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Analysis Methods Overview

Solving Linear Equations

Nodal Analysis

Supernodes (Nodal Analysis with Voltage Sources)

Mesh Analysis

Supermeshes (Mesh Analysis with Current Sources)

This is a very important chapter.

J. McNames Portland State University ECE 221 Analysis Methods Ver. 1.68 1

Review of Basic Concepts: Current

i4 i5i3i2i1

What goes in, has to come out

Kirchhoffs current law

Similar to conservation of mass

Conservation of electrons


Review of Basic Concepts: Voltage

10 V-

+

-

++ - + -

2 k2 k

5 k 7 kv1 v2

v3 v4

The voltage drop from one node to another is the same, nomatter what path is chosen

Kirchhoffs voltage law


Resistors in Parallel with Voltage Sources

CircuitRVs vo-

+

CircuitVs vo-

+

What is vo in each case?

What effect does the resistor have on the current pumped into thecircuit?


Resistors in Series with Current Sources

CircuitIs CircuitIs

Rio

io

What is io in each case?

What effect does the resistor have on the voltage seen by thecircuit?


Solving Linear Equations

Much of our circuit analysis will focus on finding a set of linearequations and solving these equations

Need as many equations as there are unknowns

Three possible approaches

Algebra (elimination, substitution, etc.)

Cramers rule

Linear algebra

Last is easiest and least susceptible to errors

Requires use your scientific calculators


Example 1: Solving Linear Equations

i1 = i2 + i3

i4 = i3 + 2m

10 = (1k)i1 + (5k)i2

(5k)i2 = (2k)i3 + (10k)i4

Rewrite so variables are in consistent order on left side and constantsare on the right side

i1 i2 i3 = 0 i3 + i4 = 2m

(1k)i1 + (5k)i2 = 10+ (5k)i2 (2k)i3 (10k)i4 = 0


Example 1: Continued (1)

i1 i2 i3 = 0 i3 + i4 = 2m

(1k)i1 + (5k)i2 = 10+ (5k)i2 (2k)i3 (10k)i4 = 0

In Matrix form this becomes

1 1 1 00 0 1 11k 5k 0 00 5k 2k 10k

i1

i2

i3

i4

=

02m100

or

Ai = b



Ai = b where

A =

1 1 1 00 0 1 11k 5k 0 00 +5k 2k 10k

i =

i1

i2

i3

i4

b =

02m100

Your calculator should be able to solve this directly

You should only need to enter A and b

Your calculator will return a vector i

Simultaneously solves for all the unknown variables

Much faster than Cramers rule or brute-force algrebra

Read the manuals for your calculators

This will save you time (homework & exams) and reduce errors



Linear Equations:1 1 1 00 0 1 11k 5k 0 00 5k 2k 10k

i1

i2

i3

i4

=

02m100

Calculator should return:

i1

i2

i3

i4

=

+0.909+1.8180.909+1.091

mA


Network Terminology

Planar Circuit A circuit that can be drawn on a plane with nocrossing branches

Node Point or portion of a circuit where 2 or more elements arejoined

Essential Node Point or portion of a circuit where 3 or moreelements are joined

Branch Path that connects 2 nodes

Essential Branch Path that connects 2 essential nodes w/o passingthrough an essential node

Loop Path with last node same as starting node that does not crossitself

Mesh Loop that does not enclose any other loops


Example 2: Terminology

20 V 2 A

R1 R2

R3 R4 R4

R6 R7 R8

35ip

ip

Identify the following informationNodes: Essential Nodes:Branches: Essential Branches:EBs with Unknown Current: Meshes:


Example 3: Circuit Analysis The Hard Way

10 V

i1 i3

i2 i42 mA

1 k 2 k

5 k 10 k

Can solve with KCL & KVL. Four unknowns.


Nodal Analysis: Introduction

There is an another way to solve for currents and voltages

Easier

More methodical

Still based on Ohms law, KVL, & KCL

Nodal analysis is one of two key methods

Mesh analysis is the other

We will discuss nodal analysis first

Based on KCL

Must understand terminology introduced earlier

Use to solve for voltages

All voltages have a common reference point


Nodal Analysis Steps

1. Identifiy essential nodes

2. Pick a reference node

3. Label all other essential nodes

4. Apply KCL to all labelled nodes

5. Solve linear equations for all node voltages

6. Solve for variables of interest


Nodal Analysis: Step 1 Identify Essential Nodes

10 V 2 mA

1 k 2 k

5 k 10 k

Some essential nodes may include portions of the circuit (pieces ofwire)

Circle the entire node to prevent errors


Nodal Analysis: Step 2 Pick a Reference

10 V 2 mA

1 k 2 k

5 k 10 k

Second step is to pick a reference node

Is often easiest to choose the node that interconnects the mostbranches

Must be an essential node

Usually is at bottom of circuit

Label with the same symbol used for ground


Nodal Analysis: Step 3 Label Other Essential Nodes

10 V 2 mA

1 k 2 k

5 k 10 k

Also a bit easier if voltages are labeled

All voltages are measured relative to the reference node


Nodal Analysis: Step 4 Apply KCL All Labeled Nodes

10 V 2 mA

1 2

-

+

v2-

+

v1

1 k 2 k

5 k 10 k

50 k


Nodal Analysis: Step 5 Solve Linear Equations

Linear Equations:

