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

Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected]

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

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

- 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.

- 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 TechniqueSUPERNODE

- But Have Ckt V-Src Reln

- More Efficient solution:
- Enclose The Source, And All Elements In Parallel, Inside A Surface.
- Call That a SuperNode

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

- 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

- 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

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

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

- Eqn Bookkeeping:
- [email protected] V3
- [email protected] SuperNode,
- 2 Constraint Equations
- One Known Node

- 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

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

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

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

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