Mesh Analysis

Summary

This educational video script introduces mesh analysis, a method for analyzing electrical networks to determine unknown currents. It explains the concept of a mesh as a loop without any inner loops and outlines the four steps for performing mesh analysis: identifying meshes, assigning mesh currents, developing KVL equations for each mesh, and solving these equations. The script also highlights that mesh analysis is applicable only to planar networks and emphasizes the importance of choosing the direction of mesh currents. It concludes with an example demonstrating how to apply these steps to calculate power loss in a resistor.

Takeaways

• 🔍 Mesh analysis is a method used to analyze electrical networks by determining the unknown currents flowing through various elements.
• 🎯 The primary purpose of mesh analysis is to calculate the power delivered or absorbed by different components within an electrical network.
• 🔌 A mesh is defined as a loop in a circuit that does not contain any other loops within it, and it is a fundamental concept in mesh analysis.
• 📚 Mesh analysis is applicable only to planar networks, where no branch crosses over another, ensuring the network can be drawn on a single plane without any intersections.
• ✏️ The process of mesh analysis involves four main steps: identifying meshes, assigning mesh currents, developing KVL equations for each mesh, and solving those equations.
• 🔄 The direction of mesh current can be chosen as clockwise or counterclockwise, but a clockwise direction is often preferred for convenience and consistency with typical current flow from sources.
• 📉 The number of equations needed for mesh analysis is equal to the number of meshes, which can be calculated using the formula: number of meshes = (number of branches - number of nodes + 1).
• 🧮 Once the mesh currents are determined, further calculations such as power loss in resistors can be performed using the values obtained from the mesh analysis.
• 📝 When developing KVL equations, the mesh current of the particular mesh being considered is given priority in the calculation of net currents through shared resistors.
• 🔑 A key takeaway is the importance of considering the mesh current as the dominant current when formulating KVL equations for a specific mesh, which affects the calculation of voltage drops across resistors.

Q & A

What is the primary purpose of mesh analysis in electrical networks?

The primary purpose of mesh analysis is to determine the unknown currents in an electrical network, which is essential for calculating the power delivered or absorbed by different electrical elements.

What is a mesh in the context of electrical networks?

A mesh is a loop in an electrical network that does not contain any other loops within it. It is characterized by having the same first and last node.

How does mesh analysis help in obtaining the values of unknown currents?

Mesh analysis provides a systematic approach to set up equations based on Kirchhoff's Voltage Law (KVL) for each mesh, which when solved, yield the values of the unknown currents flowing through the network.

What are the steps involved in performing mesh analysis?

The steps involved in performing mesh analysis are: 1) Identify the total number of meshes in the network, 2) Assign mesh currents to each mesh, 3) Develop KVL equations for each mesh, and 4) Solve the KVL equations to find the mesh currents.

Why is mesh analysis only applicable to planar networks?

Mesh analysis is only applicable to planar networks because it relies on the ability to draw the network without any branches crossing each other. Non-planar networks, where branches cross, cannot be analyzed using mesh analysis due to the complexity introduced by the crossings.

Can the direction of mesh current be chosen arbitrarily?

Yes, the direction of mesh current can be chosen arbitrarily, but it is conventionally chosen as clockwise for convenience and to align with the typical current flow from the source on the left in the network.

What is the relationship between the number of meshes and the number of equations required in mesh analysis?

The number of equations required to solve an electrical network using mesh analysis is equal to the number of meshes, which can be calculated using the formula: number of meshes = number of branches - number of nodes + 1.

How do you handle the situation where multiple mesh currents pass through the same resistor?

When multiple mesh currents pass through the same resistor, the net current through the resistor is represented as the difference of the mesh currents, with the priority given to the mesh current for which the KVL equation is being written.

What is the significance of assigning clockwise direction to mesh currents?

Assigning a clockwise direction to mesh currents is significant because it typically aligns with the direction of current flow from the source, which helps in obtaining positive values for the currents and simplifies the analysis.

How can mesh analysis be used to calculate power loss in a resistor?

Once the mesh currents are determined through mesh analysis, the power loss in a resistor can be calculated using the formula: power loss = (current^2) * resistance, where 'current' is the current flowing through the resistor.

