**4/19/10–Wave Guide Simulation (HW#10)**

Here’s the accompanying Mathematica simulation to Griffiths problem 9.30 of the TM mode in a rectangular wave guide. I changed most of the code accordingly, but ran into some trouble adjusting the contour plots from the the original file.

**4/18/10– Blog Project**

**INDUCTION HEATING
**

**http://www.youtube.com/watch?v=y_-Iv9FddOU**

Induction heating is the process in which a conductive material is heated by current and hysteresis power losses caused by electromagnetic induction.

Remember EM induction?

Original Video – More videos at TinyPic

(Background Image from Google Images*)

I created this animation from PowerPoint slides to remind us about the principles of EM induction and how they relate to induction heating. By **Faraday’s law**, a changing magnetic flux (generated here by a time-variant current in a solenoid) induces an EMF (voltage)/field. Moreover, **Lenz’s law **tells us that the induced field opposes the change in flux. We learned in introductory physics that a current is also induced in a wire loop placed in the field, by **Ohm’s law**. Similarly, when a conductive material is placed in the field, electrons within the material flow in the direction of the induced field, called *eddy currents*. The **Joule heating law **associates current with power, and thus eddy current power losses contribute to heating in the material. Magnetic materials below the *Curie point *are further heated by hysteresis losses, a result of magnetic domains within the material resisting the changing field. As the first video showed, induction heating can be used to melt, solder, harden, weld, and even cook materials!

**Considerations**

*Workpiece: *The material being heated. It must be conductive and is usually a metal.

*Eddy Currents: *Currents induced in a conductive material by a variant magnetic field.

*Joule effect: *Flowing currents in a conductor have an associated power which produces heat ( P = IV = I^{2}R ).

*Hysteresis: *“Memory” of the magnetic material, such that an applied current produces a different magnetization based on the material’s magnetic history. Small patches of similarly aligned dipoles within a magnetic, called *domains*, resist the applied field and are responsible for the lag.* Hysteresis losses *are the power dissipated, a result of the domains resisting the breaking and building of the field.

*Curie Point: *The temperature above which a ferromagnet (permanent magnet, needs no magnetic field to sustain magnetization) becomes paramagnetic (magnetization parallel to the magnetic field).

*Skin Depth: Skin effect* describes the phenomenon in which the induced current is concentrated in the surface layers of the workpiece, as both the field and currents are quickly damped in the material. Skin depth or *depth of penetration* is the depth at which eddy current density has decreased to *e ^{–1}*, roughly one third. Beyond this depth the current density rapidly falls.

–> Because the magnetic field, and thus the current (since area and angle- also variables in magnetic flux- are less easily altered), must be constantly pulsating in magnitude, only AC current is used to supply induction heaters

**Why Heat by Induction?**

There exist other methods of heating materials. Conduction heating, for example, is a similar process in which current flows in the workpiece itself, directly heating it by the Joule effect. Of course, industrial furnaces are common. What benefits, then, are there to induction heating?** **

Furnaces tend to be large, have long start-up and shut-down times, and emit fumes and byproducts of combustion, both a pollutant and a potential safety hazard. The induction heater can be **small **and, as all electric devices, is **immediately turned on and off**. It is a “**clean**” process and safer for those operating the system (as the video demonstrated!). It also has **fewer maintenance costs**.

As with conduction heating, induction heating has the benefit that **all of the power supplied goes directly into the workpiece** and **heating times are short**. But induction heaters are more feasible in practice; the **material can be thermally insulated from the wire** and still experience heating, trapping thermal radiation and protecting the device. They **fit well into automated production methods**, are easily controlled, and the process is highly **repeatable**.

There are some surprising benefits to induction heating. For example, alloys are easily mixed in induction heating processes because the induced field automatically **stirs** the melted metal! Also, **special techniques**— precision melting, hardening of surface— can be implemented in the process.

