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<h1 class="title">B.Sc - Electric Drives - Exercise Python Solutions</h1>
<div id="table-of-contents" role="doc-toc">
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#org972b950">1. Programming Preamble</a>
<ul>
<li><a href="#org3ab32b4">1.1. Import Necessary Modules</a></li>
<li><a href="#orgaa20bb0">1.2. Define Common Functions</a></li>
</ul>
</li>
<li><a href="#org6033e76">2. Magnetic Circuits and Materials</a></li>
<li><a href="#org22d6e14">3. Transformers</a></li>
<li><a href="#org44f654d">4. Electromechanical Energy Converion Principles</a></li>
<li><a href="#org4b10002">5. Introduction to Rotating Machines</a></li>
<li><a href="#org55f5bf2">6. Synchronous Drives</a></li>
<li><a href="#org0e20416">7. Polyphase Induction Drives</a>
<ul>
<li><a href="#org45e6fb0">7.1. Analysis of Poly-phase Equivalent Circuit</a></li>
</ul>
</li>
<li><a href="#orgb03a627">8. Direct Current Machines</a>
<ul>
<li><a href="#org831d1bf">8.1. Motor Analysis</a></li>
<li><a href="#org5dce1cc">8.2. Magnetic Circuit Analysis</a></li>
<li><a href="#org438247a">8.3. Steady-state Analysis</a></li>
<li><a href="#orgb595242">8.4. Motor Analysis</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org972b950" class="outline-2">
<h2 id="org972b950"><span class="section-number-2">1.</span> Programming Preamble</h2>
<div class="outline-text-2" id="text-1">
<p>
Before diving into electrical machines and the electromagnetic energy conversion,
we need to do some python preambling of importing the necessary modules and definining
frequently used functions.
</p>
</div>
<div id="outline-container-org3ab32b4" class="outline-3">
<h3 id="org3ab32b4"><span class="section-number-3">1.1.</span> Import Necessary Modules</h3>
<div class="outline-text-3" id="text-1-1">
<p>
As with all python programming, or programming in general, we need some modules to
borrow the necessary functionality and not to <font color=#ef9f76>reinvent the wheel</font>.
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-keyword">import</span> numpy <span class="org-keyword">as</span> np <span class="org-comment-delimiter"># </span><span class="org-comment">For all mathematical operations</span>
<span class="org-keyword">import</span> math <span class="org-comment-delimiter"># </span><span class="org-comment">Additional math functions from the core </span>
<span class="org-keyword">from</span> units <span class="org-keyword">import</span> unit <span class="org-comment-delimiter"># </span><span class="org-comment">Access unit operations and manipulations</span>
<span class="org-keyword">import</span> sympy <span class="org-keyword">as</span> sy <span class="org-comment-delimiter"># </span><span class="org-comment">For doing symbolic calculations</span>
<span class="org-keyword">from</span> sympy.plotting <span class="org-keyword">import</span> plot <span class="org-comment-delimiter"># </span><span class="org-comment">For plotting symbolic plotting</span>
<span class="org-keyword">import</span> matplotlib <span class="org-comment-delimiter"># </span><span class="org-comment">imports plotting functions for non-symbolic functions</span>
<span class="org-keyword">import</span> matplotlib.pyplot <span class="org-keyword">as</span> plt <span class="org-comment-delimiter"># </span><span class="org-comment">Access axis and fig properties</span>
<span class="org-keyword">from</span> prettytable <span class="org-keyword">import</span> PrettyTable <span class="org-comment-delimiter"># </span><span class="org-comment">To pretty print results</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgaa20bb0" class="outline-3">
<h3 id="orgaa20bb0"><span class="section-number-3">1.2.</span> Define Common Functions</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Some functions are used more than once, and to make sure debugging is kept at
a minimum, they are defined below.
</p>
</div>
<ol class="org-ol">
<li><a id="org6ac11d4"></a><font color=#e5c890>Complex-to-Polar Conversion</font><br />
<div class="outline-text-6" id="text-1-2-0-0-1">
<p>
As we are working with AC circuits, we will work closely with phasor values, which is
described below. This function merely takes a complex value in cartesian coordinates and
converts it into a polar coordinate value.
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-keyword">def</span> <span class="org-function-name">complex2polar</span><span class="org-rainbow-delimiters-depth-1">(</span>Z<span class="org-rainbow-delimiters-depth-1">)</span>:
<span class="org-doc">"""</span>
<span class="org-doc"> Produces the phasor and the angle (in radians).</span>
<span class="org-doc"> """</span>
<span class="org-variable-name">X</span> <span class="org-operator">=</span> Z.real <span class="org-comment-delimiter"># </span><span class="org-comment">get the real value of the impedance</span>
<span class="org-variable-name">Y</span> <span class="org-operator">=</span> Z.imag <span class="org-comment-delimiter"># </span><span class="org-comment">get the imaginary value of the impedance</span>
<span class="org-variable-name">R</span> <span class="org-operator">=</span> math.hypot<span class="org-rainbow-delimiters-depth-1">(</span>X,Y<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">get the radius of the phasor</span>
<span class="org-variable-name">theta</span> <span class="org-operator">=</span> math.atan2<span class="org-rainbow-delimiters-depth-1">(</span>Y,X<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">get the angle of the phasor</span>
<span class="org-keyword">return</span> <span class="org-rainbow-delimiters-depth-1">[</span>R, theta <span class="org-operator">*</span> 180 <span class="org-operator">/</span> np.pi<span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-comment-delimiter"># </span><span class="org-comment">return the phasor and angle as an array</span>
</pre>
</div>
</div>
</li>
<li><a id="org52076c4"></a><font color=#e5c890>Magnitude of a Complex Number</font><br />
<div class="outline-text-6" id="text-1-2-0-0-2">
<p>
Calculates the magnitude of a complex number. The input is of cartesian coordinate system.
