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| The '''transmission-line matrix (TLM) method''' is a space and time discretising method for computation of [[electromagnetic fields]]. It is based on the analogy between the electromagnetic field and a mesh of [[transmission line]]s. The TLM method allows the computation of complex three-dimensional electromagnetic structures and has proven to be one of the most powerful time-domain methods along with the finite difference time domain ([[FDTD]]) method.
| | by Nas, is very fitting and the film agrees with it. Medical word press themes give you the latest medical designs. 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. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br>These websites can be easily customized and can appear in the top rankings of the major search engines. If you treasured this article therefore you would like to collect more info pertaining to [http://shortener.us/wordpress_backup_3166710 backup plugin] please visit the webpage. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. This plugin is a must have for anyone who is serious about using Word - Press. You can up your site's rank with the search engines by simply taking a bit of time with your site. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links. <br><br>It is very easy to install Word - Press blog or website. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. Enough automated blog posts plus a system keeps you and your clients happy. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <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. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. 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. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. An ease of use which pertains to both internet site back-end and front-end users alike. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order. |
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| == Basic principle ==
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| [[File:SingleNode2DTLM.png|thumb|500px|right|2D TLM example: an incident voltage pulse in two consecutive scattering events.]] The TLM method is based on [[Huygens Principle|Huygens' model of wave propagation]] and scattering and the analogy between field propagation and transmission lines. Therefore it considers the computational domain as a mesh of transmission lines, interconnected at nodes. In the figure on the right is considered a simple example of a 2D TLM mesh with a voltage pulse of amplitude 1 V incident on the central node. This pulse will be partially reflected and transmitted according to the transmission-line theory. If we assume that each line has a characteristic impedance <math>Z</math>, then the incident pulse sees effectively three transmission lines in parallel with a total impedance of <math>Z/3</math>. The reflection coefficient and the transmission coefficient are given by
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| : <math>R = \frac{Z/3-Z}{Z/3+Z} = -0.5</math>
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| : <math>T = \frac{2(Z/3)}{Z/3+Z} = 0.5</math>
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| The energy injected into the node by the incident pulse and the total energy of the scattered pulses are correspondingly
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| : <math>E_I = vi\,\Delta t = 1 \left(1/Z\right)\Delta t = \Delta t/Z</math> | |
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| : <math>E_S = \left[0.5^2+0.5^2+0.5^2+(-0.5)^2\right](\Delta t/Z) = \Delta t/Z</math>
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| Therefore the [[energy conservation law]] is fulfilled by the model.
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| The next scattering event excites the neighbouring nodes according to the principle described above. It can be seen that every node turns into a secondary source of spherical wave. These waves combine to form the overall waveform. This is in accordance with Huygens principle of light propagation.
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| In order to show the TLM schema we will use time and space discretisation. The time-step will be denoted with <math>\Delta t</math> and the space discretisation intervals with <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math>. The absolute time and space will therefore be <math>t = k\,\Delta t</math>, <math>x = l\,\Delta x</math>, <math>y = m\,\Delta y</math>, <math>z = n\,\Delta z</math>, where <math>k=0,1,2,\ldots</math> is the time instant and <math>m,n,l</math> are the cell coordinates. In case <math>\Delta x = \Delta y = \Delta z</math> the value <math>\Delta l</math> will be used, which is the [[lattice constant]]. In this case the following holds:
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| : <math>\Delta t=\frac{\Delta l}{c_0},</math>
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| where <math>c_0</math> is the free space speed of light.