Electricity costs to run induction heaters may be higher than costs for other fuels but Davies argues that the difference is offset by **higher efficiencies**. And, as he makes note, fossil fuels are not sustainable– **electricity** could soon become the major source of industrial energy, heightening the importance of induction heating.

**Modeling the Phenomenon**

Now that we understand the physics of induction heating, on to the models…* *

*Objective and Methods*

I aimed to model various aspects of induction heating using Wolfram Mathematica (version 7). I wanted to create interactive demonstrations that show how differences in the workpiece material (size, magnetic permeability, conductivity) and the applied field (operating frequency, initial strength) affect the induced field, intensity, eddy currents, and power losses. I was interested in comparing power losses due to the Joule effect to those due to hysteresis losses. * *

*Theory*

Fundamental electromagnetism equations and theory are drawn from Griffiths, while the theory of induction heating is drawn from the *Induction Heating Handbook *(Davies and Simpson) chapter 12 and *Conduction and Induction Heating* (E.J. Davies). Research on hysteresis was drawn from *Hysteresis in Magnetism *(Bertotti).

The ranges for frequency and characteristics of the material in my models were chosen to accurately reflect real-world values. Frequency (f) values were obtained from Ameritherm, an actual purveyor of induction heating systems. Resistivity (rho) and magnetic permeability (mu) of the material reflect the range of typical materials used in induction heating. There are handbooks that list these values; the books or the following websites (among many) can be used as resources when picking values in the simulation:

The shape of the workpiece affects how one derives the current density, induced fields, etc. The simplest configuration to examine is that of a rectangular slab, and I have limited my models to this shape.

One begins with the most fundamental model, the “semi-infinite” slab (which is theoretically the side of a cylinder of infinite radius). Thus, in this model, there is only one uniform eddy current running in a single direction (I took to be the x) parallel to the edge of the slab.

The** ***diffusion equations *describe the auxialiary field, electric field, and current density at induction heating frequencies. They are given by the Laplacian of {H, E, J} = the time-derivative of {H, E J} * (mu/rho). Solving the diffusion equation for H, we obtain a differential equation that can be solved under the boundary conditions of the problem, yielding H(y,t)=H_{0}*exp[-alpha*y]*cos(omega*t-alpha*y), where alpha is a constant from the DE equal to Sqrt[(mu*omega)/(2*rho)]. Notice that the solution is exponential; the current rapidly decreases into the surface of the workpiece! Also note that skin depth/depth of penetration occurs when y=1/alpha.

From the auxiliary field, the magnetic field is easily obtained by the relation B=mu*H, with the intensity proportional to B^{2}. Furthermore, we employ the relation J=curl[H] to obtain an expression for the current density. Electric field is then obtained from E=rho*J, an expression of Ohm’s law. These are the equations modeled in the first Mathematica file available below.

The power losses are calculated for a rectangular slab of finite dimensions (for which the solution to the diffusion equation slightly differs from that of the semi-infinite slab, and is solved in the references). Power loss per square meter due to the Joule effect (eddy current power losses) are obtained by integrating rho*J^{2} over a period of time. To get the average power loss per square meter, one employs the same trick the class learned that the time-average of the square of a sine or cosine is 1/2. Multiplying the power loss per square meter by the dimensions of the block then gives the average power loss, which is modeled in the second Mathematica file below.

I found hysteresis losses more difficult to model, and found I had to vastly simplify the problem in order to solve and model it on Mathematica. Bertotti derives the formula that the dissipated power due to hysteresis effects is given by conductivity of the material times the square of the change in magnetic flux through the material in the time interval of a *Barkhausen jump *(a subject I would like to do more research on; related to the changing domains of a ferromagnet in a time-variant magnetic field). Of course conductivity is the inverse of the resistivity (rho), and I separated flux into the field times the volume of the shape (while flux is through the cross-sectional area, it is strengthened proportionally to the width). I once again estimated that the time average of the square of a trigonometric function is ½, multiplied by the original strength of the field. I think that these calculations may be slightly over-simplified, but they provide a first comparison to the current power losses that can be better adjusted as I further research the subject.