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-keyword">def</span> <span class="org-function-name">magnitude</span><span class="org-rainbow-delimiters-depth-1">(</span>Z<span class="org-rainbow-delimiters-depth-1">)</span>:
<span class="org-doc">"""</span>
<span class="org-doc"> Calculates the magnitude of a complex value.</span>
<span class="org-doc"> """</span>
<span class="org-keyword">return</span> np.sqrt<span class="org-rainbow-delimiters-depth-1">(</span>Z.real<span class="org-operator">**</span>2 <span class="org-operator">+</span> Z.imag<span class="org-operator">**</span>2<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">Returns magnitude</span>
</pre>
</div>
</div>
</li>
<li><a id="org332ebec"></a><font color=#e5c890>Polar-to-Complex Conversion</font><br />
<div class="outline-text-6" id="text-1-2-0-0-3">
<p>
A simple function to calculate the complex number based on its polar coordinate values.
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-keyword">def</span> <span class="org-function-name">polar2complex</span><span class="org-rainbow-delimiters-depth-1">(</span>angles, radii<span class="org-rainbow-delimiters-depth-1">)</span>:
<span class="org-doc">"""</span>
<span class="org-doc"> Calculate the complex number based on the polar coordinate values</span>
<span class="org-doc"> """</span>
<span class="org-keyword">return</span> radii <span class="org-operator">*</span> np.exp<span class="org-rainbow-delimiters-depth-1">(</span>1j<span class="org-operator">*</span>angles<span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
</div>
</li>
<li><a id="org2f2294b"></a><font color=#e5c890>Complex Rounder</font><br />
<div class="outline-text-6" id="text-1-2-0-0-4">
<p>
Rounds complex numbers to some arbitrary presicion.
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-keyword">def</span> <span class="org-function-name">cRound</span><span class="org-rainbow-delimiters-depth-1">(</span>Z, presicion<span class="org-rainbow-delimiters-depth-1">)</span>:
<span class="org-doc">"""</span>
<span class="org-doc"> Rounds complex numbers to an arbitrary precision.</span>
<span class="org-doc"> """</span>
<span class="org-keyword">return</span> <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-1">(</span>Z.real, presicion<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">+</span> <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-1">(</span>Z.imag, presicion<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">*</span> 1j
</pre>
</div>
</div>
</li>
<li><a id="org8b7e58a"></a><font color=#e5c890>MACRO: Matplotlib Boilerplate</font><br />
<div class="outline-text-6" id="text-1-2-0-0-5">
<p>
It is never a good practice to write <font color=#ef9f76>the same code more than once</font>, and this
includes all the redundant matplotlib config commands, which are presented here.
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-keyword">def</span> <span class="org-function-name">mplBoilerplate</span><span class="org-rainbow-delimiters-depth-1">()</span>:
matplotlib.use<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Agg'</span><span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">Use this to write to a file</span>
<span class="org-variable-name">fig</span>, <span class="org-variable-name">ax</span> <span class="org-operator">=</span> plt.subplots<span class="org-rainbow-delimiters-depth-1">(</span>1,1<span class="org-rainbow-delimiters-depth-1">)</span>
ax.spines<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">'top'</span><span class="org-rainbow-delimiters-depth-1">]</span>.set_visible<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-constant">False</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.spines<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">'right'</span><span class="org-rainbow-delimiters-depth-1">]</span>.set_visible<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-constant">False</span><span class="org-rainbow-delimiters-depth-1">)</span>
fig.patch.set_facecolor<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-3"><span class="custom-4">#363a4f</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.set_facecolor<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-3"><span class="custom-4">#363a4f</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.xaxis.label.set_color<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.yaxis.label.set_color<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-keyword">for</span> axis <span class="org-keyword">in</span> <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">'top'</span>,<span class="org-string">'bottom'</span>,<span class="org-string">'left'</span>,<span class="org-string">'right'</span><span class="org-rainbow-delimiters-depth-1">]</span>:
ax.spines<span class="org-rainbow-delimiters-depth-1">[</span>axis<span class="org-rainbow-delimiters-depth-1">]</span>.set_linewidth<span class="org-rainbow-delimiters-depth-1">(</span>3<span class="org-rainbow-delimiters-depth-1">)</span>
ax.tick_params<span class="org-rainbow-delimiters-depth-1">(</span>width<span class="org-operator">=</span>3<span class="org-rainbow-delimiters-depth-1">)</span>
ax.spines<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">'bottom'</span><span class="org-rainbow-delimiters-depth-1">]</span>.set_color<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.spines<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">'top'</span><span class="org-rainbow-delimiters-depth-1">]</span>.set_color<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.spines<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">'left'</span><span class="org-rainbow-delimiters-depth-1">]</span>.set_color<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.spines<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">'right'</span><span class="org-rainbow-delimiters-depth-1">]</span>.set_color<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
ax.tick_params<span class="org-rainbow-delimiters-depth-1">(</span>axis<span class="org-operator">=</span><span class="org-string">'both'</span>, colors<span class="org-operator">=</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
plt.grid<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-constant">True</span>, <span class="org-string">"both"</span>, color<span class="org-operator">=</span><span class="org-string">'</span><span class="custom"><span class="custom-4">#414559</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
</div>
</li>
<li><a id="orgebfdcbb"></a><font color=#e5c890>MACRO: Results Print Header</font><br />
<div class="outline-text-6" id="text-1-2-0-0-6">
<p>
A MACRO which stores printing headers. This is used predominantly
used in output of the solutions.