Outlines

00:00

🔍 Introduction to Mesh Analysis

The paragraph introduces mesh analysis as a method for analyzing electrical networks. It emphasizes the importance of understanding mesh analysis for determining the power delivered or absorbed by different electrical elements. The concept of a mesh is explained as a loop without any inner loops, and the steps for performing mesh analysis are outlined: identifying the number of meshes, assigning mesh currents, developing KVL equations for each mesh, and solving these equations. The paragraph also highlights that mesh analysis is only applicable to planar networks where no branch crosses another.

05:01

🧭 Assigning Mesh Currents and Developing KVL Equations

This paragraph delves into the specifics of assigning mesh currents, with a preference for the clockwise direction due to its convenience and alignment with the conventional current flow from the source. It explains the relationship between the number of meshes, branches, and nodes in a network, providing a formula to calculate the number of meshes. The process of developing KVL equations for each mesh is discussed, with an example illustrating how to account for the net current through shared resistors by prioritizing the mesh current of the mesh being analyzed.

10:04

🔧 Solving KVL Equations and Calculating Power Loss

The final paragraph demonstrates the application of mesh analysis through an example. It shows how to develop KVL equations for two meshes, taking care to prioritize the mesh current for the mesh being analyzed. The equations are then solved to find the mesh currents, and the power loss in a resistor is calculated using the current through it. The paragraph concludes with a summary of the key points to remember, such as considering the mesh current as the largest when drawing KVL equations, and emphasizes that more examples will be solved in future lectures for better clarity.

Keywords

💡Mesh Analysis

Mesh analysis is a method used in electrical engineering to analyze circuits, specifically to determine the current flowing through each mesh (loop) within a network. It is a systematic approach that involves applying Kirchhoff's Voltage Law (KVL) to each mesh. In the context of the video, mesh analysis is the central theme, with the script detailing its steps and applications. The video explains how mesh analysis is used to find unknown currents in a network, which is essential for calculating power delivered or absorbed by electrical elements.

💡Kirchhoff's Voltage Law (KVL)

Kirchhoff's Voltage Law (KVL) states that the algebraic sum of the voltages around any closed loop in a network is zero. This fundamental principle is used in mesh analysis to develop equations for each mesh. The video script illustrates how KVL is applied to each mesh to create equations that are then solved to find the mesh currents, showcasing its critical role in the analysis process.

💡Mesh Current

In the context of the video, mesh current refers to the current that flows around the perimeter of a mesh, which is a loop in a circuit that does not contain any other loops inside it. The script emphasizes that each mesh has its own unique mesh current, and these currents are essential for developing KVL equations. Assigning mesh currents is the second step in performing mesh analysis, as highlighted in the script.

💡Planar Network

A planar network is a type of electrical network where no branch crosses another, and the entire network can be drawn on a plane without any intersections. The video script specifies that mesh analysis is only applicable to planar networks, as non-planar networks with crossing branches complicate the analysis by not allowing for straightforward identification of meshes. This concept is crucial for determining the applicability of mesh analysis to a given network.

💡Loop

A loop, as mentioned in the script, is a part of an electrical network where the starting and ending nodes are the same. A mesh is a special type of loop that contains no other loops within it. Understanding loops is fundamental to mesh analysis because it helps in identifying the meshes within a network, which is the first step in the analysis process.

💡Voltage

Voltage is the electrical potential difference between two points in a circuit and plays a critical role in the operation of electrical networks. In the video, voltage is essential for calculating power, which is the product of voltage and current. Mesh analysis helps in determining the unknown currents in a network, which, along with voltage, allows for the calculation of power delivered or absorbed.

💡Current

Current is the flow of electric charge in a circuit, and it is a fundamental quantity in electrical engineering. The video script discusses how mesh analysis is used to find the unknown currents in a network. Current is essential for calculating the power delivered or absorbed by electrical elements, which is a primary goal of network analysis.

💡Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. Although not explicitly mentioned in the script, Ohm's Law is implicitly used in mesh analysis when calculating voltage drops across resistors, which is part of setting up the KVL equations for each mesh.

💡Power

Power in an electrical context refers to the rate at which electrical energy is transferred by an electric circuit. The video script discusses the importance of obtaining power delivered or absorbed by different electrical elements, which is why the analysis of currents and voltages in a network is crucial. Power calculations often follow the determination of currents using mesh analysis.

💡Resistor

A resistor is a passive component in an electrical network that opposes the flow of current. In the video script, resistors are used in the example circuit to create voltage drops, which are then used in the KVL equations. The resistance values are essential for calculating the voltage drop across each resistor and for determining the power dissipated in the resistors using the currents found through mesh analysis.

