# Mesh Analysis

Mesh analysis is a technique used to find **currents** circulating around a loop or network in any closed path of a circuit.

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While Kirchhoff Laws give us the basic method for analyzing any complex electrical circuitry, there are different ways to improve this method when it comes to large networks and using **Mesh Current Analysis** or **Node Voltage Analysis,** which results in a decrease in the relevant mathematics. this decrease in mathematics can be a great advantage.

For example, consider the example of an electrical circuit in the previous section.

### Mesh Current Analysis Circuit

A simple method of reducing the amount of relevant mathematics is to analyze the circuit using Kirchhoff's Current Law equations to determine the currents I _{1} and I _{2} flowing in two resistances.Then there is no need to calculate I _{3} only as the sum of I _{1} and I _{2.}So Kirchhoff's second voltage law simply goes like this:

- Equation No 1: 10 = 50I
_{1}+ 40I_{2} - Equation No 2 : 20 = 40I
_{1}+ 60I_{2}

therefore, a row math calculation was recorded.

## Mesh Current Analysis

An easier method of solving the above circuit is to use **Mesh** Current Analysis or Cycle **Analysis,** sometimes referred to as **Maxwell's Circulatory Currents** method.Instead of tagging branch currents, we need to label each "closed loop" with a circulating current.

As a general rule, label internal loops clockwise only with circulating currents, since the goal is to cover all elements of the circuit at least once.Any required branch current can be found from the appropriate loops or network currents, as before using Kirchhoff's method.

For example: : i _{1} = I _{1} , ben _{2} = -I _{2} and I _{3} = I _{1} – I _{2}

Now we write Kirchhoff's equation of voltage law in the same way as before to solve them, but the advantage of this method is that the information obtained from circuit equations ensures that it is the minimum required to solve the circuit, since the information is more general and can be more general. can be easily put into a matrix form.

For example, consider the circuit in the previous section.

These equations can be solved quite quickly using a single network impedance matrix Z.Each element on the main diagonal will be "positive" and is the total impedance of each network.In cases where each element is OFF, the main diagonal will be either "zero" or "negative" and represents the circuit element that connects all the appropriate networks.

First, when dealing with matrices, we need to understand that the division of two matrices is the same as multiplying one matrix by the other, as shown.

As R, V/R is the same as V R x ^{-1,} which can now be used for two circulation currents.

Where:

- [ V ] gives the total battery voltage for loop 1 and then cycle 2
- [ I ] specifies the names of the loop currents we are trying to find
- [ R ] is the resistance matrix
- [R-
^{1}] is the inverse of the matrix [R]

and this gives I _{1} as -0.143 Amperage and I as _{2} as -0.429 Amperage

As : I _{3} = I _{1} – I _{2}

Therefore, the combined current of I _{3} is given as follows: -0.143 – (-0.429) = 0.286 Amperage

This is the same value of the 0.286 amp current that we found in the Kirchhoffs circuit law tutorial earlier.

## Summarize

This method of circuit analysis is probably the best of all circuit analysis methods, together with the basic procedure for solving **Mesh Current Analysis** equations:

**1.**Label all internal loops with circulating currents.( I_{1}, I_{2}, … I_{L}etc.)**2.**Type the column matrix [ V ] that gives the sum of all voltage sources in each loop.**3.**For all resistors in the circuit, type the [ L x L ] matrix, [ R ] as follows:- R
_{11}= total resistance in the first cycle. - R
_{nn}= N. total resistance in the loop. - Resistance that connects the R
_{JK}= J loop directly to the K loop.

- R
**4.**Type the matrix or vector equation [ V] = [R] x [I], where [I] is a list of currents to find.

In addition to using **Mesh Analysis** , we can also use node analysis to calculate the voltages around loops, which again reduces the amount of mathematics required using only Kirchoff laws.In the next lesson on DC circuit theory, we will look at Node Voltage Analysis to do just that.