*Results*

See the following Mathematica notebooks (hosted through Vspace) to view my computational models and accompanying code (screenshots shown):

Comparing hysteresis and current power losses in finite slab

**Discussion**

In the process of creating these models I strengthened my knowledge of electromagnetic induction and the fundamentals of electromagnetism, improved my computational skills with Mathematica, examined graduate-level texts, and learned about an interesting real-world application of our class material.

It was interesting to read very specified texts which come with new terminology and practices that must be learned. The induction heating process needs to be performed in the real world, and I was surprised at how many constraints were thus put on the problem (operating frequencies, actual materials). I now realize the importance of simplifying and making assumptions to work out your theoretical problem. The problems that arose as I undertook this project, such as limited resources on hysteresis losses and lack of computer programming experience, further exemplified the issues that may arise in scientific research and reinforced the importance of making simplifications (which can be later corrected for increased accuracy).

I found the computational work particularly enlightening, and feel more competent with Mathematica having created these models. The “manipulate” command, which allows for the interactive panels in my models, is a great tool and I plan to use this feature in future projects.

I envision a number of potential extensions of this project for future modeling and research. The theory, for example, could be developed for a number of workpiece shapes with varying results (my references extend their theory to cylinders; perhaps in practice, only a few geometries are used or perhaps these models approximate a number of shapes adequately). For a more multidisciplinary approach, one could connect the electromagnetism to the thermodynamics of induction heating, where there are a wealth of topics to explore– heat flow through the material, phases changes in the metal, etc. Were I to continue this work, I would focus on the hysteresis losses, for which it seems models are needed. I also think it would be interesting to compare aspects of induction heating with those of conduction heating, furnaces, and other alternative heating methods.

As a final note, the second video explaining electromagnetic induction is also available through Vspace and could possibly be used as a demonstration tool. I would also like to thank Professor Magnes for her assistance with the modeling and the direction of the project.

**References**

Ameritherm: Precision Induction Heating. (n.d.). *Ameritherm Induction Heating: Applications. Your Solutions.* Retrieved April 4, 2010, from http://www.ameritherm.com

Bertotti, B. (1998). *Hysteresis In Magnetism*. New York: Academic.

Davies, E. J. (1990). *Conduction and Induction Heating*. London: Peregrinus.

Davies, J., & Simpson, P. (1979). *Induction Heating Handbook*. London: McGraw-Hill

Griffiths, D. J. (1999). *Introduction to Electrodynamics* (Third ed.). Upper Saddle River: Pearson.

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**4/11/01(+)– Project Resources**

In addition to Griffiths, I found two books through ILL on induction heating and its electromagnetic theory to use for my project. I’ve also used many helpful websites, which I’d like to save on this blog:

Here’s an interesting site that explains depth of penetration of eddy currents. It includes an interesting java app; I could try something similar on my project.

I will try to update this post if I find more helpful sites…

Here is a great compilation of the definitions of *Induction Heating Fundamentals*, including hysteresis heating. It is provided by a private company, “the leading manufacturer of RF induction heating systems.”

**3/28/10– Eddy currents and heat induction**

In researching my project, I’ve come across a number of accessible, but useful, resources. Here’s a video that visualizes and demos eddy currents that arise from simple magnetic induction as well as an article explaining the special currents. Wikipedia actually has some good pages on radio frequency heating, induction heating, and eddy currents that can briefly sketch each topic. This site succinctly explains the fundamentals of induction heating. I’m still on the look out for more advanced texts to help model the phenomena.

**3/25/10– Electromagnetic Waves**

Here is a worksheet completed in class that explores and simulates electromagnetic wave propagation.

