The Node Voltage Method - Iowa State University
The node voltage method Equivalent resistance Voltage / current dividers Source transformations Node voltages Mesh currents SuperpositionNot every circuit lends itself to “short-cut” methods. Sometimes weneed a formal approach that does not rely on seeing a trick that can beused. The node-voltage is the first (and maybe most used) of our threeformal methods.The node-voltage method allows for the calculation of the voltages ateach node of the circuit, relative to a reference node. Once the nodevoltages are known, all currents in the circuit can be determined easily.The method leads to a set of simultaneous that must be solved. Biggercircuits will have more nodes and require more equations (more math).EE 201node-voltage method – 1
Recall:ISiVS1IS2R1aVS1 –iR1bR2R3iR2iR3ciVS2 VS2–dUsing KCL at each node:a:b:c:d:EE 201 4 equations, but 5 unknowns (iVS1, iVS2,iR1, iR2, and iR3).Need more info. Or a better method. node-voltage method – 2
The node voltage methodLook at an example.R1VS10 V –10 !R25!IS 1 AStep 1 - Identify all of the nodes in the circuit.aVS –R1bR2IScThis small circuit has three nodes. In principle, three unknownvoltages, but we will try to reduce this.EE 201node-voltage method – 3
Step 2 - Choose one node to be the reference, usually called ground.Since voltage is a relative quantity (only voltage differencesmatter), we can choose one point in the circuit to be V 0. Thevoltages at the other nodes will be with respect this ground node.aVS –R1bR2ISvc 0In general, we can choose any node as the reference, but somechoices are better than others. Typically, we should choose anode that is connected to the positive or negative terminal of avoltage source.For this circuit, that implies node a or node c. This time, wechoose node c, so we can now say vc 0.EE 201node-voltage method – 4
Step 3 - Identify any other nodes for which the voltages (with respectto ground) are known.va V SVS –R1bR2ISvc 0For this circuit, the voltage source tells us that node a is VS higherin voltage than the ground node. Therefore va VS.Since the voltage at a is now known, we have reduced thenumber of unknowns down to one – the voltage at node b.EE 201node-voltage method – 5
Step 4 - Look for other ways (like resistor reductions) that could beused to reduce the number of unknown voltages further.It is not necessary to calculate the voltage at every node. If wecan eliminate non-essential nodes, we should do so.In this example, we are already down to one node – not much morewe can do.Step 5 - Assign voltage variables to each of the remaining unknown nodes.va VSVS –R1vbR2ISFor this example, thereis only one unknownnode voltage.vc 0EE 201node-voltage method – 6
Step 6 - Assign currents to all the branches connected to each of theunknown nodes. You are at liberty to choose whatever directionsyou want for the current arrows.R1va VSVS –iR1ISvbR2iR2ISvc 0Step 7 - Use KCL to write equations balancing the currents at eachunknown node.iR1 IS iR2Step 8 - Use Ohm’s law to express the resistor currents in terms of thenode voltages on either side of the resistor. Pay attention to polarity! EE 201 node-voltage method – 7
Step 9 - Substitute the Ohm’s law expressions for the resistor currentsinto the KCL equations to form the node-voltage equations. The circuit analysis is done. The rest is just math.Step 10 - Solve the equation(s). ( )() .Finally, the resistor voltages and currents (using Ohm’s law) can be calculated. EE 201 . . . . . .node-voltage method – 8
node-voltage method – summary1. Identify all of the nodes in the circuit.2. Choose one node to be ground.3. Identify nodes for which the voltage is known due to sources.4. Use resistor reductions to eliminate any other non-essential nodes.5. Assign variables for the voltages at the remaining unknown nodes.6. Assign currents to all of the branches connected to the nodes.7. Write KCL equations balancing the currents at each of the nodes.8. Use Ohm’s law to express resistor currents in terms of the (unknown)node voltages on either side of the resistor. (Be sure to get the correctpolarity!)9. Substitute the resistor currents into the KCL equations to form thenode-voltage equations. (Set of equations relating the unknown nodevoltages.)10.Do the math to solve the equations and determine the node voltages.Determine currents, powers, etc., if needed.EE 201node-voltage method – 9
ExampleSame circuit, but choose a different ground.1. Identify the nodes.aVS10 V –R1b10 !R25!IS 1 Ac2. Choose one to be ground. This time, choose node a.va 0VS –R1bR2IScEE 201node-voltage method – 10
