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# Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected] - PowerPoint PPT Presentation

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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Chp 3.1bNodal Analysis

Bruce Mayer, PE

Registered Electrical & Mechanical [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...

• 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

• 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

*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

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

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

• Use The V-Source Rln Eqn to Find V2

SUPERNODE

• Mult Eqn-1 by LCD (12 kΩ)

• Add Eqns to Elim V2

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!!

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

Known Node Voltages

SUPERNODE

Example

• The SuperNode V-Constraint

• Now KCL at SuperNode

• Or

• 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

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

Set UP

Identify all nodes

Select a reference

Label All nodes

supernode

Complex SuperNode

• Nodes Connected To Reference Through A Voltage Source

• Voltage Sources In Between Nodes And Possible Supernodes

• Choose to Connect V2 & V4

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

• 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

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

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

• By Ohm’s Law

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

IX

WhiteBoard Work

• Let’s Work This Problem

• Find the OutPut Voltage, VO