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| {{Use dmy dates|date=August 2013}}
| | by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. If you are using videos on your site then this is the plugin to use. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. This plugin allows a blogger get more Facebook fans on the related fan page. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. To turn the Word - Press Plugin on, click Activate on the far right side of the list. There are plenty of tables that are attached to this particular database. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. See a product, take a picture, and it gives you an Amazon price for that product, or related products. <br><br>Many developers design websites and give them to the clients, but still the client faces problems to handle the website. s ability to use different themes and skins known as Word - Press Templates or Themes. When you have any queries concerning wherever along with the way to use [http://GET7.pw/backup_plugin_353050 wordpress backup plugin], it is possible to e mail us at our own page. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. Word - Press is an open source content management system which is easy to use and offers many user friendly features. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and full user registration plug-ins. |
| Miniaturizing components has always been a primary goal in the [[semiconductor]] industry because it cuts production cost and lets companies build smaller computers and other devices. Miniaturization, however, has increased dissipated power per unit area and made it a key limiting factor in [[integrated circuits|integrated circuit]] performance. Temperature increase becomes relevant for relatively small-cross-sections wires, where it may affect normal semiconductor behavior. Besides, since the generation of heat is proportional to the frequency of operation for switching circuits, fast computers have larger heat generation than slow ones, an undesired effect for chips manufacturers. This article summaries physical concepts that describe the generation and conduction of heat in an integrated circuit, and presents numerical methods that model heat transfer from a macroscopic point of view.
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| ==Generation and transfer of heat==
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| ===Fourier's law===
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| At macroscopic level, [[Fourier's law]] states a relation between the transmitted heat per unit time per unit area and the gradient of temperature:
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| : <math>q= - \kappa \nabla T </math>
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| Where <math>\kappa</math> is the thermal conductivity, <nowiki>[</nowiki>W·m<sup>−1</sup> K<sup>−1</sup><nowiki>]</nowiki>.
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| ===Joule heating===
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| Electronic systems work based on current and voltage signals. Current is the a flow of charged particles through the material and these particles (electrons or holes), interact with the lattice of the crystal losing its energy which is released in form of heat. [[Joule Heating]] is a predominant mechanism for heat generation in integrated circuits<ref name="test">T. Bechtold, E. V. Rudnyi and J. G Korvink, "Dynamic electro-thermal simulation of microsystems—a review," Journal of Micromechanics and Microengineering. vol. 15, pp. R17–R31, 2005</ref> and is an undesired effect in most of the cases. For an ohmic material, it has the form:
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| : <math> Q = j^{2} \rho </math>
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| Where <math> j </math> is the current density in <nowiki>[</nowiki>A·m<sup>−2</sup><nowiki>]</nowiki>, <math> \rho </math> is the specific electric resistivity in <nowiki>[</nowiki><math>{\Omega}</math>·m<nowiki>]</nowiki> and <math> Q </math> is the generated heat per unit volume in <nowiki>[</nowiki>W·m<sup>−3</sup><nowiki>]</nowiki>.<ref name="test"/>
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| ===Heat-transfer equation===
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| The governing equation of the physics of the [[heat transfer]] problem relates the flux of heat in space, its variation in time and the generation of power by the following expression:
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| : <math>\nabla\left(\kappa\left( T\right) \nabla T\right)+g=\rho C\frac{\partial T}{\partial t} </math>
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| Where <math>\kappa</math> is the thermal conductivity, <math>\rho</math> is the density of the medium, <math>C</math> is the specific heat, <math>k=\frac{\kappa}{\rho C}</math>, the thermal diffusivity and <math>g</math> is the rate of heat generation per unit volume. Heat diffuses from the source following the above equation and solution in an homogeneous medium follows a Gaussian distribution.