3. Identify nodes for which the voltage is known.Clearly, va – vc VS. Therefore, vc – VS.va 0VSR1 –bR2ISvc –VS5&6. Assign variables for the unknown voltages. Assign currents foreach branch connected to the nodes.va 0VSEE 201 –R1iR1ISvbR2iR2vc –VSISnode-voltage method – 11
7. Write KCL equations balancing the currents at each node.va 0VS –R1ISvbiR1R2ISiR2vc –VSiR1 IS iR28. Use Ohm’s law to express the resistor currents in terms of thenode voltages. ()9. Substitute resistor current equations into the KCL equations toform the node-voltage equations. EE 201 node-voltage method – 12
10. Solve it. ()() .At first glance, this seems wrong – vb here is different from itsvalue in the previous calculation.But remember that only voltage differences matter. In choosing adifferent reference node, all other node voltages will be shiftedaccordingly.vR1 VS – vb 3.33 V and vR2 vb – 0 6.67 V, in both cases. Thecorresponding currents will be the same, as well.EE 201node-voltage method – 13
another exampleVS1 –10 VR1R31 k!5 k!R23 k!IS5 mA –R3cVS220 V1. Identify the nodes.aVS1 –R1bR2IS –VS2dEE 201node-voltage method – 14
2. Choose one to be ground. Since there are voltage sourcesconnected to d, that would seem to be a good choice.3. Identify nodes for which the voltage is known. With d as ground,then we see that va VS1 and vc VS2.va VS1VS1 –R1R3bR2ISvc VS2 –VS2Only one node left, so only oneunknown voltage to be found.EE 201node-voltage method – 15
5&6. Assign a variable for the unknown voltage. Assign currents foreach branch connected to the node.va VS1R1VS1 –iR1R3vbR2iR2ISiR3vc VS2 –VS27. Write KCL equations balancing the currents at the node. 8. Express the resistor currents in terms of the node voltage. EE 201 node-voltage method – 16
9. Substitute resistor currents into the KCL equation to form thenode-voltage equation. 10. Solve it. () ( ( . EE 201 ) ( )( ) ).node-voltage method – 17
ExampleIn the circuit below, we might like to find the currents of theresistors. The resistor that bridges across the top makes the short-cutmethods unusable.R3 30 !R110 !R2R420 !40 !VS –100 V1. Identify the nodes.xR1EE 201R550 !R3R2yVS –gR4zR5node-voltage method – 18
2. Choose one to be ground. Due to the single voltage source, thebottom node seems to be a likely choice.3. Identify nodes for which the voltage is known. With g as ground,then we see that vy VS.R3xR1R2vy VSR4zR5VS –vg 0EE 201node-voltage method – 19
5&6. Assign a variable for the unknown voltage. Assign currents foreach branch connected to the node.R3R2vxR1iR2iR1iR3VS –R4iR4vz2 nodes this time!R5iR57. Write KCL equations balancing the currents at the node.x: z: 8. Express the resistor currents in terms of the node voltages. EE 201 node-voltage method – 20
9. Substitute resistor currentsinto the KCL equations to formthe node-voltage equations.x: z: 10. Solve the equations 3.667vx – 0.667vz 100 V EE 201 -1.333vx 3.133vz 100 Vvx 35.85 V and vz 47.175 Vnode-voltage method – 21
Final example(a big one)Again, we might like to know how much power is being generatedand dissipated in the various elements for the circuit below. Ourshort-cut methods are useless here, but the node-voltage works inexactly the same manner as the previous examples. The math is a bitmore tedious for a bigger circuit – there will be 3 equations in 3unknowns in this case – but the method is still straight forward.R3 50 !R1 10 !VSEE 201 15 V–R4 20 !R2 20 ! IS10.25 AR5 30 !R640 !IS20.5 Anode-voltage method – 22
1. Identify the nodes.2. Choose one to be ground. (The bottom one is good, again.)3. Identify known node voltages. (The left-most node is obviously at VS.)4. Assign variable names to the remaining, unknown nodes.5. Assign resistor currents in each branch.R3va V SaVS –vb R4 iR4vc R5bcR1iR1iR3R2iR2vdiR5IS1edR6iR6IS2ve 0EE 201node-voltage method – 23
iR3R3vb R4 iR4R1VSiR1 –R2vc R5iR5IS1iR2vdR6IS2iR67. Write KCL equations balancing the currents at the node. 8. Express the resistor currents in terms of the node voltages.9. Substitute resistor currents into the KCL equations. EE 201 node-voltage method – 24
10. Solve the equations . . . . Two web-site solvers:EE 201 . vb 9.554 Vvc 8.217 V"vd 13.709 Vhttp://math.bd.psu.edu/ unknwn3.htmnode-voltage method – 25
Sep 19, 2014 · The node-voltage is the first (and maybe most used) of our three formal methods. The node-voltage method allows for the calculation of the voltages at each node of the circuit, relative to a reference node. Once the node voltages are kno
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