microscale Thermophysics". New York: Cambridge University Press, 1999

Sites:

Wikipedia:

• http://en.wikipedia.org/wiki/Magnets
• http://en.wikipedia.org/wiki/AC_motor
• http://en.wikipedia.org/wiki/AC_induction_motor
• http://en.wikipedia.org/wiki/Law_of_Dulong_Petit
• http://en.wikipedia.org/wiki/Debye_model
• http://en.wikipedia.org/wiki/Einstein_solid
• http://en.wikipedia.org/wiki/Maxwell%27s_equations
• http://en.wikipedia.org/wiki/Magnetic_field
• http://en.wikipedia.org/wiki/Electromagnetism
• http://en.wikipedia.org/wiki/Electromagnets

Law of Dulong and Petit

The Dulong-Petit law, a chemical law proposed in 1819 by French physicists and chemists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations.

The result is extremely simple; regardless of the nature of the crystal, the specific heat capacity:

(1)

\begin{align} c_v=\frac{3N_A k_B}{M} \end{align}

c_v = Specific Heat
N_A = Avogadro's Number
k_B = Boltzmann Constant
M = Atomic Weight

Despite its simplicity, Dulong-Petit law offers fairly good prediction for the specific heat capacity of solids with relatively simple crystal structure at high temperatures. It fails, however, in the low temperature region, where the quantum mechanical nature of the solid manifests itself. There, the Debye model works well.

Einstein's Model

The original theory proposed by Einstein in 1907 had a great historical relevance. The heat capacity of solids as predicted by the empirical Dulong-Petit law was known to be consistent with classical mechanics. However, experimental observations at low temperatures showed heat capacity vanished at absolute zero and grew monotonously towards the Dulong and Petit prediction at high temperature. By employing Planck's quantization assumption Einstein was able to predict the observed experimental trend for the first time. Together with the photoelectric effect, this became one of the most important evidence for the need of quantization (remarkably Einstein was solving the problem of the quantum mechanical oscillator many years before the advent of modern quantum mechanics). Despite its success, the approach towards zero is predicted to be exponential, whereas the correct behavior is known to follow a T3 power law.

The Einstein solid is a model of a solid based on three assumptions:

* Each atom in the lattice is a 3D quantum harmonic oscillator
* Atoms do not interact with each other
* All atoms vibrate with the same frequency

Einstein's Model for Specific Heat:

(2)

\begin{align} c_v=\frac{3N_A k_B}{M} (\frac{\theta_E}{T})^2 \frac{e^\frac{-\theta_E}{T}}{(1-e^\frac{-\theta_E}{T})^2} \end{align}

(3)

\begin{align} \theta_E=\frac{h v_E}{k_B} \end{align}

c_v = Specific Heat
N_A = Avogadro's Number
k_B = Boltzmann Constant
M = Atomic Weight
T = Temperature
θ_E = Einstein Temperature
v_E = Einstein Characteristic Frequency

The Debye Model

In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to T3. Just like the Einstein model, it also recovers the Dulong-Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical.

Debye Model for Specific Heat:

(4)

\begin{align} c_v=\frac{9N_A k_B}{M} (\frac{T}{\theta_E}) \int_{0}^{\frac{\theta_D}{T}} \frac{x^4 e^x}{(e^x - 1)^2} dx \end{align}

(5)

\begin{align} \theta_D=\frac{h v_D}{k_B} \end{align}

(6)

\begin{align} v_D=(\frac{3N}{4 \pi V})^\frac{1}{3} \upsilon_m \end{align}

c_v = Specific Heat
N_A = Avogadro's Number
k_B = Boltzmann Constant
M = Atomic Weight
T = Temperature
θ_E = Debye Temperature
v_E = Debye cutoff frequency

Electromagnets

Danish scientist Hans Christian Ørsted discovered in 1820 that electric currents create magnetic fields. British scientist William Sturgeon invented the electromagnet in 1823. His first electromagnet was a horseshoe-shaped piece of iron that was wrapped with about 18 turns of bare copper wire (insulated wire didn't exist yet). The iron was varnished to insulate it from the windings. When a current was passed through the coil, the iron became magnetized and when the current was stopped, it was de-magnetized. Sturgeon displayed its power by showing that although it only weighed seven ounces, it could lift nine pounds when the current of a single-cell battery was applied. However, Sturgeon's magnets were weak because the uninsulated wire he used could only be wrapped in a single spaced out layer around the core, limiting the number of turns. Beginning in 1827, American scientist Joseph Henry systematically improved and popularized the electromagnet. By using wire insulated by silk thread he was able to wind multiple layers of wire on cores, creating powerful magnets with thousands of turns of wire, including one that could support 2063 pounds. The first major use for electromagnets was in telegraph sounders.

The magnetic field of electromagnets in the general case is given by Ampere's Law:

(7)

\begin{align} \int \mathbf{J}\cdot d\mathbf{A} = \oint \mathbf{H}\cdot d\mathbf{l} \end{align}

which says that the integral of the magnetizing field H around any closed loop of the field is equal to the sum of the current flowing through the loop. Computing the magnetic field and force exerted by ferromagnetic materials is difficult for two reasons. First, because the geometry of the field is complicated, particularly outside the core and in air gaps, where fringing fields and leakage flux must be considered. Second, because the magnetic field B and force are nonlinear functions of the current, depending on the nonlinear relation between B and H for the particular core material used. For precise calculations the finite element method is used.

