EGR 2201 Unit 4 - PowerPoint Presentation

Slide1

EGR 2201 Unit 4Mesh Analysis

Read Alexander &

Sadiku

, Sections

3.4 to 3.10.

Homework #4 and Lab #4 due next week.

Quiz next week.Slide2

Mesh Analysis

We’ve seen that nodal analysis is

a systematic method for analyzing circuits.

It’s based on Kirchhoff’s

current law (KCL

).

It gives us the node voltages in

a circuit

.

Once we have these node voltages, we can find any other voltage or current.

Mesh analysis

is another

systematic method for analyzing circuits.

It’s

based on Kirchhoff’s

voltage law

(

KVL)

.

It

gives us the

mesh currents in

a circuit

.

Once we have these

mesh currents

, we can find any other

current or voltage.Slide3

Meshes Versus Loops

Recall that a

loop

is any closed path in a circuit.

Example: This circuit has six loops.

A mesh is a loop that does not contain any other loop within it.Example: The circuit above has three meshes.Slide4

Mesh Currents Versus Branch Currents

A

mesh current

is a current that we imagine to travel around a mesh.

You can imagine them to travel in either direction, but most people assume clockwise.

A branch current is a current that passes through a branch (i.e., an element).If we know the values of all the mesh currents in a circuit, we can compute any branch current.In many diagrams, our textbook uses:Lowercase i and a looping arrow for mesh currents

Uppercase

I

and a

straight arrow

for

branch currentsSlide5

Example: Mesh Currents Versus Branch Currents

In this circuit, the

mesh currents are

labeled

i

1 and i2. The branch currents

are labeled

I

1

,

I

2

,

and

I

3

.

Suppose you were

given the values of

the mesh currents.

Then you could easily

compute the branch currents, since:

I

1

=

i

1

and

I

2

=

i

2

and

I

3

=

i

1



i

2

Slide6

Steps in Performing Mesh Analysis on a Circuit with No Current Sources

Given

a circuit with

n

meshes, without current sources, follow these steps:Assign mesh currents i1, i2, …, in to the n meshes.Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of mesh currents. Then simplify the equations.Solve the resulting n simultaneous equations to obtain the unknown mesh currents.Slide7

Example: Step 1 (Assign the Mesh Currents)

Consider this circuit

from the book’s

Example 3.5.

Step 1 has

already been performed for us, since the mesh currents are labeled i1 and i2.If an assumed current direction is wrong, that’s no problem. The math will still work out.Slide8

Apply KVL (and

Ohm’s law) to

mesh 1:

15 + 10

i

2 = 5i1 + 10i1 + 10Apply KVL (and Ohm’s law) to mesh 2: 10 + 10i1 = 10i2 + 6i2 + 4i2Example: Step 2 (Apply KVL)Part 1 of 2Slide9

Next we use algebra to simplify our equations.

For mesh 1:

15

+ 10

i

2 = 5i1 + 10i1 + 10 becomes For mesh 2: 10 + 10i1 = 10i2 + 6i2 + 4i2 becomes

We now have our two equations in two variables.

Example: Step 2 (Apply KVL)

Part

2

of

2Slide10

Next we use any of our methods—substitution, Cramer’s rule, matrix inversion, MATLAB—to solve our two

equations in two variables.

Using MATLAB, the solution to

is

i

1 = 1 A and i2 = 1 A Example: Step 3 (Solve)Slide11

Mesh analysis has

given us the values

of the mesh

currents

i

1 and i2. We can find all other currentsand voltages in the circuit once we know these mesh currents.Example: Knowing that i1 = 1 A and i2 = 1 A, how would we find I3?Example: Extending the AnalysisSlide12

Review: Steps in Performing Mesh Analysis on a Circuit with No Current Sources

Given

a circuit with

n

meshes, without current sources, follow these steps:Assign mesh currents i1, i2, …, in to the n meshes.Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of mesh currents. Then simplify the equations.Solve the resulting n simultaneous equations to obtain the unknown mesh currents.Slide13

MATLAB’s

format

command

MATLAB’s

format

command lets you control the way MATLAB displays answers (for example, whether to use engineering notation in answers).For a list of options, type help format in MATLAB or see this web page.By default, the format is set to short.Usually this works well, but sometimes format shortg or format shorteng works better. See example on next slide.Slide14

Much nicer!