Highlights

Mesh analysis is used to determine unknown currents in electrical networks.

The main aim of analyzing an electrical network is to obtain power delivered or absorbed by different elements.

Mesh is defined as a loop without any inner loops.

Mesh analysis involves identifying the total number of meshes in a network.

Each mesh is assigned a mesh current that flows only around the perimeter of the mesh.

KVL equations are developed for each mesh to find the mesh currents.

Mesh analysis is applicable only for planar networks where no branch crosses another.

The direction of mesh current can be chosen as clockwise or anticlockwise, but clockwise is preferred.

The number of equations required for mesh analysis equals the number of meshes, which is calculated as the branch number minus node number minus one.

An example problem demonstrates how to identify meshes, assign currents, develop KVL equations, and solve for mesh currents.

The outer loop of a network is not considered a mesh if it contains inner loops.

When developing KVL equations, the net current through a shared resistor is the difference between the mesh currents.

The priority is given to the mesh current of the mesh for which the KVL equation is being written.

Solving the KVL equations yields the mesh currents, which can be used to calculate power loss in resistors.

The power dissipated in a resistor is calculated as the square of the current through it times the resistance.

Mesh analysis provides a systematic approach to solving complex electrical network problems.

Transcripts

we have completed KVL and KCl and now we

are going to understand what is mesh

analysis and how to perform the mesh

analysis and the first question is why

we use mesh analysis what is the use of

mesh analysis and to get the answer of

this question we need to understand the

very basic requirement to analyze any

electrical network when we analyze any

electrical network our main aim is to

obtain the power delivered or the power

absorbed by different electrical

elements and to have the power delivered

or the power absorbed we need the

voltage and we need the current and

using the mesh analysis we can have the

unknown currents in the electrical

Network so to obtain the values of

unknown currents in the electrical

Network we perform the mesh analysis so

let's begin our discussion and first we

will understand what is mesh and then I

will give you all these steps required

to perform the mesh analysis mesh is a

loop mesh is a loop and this loop is not

a normal loop this loop does not contain

any inner loop so whenever you have a

loop having no loops inside then you

will call the loop a mesh and we know

what is a loop loop is a part having the

first node and the last node same and I

hope from this definition you now know

what is a mesh and you can identify mesh

in a given network now we will move on

to the steps required to perform the

mesh analysis step number one is to

identify the total number of meshes you

will be given in network and in that

network you need to identify total

number of

Masha's we know what is a mesh it is a

loop having no loops inside so you

simply need to identify the total number

of such loops in the given network and

once we are done with identifying the

total number of meshes in step number

two we assigned the mesh currents mesh

current is the current that flows only

around the perimeter of a mesh so all

the meshes which we have identified will

have their individual mesh currents so

this is all four first step and the

second step let's move on to the third

step in the third step we are required

to develop the KVL equation for each

mesh and in the fourth step we need to

solve the KVL equation which we have

developed to find the mesh currents

which we assigned in step number two so

these are the four steps involved in

performing the mesh analysis and once we

have the mesh currents we can perform

the required calculations according to

the given question now before solving

one example and before implementing all

these steps there are few important

points which you need to know the first

point is mesh analysis is only

applicable for planar networks now what

is a planar network for example here we

have a network and in this network you

can see that this particular branch is

crossing this branch so this network is

non planar why because we don't have the

whole network placed on a plane why

because this branch is crossing this

branch so no branch should cross another

branch and when this happens we will

call the network cleaner network this

network and this network is same instead

of having

this particular branch crossing this

branch we have drawn it like this and

therefore we have a planar Network and

in this network we cannot obtain

different currents by using the mesh

analysis mesh analysis is not applicable

in this network but mesh analysis is

applicable in this network so this is

what do we mean by mesh analysis is only

applicable for planar networks this is

all for the first point according to the

second point the direction of the mesh

current can be clockwise or it can be

anti clock wise it is up to you which

direction you want to choose but I will

select the clock wise direction I will

not select the anti-clockwise direction

I will prefer clockwise direction there

are two reasons the first reason is it

is psychological having a clock wise

direction is more convenient to look at

and to handle as compared to a current

which is anti-clockwise and the second

important reason is generally in the

network for example in this network the

source is located on the left hand side

and therefore current will have this