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-keyword">def</span> <span class="org-function-name">headerPrint</span><span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>:
<span class="org-doc">"""</span>
<span class="org-doc"> A MACRO for printing results.</span>
<span class="org-doc"> """</span>
Results.<span class="org-variable-name">align</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">"Definition"</span><span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-operator">=</span> <span class="org-string">"r"</span>
Results.<span class="org-variable-name">align</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">"Symbol"</span><span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-operator">=</span> <span class="org-string">"l"</span>
Results.<span class="org-variable-name">align</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">"Value"</span><span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-operator">=</span> <span class="org-string">"l"</span>
Results.<span class="org-variable-name">align</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-string">"Unit"</span><span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-operator">=</span> <span class="org-string">"l"</span>
<span class="org-keyword">return</span> Results
</pre>
</div>
</div>
</li>
</ol>
</div>
</div>
<div id="outline-container-org6033e76" class="outline-2">
<h2 id="org6033e76"><span class="section-number-2">2.</span> Magnetic Circuits and Materials</h2>
<div class="outline-text-2" id="text-2">
</div>
<ol class="org-ol">
<li><a id="org8fd7fcb"></a><font color=#e78284>QUESTION - A SIMPLE MAGNETIC CIRCUIT</font><br />
<div class="outline-text-5" id="text-2-0-0-1">
<p>
The magnetic circuit shown in the figure above, having dimensions of \(A_c = A_g = 9\) cm<sup>2</sup>,
\(g = 0.050\) cm, \(l_c\) = 30 cm, and \(N\) = 500 tums. Assume the value \(\mu_r\) = 70,000 (i.e., Mu-metal)
for core material.
</p>
<ol class="org-ol">
<li>Find the \(\mathcal{R}_c\) and \(\mathcal{R}_g\).</li>
<li>For the condition that the magnetic circuit iso perating with \(B_c\) = 1.0 T, find the flux \(\phi\) and the current $i$h.</li>
</ol>
<p>
<font color=#8caaee>NOTE: </font> The Reluctance (\(\mathcal{R}\)), or magnetic resistance, used in the analysis of magnetic circuits.
It is defined as the ratio of magnetomotive force (mmf) to magnetic flux (i.e., \(\mathcal{F} / \Phi\)).
It represents the opposition to magnetic flux, and depends on the geometry and composition of an object. An analogy would be
<font color=#ef9f76>resistance in an electrical circuit</font>.
</p>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
<div class="org-src-container">
<pre class="src src-python">
<span class="org-variable-name">mu0</span> <span class="org-operator">=</span> 4 <span class="org-operator">*</span> np.pi <span class="org-operator">*</span> 1e<span class="org-operator">-</span>7 <span class="org-comment-delimiter"># </span><span class="org-comment">(H/m) permeability of free space</span>
<span class="org-variable-name">lc</span> <span class="org-operator">=</span> 0.3 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) length</span>
<span class="org-variable-name">Ac</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">Ag</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">mur</span> <span class="org-operator">=</span> 7e5 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) relative permeability</span>
<span class="org-variable-name">g</span> <span class="org-operator">=</span> 5e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) air gap length</span>
<span class="org-variable-name">N</span> <span class="org-operator">=</span> 500 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) number of turns</span>
<span class="org-variable-name">Bc</span> <span class="org-operator">=</span> 1 <span class="org-comment-delimiter"># </span><span class="org-comment">(T) Core magnetic field density</span>
<span class="org-variable-name">Rel_c</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>lc<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mur <span class="org-operator">*</span> mu0 <span class="org-operator">*</span> Ac<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) core reluctance</span>
<span class="org-variable-name">Rel_g</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mu0 <span class="org-operator">*</span> Ag<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) air-gap reluctance</span>
<span class="org-variable-name">Phi</span> <span class="org-operator">=</span> Bc <span class="org-operator">*</span> Ac <span class="org-comment-delimiter"># </span><span class="org-comment">(Wb) Magnetic core flux</span>
<span class="org-variable-name">MMF</span> <span class="org-operator">=</span> Phi <span class="org-operator">*</span> <span class="org-rainbow-delimiters-depth-1">(</span>Rel_c <span class="org-operator">+</span> Rel_g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns) Magnetomotive force</span>
<span class="org-variable-name">i</span> <span class="org-operator">=</span> MMF <span class="org-operator">/</span> N <span class="org-comment-delimiter"># </span><span class="org-comment">(A) Current</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">-- Printing Results</span>
<span class="org-variable-name">Results</span> <span class="org-operator">=</span> PrettyTable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Definition"</span>, <span class="org-string">"Symbol"</span>, <span class="org-string">"Value"</span>, <span class="org-string">"Unit"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
headerPrint<span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Core Reluctance"</span>, <span class="org-string">"Rel_c"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>Rel_c, 3<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-string">"Ampere-Turns/Weber"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Air-gap Reluctance"</span>, <span class="org-string">"Rel_g"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>Rel_g, 3<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-string">"Ampere-Turns/Weber"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"***********************"</span>, <span class="org-string">"************"</span>, <span class="org-string">"****"</span>, <span class="org-string">"****"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Core Flux"</span>, <span class="org-string">"Phi"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>Phi, 3<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-string">"Weber"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Magneto-motive Force"</span>, <span class="org-string">"F or MMF"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>MMF, 3<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-string">"Ampere-Turns"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Current"</span>, <span class="org-string">"i"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>i, 3<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-string">"A"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">--</span>