Solution (from calculator):


Nodal Analysis: Step 6 Solve for Variables of Interest

10 V 2 mA

1 2

-

+

v2-

+

v1

i1 i3

i2 i4

1 k 2 k

5 k 10 k

50 k

i1 =

i2 =

i3 =

i4 =


Nodal Analysis: Review of Steps

1. Identify essential nodes

2. Pick a reference

Must be an essential node

Always label with the ground symbol

Best to pick essential node with most branches

Often at the bottom of the circuit diagram

3. Label other essential nodes

4. Apply KCL to all labelled nodes except reference node

5. Solve linear equations

Generates voltage at each node (relative to reference node)

6. Solve for variables of interest

Usually easy after Step 5


Nodal Analysis: Use of Laws

All three laws are used

KCL is applied at each labelled node except the reference node

Ohms law is used to determine the current in branches thatcontain resistors

KVL is used to determine the voltage drop across the resistors


Example 4: Nodal Analysis

144 V-

+

v2-

+

v1 3 A

4

5 10

80

Solve for v1 and v2.


Example 4: Workspace



20 mA-

+

v2-

+

v1-

+

v3 5 V2 k

2.7 k2.7 k

3.3 k

4.7 k

10 k

Solve for v1, v2, and v3.


Example 6: Dependent Voltage Source

50 V-

+

10

10

30 39 78

80 k

v/5

v

Solve for v.

What effect does the 10 resistor have on the circuit?

What is the current flowing through the dependent source?

How can we apply KCL at the essential nodes without thisinformation?

Ans: One extra variable

Implies we need an extra equation


Example 6: Continued

50 V-

+

10

10

30 39 78

100 k

v/5

v

Solve for v.


Nodal Analysis and Supernodes

Supernodes eliminate the need to introduce an extra variable(unknown current)

Necessary when a voltage source is between two labeled nodes(excluding reference node)

Still need to use voltage source to generate one of the equations


Example 7: Dependent Source Continued

50 V-

+

10

10

30 39 78

160 k

v/5

v

Solve for v. Use a supernode.


Example 8: Dependent Voltage Source

20 V

+ -

1 2 4

20 40 80 3.125v

v

35i

i

Find the power developed by the 20 V source.



11 mA

i1

20 Vi2

10 Vi3

250

500

1 k

25 k

Solve for i1, i2, and i3.



1 A

3i

i

-

+

v

1

1

2

2 4

Solve for v.


Mesh Analysis: Introduction

Recall: There is an easier way to solve for currents and voltagesthan applying KVL and KCL directly

Nodal analysis is one of two key methods

Mesh analysis is the other

Applies KVL to solve for currents

More abstract

Work with imaginary currents

Only applies to planar circuits


Mesh Analysis: Step 1 Label Meshes

40 V 64 V

ia

ic

ib

1.5 2

3 4

45

Find the branch currents ia, ib, and ic.

Recall: A mesh is a loop that does not enclose any other loops


Mesh Analysis: Step 2 Apply KVL to Each Mesh

40 V 64 V

ia

ic

ib

1.5 2

3 4

45


Mesh Analysis: Step 3 Solve Linear Equations[50 4545 50.5

] [i1

i2

]=

[4064

]

i1 = 9.8 A

i2 = 10 A


Mesh Analysis: Step 4 Solve for Variables of Interest

40 V 64 V

ia

ic

ib

1.5 2

3 4

45

ia =

ib =

ic =


Mesh Analysis: Review of Steps

Step 1 Label Meshes

Step 2 Apply KVL to Each Mesh

Step 3 Solve Linear Equations

Step 4 Solve for Variables of Interest

Usually easy after Step 3

Limitation: Only works with planar circuits


Example 11: Mesh Analysis

12 V

110 V 70V

2

3

4

6

10 12

Find the total power developed in the circuit.



18 V 15 V3 A

2

3

6

9

Find the total power dissipated.

Problem: What is the voltage across the 3 A source?

Solutions

1 Add it as a variable

2 Use a supermesh

Second option requires less work



18 V 15 V3 A

2

3

6

9

Find the total power dissipated. Add a variable.



18 V 15 V3 A

2

3

6

9

Find the total power dissipated. Use a supermesh.



200 V

4.3 id

ie

ib

id

ia

ic

10

10

25

50

100

Find the branch currents ia ie.



1.5 mA

8 V

2 k

3 k

4 k

4 k 4 k

5 k

7 k

3i

i

Solve for i


Nodal versus Mesh Analysis

You should know how to do both

Which is more efficient depends on the problem

Will learn which to use with experience

Nodal analysis used more often

On exams, I will specify which method to use

Concise Summary:

Nodal Analysis Mesh AnalysisMethod KCL KVLSolve For Node Voltages Mesh CurrentsSuper Conditions Voltage Sources Current Sources


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