CCT Analysis Methods

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CCT Analysis Methods

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<ul><li><p>Analysis Methods Overview</p><p> Solving Linear Equations</p><p> Nodal Analysis</p><p> Supernodes (Nodal Analysis with Voltage Sources)</p><p> Mesh Analysis</p><p> Supermeshes (Mesh Analysis with Current Sources)</p><p>This is a very important chapter.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 1</p></li><li><p>Review of Basic Concepts: Current</p><p>i4 i5i3i2i1</p><p> What goes in, has to come out</p><p> Kirchhoffs current law</p><p> Similar to conservation of mass</p><p> Conservation of electrons</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 2</p></li><li><p>Review of Basic Concepts: Voltage</p><p>10 V-</p><p>+</p><p>-</p><p>++ - + -</p><p>2 k2 k</p><p>5 k 7 kv1 v2</p><p>v3 v4</p><p> The voltage drop from one node to another is the same, nomatter what path is chosen</p><p> Kirchhoffs voltage law</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 3</p></li><li><p>Resistors in Parallel with Voltage Sources</p><p>CircuitRVs vo-</p><p>+</p><p>CircuitVs vo-</p><p>+</p><p> What is vo in each case?</p><p> What effect does the resistor have on the current pumped into thecircuit?</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 4</p></li><li><p>Resistors in Series with Current Sources</p><p>CircuitIs CircuitIs</p><p>Rio</p><p>io</p><p> What is io in each case?</p><p> What effect does the resistor have on the voltage seen by thecircuit?</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 5</p></li><li><p>Solving Linear Equations</p><p> Much of our circuit analysis will focus on finding a set of linearequations and solving these equations</p><p> Need as many equations as there are unknowns</p><p> Three possible approaches</p><p> Algebra (elimination, substitution, etc.)</p><p> Cramers rule</p><p> Linear algebra</p><p> Last is easiest and least susceptible to errors</p><p> Requires use your scientific calculators</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 6</p></li><li><p>Example 1: Solving Linear Equations</p><p>i1 = i2 + i3</p><p>i4 = i3 + 2m</p><p>10 = (1k)i1 + (5k)i2</p><p>(5k)i2 = (2k)i3 + (10k)i4</p><p>Rewrite so variables are in consistent order on left side and constantsare on the right side</p><p>i1 i2 i3 = 0 i3 + i4 = 2m</p><p>(1k)i1 + (5k)i2 = 10+ (5k)i2 (2k)i3 (10k)i4 = 0</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 7</p></li><li><p>Example 1: Continued (1)</p><p>i1 i2 i3 = 0 i3 + i4 = 2m</p><p>(1k)i1 + (5k)i2 = 10+ (5k)i2 (2k)i3 (10k)i4 = 0</p><p>In Matrix form this becomes</p><p>1 1 1 00 0 1 11k 5k 0 00 5k 2k 10k</p><p>i1</p><p>i2</p><p>i3</p><p>i4</p><p> =</p><p>02m100</p><p>or</p><p>Ai = b</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 8</p></li><li><p>Example 1: Continued (2)</p><p>Ai = b where</p><p>A =</p><p>1 1 1 00 0 1 11k 5k 0 00 +5k 2k 10k</p><p> i =</p><p>i1</p><p>i2</p><p>i3</p><p>i4</p><p> b =</p><p>02m100</p><p> Your calculator should be able to solve this directly</p><p> You should only need to enter A and b</p><p> Your calculator will return a vector i</p><p> Simultaneously solves for all the unknown variables</p><p> Much faster than Cramers rule or brute-force algrebra</p><p> Read the manuals for your calculators</p><p> This will save you time (homework &amp; exams) and reduce errors</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 9</p></li><li><p>Example 1: Continued (3)</p><p>Linear Equations:1 1 1 00 0 1 11k 5k 0 00 5k 2k 10k</p><p>i1</p><p>i2</p><p>i3</p><p>i4</p><p> =</p><p>02m100</p><p>Calculator should return:</p><p>i1</p><p>i2</p><p>i3</p><p>i4</p><p> =</p><p>+0.909+1.8180.909+1.091</p><p> mA</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 10</p></li><li><p>Network Terminology</p><p>Planar Circuit A circuit that can be drawn on a plane with nocrossing