<span class="org-builtin">print</span><span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
<pre class="example" id="orga31ab53">
+-------------------------+--------------+------------+--------------------+
| Definition | Symbol | Value | Unit |
+-------------------------+--------------+------------+--------------------+
| Core Reluctance | Rel_c | 378.94 | Ampere-Turns/Weber |
| Air-gap Reluctance | Rel_g | 442097.064 | Ampere-Turns/Weber |
| *********************** | ************ | **** | **** |
| Core Flux | Phi | 0.001 | Weber |
| Magneto-motive Force | F or MMF | 398.228 | Ampere-Turns |
| Current | i | 0.796 | A |
+-------------------------+--------------+------------+--------------------+
</pre>
</div>
</li>
<li><a id="orgd853479"></a><font color=#e78284>QUESTION - A SIMPLE MAGNETIC CIRCUIT WITH DIFFERENT TURNS</font><br />
<div class="outline-text-5" id="text-2-0-0-2">
<p>
Find the flux (\(\Phi\)) and current for the previously given question if:
</p>
<ol class="org-ol">
<li>the number of turns is doubled to \(N\) = 1000 turns while the circuit dimensions remain the same and</li>
<li>the number of turns is equal to \(N\) = 500 and the gap is reduced to \(g\) = 0.040 cm.</li>
<li>If the number of turns is doubled to \(N\) = 1000 turns:</li>
</ol>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
<div class="org-src-container">
<pre class="src src-python">
<span class="org-variable-name">mu0</span> <span class="org-operator">=</span> 4 <span class="org-operator">*</span> np.pi <span class="org-operator">*</span> 1e<span class="org-operator">-</span>7 <span class="org-comment-delimiter"># </span><span class="org-comment">(H/m) permeability of free space</span>
<span class="org-variable-name">lc</span> <span class="org-operator">=</span> 0.3 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) length</span>
<span class="org-variable-name">Ac</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">Ag</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">mur</span> <span class="org-operator">=</span> 7e5 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) relative permeability</span>
<span class="org-variable-name">g</span> <span class="org-operator">=</span> 5e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) air gap length</span>
<span class="org-variable-name">N</span> <span class="org-operator">=</span> 1000 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) number of turns</span>
<span class="org-variable-name">Bc</span> <span class="org-operator">=</span> 1 <span class="org-comment-delimiter"># </span><span class="org-comment">(T) Core magnetic field density</span>
<span class="org-variable-name">Rel_c</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>lc<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mur <span class="org-operator">*</span> mu0 <span class="org-operator">*</span> Ac<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) core reluctance</span>
<span class="org-variable-name">Rel_g</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mu0 <span class="org-operator">*</span> Ag<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) air-gap reluctance</span>
<span class="org-variable-name">Phi</span> <span class="org-operator">=</span> Bc <span class="org-operator">*</span> Ac <span class="org-comment-delimiter"># </span><span class="org-comment">(Wb) Magnetic core flux</span>
<span class="org-variable-name">MMF</span> <span class="org-operator">=</span> Phi <span class="org-operator">*</span> <span class="org-rainbow-delimiters-depth-1">(</span>Rel_c <span class="org-operator">+</span> Rel_g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns) Magnetomotive force</span>
<span class="org-variable-name">i</span> <span class="org-operator">=</span> MMF <span class="org-operator">/</span> N <span class="org-comment-delimiter"># </span><span class="org-comment">(A) Current</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">-- Printing Results</span>
<span class="org-variable-name">Results</span> <span class="org-operator">=</span> PrettyTable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Definition"</span>, <span class="org-string">"Symbol"</span>, <span class="org-string">"Value"</span>, <span class="org-string">"Unit"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
headerPrint<span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Core Flux"</span>, <span class="org-string">"Phi"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>Phi <span class="org-operator">*</span> 1e3, 3<span class="org-rainbow-delimiters-depth-3">)</span> , <span class="org-string">"milli-Weber"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Current"</span>, <span class="org-string">"i"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>i, 3<span class="org-rainbow-delimiters-depth-3">)</span> , <span class="org-string">"A"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">--</span>
<span class="org-builtin">print</span><span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
<pre class="example" id="org42d6268">
+------------+--------+-------+-------------+
| Definition | Symbol | Value | Unit |
+------------+--------+-------+-------------+
| Core Flux | Phi | 0.9 | milli-Weber |
| Current | i | 0.398 | A |
+------------+--------+-------+-------------+
</pre>
<ol class="org-ol">
<li>If the number of turns is equal to \(N\) = 500 and the gap is reduced to \(g\) = 0.040 cm.</li>
</ol>
<div class="org-src-container">
<pre class="src src-python">
<span class="org-variable-name">lc</span> <span class="org-operator">=</span> 0.3 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) length</span>
<span class="org-variable-name">Ac</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">Ag</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">mur</span> <span class="org-operator">=</span> 7e5 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) relative permeability</span>
<span class="org-variable-name">g</span> <span class="org-operator">=</span> 4e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) air gap length</span>
<span class="org-variable-name">N</span> <span class="org-operator">=</span> 500 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) number of turns</span>
<span class="org-variable-name">Bc</span> <span class="org-operator">=</span> 1 <span class="org-comment-delimiter"># </span><span class="org-comment">(T) Core magnetic field density</span>
