Engineering 43
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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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Bruce mayer pe registered electrical mechanical engineer bmayer chabotcollege

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


Example

Example

  • 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

*2k

*2k

*2k

Example cont.

(1)

(2)

(3)

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

(4)

  • 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

SUPERNODE

  • 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.

SUPERNODE

  • 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

    SUPERNODE

    • Mult Eqn-1 by LCD (12 kΩ)

    • Add Eqns to Elim V2


    Bruce mayer pe registered electrical mechanical engineer bmayer chabotcollege

    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

    BASED ON PASSIVE SIGN CONVENTION THE

    POWER IS ABSORBED BY THE SOURCE!!


    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


    Example1

    Find Io

    Known Node Voltages

    SUPERNODE

    Example

    • 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

    SUPERNODE

    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

    supernode

    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

    supernode

    Complex SuperNode cont.

    Vs2

    Vs3

    • Now KCL at Supernode

      • Take Care Not to Omit Any Currents

    Vs1

    • 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

    IX

    WhiteBoard Work

    • Let’s Work This Problem

    • Find the OutPut Voltage, VO


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