branches</p><p>Node Point or portion of a circuit where 2 or more elements arejoined</p><p>Essential Node Point or portion of a circuit where 3 or moreelements are joined</p><p>Branch Path that connects 2 nodes</p><p>Essential Branch Path that connects 2 essential nodes w/o passingthrough an essential node</p><p>Loop Path with last node same as starting node that does not crossitself</p><p>Mesh Loop that does not enclose any other loops</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 11</p></li><li><p>Example 2: Terminology</p><p>20 V 2 A</p><p>R1 R2</p><p>R3 R4 R4</p><p>R6 R7 R8</p><p>35ip</p><p>ip</p><p>Identify the following informationNodes: Essential Nodes:Branches: Essential Branches:EBs with Unknown Current: Meshes:</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 12</p></li><li><p>Example 3: Circuit Analysis The Hard Way</p><p>10 V</p><p>i1 i3</p><p>i2 i42 mA</p><p>1 k 2 k</p><p>5 k 10 k</p><p>Can solve with KCL &amp; KVL. Four unknowns.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 13</p></li><li><p>Nodal Analysis: Introduction</p><p> There is an another way to solve for currents and voltages</p><p> Easier</p><p> More methodical</p><p> Still based on Ohms law, KVL, &amp; KCL</p><p> Nodal analysis is one of two key methods</p><p> Mesh analysis is the other</p><p> We will discuss nodal analysis first</p><p> Based on KCL</p><p> Must understand terminology introduced earlier</p><p> Use to solve for voltages</p><p> All voltages have a common reference point</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 14</p></li><li><p>Nodal Analysis Steps</p><p>1. Identifiy essential nodes</p><p>2. Pick a reference node</p><p>3. Label all other essential nodes</p><p>4. Apply KCL to all labelled nodes</p><p>5. Solve linear equations for all node voltages</p><p>6. Solve for variables of interest</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 15</p></li><li><p>Nodal Analysis: Step 1 Identify Essential Nodes</p><p>10 V 2 mA</p><p>1 k 2 k</p><p>5 k 10 k</p><p> Some essential nodes may include portions of the circuit (pieces ofwire)</p><p> Circle the entire node to prevent errors</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 16</p></li><li><p>Nodal Analysis: Step 2 Pick a Reference</p><p>10 V 2 mA</p><p>1 k 2 k</p><p>5 k 10 k</p><p> Second step is to pick a reference node</p><p> Is often easiest to choose the node that interconnects the mostbranches</p><p> Must be an essential node</p><p> Usually is at bottom of circuit</p><p> Label with the same symbol used for ground</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 17</p></li><li><p>Nodal Analysis: Step 3 Label Other Essential Nodes</p><p>10 V 2 mA</p><p>1 k 2 k</p><p>5 k 10 k</p><p> Also a bit easier if voltages are labeled</p><p> All voltages are measured relative to the reference node</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 18</p></li><li><p>Nodal Analysis: Step 4 Apply KCL All Labeled Nodes</p><p>10 V 2 mA</p><p>1 2</p><p>-</p><p>+</p><p>v2-</p><p>+</p><p>v1</p><p>1 k 2 k</p><p>5 k 10 k</p><p>50 k</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 19</p></li><li><p>Nodal Analysis: Step 5 Solve Linear Equations</p><p>Linear Equations:</p><p>Solution (from calculator):</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 20</p></li><li><p>Nodal Analysis: Step 6 Solve for Variables of Interest</p><p>10 V 2 mA</p><p>1 2</p><p>-</p><p>+</p><p>v2-</p><p>+</p><p>v1</p><p>i1 i3</p><p>i2 i4</p><p>1 k 2 k</p><p>5 k 10 k</p><p>50 k</p><p>i1 =</p><p>i2 =</p><p>i3 =</p><p>i4 =</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 21</p></li><li><p>Nodal Analysis: Review of Steps</p><p>1. Identify essential nodes</p><p>2. Pick a reference</p><p> Must be an essential node</p><p> Always label with the ground symbol</p><p> Best to pick essential node with most branches</p><p> Often at the bottom of the circuit diagram</p><p>3. Label