Example: MATLAB’s

format

command

This is telling you to multiply each number displayed below by

0.001. Ugly! Nicer! Slide15

As described above, our mesh-analysis procedure applies only to circuits

without

current sources.

But it’s not hard to extend the procedure to circuits

with

current sources.The way you handle a current source depends on whether the source is located in only one mesh or is shared by two meshes….What About Circuits with Current Sources?Slide16

Case 1: A Current Source Located in Only One Mesh

A current source

located in only

one mesh

is easy to

handle, because itimmediately reveals the mesh current in that mesh.

Example: In the circuit shown, we can immediately see that

i

2

=

5

A.Slide17

Case 2: A Current Source Shared by Two Meshes

A current source

shared by two

meshes is

trickier.

To handle it,

we create a

supermesh

by excluding the

current source

and any elements

in series with it.Slide18

We apply KVL

and

KCL to the super-

mesh to get two

equations.Example: Here, KVL around the

supermesh

gives

20 =

6

i

1

+ 10

i

2

+ 4

i

2

And KCL

gives

i

2 = i1 +6How to Handle a SupermeshSlide19

We Still Get

Enough Equations

If this circuit did

not have a super-

mesh, we would

get one equation by applying KVLto mesh 1 and another by applying KVL to mesh 2.

With the

supermesh

, we get one equation

by applying

KCL and another

by applying

KVL

to

the

supermesh

.Slide20

Nodal Analysis and Mesh Analysis “By Inspection”

With practice, you’ll become good at writing down the set of simultaneous equations that describe a circuit using either nodal or mesh analysis.

As discussed in Section 3.6, there is a shortcut way to write down the equations quickly by looking at a circuit without even thinking in terms of KCL or KVL.

I won’t expect you to learn this shortcut method, but you can use it if you wish.Slide21

Which Should You Use: Nodal Analysis or Mesh Analysis?

Most circuits can be analyzed using either method, and

the

results from the two methods will agree with each other.

But as discussed in Section 3.7, in some cases you’ll get the answer with less work if you’re smart about picking the better method for your circuit.

See next slide for example.Slide22

Examples: Should You Use Nodal Analysis or Mesh Analysis?

Recommendation:

Use nodal analysis for circuits with fewer nodes than meshes, and use mesh analysis

for circuits with fewer

meshes than nodes.

How many nodes does this circuit have?

How many meshes?Slide23

Dot Convention to Show Intersections

Before we look at the next example, note that our textbook usually does not follow the widespread convention of using a dot to show intersection points between wires.

Examples from

Multisim

:

If you saw this in Multisim, you would know that these two wires cross without intersecting.

If you saw this in

Multisim

, you would know that

these wires intersect.Slide24

More Examples: Should You Use Nodal Analysis or Mesh Analysis?

As also noted in Section 3.7, mesh analysis cannot be applied to

nonplanar

circuits.

A circuit is

planar if it can be drawn on a plane with no branches crossing each other; otherwise it is nonplanar.We will only deal with planar circuits in this course.Slide25

Example of a Planar Circuit

This circuit, in which branches cross, can be redrawn with no crossing branches (as on the right below), so it is a planar circuit.

Crossing branches:

No intersections here.Slide26

Example of a

Nonplanar

Circuit

There is no way

to

redraw this circuit without crossing branches, so it is a nonplanar circuit. So you cannot use mesh analysis on this circuit.Crossing branches:No intersections here.

Crossing branches:

No intersections here.Slide27

The Rest of Today’s Class

Use the remaining time today to:

Finish Lab #4.

Review

Homework #3

(particularly the super-node problems, Problems 3.13 and 3.18) and make sure you understand how to do them.Work on Homework #4, which is due at our next class.In preparation for the Midterm Exam, review earlier homeworks and practice sheets, and work on your crib sheet.