Paramagnetism

Paramagnetism is a form of magnetism which occurs only in the presence of an externally applied magnetic field. Paramagnetic materials are attracted to magnetic fields, hence have a relative magnetic permeability greater than one (or, equivalently, a positive magnetic susceptibility). The force of attraction generated by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect. Unlike ferromagnets, paramagnets do not retain any magnetization in the absence of an externally applied magnetic field, because thermal motion causes the spins to become randomly oriented without it. Thus the total magnetization will drop to zero when the applied field is removed. Even in the presence of the field there is only a small induced magnetization because only a small fraction of the spins will be oriented by the field. This fraction is proportional to the field strength and this explains the linear dependency. The attraction experienced by ferromagnets is non-linear and much stronger, so that it is easily observed, for instance, in magnets on one's refrigerator.

For low levels of magnetisation, the magnetisation of paramagnets follows Curie's law to good approximation:

(8)

\begin{align} \boldsymbol{M} = \chi \cdot\boldsymbol{H}=C\cdot \frac{\boldsymbol{H}}{T} \end{align}

where

M is the resulting magnetization
χ is the magnetic susceptibility
H is the auxiliary magnetic field, measured in amperes/meter
T is absolute temperature, measured in kelvins
C is a material-specific Curie constant

This law indicates that the susceptibility χ of paramagnetic materials is inversely proportional to their temperature. Curie's law is only valid under conditions of low magnetisation, since it does not consider the saturation of magnetisation that occurs when the atomic dipoles are all aligned in parallel. After everything is aligned, increasing the external field will not increase the total magnetisation since there can be no further alignment. However such saturation typically requires very strong magnetic fields.

Squirrel Cage Induction Motors

In 1882 Serb inventor Nikola Tesla identified the rotating magnetic induction field principle and pioneered the use of this rotating and inducting electromagnetic field force to generate torque in rotating machines. He exploited this principle in the design of a poly-phase induction motor in 1883. In 1885, Galileo Ferraris independently researched the concept. In 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.

Introduction of Tesla's motor from 1888 onwards initiated what is sometimes referred to as the Second Industrial Revolution, making possible both the efficient generation and long distance distribution of electrical energy using the alternating current transmission system, also of Tesla's invention (1888). Before widespread use of Tesla's principle of poly-phase induction for rotating machines, all motors operated by continually passing a conductor through a stationary magnetic field.

Initially Tesla suggested that the commutators from a machine could be removed and the device could operate on a rotary field of electromagnetic force. Professor Poeschel, his teacher, stated that would be akin to building a perpetual motion machine. This was because Tesla's teacher had only understood one half of Tesla's ideas. Professor Poeschel had realized that the induced rotating magnetic field would start the rotor of the motor spinning, but he did not see that the counter electromotive force generated would gradually bring the machine to a stop. Tesla would later obtain U.S. Patent 0,416,194 , Electric Motor (December 1889), which resembles the motor seen in many of Tesla's photos. This classic alternating current electro-magnetic motor was an induction motor.

Michail Osipovich Dolivo-Dobrovolsky later invented a three-phase "cage-rotor" in 1890. This type of motor is now used for the vast majority of commercial applications.

Squirrel Cage Rotor
Most common AC motors use the squirrel cage rotor, which will be found in virtually all domestic and light industrial alternating current motors. The squirrel cage takes its name from its shape - a ring at either end of the rotor, with bars connecting the rings running the length of the rotor. It is typically cast aluminum or copper poured between the iron laminates of the rotor, and usually only the end rings will be visible. The vast majority of the rotor currents will flow through the bars rather than the higher-resistance and usually varnished laminates. Very low voltages at very high currents are typical in the bars and end rings; high efficiency motors will often use cast copper in order to reduce the resistance in the rotor.

In operation, the squirrel cage motor may be viewed as a transformer with a rotating secondary. When the rotor is not rotating in sync with the magnetic field, large rotor currents are induced; the large rotor currents magnetize the rotor and interact with the stator's magnetic fields to bring the rotor into synchronization with the stator's field. An unloaded squirrel cage motor at synchronous speed will consume electrical power only to maintain rotor speed against friction and resistance losses; as the mechanical load increases, so will the electrical load - the electrical load is inherently related to the mechanical load. This is similar to a transformer, where the primary's electrical load is related to the secondary's electrical load.

This is why, for example, a squirrel cage blower motor may cause the lights in a home to dim as it starts, but doesn't dim the lights when its fanbelt (and therefore mechanical load) is removed. Furthermore, a stalled squirrel cage motor (overloaded or with a jammed shaft) will consume current limited only by circuit resistance as it attempts to start. Unless something else limits the current (or cuts it off completely) overheating and destruction of the winding insulation is the likely outcome.

In order to prevent the currents induced in the squirrel cage from superimposing itself back onto the supply, the squirrel cage is generally constructed with a prime number of bars, or at least a small multiple of a prime number (rarely more than 2). There is an optimum number of bars in any design, and increasing the number of bars beyond that point merely serves to increase the losses of the motor particularly when starting.

Squirrel Cage Induction Motor

Formulas

The relationship between the supply frequency, f, the number of pole pairs, p, and the synchronous speed (speed of rotating field), n_s, is given by:

(9)

\begin{align} f = \frac{pn_s}{60}. \end{align}

From this relationship:

(10)

\begin{align} \mbox{Synchronous speed, }n_s = \frac{60f}{p}\quad[\mbox{rev/min}] \end{align}

The rotor speed is:

(11)

\begin{align} \mbox{Rotor speed, }n_r = n_s(1-s)\,\! \end{align}

where: s is the slip.

Slip is calculated using:

(12)

\begin{align} s = \frac{n_s-n_r}{n_s} \end{align}

Torque is calculated using:

(13)

\begin{align} T = \frac{(HP) (5255)}{n_r} \end{align}

Horsepower is calculated using:

(14)

\begin{align} HP = \frac{(V) (I) (Eff)}{746} \end{align}

V = Voltage
I = Current
Eff = Efficiency