**3/23/10– Project Proposal: Exploring Induction Heating
**

I will research heat induction, the process by which a conductor is heated via electromagnetic induction. The heat is both generated by eddy currents created by a changing magnetic field in addition to magnetic hysteresis loss. This topic thus presents a number of entities to model, such as the time-dependent magnetic field and resultant eddy currents. Guiding research questions include: what is the primary cause of the heat? How much energy do the eddy currents/hysteresis provide? How do the currents flow in the material?

Induction heating is directly connected to our electromagnetism course, particularly the topics in Griffiths Chapter 7 (on induction). There are a number of direct applications of induction heating, including induction furnaces and induction cooking, which can be researched to supplement my models.

**3/20/10– Class Journal Summary**

My class journal has an assortment of materials and thoughts I’ve recorded over the semester. In it I have tried to note all of the references that have helped me complete homework assignments and study for quizzes and the midterm. Particularly, I have used the journal to organize any helpful websites I have found, from Wikipedia, Wolfram Alpha, and the Wolfram Mathematica Online Integrator to more content-specific websites like HyperPhysics and Physics Forums . I’ve also noted times that I downloaded lectures from electromagnetism courses at other institutions for more examples and instruction, which was a great study aid for the midterm as a supplement to the class notes. By organizing these materials, I have created a handy guide with a list of study tools that can be easily referenced for future homework assignments and the final.

Additionally, I’ve tried to use the journal as a place to record my thoughts regarding our coursework and my experience as a student in the course. It is clear that working with classmates on homework assignments and in study groups has been a great help for me in electromagnetism. The journal also evidences a progression during the dielectrics chapters from confusion to understanding, which makes me feel more accomplished in the course. However, recording my thoughts didn’t help me pick out my project topic and was not useful as a study aid. Perhaps, rather than record my experiences in the journal, it would be more helpful to post to the class blog when I am confused on a topic so that others in the same situation can connect and form a study group such that we all benefit.

**2/15/10– “Snell’s Law for Dielectrics”
**

Here is a link to my Mathematica simulation exploring eq. 4.86 in Griffiths, or “Snell’s Law for Dielectrics.” I plotted the equation for several incident angles in three cases: equal permittivites, e1>e2, and e1<e2. It seems that there is no analogous critical angle/total internal reflection in the dielectric version of Snell’s law.

**2/8/10– Griffiths 4.10 Visualization
**

Here’s my Mathematica visualization of the electric field inside of the sphere from problem 4.10 in Griffiths EM. I left out the constant k from the answer, as it would only scale the vectors and the overall field will look the same. Though I assumed a positive k, if k is negativethe field will point radially outward from the origin instead of inward. Of course, the field only looks like this up to r, the radius of the sphere, at which point it becomes 0 everywhere.

**2/4/10– Online Resource**

I’ve found this website pretty helpful for homework/studying. Hopefully other people will too! If you go to the main page it also has information about other areas of physics that might be come in handy in other courses.

WmFlanigan[we learned of your work from our website referrer logs; thanks for the link]

You have clearly grasped this subject, one which forces us to reconcile all those pesky EM laws we dreaded in Phy101. If you are ever in the upstate NY area and would like to see induction heating put to work in our labs, please contact me.

WmFlanigan | Website Manager | Ameritherm

mafaginThe user interface is great! Are hysteresis losses and eddy currents the only significant sources of heating? If so, it might be interesting to give the user the ability to adjust the mass, heat capacity and melting point of the work piece, and then have the program display how long it will take to melt the work piece given the amount of energy the induction furnace is putting out.

makinnebergThe heat by induction part IS the blog project; everything else on the page is part of other blog assignments. EM induction is a part of heat by induction, and I was just interested in learning the theory/modeling aspects of heat by induction, not comparing it to EM induction.

zhxiewow margo, your research is super long and professional! i am a little confused when i read the part for heat by induction: are you interested in comparing heat by induction with electromagnetic induction?