<span class="org-variable-name">Rel_c</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>lc<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mur <span class="org-operator">*</span> mu0 <span class="org-operator">*</span> Ac<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) core reluctance</span>
<span class="org-variable-name">Rel_g</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mu0 <span class="org-operator">*</span> Ag<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) air-gap reluctance</span>
<span class="org-variable-name">Phi</span> <span class="org-operator">=</span> Bc <span class="org-operator">*</span> Ac <span class="org-comment-delimiter"># </span><span class="org-comment">(Wb) Magnetic core flux</span>
<span class="org-variable-name">MMF</span> <span class="org-operator">=</span> Phi <span class="org-operator">*</span> <span class="org-rainbow-delimiters-depth-1">(</span>Rel_c <span class="org-operator">+</span> Rel_g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns) Magnetomotive force</span>
<span class="org-variable-name">i</span> <span class="org-operator">=</span> MMF <span class="org-operator">/</span> N <span class="org-comment-delimiter"># </span><span class="org-comment">(A) Current</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">-- Printing Results</span>
<span class="org-variable-name">Results</span> <span class="org-operator">=</span> PrettyTable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Definition"</span>, <span class="org-string">"Symbol"</span>, <span class="org-string">"Value"</span>, <span class="org-string">"Unit"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
headerPrint<span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Core Flux"</span>, <span class="org-string">"Phi"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>Phi <span class="org-operator">*</span> 1e3, 3<span class="org-rainbow-delimiters-depth-3">)</span> , <span class="org-string">"milli-Weber"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Current"</span>, <span class="org-string">"i"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>i, 3<span class="org-rainbow-delimiters-depth-3">)</span> , <span class="org-string">"A"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">--</span>
<span class="org-builtin">print</span><span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
<pre class="example" id="org1f6db95">
+------------+--------+-------+-------------+
| Definition | Symbol | Value | Unit |
+------------+--------+-------+-------------+
| Core Flux | Phi | 0.9 | milli-Weber |
| Current | i | 0.637 | A |
+------------+--------+-------+-------------+
</pre>
</div>
</li>
<li><a id="orga39fdaa"></a><font color=#e78284>QUESTION - MAGNETIC CIRCUIT OF A SYNCHRONOUS DRIVE</font><br />
<div class="outline-text-5" id="text-2-0-0-3">
<p>
The magnetic structure of a synchronous drive is shown schematically in the Figure above.
Assuming that rotor and stator iron have infinite permeability (\(\mu \rightarrow \infty\)),
find the air-gap flux band flux density \(B_g\).
</p>
<p>
For this example the given values are \(I\) = 10 A, \(N\) = 1000 turns, \(g\) = 1 cm, and \(A_g\) = 2000 cm<sup>2</sup>.
</p>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
<div class="org-src-container">
<pre class="src src-python">
<span class="org-variable-name">mu0</span> <span class="org-operator">=</span> 4 <span class="org-operator">*</span> np.pi <span class="org-operator">*</span> 1e<span class="org-operator">-</span>7 <span class="org-comment-delimiter"># </span><span class="org-comment">(H/m) permeability of free space</span>
<span class="org-variable-name">N</span> <span class="org-operator">=</span> 1000 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) number of turns</span>
<span class="org-variable-name">I</span> <span class="org-operator">=</span> 10 <span class="org-comment-delimiter"># </span><span class="org-comment">(A) current</span>
<span class="org-variable-name">g</span> <span class="org-operator">=</span> 1e<span class="org-operator">-</span>2 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) air-gap</span>
<span class="org-variable-name">Ag</span> <span class="org-operator">=</span> 0.2 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) air-gap cross-sectional area</span>
<span class="org-variable-name">Phi</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>N <span class="org-operator">*</span> I <span class="org-operator">*</span> mu0 <span class="org-operator">*</span> Ag<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>2 <span class="org-operator">*</span> g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Weber) Air-gap flux</span>
<span class="org-variable-name">Bg</span> <span class="org-operator">=</span> Phi <span class="org-operator">/</span> Ag <span class="org-comment-delimiter"># </span><span class="org-comment">(Tesla) Air-gap magnetic flux density</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">-- Printing Results</span>
<span class="org-variable-name">Results</span> <span class="org-operator">=</span> PrettyTable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Definition"</span>, <span class="org-string">"Symbol"</span>, <span class="org-string">"Value"</span>, <span class="org-string">"Unit"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
headerPrint<span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Air-gap Flux"</span>, <span class="org-string">"Phi"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>Phi <span class="org-operator">*</span> 1e3, 3<span class="org-rainbow-delimiters-depth-3">)</span> , <span class="org-string">"Weber"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
Results.add_row<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-string">"Air-gap Magnetic Flux Density"</span>, <span class="org-string">"Bg"</span>, <span class="org-builtin">round</span><span class="org-rainbow-delimiters-depth-3">(</span>Bg, 3<span class="org-rainbow-delimiters-depth-3">)</span> , <span class="org-string">"Tesla"</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">--</span>
<span class="org-builtin">print</span><span class="org-rainbow-delimiters-depth-1">(</span>Results<span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
<pre class="example" id="org9f44531">
+-------------------------------+--------+---------+-------+
| Definition | Symbol | Value | Unit |
+-------------------------------+--------+---------+-------+
| Air-gap Flux | Phi | 125.664 | Weber |
| Air-gap Magnetic Flux Density | Bg | 0.628 | Tesla |
+-------------------------------+--------+---------+-------+
</pre>
</div>
</li>
<li><a id="orgc99403e"></a><font color=#e78284>QUESTION - PLOTTING INDUCTANCE VS. RELATIVE PERMEABILITY</font><br />
<div class="outline-text-5" id="text-2-0-0-4">
<p>
Using Python, please plot the inductance (\(L\)) of the magnetic circuit of Question 1 and Question 2 as a function of
core permeability (\(\mu_r\)) over the range 100 \(\leq \mu_r \leq\) 100,000.
