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Objective of Lecture Provide step-by-step instructions for nodal analysis, which is a method to calculate node voltages and currents that flow through components in a circuit. Chapter 3.1 Basic Engineering Circuit Analysis by J.D. Irwin and R.M. Nelms Nodal Analysis Technique to find currents at a node using Ohm’s Law, Kirchhoff’s Current Law, and the potential differences between nodes. First result from nodal analysis is the determination of node voltages (voltage at nodes referenced to ground). These voltages are not equal to the voltage dropped across the resistors. Second result is the calculation of the currents This is the technique that is employed by PSpice. Steps in Nodal Analysis Vin Steps in Nodal Analysis Pick one node as a reference node Its voltage will be arbitrarily defined to be zero Vin Step 1 Pick one node as a reference node Its voltage will be arbitrarily defined to be zero Vin Step 2 Label the voltage at the other nodes Vin Step 2 Label the voltage at the other nodes Vin Step 3 Label the currents flowing through each of the components in the circuit Step 4 Use Kirchhoff’s Current Law I 7 I1 I 2 I 6 I 2 I3 I 4 I 4 I5 I3 I5 I6 Step 5 Use Ohm’s Law to relate the voltages at each node to the currents flowing in and out of them. Current flows from a higher potential to a lower potential in a resistor The difference in node voltage is the magnitude of electromotive force that is causing a current I to flow. I Va Vb R Step 5 We do not write an equation for I 7 as it is equal to I1 I1 V1 V2 R1 I 2 V2 V3 R2 I 3 V3 V5 R3 I 4 V3 V4 R4 I 5 V4 V5 R5 I 6 V5 0V R6 Step 6 Solve for the node voltages In this problem we know that V1 = Vin Step 6 Substitute the equations obtained using Ohm’s Law into the equations obtained using KCL. Vin V2 R1 V2 V3 R2 V5 R6 V2 V3 R2 V3 V5 R3 V3 V4 R4 V3 V4 R 4 V4 V5 R5 V3 V5 R3 V3 V4 R4 V5 R6 Step 7 Once the node voltages are known, calculate the currents. From Previous Slides I 7 I1 I 2 I 6 I 2 I3 I 4 I 4 I5 I3 I5 I6 V 1 Vin I1 V1 V2 R1 I 2 V2 V3 R2 I 3 V3 V5 R3 I 4 V3 V4 R4 I 5 V4 V5 R5 I 6 V5 0V R6 Substituting in Numbers I 7 I1 I 2 I 6 I 2 I3 I 4 I 4 I5 I3 I5 I6 V 1 10V I1 10V V2 9k I 2 V2 V3 2k I 3 V3 V5 5k I 4 V3 V4 3k I 5 V4 V5 1k I 6 V5 0V 7k Substituting the results from Ohm’s Law into the KCL equations 10V V2 9k V2 V3 2k V5 7k V2 V3 2k V3 V5 5k V3 V4 3k V3 V4 3k V4 V5 1k V3 V5 5k V4 V5 1k V5 7k Chugging through the Math Node Voltages (V) V1 10 V2 5.55 V3 4.56 V4 3.74 V5 3.46 Node voltages must have a magnitude less than the sum of the voltage sources in the circuit One or more of the node voltages may have a negative sign This depends on which node you chose as your reference node. Chugging through the Math Voltage across resistors (V) VR1 = (V1 – V2) VR2 = (V2 – V3) 4.45 0.990 VR3 = (V3 – V5) 1.10 VR4 = (V3 – V4) VR5 = (V4 – V5) VR6 = (V5 – 0V) 0.824 0.274 3.46 The magnitude of any voltage across a resistor must be less than the sum of all of the voltage sources in the circuit. In this case, no voltage across a resistor can be greater than 10V. Chugging through More Math Currents (mA) I1 I2 495 495 I3 I4 220 275 I5 275 I6 I7 495 495 Check None of the currents should be larger than the current that flows through the equivalent resistor in series with the 10V supply. Note that this check is only valid if there is one voltage source in the circuit. Req 9k 2k 5k 3k 1k 7k Req 20.2k I eq 10V Req 495mA 0.495mA Warning I know that you can find the solutions for nodal analysis using PSpice. I know that you can use a graphing calculator, MATLAB, or other tool to find the solutions. You will NOT have access to PSpice, MATLAB, your graphing calculator, or other tools during the exam at the end of the semester. Make sure that you can solve for the node voltages by hand using algebra. Summary Steps in Nodal Analysis 1. Pick one node as a reference node 2. Label the voltage at the other nodes 3. Label the currents flowing through each of the components in the circuit 4. Use Kirchhoff’s Current Law 5. Use Ohm’s Law to relate the voltages at each node to the currents flowing in and out of them. 6. Solve for the node voltage 7. Once the node voltages are known, calculate the currents.