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| == The 2D TLM node ==
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| === The scattering matrix of an 2D TLM node ===
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| [[File:SeriesTlmNode.png|thumb|400px|right|A 2D series TLM node]]
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| If we consider an electromagnetic field distribution, in which the only non-zero components are <math>E_x</math>, <math>E_y</math> and <math>H_z</math> (i.e. a TE-mode distribution), the Maxwell's equations in [[Cartesian coordinates]] reduce to
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| : <math>\frac{\partial{H_z}}{\partial{y}} = \varepsilon\frac{\partial{E_x}}{\partial{t}}</math>
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| : <math>-\frac{\partial{H_z}}{\partial{x}} = \varepsilon\frac{\partial{E_y}}{\partial{t}}</math>
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| : <math>\frac{\partial{E_y}}{\partial{x}}-\frac{\partial{E_x}}{\partial{y}} = -\mu\frac{\partial{H_z}}{\partial{t}}</math>
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| We can combine these equations to obtain
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| : <math>\frac{\partial^2H_z}{\partial{x}^2}+\frac{\partial^2{H_z}}{\partial{y}^2} = \mu\varepsilon\frac{\partial^2{H_z}}{\partial{t}^2}</math>
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| The figure on the right presents a structure, referred to as a ''series node''. It describes a block of space dimensions <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math> and consists of four ports. <math>L'</math> and <math>C'</math> are the distributed inductance and capacitance of the transmission lines. It is possible to show that a series node is equivalent to a TE-wave, more precisely the mesh current ''I'', the ''x''-direction voltages (ports 1 and 3) and the ''y''-direction voltages (ports 2 and 4) may be related to the field components <math>H_z</math>, <math>E_x</math> and <math>E_y</math>. If the voltages on the ports are considered, <math>L_x = L_y</math>, and the polarity from figure holds, than the following is valid
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| : <math>-V_1+V_2+V_3-V_4 = 2L'\,\Delta l\frac{\partial{I}}{\partial{t}}</math>
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| where <math>\Delta x = \Delta y = \Delta l</math>. | |
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| : <math>\left(V_3 - V_1\right)-\left(V_4-V_2\right) = 2L'\,\Delta l\frac{\partial I}{\partial t}</math>
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| : <math>\left[E_x(y+\Delta y)-E_x(y)\right]\,\Delta x-[E_y(x+\Delta x)-E_y(x)]\Delta y = 2L'\,\Delta l\frac{\partial{I}}{\partial{t}}</math>
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| and dividing both sides by <math>\Delta x \Delta y</math> | |
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| : <math>\frac{E_x(y+\Delta y)-E_x(y)}{\Delta y}-\frac{E_y(x+\Delta x)-E_y(x)}{\Delta x} = 2L'\,\Delta l\frac{\partial{I}}{\partial{t}}\frac{1}{\Delta x \, \Delta y}</math>
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| Since <math>\Delta x = \Delta y = \Delta z = \Delta l</math> and substituting <math>I = H_z \,\Delta z</math> gives
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| : <math>\frac{\Delta E_x}{\Delta y} - \frac{\Delta E_y}{\Delta x} = 2L'\frac{\partial H_z}{\partial t}</math>
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| This reduces to the Maxwell's equation when <math>\Delta l \rightarrow 0</math>.
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| Similarly, using the conditions across the capacitor on ports 1 and 4, it can be shown that the corresponding to the other two Maxwell equations are the following:
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| : <math>\frac{\partial{H_z}}{\partial{y}} = C'\frac{\partial{E_x}}{\partial{t}}</math>
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| : <math>-\frac{\partial{H_z}}{\partial{x}} = C'\frac{\partial{E_y}}{\partial{t}}</math>
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| Having these results it is possible to compute the scattering matrix of a shunt node. The incident voltage pulse on port 1 at time-step ''k'' is denoted as <math>_kV^i_1</math>. Replacing the four line segments from figure with their [[Thevenin equivalent]] it is possible to show that the following equation for the reflected voltage pulse holds:
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| : <math>_kV^r_1 = 0.5\left(_kV^i_1 + _kV^i_2 + _kV^i_3 - _kV^i_4\right)</math>
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| If all incident waves are summarised in one vector as well as all reflected waves, this equation may be written down for all ports in matrix form:
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| : <math>_k\mathbf{V}^r=\mathbf{S}_k\mathbf{V}^i</math>
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| where <math>_k\mathbf{V}^i</math> and <math>_k\mathbf{V}^r</math> are the incident and the reflected pulse amplitude vectors. | |
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| For a series node the scattering matrix '''S''' has the following form
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| : <math>
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| \mathbf{S}=\frac12\left[
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| \begin{array}{cccc}
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| 1& 1& 1& -1\\
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| 1& 1& -1& 1\\
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| 1& -1& 1& 1\\
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| -1& 1& 1& 1
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| \end{array}
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| \right]
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| </math>
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| === Connection between TLM nodes ===
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| [[File:2DTLMmes.png|272px|thumb|right|A 2D series TLM node]]
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| In order to describe the connection between adjacent nodes the mesh of series nodes look at the figure on the right. As the incident pulse in timestep ''k+1'' on a node is the scattered pulse from an adjacent node in timestep ''k'' the following connection equations are derived:
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| : <math>_{k+1}V^i_1(x,y)=_kV^r_3(x,y-1)</math>
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| : <math>_{k+1}V^i_2(x,y)=_kV^r_4(x-1,y)</math>
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| : <math>_{k+1}V^i_3(x,y)=_kV^r_1(x,y+1)</math>
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| : <math>_{k+1}V^i_4(x,y)=_kV^r_2(x+1,y)</math>
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| By modifying the scattering matrix <math>\textbf{S}</math> inhomogeneous and lossy materials can be modelled. By adjusting the connection equations it is possible to simulate different boundaries.