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| ==Techniques to solve heat equation ==
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| ===Kirchhoff transformation===
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| To get rid of the temperature dependence of <math> \kappa </math>, Kirchhoff transformation can be performed <ref name="baty">W. Batty, C. E. Christoffersen, A. J. Panks, S. David, C. M. Snowden, M. B. Steer, “Electrothermal CAD of Power Devices and Circuits With Fully Physical Time- Dependent Compact Thermal Modeling of Complex Nonlinear 3-d Systems,” IEEE Trans. Comp. and Pack. Technologies, vol. 24, no. 4, pp. 566–590, 2001.</ref>
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| : <math> \theta=T_{s}+\frac{1}{\kappa_{s}}\int_{T_{s}}^{T}\kappa(T)dT </math>
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| where <math>\kappa_{s} = \kappa\left( T_{s}\right) </math> and <math>T_{s}</math> is the heat sink temperature. When applying this transformation, the heat equation becomes:
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| : <math> k\nabla^{2}\theta+\frac{k}{\kappa_{s}}g=\frac{\partial\theta}{\partial t} </math>
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| where <math> k=\frac{\kappa}{\rho C} </math> is called the diffusivity,<ref name="baty"/> which also depends on the temperature. To completely linearize the equation, a second transformation is employed:
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| <math> k_{s}\tau=\int_{0}^{t}k(\theta)dt </math>
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| yielding the expression:
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| <math> \nabla^{2}\theta-\frac{1}{k_{s}}\frac{\partial \theta}{\partial \tau}=-\frac{g}{\kappa_{s}} </math>
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| Simple, direct application of this equation requires approximation. Additional terms arising in the transformed Laplacian are dropped, leaving the Laplacian in its conventional form.<ref name="baty"/>
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| ===Analytical solutions===
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| Although analytical solutions can only be found for specific and simple cases, they give a good insight to deal with more complex situations. Analytical solutions for regular subsystems can also be combined to provide detailed descriptions of complex structures. In Prof. Batty's work,<ref name="baty"/> a Fourier series expansion to the temperature in the Laplace domain is introduced to find the solution to the linearized heat equation.
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| ====Example====
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| This procedure can be applied to a simple but nontrivial case: an homogeneous cube die made out of GaAs, L=300 um. The goal is to find the temperature distribution on the top surface. The top surface is discretized into smaller squares with index i=1...N. One of them is considered to be the source.
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| Taking the Laplace transform to the heat equation:
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| : <math> \nabla^{2}\bar{\Theta}-\frac{s}{k_{s}}\bar{\Theta}=0</math>
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| where <math>\overline{\Theta}=s\theta-\theta\left(\tau=0\right)</math>
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| Function <math>\overline{\Theta}</math> is expanded in terms of cosine functions for the <math>x</math> and <math>y</math> variables and in terms of hyperbolic cosines and sines for <math>z</math> variable. Next, by applying adiabatic boundary conditions at the lateral walls and fix temperature at the bottom (heat sink temperature), thermal impedance matrix equation is derived:
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| : <math>\Delta\theta_{i}=\sum_{j=1}^{N}R_{TH_{ij}}(t)P_{j}(t)</math>
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| Where the index <math>j</math> accounts for the power sources, while the index <math>i</math> refers to each small area.
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| For more details about the derivation, please see Prof. Batty's paper,.<ref name="baty"/>
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| The below figure shows the steady state temperature distribution of this analytical method for a cubic die, with dimensions 300 um. A constant power source of 0.3W is applied over a central surface of dimension 0.1L x 0.1L. As expected, the distribution decays as it approaches to the boundaries, its maximum its located at the center and almost reaches 400K
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| <center>
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| [[File:battysup.png]]
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| </center>
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| ===Numerical solutions===
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| Numerical solutions use a mesh of the structure to perform the simulation. The most popular methods are: Finite difference time-domain (FDTD) method, Finite element method (FEM) and method of moments (MoM).
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| The finite-difference time-domain (FDTD) method is a robust and popular technique that consists in solving differential equations numerically as well as certain boundary conditions defined by the problem. This is done by discretizing the space and time, and using finite differencing formulas, thus the partial differential equations that describe the physics of the problem can be solved numerically by computer programs.
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|
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| The FEM is also a numerical scheme employed to solve engineering and mathematical problems described by differential equations as well as boundary conditions. It discretizes the space into smaller elements for which basis functions are assigned to their nodes or edges. Basis functions are linear or higher order polynomials. Applying the differential equation and the boundary conditions of the problem to the basis functions, a system of equations is formulated using either the [[Ritz method|Ritz]] or [[Galerkin method]]. Finally, a direct or iterative method is employed to solve the system of linear equations.<ref name="jin">J.-M. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 2nd ed., 2002</ref> For the thermal case, FEM method is more suitable due to the nonlinearity nature of the thermal properties.
