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Engineering 43. Chp 3.1b Nodal Analysis. Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected] Need Only ONE KCL Eqn. Ckts with Voltage Sources. The Remaining Eqns From the Indep Srcs. 3 Nodes Plus the Reference. In Principle Need 3 Equations...

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Engineering 43

Chp 3.1bNodal Analysis

Bruce Mayer, PE

Registered Electrical & Mechanical [email protected]

Ckts with voltage sources

Need Only ONE KCL Eqn

Ckts with Voltage Sources

  • The Remaining Eqns From the Indep Srcs

  • 3 Nodes Plus the Reference. In Principle Need 3 Equations...

    • But two nodes are connected to GND through voltage sources. Hence those node voltages are KNOWN

  • Solving The Eqns


  • Find Vo

  • To Start

    • Identify & Label All Nodes

    • Write Node Equations

    • Examine Ckt to Determine Best Solution Strategy

R1 = 1k; R2 = 2k, R3 = 1k, R4 = 2k

Is1 =2mA, Is2 = 4mA, Is3 = 4mA,

Vs1 = 12 V

  • Notice

  • Need Only V1 and V2to Find Vo

  • Now KCL at Node 1

  • Known Node Potential

Example cont
Example cont.

  • At Node 2

  • At Node 4

R1 = 1k; R2 = 2k, R3 = 1k, R4 = 2k

Is1 =2mA, Is2 = 4mA, Is3 = 4mA,

Vs = 12 V

  • To Solve the System of Equations Use LCD-multiplication and Gaussian Elimination

Example cont1

The LCDs




Example cont.




  • Now Add Eqns (2) & (3) To Eliminate V4


  • Now Add Eqns (4) & (1) To Eliminate V2

  • BackSub into (4) To Find V2

  • Find Vo by Difference Eqn

Supernode technique

Consider ThisExample

Conventional Node Analysis Requires All Currents At A Node

SuperNode Technique


  • But Have Ckt V-Src Reln

  • More Efficient solution:

    • Enclose The Source, And All Elements In Parallel, Inside A Surface.

      • Call That a SuperNode

  • 2 eqns, 3 unknowns...Not Good

    • Recall: The Current thru the Vsrc is NOT related to the Potential Across it

Supernode cont

Apply KCL to the Surface

Supernode cont.


  • The Source Current Is interior To The Surface And Is NOT Required

  • Still Need 1 More Equation – Look INSIDE the Surface to Relate V1 & V2

    • Now Have 2 Equations in 2 Unknowns

    • Then The Ckt Solution Using LCD Technique

      • See Next Slide

    Now apply gaussian elim

    The Equations

    Now Apply Gaussian Elim

    • Use The V-Source Rln Eqn to Find V2


    • Mult Eqn-1 by LCD (12 kΩ)

    • Add Eqns to Elim V2

    Find the node voltages

    And the power supplied

    By the voltage source

    To compute the power supplied by the voltage source We must know the current through it: @ node-1



    Illustration using conductances

    Write the Node Equations

    KCL At v1

    Illustration using Conductances

    • At The SuperNode Have V-Constraint

      • v2−v3= vA

    • KCL Leaving Supernode

    • Now Have 3 Eqnsin 3 Unknowns

      • Solve Using Normal Techniques


    Find Io

    Known Node Voltages



    • The SuperNode V-Constraint

    • Now KCL at SuperNode

    • Or

    Student exercise
    Student Exercise

    • Lets Turn on the Lights for 5-7 min

    • Students are invited to Analyze the following Ckt

      • Hint: Use SuperNode

    • Determine the OutPut Current, IO

    Numerical example

    Find Io Using Nodal Analysis

    Known Voltages for Sources Connected to GND


    Numerical Example

    • Now Notice That V2 is NOT Needed to Find Io

      • 2 Eqns in 2 Unknowns

    • The Constraint Eqn

    • Now KCL at SuperNode

    • By Ohm’s Law

    Complex supernode

    Write the Node Eqns

    Set UP

    Identify all nodes

    Select a reference

    Label All nodes


    Complex SuperNode

    • Nodes Connected To Reference Through A Voltage Source

    • Eqn Bookkeeping:

      • KCL@ V3

      • KCL@ SuperNode,

      • 2 Constraint Equations

      • One Known Node

    • Voltage Sources In Between Nodes And Possible Supernodes

    • Choose to Connect V2 & V4

    Complex supernode cont

    Now KCL at Node-3


    Complex SuperNode cont.



    • Now KCL at Supernode

      • Take Care Not to Omit Any Currents


    • Constraints Due to Voltage Sources

    • 5 Equations 5 Unknowns → Have to Sweat Details

    Dependent sources
    Dependent Sources

    • Circuits With Dependent Sources Present No Significant Additional Complexity

    • The Dependent Sources Are Treated As Regular Sources

    • As With Dependent CURRENT Sources Must Add One Equation For Each Controlling Variable

    Numerical example dep i src

    Find Io by Nodal Analysis

    Notice V-Source Connected to the Reference Node

    Numerical Example – Dep Isrc

    • Sub Ix into KCL Eqn

    • KCL At Node-2

    • Mult By 6 kΩ LCD

    • Controlling Variable In Terms of Node Potential

    • Then Io

    Dep v source example

    Find Io by Nodal Analysis

    Notice V-Source Connected to the Reference Node

    Dep V-Source Example

    • SuperNode Constraint

    • KCL at SuperNode

    • Controlling Variable in Terms of Node Voltage

    • Mult By 12 kΩ LCD

    Dep v source example cont

    Simplify the LCD Eqn

    Dep V-Source Example cont

    • By Ohm’s Law

    Current controlled v source

    Find Io

    Supernode Constraint

    Current Controlled V-Source

    • Controlling Variable in Terms of Node Voltage

    • Multiply by LCD of 2 kΩ

    • Recall

    • Then

    • KCL at SuperNode

    • So Finally

    Whiteboard work


    WhiteBoard Work

    • Let’s Work This Problem

    • Find the OutPut Voltage, VO