</p>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
<div class="org-src-container">
<pre class="src src-python">
<span class="org-variable-name">mu0</span> <span class="org-operator">=</span> 4 <span class="org-operator">*</span> np.pi <span class="org-operator">*</span> 1e<span class="org-operator">-</span>7 <span class="org-comment-delimiter"># </span><span class="org-comment">(H/m) permeability of free space</span>
<span class="org-variable-name">lc</span> <span class="org-operator">=</span> 0.3 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) length</span>
<span class="org-variable-name">Ac</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">Ag</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">g</span> <span class="org-operator">=</span> 5e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) air gap length</span>
<span class="org-variable-name">N</span> <span class="org-operator">=</span> 500 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) number of turns</span>
<span class="org-variable-name">Bc</span> <span class="org-operator">=</span> 1 <span class="org-comment-delimiter"># </span><span class="org-comment">(T) Core magnetic field density</span>
<span class="org-variable-name">Rel_g</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mu0 <span class="org-operator">*</span> Ag<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) air-gap reluctance</span>
<span class="org-variable-name">mur</span> <span class="org-operator">=</span> np.zeros<span class="org-rainbow-delimiters-depth-1">(</span>1000<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">Relative Permeability emtpy array for looping</span>
<span class="org-keyword">for</span> i <span class="org-keyword">in</span> <span class="org-builtin">range</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-builtin">len</span><span class="org-rainbow-delimiters-depth-2">(</span>mur<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>:
<span class="org-variable-name">mur</span><span class="org-rainbow-delimiters-depth-1">[</span>i<span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-operator">=</span> 100 <span class="org-operator">+</span> <span class="org-rainbow-delimiters-depth-1">(</span>100000 <span class="org-operator">-</span> 100<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">*</span> <span class="org-rainbow-delimiters-depth-1">(</span>i <span class="org-operator">-</span> 1<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-operator">/</span>1000
<span class="org-variable-name">Rel_c</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>lc<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mur <span class="org-operator">*</span> mu0 <span class="org-operator">*</span> Ac<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) core reluctance</span>
<span class="org-variable-name">R_tot</span> <span class="org-operator">=</span> Rel_c <span class="org-operator">+</span> Rel_g <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/Weber) total reluctance</span>
<span class="org-variable-name">L</span><span class="org-operator">=</span> N<span class="org-operator">**</span>2 <span class="org-operator">/</span> R_tot <span class="org-comment-delimiter"># </span><span class="org-comment">(H) Inductance</span>
mplBoilerplate<span class="org-rainbow-delimiters-depth-1">()</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">-- Plotting</span>
plt.xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Core Relative Permeability ($\mu_r$) [H/m]'</span><span class="org-rainbow-delimiters-depth-1">)</span>
plt.ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Inductance [H]'</span><span class="org-rainbow-delimiters-depth-1">)</span>
plt.xlim<span class="org-rainbow-delimiters-depth-1">(</span>0, 0.8e5<span class="org-rainbow-delimiters-depth-1">)</span>
plt.ylim<span class="org-rainbow-delimiters-depth-1">(</span>0, 1<span class="org-rainbow-delimiters-depth-1">)</span>
plt.title<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">"Core Relative Permeability vs. Inductance Graph"</span>, color<span class="org-operator">=</span><span class="org-string">'</span><span class="custom-1"><span class="custom-2">#c6d0f5</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>
plt.plot<span class="org-rainbow-delimiters-depth-1">(</span>mur, L,<span class="org-string">'</span><span class="custom"><span class="custom-2">#ef9f76</span></span><span class="org-string">'</span>, linewidth<span class="org-operator">=</span>5<span class="org-rainbow-delimiters-depth-1">)</span>
fig.tight_layout<span class="org-rainbow-delimiters-depth-1">()</span>
plt.savefig<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'images/python-matplot-fig.svg'</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-comment-delimiter"># </span><span class="org-comment">--</span>
</pre>
</div>
<div id="org31ba942" class="figure">
<p><img src="images/python-matplot-fig.svg" alt="python-matplot-fig.svg" class="org-svg" />
</p>
</div>
</div>
</li>
<li><a id="org95d6021"></a><font color=#e78284>QUESTION - DC MAGNETISATION CURVE - I</font><br />
<div class="outline-text-5" id="text-2-0-0-5">
<p>
Assume that the core material in Question 1 is M-5 electrical steel, which has the DC magnetisation
curve given in the figure above. Find the current \(I\) required to produce \(B_e\) = 1 T.