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| ====Example====
| |
| The previous example can be solved with a numerical method. For this case, the cube can by discretized into rectangular elements. Its basis functions can be chosen to be a first order approximation (linear):
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| <math> N_{i}^{e}=\frac{1}{2}\xi\left(1\mp\zeta\right),\qquad i=1,4 </math> | |
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| <math> N_{i}^{e}=\frac{1}{2}\eta\left(1\mp\zeta\right),\qquad i=2,5 </math>
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| <math> N_{i}^{e}=\frac{1}{2}\left(1-\xi-\eta\right)\left(1\mp\zeta\right),\qquad i=3,6 </math>
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| where <math>\zeta=2(z-z_{c})/h_{z} </math>. If <math>z_{c}=0</math>, then <math>\zeta=2z/h_{z}</math>. | |
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| Using this basis functions and after applying Galerkin's method to the heat transfer equation, a matrix equation is obtained:
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| <math> \left[S\right]\left\{ \theta\right\} +\left[R\right]\frac{d}{dt}\left\{ \theta\right\} =\left\{ B\right\} </math>
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| where,
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| : <math> R_{ij}=\int_{v}N_{j} N_{i}dV </math>
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| : <math> S_{ij}=k\int_{v}\nabla N_{j}.\nabla N_{i}dV </math>
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| : <math> B_{i}=\frac{k}{\kappa_{s}}\int_{\Omega_{1}}N_{i}p(x,y)d\Omega+\frac{k}{\kappa_{s}}\int_{v}N_{i}gdV-kT_{o}\sum_{j=0}^{N_{D}}\int_{v}\nabla N_{j}^{D}.\nabla N_{i}dV</math>.
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| This expressions can be evaluated by using a simple FEM code. For more details, please see.<ref name="jin"/> The figure below shows the temperature distribution for the numerical solution case. This solution shows very good agreement with the analytical case, its peak also reaches 390 K at the center. The apparent lack of smoothness of the distribution comes from the first order approximation of the basis functions and this can be solved by using higher order basis functions. Also, better results might be obtained by employing a denser mesh of the structure; however, for very dense meshes the computation time increases a lot, making the simulation non-practical.
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| <center> | |
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| </center> | |
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| The next figure shows a comparison of the peak temperature as a function of time for both methods. The system reaches steady state in approximately <math>1 ms</math>.
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| <center>
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| [[File:Analithic&FEM1.PNG|400px]]
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| </center>
| |
| | |
| ===Model order reduction===
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| The numerical methods such as FEM or FDM derive a matrix equation as shown in the previous section. To solve this equation faster, a method called Model Order Reduction can be employed to find an approximation of lower order. This method is based on the fact that a high dimensional state vector belongs to a low dimensional subspace [http://modelreduction.com/ModelReduction/].
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| Figure below shows the concept of the MOR approximation: finding matrix V, the dimension of the system can be reduced to solve a simplified system.
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| <center>
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| [[File:DiagramMOR2.png|400px]]
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| </center>
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| Therefore the original system of equation:
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| : <math>C\left\{ x\right\} '+K\left\{ x\right\} =F\left\{ u\right\}</math>
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| becomes:
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| : <math>V^{T}CV\left\{ z\right\} '+V^{T}KV\left\{ z\right\} =V^{T}F\left\{ u\right\}</math>
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| Whose order is much lower than the original making the computation much less expensive. Once the solution is obtained, the original vector is found by taking the product with V.
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| ==Conclusion==
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| The generation of heat is mainly produced by joule heating, this undesired effect has limited the performance of integrated circuits. In the preset article heat conduction was described and analytical and numerical methods to solve a heat transfer problem were presented. Using this methods, steady state temperature distribution was computed as well as the peak temperature as a function of time for a cubic die. For an input power of <math>0.3 W </math> (or <math> 3.333e8 W/m_{2} </math>) applied over a single surface source on the top of a cubic die a peak increment of temperature in the order of 100 K was computed. Such increase in temperature can affect the behavior of surrounding semiconductor devices. Important parameters like mobility change drastically. That is why the heat dissipation is a relevant issue and must be considered for circuit designing.
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| ==See also==
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| *[[Heat generation in integrated circuits]]
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| ==References==
| |
| {{reflist|colwidth=30em}}
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| | |
| {{DEFAULTSORT:Thermal Simulations For Integrated Circuits}}
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| [[Category:Integrated circuits]]
| |
by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. If you are using videos on your site then this is the plugin to use. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have.
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