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| === The shunt TLM node ===
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| Apart from the series node, described above there is also the ''shunt TLM node'', which represents a TM-mode field distribution. The only non-zero components of such wave are <math>H_x</math>, <math>H_y</math>, and <math>E_z</math>. With similar considerations as for the series node the scattering matrix of the shunt node may be derived.
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| == 3D TLM models ==
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| [[File:SymmetricCondensedNode.png|thumb|320px|right|A 3D symmetric condensed node]]
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| Most problems in electromagnetics require a three-dimensional computing. As we have structures, that describe TE and TM-field distributions, intuitively it seem possible to provide a combination of shunt and series nodes, which will provide a full description of the electromagnetic field. Such attempts have been made, but they proved not very useful because of the complexity of the resulting structures. Using the normal analogy, presented above, leads to calculation of the different field components at physically separated points. This causes difficulties in simple and efficient boundary definition. A solution to these problems was provided by Johns in 1987, when he proposed the structure, known as the '''symmetrical condensed node''' (SCN), presented in the figure. It consists of 12 ports, because two field polarisations are to be assigned to each of the 6 sides of a mesh cell.
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| The topology of the SCN can not be analysed using Thevenin equivalent circuits. More general energy and charge conservation principles are to be used.
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| The electric and the magnetic fields on the sides of the SCN node number ''(l,m,n)'' at time instant ''k'' may be summarised in 12-dimensional vectors
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| : <math>_k\mathbf{E}_{l,m,n}=_k\left[E_1,E_2,\ldots,E_{11},E_{12}\right]^T_{l,m,n}</math>
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| : <math>_k\mathbf{H}_{l,m,n}=_k\left[H_1,H_2,\ldots,H_{11},H_{12}\right]^T_{l,m,n}</math>
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| They can be linked with the incident and scattered amplitude vectors via
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| : <math>_k\mathbf{a}_{l,m,n} = \frac{1}{2\sqrt{Z_F}}{_k\mathbf{E}}_{l,m,n}+\frac{\sqrt{Z_F}}{2}{_k\mathbf{H}}_{l,m,n}</math>
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| : <math>_k\mathbf{b}_{l,m,n}=\frac{1}{2\sqrt{Z_F}}{_k\mathbf{E}}_{l,m,n}-\frac{\sqrt{Z_F}}{2}{_k\mathbf{H}}_{l,m,n}</math>
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| where <math>Z_F = \sqrt{\frac{\mu}{\varepsilon}}</math> is the field impedance, <math>_k\mathbf{a}_{l,m,n}</math> is the vector of the amplitudes of the incident waves to the node, and <math>_k\mathbf{b}_{l,m,n}</math> is the vector of the scattered amplitudes. The relation between the incident and scattered waves is given with the matrix equation
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| : <math>_k\mathbf{b}_{l,m,n} = \mathbf{S}_k\mathbf{a}_{l,m,n}</math>
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| The scattering matrix '''S''' may be calculated. For the symmetrical condensed node with ports defined as in the figure the following result is obtained
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| : <math>\mathbf{S} = \left[
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| \begin{array}{ccc}
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| 0& \mathbf{S}_0& \mathbf{S}^T_0\\
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| \mathbf{S}^T_0& 0& \mathbf{S}_0\\
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| \mathbf{S}_0& \mathbf{S}^T_0& 0
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| \end{array}\right]</math>
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| where the following matrix was used
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| : <math>\mathbf{S} = \frac{1}{2}\left[
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| \begin{array}{cccc}
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| 0& 0& 1& -1\\
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| 0& 0& -1& 1\\
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| 1& 1& 0& 0\\
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| 1& 1& 0& 0
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| \end{array}\right]</math>
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| The connection between different SCNs is done in the same manner as for the 2D nodes.
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| == References ==
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| <references/>
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| * C. Christopoulos, ''The Transmission Line Modeling Method: TLM'', Piscataway, NY, IEEE Press, 1995. ISBN 978-0-19-856533-8
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| * Russer, P., Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering, Second edition, Artec House, Boston, 2006, ISBN 978-1-58053-907-4
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| * P. B. Johns and M.O'Brien. "Use of the transmission line modelling (t.l.m) method to solve nonlinear lumped networks", The Radio Electron and Engineer. 1980.
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| * J. L. Herring, Developments in the Transmission-Line Modelling Method for Electromagnetic Compatibility Studies, [http://www.nottingham.ac.uk/ggiemr/publications/JLH_thesis.htm PhD thesis], University of Nottingham, 1993.
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| [[Category:Numerical differential equations]]
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| [[Category:Computational science]]
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| [[Category:Computational physics]]
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| [[Category:Electromagnetism]]
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| [[Category:Electrodynamics]]
| |
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