</p>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
<div class="org-src-container">
<pre class="src src-python"><span class="org-variable-name">mu0</span> <span class="org-operator">=</span> 4 <span class="org-operator">*</span> np.pi <span class="org-operator">*</span> 1e<span class="org-operator">-</span>7 <span class="org-comment-delimiter"># </span><span class="org-comment">(H/m) permeability of free space</span>
<span class="org-variable-name">lc</span> <span class="org-operator">=</span> 0.3 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) length</span>
<span class="org-variable-name">Ac</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">Ag</span> <span class="org-operator">=</span> 9e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m2) cross-sectional area</span>
<span class="org-variable-name">g</span> <span class="org-operator">=</span> 5e<span class="org-operator">-</span>4 <span class="org-comment-delimiter"># </span><span class="org-comment">(m) air gap length</span>
<span class="org-variable-name">N</span> <span class="org-operator">=</span> 500 <span class="org-comment-delimiter"># </span><span class="org-comment">(-) number of turns</span>
<span class="org-variable-name">Bc</span> <span class="org-operator">=</span> 1 <span class="org-comment-delimiter"># </span><span class="org-comment">(T) Core magnetic field density</span>
<span class="org-variable-name">Hc</span> <span class="org-operator">=</span> 11 <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns/m)</span>
<span class="org-variable-name">Ic</span> <span class="org-operator">=</span> 0.3 <span class="org-comment-delimiter"># </span><span class="org-comment">(I) </span>
<span class="org-variable-name">Fc</span> <span class="org-operator">=</span> Hc <span class="org-operator">*</span> Ic <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns) </span>
<span class="org-variable-name">Fg</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>Bg <span class="org-operator">*</span> g<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> mu0 <span class="org-comment-delimiter"># </span><span class="org-comment">(Ampere-Turns)</span>
<span class="org-variable-name">i</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>Fc <span class="org-operator">+</span> Fg<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">/</span> N <span class="org-comment-delimiter"># </span><span class="org-comment">(A) Current</span>
</pre>
</div>
</div>
</li>
<li><a id="org2c67b50"></a><font color=#e78284>QUESTION - DC MAGNETISATION CURVE - II</font><br />
<div class="outline-text-5" id="text-2-0-0-6">
<p>
Using the previous DC magnetisation curve in the previous question, find the current \(I\) for \(B_c\) = 1.6 T.
</p>
<p>
By what factor does the current have to be increased to result in this factor of 1.6 increase in flux density?
</p>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
</div>
</li>
<li><a id="org50fdf1b"></a><font color=#e78284>QUESTION - MAGNETIC CORE OF A TRANSFORMER</font><br />
<div class="outline-text-5" id="text-2-0-0-7">
<p>
The magnetic core in a transformer is made from laminations of M-5 grain-oriented electrical steel.
The winding is excited with a 60 Hz voltage to produce a flux density in the steel of B = 1.5 \(\sin \omega t\) T,
where \(\omega = 2\pi f\) which is 377 rad/s.
</p>
<p>
The steel occupies 0.94 of the core cross-sectional area. The mass-density of the steel is 7.65 g/cm<sup>3</sup>.
</p>
<p>
Using this information find:
</p>
<ol class="org-ol">
<li>the applied voltage,</li>
<li>the peak current,</li>
<li>the rms exciting current, and</li>
<li>the core loss.</li>
</ol>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
</div>
</li>
<li><a id="orgbaf1a1c"></a><font color=#e78284>QUESTION - A MAGNETIC CIRCUIT OF HIGH PERMEABILITY</font><br />
<div class="outline-text-5" id="text-2-0-0-8">
<p>
A magnetic circuit consists of a core of high permeability (\(\mu \rightarrow \infty\)),
an air gap of length \(g\) = 0.2 cm, and a section of magnetic material of length \(l_m\) = 1.0 cm.
The cross-sectional area of the core and gap is equal to \(A_m\) = \(A_g\) = 4 cm<sup>2</sup>.
Calculate the flux density \(B_s\) in the air gap if the magnetic material is:
</p>
<ol class="org-ol">
<li>Alnico 5,</li>
<li>Electrical steel.</li>
</ol>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
</div>
</li>
<li><a id="org242aa79"></a><font color=#e78284>QUESTION - MINIMUM MAGNET VOLUME</font><br />
<div class="outline-text-5" id="text-2-0-0-9">
<p>
A magnetic circuit ofFig.1.17ismodifiedsothattheair-gapareaisreducedtoAg= 2.0cm2, as shown in Fig. 1.18.
Find the minimum magnet volume required to achieve an air-gap flux density of 0.8 T.
</p>
<p>
<font color=#a6d189>ANSWER:</font> <p>The following part describes the answer to the aforementioned Question. </p>
</p>
</div>
</li>
<li><a id="org8022f6b"></a><font color=#e78284>QUESTION - RINGS OF MAGNETIC MATERIALS</font><br />
<div class="outline-text-5" id="text-2-0-0-10">
<p>
The magnetic circuit consists of rings of magnetic material in a stack of height h.
The rings have inner radius \(R_i\) and outer radius \(R_o\).