other essential nodes</p><p>4. Apply KCL to all labelled nodes except reference node</p><p>5. Solve linear equations</p><p> Generates voltage at each node (relative to reference node)</p><p>6. Solve for variables of interest</p><p> Usually easy after Step 5</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 22</p></li><li><p>Nodal Analysis: Use of Laws</p><p> All three laws are used</p><p> KCL is applied at each labelled node except the reference node</p><p> Ohms law is used to determine the current in branches thatcontain resistors</p><p> KVL is used to determine the voltage drop across the resistors</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 23</p></li><li><p>Example 4: Nodal Analysis</p><p>144 V-</p><p>+</p><p>v2-</p><p>+</p><p>v1 3 A</p><p>4 </p><p>5 10 </p><p>80 </p><p>Solve for v1 and v2.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 24</p></li><li><p>Example 4: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 25</p></li><li><p>Example 5: Nodal Analysis</p><p>20 mA-</p><p>+</p><p>v2-</p><p>+</p><p>v1-</p><p>+</p><p>v3 5 V2 k</p><p>2.7 k2.7 k</p><p>3.3 k</p><p>4.7 k</p><p>10 k</p><p>Solve for v1, v2, and v3.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 26</p></li><li><p>Example 5: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 27</p></li><li><p>Example 6: Dependent Voltage Source</p><p>50 V-</p><p>+</p><p>10 </p><p>10 </p><p>30 39 78 </p><p>80 k</p><p>v/5</p><p>v</p><p>Solve for v.</p><p> What effect does the 10 resistor have on the circuit?</p><p> What is the current flowing through the dependent source?</p><p> How can we apply KCL at the essential nodes without thisinformation?</p><p> Ans: One extra variable</p><p> Implies we need an extra equation</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 28</p></li><li><p>Example 6: Continued</p><p>50 V-</p><p>+</p><p>10 </p><p>10 </p><p>30 39 78 </p><p>100 k</p><p>v/5</p><p>v</p><p>Solve for v.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 29</p></li><li><p>Example 6: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 30</p></li><li><p>Nodal Analysis and Supernodes</p><p> Supernodes eliminate the need to introduce an extra variable(unknown current)</p><p> Necessary when a voltage source is between two labeled nodes(excluding reference node)</p><p> Still need to use voltage source to generate one of the equations</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 31</p></li><li><p>Example 7: Dependent Source Continued</p><p>50 V-</p><p>+</p><p>10 </p><p>10 </p><p>30 39 78 </p><p>160 k</p><p>v/5</p><p>v</p><p>Solve for v. Use a supernode.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 32</p></li><li><p>Example 7: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 33</p></li><li><p>Example 8: Dependent Voltage Source</p><p>20 V</p><p>+ -</p><p>1 2 4 </p><p>20 40 80 3.125v</p><p>v</p><p>35i</p><p>i</p><p>Find the power developed by the 20 V source.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 34</p></li><li><p>Example 8: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 35</p></li><li><p>Example 9: Nodal Analysis</p><p>11 mA</p><p>i1</p><p>20 Vi2</p><p>10 Vi3</p><p>250 </p><p>500 </p><p>1 k</p><p>25 k</p><p>Solve for i1, i2, and i3.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 36</p></li><li><p>Example 9: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 37</p></li><li><p>Example 10: Nodal Analysis</p><p>1 A</p><p>3i</p><p>i</p><p>-</p><p>+</p><p>v</p><p>1 </p><p>1 </p><p>2 </p><p>2 4 </p><p>Solve for v.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 38</p></li><li><p>Example 10: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 39</p></li><li><p>Mesh Analysis: Introduction</p><p> Recall: There is an easier way to solve for currents and voltagesthan applying KVL and KCL directly</p><p> Nodal analysis is one of two key methods</p><p> Mesh analysis is the other</p><p> Applies KVL to solve for currents</p><p> More abstract</p><p> Work with imaginary currents</p><p> Only applies to planar circuits</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 40</p></li><li><p>Mesh Analysis: Step 1 Label Meshes</p><p>40 V 64 V</p><p>ia</p><p>ic</p><p>ib</p><p>1.5 2 </p><p>3 4 </p><p>45 </p><p>Find the branch currents ia, ib, and ic.