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Lecture 6Lixuan Lu
Example (14)Determine the node equations for the circuit shown.KCL at node a:1.𝑣𝑣𝑎𝑎−𝑣𝑣𝑐𝑐𝑅𝑅1+𝑣𝑣𝑎𝑎−𝑣𝑣𝑐𝑐𝑅𝑅2+𝑣𝑣𝑎𝑎−𝑣𝑣𝑏𝑏𝑅𝑅5=𝑖𝑖1+𝑖𝑖22.1𝑅𝑅1+1𝑅𝑅2+1𝑅𝑅5𝑣𝑣𝑎𝑎1𝑅𝑅5𝑣𝑣𝑏𝑏1𝑅𝑅1+1𝑅𝑅2𝑣𝑣𝑐𝑐=𝑖𝑖1+𝑖𝑖2KCL at node ‘b’:1.𝑣𝑣𝑎𝑎−𝑣𝑣𝑏𝑏𝑅𝑅5+𝑖𝑖3=𝑣𝑣𝑏𝑏−𝑣𝑣𝑐𝑐𝑅𝑅3+𝑣𝑣𝑏𝑏𝑅𝑅4+𝑖𝑖22.𝑖𝑖3− 𝑖𝑖2=−𝑣𝑣𝑎𝑎+𝑣𝑣𝑏𝑏𝑅𝑅5+𝑣𝑣𝑏𝑏−𝑣𝑣𝑐𝑐𝑅𝑅3+𝑣𝑣𝑏𝑏𝑅𝑅43.1𝑅𝑅5𝑣𝑣𝑎𝑎+1𝑅𝑅3+1𝑅𝑅4+1𝑅𝑅5𝑣𝑣𝑏𝑏1𝑅𝑅3𝑣𝑣𝑐𝑐=𝑖𝑖3− 𝑖𝑖2Therefore, the node equation at node ‘c’ is:1𝑅𝑅1+1𝑅𝑅2𝑣𝑣𝑎𝑎1𝑅𝑅3𝑣𝑣𝑏𝑏+1𝑅𝑅1+1𝑅𝑅2+1𝑅𝑅3+1𝑅𝑅6𝑣𝑣𝑐𝑐=−𝑖𝑖12
Methods of Analysis ofR esistive Circuits3
To Write a Set of Node Equations:1.Identify the number of nodes.2.Dedicate one node to be the reference (called ground or Gnd node).3.Set the Gnd node voltage to zero.4.Express the element currents as a function of node voltages, as shown in thefigure.5.ApplyKCLat each one of the nodes of the circuit except for the reference node.4
From example 14 we can deduce that:The node voltage of interest is multiplied by the summation of the conductancesconnected to the node then subtracted by the neighboring node voltagesmultiplied by the summation of conductances connected to each neighboringnode respectively, then this is equated to all currents entering the node ofinterest.e.g.:1𝑅𝑅1+1𝑅𝑅2+1𝑅𝑅5𝑣𝑣𝑎𝑎1𝑅𝑅5𝑣𝑣𝑏𝑏1𝑅𝑅1+1𝑅𝑅2𝑣𝑣𝑐𝑐=𝑖𝑖1+𝑖𝑖25
Nodal analysis for circuits containing voltage sources connected toGND nodeNodal analysis for circuits containing voltage sources connectedbetween any two nodes (Super Node)Nodal analysis of circuits containing dependent voltage sources6
7𝑣𝑣1𝑣𝑣2𝑣𝑣3𝑣𝑣4𝑣𝑣5𝑣𝑣6𝑣𝑣7𝑣𝑣8𝑣𝑣9𝑅𝑅1𝑅𝑅2𝑅𝑅3𝑅𝑅4𝑅𝑅5𝑖𝑖1𝑖𝑖2𝑅𝑅3𝑅𝑅2𝑅𝑅5𝑅𝑅4𝑖𝑖1𝑖𝑖21𝑅𝑅2+1𝑅𝑅3+1𝑅𝑅4+1𝑅𝑅5𝑣𝑣11𝑅𝑅2𝑣𝑣21𝑅𝑅3𝑣𝑣31𝑅𝑅4𝑣𝑣41𝑅𝑅5𝑣𝑣5=−𝑖𝑖1− 𝑖𝑖2𝑣𝑣1𝑣𝑣1𝑣𝑣1=𝑣𝑣1𝑣𝑣1− 𝑣𝑣1= 0𝑣𝑣1− 𝑣𝑣1=𝑣𝑣𝑠𝑠𝑣𝑣𝑠𝑠𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒏𝒏𝒏𝒏𝒏𝒏𝒔𝒔𝑣𝑣1𝑣𝑣1Asupernodeconsists of two nodes connected by a dependent or independent voltage source.Nodal Analysis with Voltage Source not connected to reference node - Supernode1𝑅𝑅3+1𝑅𝑅5𝑣𝑣1+1𝑅𝑅2+1𝑅𝑅4𝑣𝑣11𝑅𝑅2𝑣𝑣21𝑅𝑅3𝑣𝑣31𝑅𝑅4𝑣𝑣41𝑅𝑅5𝑣𝑣5=−𝑖𝑖1− 𝑖𝑖2
Nodal Analysis with Voltage Source not connected to reference node (Super Node)Note that none of the terminals of the voltage source are connected to ground.KCL at the supernode,𝑖𝑖𝑎𝑎+𝑖𝑖𝑏𝑏=𝑖𝑖𝑠𝑠𝑣𝑣𝑎𝑎1𝑅𝑅1+𝑣𝑣𝑏𝑏1𝑅𝑅2=𝑖𝑖𝑠𝑠𝑎𝑎𝑛𝑛𝑛𝑛𝑣𝑣𝑠𝑠=𝑣𝑣𝑎𝑎− 𝑣𝑣𝑏𝑏8
Example (17a)Determine the node voltage for the circuit shown.There are 2 nodes (other than Gnd), therefore we need 2 equations for the 2 nodes:𝑣𝑣𝑎𝑎− 𝑣𝑣𝑏𝑏= 4(1)𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑛𝑛𝑆𝑆𝑛𝑛𝑆𝑆 𝑆𝑆𝑒𝑒𝑛𝑛:𝑣𝑣𝑎𝑎4+𝑣𝑣𝑏𝑏10= 3(2)1⇒ 𝑣𝑣𝑎𝑎= 4 +𝑣𝑣𝑏𝑏(3)Substituting𝑆𝑆𝑒𝑒𝑛𝑛3𝑖𝑖𝑛𝑛2 :4 +𝑣𝑣𝑏𝑏4+𝑣𝑣𝑏𝑏10= 320 + 5𝑣𝑣𝑏𝑏+ 2𝑣𝑣𝑏𝑏20= 320 +7𝑣𝑣𝑏𝑏= 60⇒ 7𝑣𝑣𝑏𝑏= 40⇒ 𝑣𝑣𝑏𝑏=407and𝑣𝑣𝑎𝑎=407+ 49V=4 v
Nodal Voltage Analysis with Dependent Sources-When a circuit contains a dependent source the controlling current or voltage ofthat dependent source must be expressed as a function of the node voltages.

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Mesh Analysis, Voltage drop, 1

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