Assume that the iron is of infinite permeability (\(\mu \rightarrow \infty\)) and neglect the effects of
magnetic leakage and fringing. For:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">SYMBOL</th>
<th scope="col" class="org-right">VALUE</th>
<th scope="col" class="org-left">UNIT</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(R_i\)</td>
<td class="org-right">3.4</td>
<td class="org-left">cm</td>
</tr>
<tr>
<td class="org-left">\(R_i\)</td>
<td class="org-right">4.0</td>
<td class="org-left">cm</td>
</tr>
<tr>
<td class="org-left">\(h\)</td>
<td class="org-right">2</td>
<td class="org-left">cm</td>
</tr>
<tr>
<td class="org-left">\(g\)</td>
<td class="org-right">0.2</td>
<td class="org-left">cm</td>
</tr>
</tbody>
</table>
<p>
Using this information, please calculate:
</p>
<ol class="org-ol">
<li>the mean core length \(l_c\) and the core cross-sectional area \(A_c\).</li>
<li>the reluctance of the core \(\mathcal{R}_c\) and that of the gap \(\mathcal{R}_g\).</li>
</ol>
<p>
For \(N\) = 65 turns, calculate:
</p>
<ol class="org-ol">
<li>the inductance \(L\).</li>
<li>current \(i\) required to operate at an air-gap flux density of \(B_g\) = 1.35T.</li>
<li>the corresponding flux linkages \(\lambda\) of the coil.</li>
</ol>
<div class="org-src-container">
<pre class="src src-python"><span class="org-comment-delimiter"># </span><span class="org-comment">Calculation of part 1</span>
<span class="org-variable-name">mu0</span> <span class="org-operator">=</span> 4 <span class="org-operator">*</span> np.pi <span class="org-operator">*</span> 1e<span class="org-operator">-</span>7 <span class="org-comment-delimiter"># </span><span class="org-comment">(H/m) permeability of free space</span>
<span class="org-variable-name">Ri</span> <span class="org-operator">=</span> 3.4e<span class="org-operator">-</span>2
<span class="org-variable-name">Ro</span> <span class="org-operator">=</span> 4.0e<span class="org-operator">-</span>2
<span class="org-variable-name">h</span> <span class="org-operator">=</span> 2.0e<span class="org-operator">-</span>2
<span class="org-variable-name">g</span> <span class="org-operator">=</span> 0.2e<span class="org-operator">-</span>2
<span class="org-variable-name">N</span> <span class="org-operator">=</span> 65
<span class="org-variable-name">lc</span> <span class="org-operator">=</span> 2 <span class="org-operator">*</span> np.pi <span class="org-operator">*</span> <span class="org-rainbow-delimiters-depth-1">(</span>Ro <span class="org-operator">-</span> Ri<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">-</span> g
<span class="org-builtin">print</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">"Mean core length is:"</span>, lc, <span class="org-string">"m"</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-variable-name">Ac</span> <span class="org-operator">=</span> <span class="org-rainbow-delimiters-depth-1">(</span>Ro <span class="org-operator">-</span> Ri<span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-operator">*</span> h
<span class="org-variable-name">Rel_g</span> <span class="org-operator">=</span> g<span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>mu0 <span class="org-operator">*</span> Ac<span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-variable-name">Rel_c</span> <span class="org-operator">=</span> 0
<span class="org-variable-name">L</span> <span class="org-operator">=</span> N<span class="org-operator">**</span>2 <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-1">(</span>Rel_c <span class="org-operator">+</span> Rel_g<span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-variable-name">I</span> <span class="org-operator">=</span>
</pre>
</div>
</div>
</li>
</ol>
</div>
<div id="outline-container-org22d6e14" class="outline-2">
<h2 id="org22d6e14"><span class="section-number-2">3.</span> Transformers</h2>
<div class="outline-text-2" id="text-3">
</div>
<ol class="org-ol">
<li><a id="org197cca6"></a><font color=#e78284>QUESTION - </font><br />
<div class="outline-text-5" id="text-3-0-0-1">
<div class="org-src-container">
<pre class="src src-python">
<span class="org-comment-delimiter"># </span><span class="org-comment">Define Given Parameters</span>
<span class="org-variable-name">X_2r</span> <span class="org-operator">=</span> 54.3e<span class="org-operator">-</span>6
<span class="org-variable-name">R_2r</span> <span class="org-operator">=</span> 9.6e<span class="org-operator">-</span>6
<span class="org-variable-name">X_m</span> <span class="org-operator">=</span> 17.7e<span class="org-operator">-</span>3
<span class="org-variable-name">N_1</span> <span class="org-operator">=</span> 5
<span class="org-variable-name">N_2</span> <span class="org-operator">=</span> 800
<span class="org-variable-name">N</span> <span class="org-operator">=</span> N_1 <span class="org-operator">/</span> N_2
<span class="org-comment-delimiter"># </span><span class="org-comment">Load Impedanceq</span>
<span class="org-variable-name">R_b</span> <span class="org-operator">=</span> 2.5
<span class="org-variable-name">X_b</span> <span class="org-operator">=</span> 0
<span class="org-variable-name">Z_br</span> <span class="org-operator">=</span> N<span class="org-operator">**</span>2 <span class="org-operator">*</span> <span class="org-rainbow-delimiters-depth-1">(</span>R_b <span class="org-operator">+</span> 1j <span class="org-operator">*</span> X_b<span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-variable-name">I_1</span> <span class="org-operator">=</span> 800
<span class="org-variable-name">I_2</span> <span class="org-operator">=</span> phasor<span class="org-rainbow-delimiters-depth-1">(</span>I_1 <span class="org-operator">*</span> N <span class="org-operator">*</span> 1j <span class="org-operator">*</span> X_m <span class="org-operator">/</span> <span class="org-rainbow-delimiters-depth-2">(</span>Z_br <span class="org-operator">+</span> R_2r <span class="org-operator">+</span> 1j <span class="org-operator">*</span> <span class="org-rainbow-delimiters-depth-3">(</span>X_2r <span class="org-operator">+</span> X_m<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
<p>
p.107
</p>
</div>
</li>
<li><a id="org1b8b4e1"></a><font color=#e78284>QUESTION - </font><br />
<div class="outline-text-5" id="text-3-0-0-2">
<div class="org-src-container">
<pre class="src src-python">
<span class="org-comment-delimiter"># </span><span class="org-comment">Given Parameters</span>
<span class="org-variable-name">R1</span> <span class="org-operator">=</span> 128
<span class="org-variable-name">X1</span> <span class="org-operator">=</span> 143
<span class="org-variable-name">Xm</span> <span class="org-operator">=</span> 163e3