</p><p> Recall: A mesh is a loop that does not enclose any other loops</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 41</p></li><li><p>Mesh Analysis: Step 2 Apply KVL to Each Mesh</p><p>40 V 64 V</p><p>ia</p><p>ic</p><p>ib</p><p>1.5 2 </p><p>3 4 </p><p>45 </p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 42</p></li><li><p>Mesh Analysis: Step 3 Solve Linear Equations[50 4545 50.5</p><p>] [i1</p><p>i2</p><p>]=</p><p>[4064</p><p>]</p><p>i1 = 9.8 A</p><p>i2 = 10 A</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 43</p></li><li><p>Mesh Analysis: Step 4 Solve for Variables of Interest</p><p>40 V 64 V</p><p>ia</p><p>ic</p><p>ib</p><p>1.5 2 </p><p>3 4 </p><p>45 </p><p>ia =</p><p>ib =</p><p>ic =</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 44</p></li><li><p>Mesh Analysis: Review of Steps</p><p> Step 1 Label Meshes</p><p> Step 2 Apply KVL to Each Mesh</p><p> Step 3 Solve Linear Equations</p><p> Step 4 Solve for Variables of Interest</p><p> Usually easy after Step 3</p><p> Limitation: Only works with planar circuits</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 45</p></li><li><p>Example 11: Mesh Analysis</p><p>12 V</p><p>110 V 70V</p><p>2 </p><p>3 </p><p>4 </p><p>6 </p><p>10 12 </p><p>Find the total power developed in the circuit.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 46</p></li><li><p>Example 11: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 47</p></li><li><p>Example 12: Mesh Analysis</p><p>18 V 15 V3 A</p><p>2 </p><p>3 </p><p>6 </p><p>9 </p><p>Find the total power dissipated.</p><p> Problem: What is the voltage across the 3 A source?</p><p> Solutions</p><p>1 Add it as a variable</p><p>2 Use a supermesh</p><p> Second option requires less work</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 48</p></li><li><p>Example 12: Mesh Analysis</p><p>18 V 15 V3 A</p><p>2 </p><p>3 </p><p>6 </p><p>9 </p><p>Find the total power dissipated. Add a variable.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 49</p></li><li><p>Example 12: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 50</p></li><li><p>Example 13: Mesh Analysis</p><p>18 V 15 V3 A</p><p>2 </p><p>3 </p><p>6 </p><p>9 </p><p>Find the total power dissipated. Use a supermesh.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 51</p></li><li><p>Example 13: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 52</p></li><li><p>Example 14: Mesh Analysis</p><p>200 V</p><p>4.3 id</p><p>ie</p><p>ib</p><p>id</p><p>ia</p><p>ic</p><p>10 </p><p>10 </p><p>25 </p><p>50 </p><p>100 </p><p>Find the branch currents ia ie.</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 53</p></li><li><p>Example 14: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 54</p></li><li><p>Example 15: Mesh Analysis</p><p>1.5 mA</p><p>8 V</p><p>2 k</p><p>3 k</p><p>4 k</p><p>4 k 4 k</p><p>5 k</p><p>7 k</p><p>3i</p><p>i</p><p>Solve for i</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 55</p></li><li><p>Example 15: Workspace</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 56</p></li><li><p>Nodal versus Mesh Analysis</p><p> You should know how to do both</p><p> Which is more efficient depends on the problem</p><p> Will learn which to use with experience</p><p> Nodal analysis used more often</p><p> On exams, I will specify which method to use</p><p>Concise Summary:</p><p>Nodal Analysis Mesh AnalysisMethod KCL KVLSolve For Node Voltages Mesh CurrentsSuper Conditions Voltage Sources Current Sources</p><p>J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 57</p></li></ul>