direction and when you choose the

clockwise direction this direction is

same as this direction therefore you

will get the positive value of the

current and if you choose the

anti-clockwise direction the current

obtained will be negative therefore we

assign the clockwise direction to

different mesh currents let's move on to

the third point according to the third

point the number of equations required

to solve an electrical Network using the

mesh analysis is equal to the number of

meshes and the number of meshes is equal

to the branch

-

number of nodes minus one so remember

this formula is the number of equations

M is the number of meshes B is the

number of branches we are having in the

network n is the number of nodes and

when you perform B minus n minus 1 you

will have the number of meshes and it is

same as the number of equations required

to solve an electrical network using

mesh analysis so I hope the Third Point

is also clear to you now it is time to

solve one example problem using the mesh

analysis and we know in step number one

we identify the total number of meshes

so let's quickly identify the total

number of meshes in this circuit we have

1 & 2 meshes you cannot consider the

outer loop as a mesh because it is

having 2 different loops inside it so

according to step number one we have

total 2 meshes and if you look at step

number 2 you will find we now need to

assign the mesh currents and we know the

direction of the mesh current is

clockwise we will take the direction has

clockwise let's say in the first mesh

current i1 is flowing and the direction

of current i1 is clockwise and in the

second mesh current i2 is flowing and

the direction of current I 2 is also

clockwise so we are done with the second

step also and in the third step we are

required to develop the KVL equation for

each mesh and in the fourth step we are

required to solve the KVL

equations to find the mesh currents so

let's develop the KVL equations for the

two meshes and we will start with the

first mesh from this point moving in the

direction of the mesh current i1 when we

move in this direction we will have +10

volts plus 10 volts then we have current

i1 flowing through resistance having the

value 5 ohms therefore we will have

minus i1 multiplied to 5 or we can write

5 times i1 and the polarity will be plus

minus now we will move further and this

time current i1 is flowing through 5 ohm

resistor but I 2 is also flowing through

this resistor I 1 is flowing in this

direction from top to bottom and high 2

is flowing in this direction from bottom

to top now what will be the net current

i1 minus i2 or I 2 minus i1 this is the

most important point in this lecture we

are developing the KVL equation for the

first mesh and the first mesh is having

its mesh current as i1 therefore while

writing down the KVL equation for the

first mesh we will give the priority to

the mesh current of the first mesh that

is I 1 therefore we will consider the

net current to be i1 minus i2 i1 is

greater than I

two therefore we have minus five times

i1 minus i2 now moving forward

we will reach to the same point

therefore we will equate this sum

with zero now when you simplify this you

will have two times i1 minus i2 equal to

two let's call this equation number one

now let's develop the KVL equation for

the second mesh and we will start from

this point and you can see that when

moving in the same direction of i2 again

we have the same condition whether to

choose I 1 minus I 2 or I 2 minus I 1 as

the net current through five ohm

resistor and like I said earlier we give

priority to the mesh current whose KVL

equation we are writing down and this

time we are writing the KVL equation for

the second mesh and therefore we will

give the priority to current i2 and not

to current i1 we will assume current I 2

is greater than current i1 therefore we

will select this to be net current and

hence we have minus 5

i2 minus i1 in the first case and this

was the polarity of the voltage drop

across this resistor and in the second

case this is the polarity of the voltage

drop across this resistor and therefore

we have minus v i2 minus i1 after this

we have minus 10 multiplied to i2 minus

10 multiplied to i2 and this sum will be

equal to 0 now when you simplify this

you will have I 1 minus 3 times i2 equal

to 0 let's call this equation number 2

and this is time to solve the KVL

equations we have obtained in step

number 3 so in step number 4 we will

perform the operation equation 1

two times equation two and this will

give us current i2 equal to two over

five amperes and now we have the current

flowing through ten ohm resistor and we

are required to calculate the power loss

in the 10 ohm resistor so there is no

need to calculate current i1 however you

can easily calculate current i1 and just

put the value of current I 2 in equation

number two or in equation number one

but we will directly calculate the power

loss or the power dissipated it is equal

to the square of the current that is I 2

square multiplied to the resistance

corresponding to which we are

calculating the power loss or power

dissipated so we have 2 over v square

multiplied to 10 watts when you solve it

you will get 1.6 watts so this is the

answer and we have obtained it using the

mesh analysis so I hope mesh analysis is

clear to you the very important thing is

this particular point you have to

remember that while drawing the KVL

equation for a particular mesh you have

to consider the mesh current to be the

largest these points will become more

clear when we will solve more questions

in the coming lectures

[Applause]

[Music]