Mesh current analysis calculates currents in closed loops of a circuit by assigning hypothetical loop currents and using Kirchhoff's Voltage Law.
Although the basis for evaluating any electrical circuit is provided by Kirchhoffs Current Law (KCL) and Voltage Law (KVL), other methods such as Mesh Current Analysis and Nodal Voltage Analysis are also available.
These techniques frequently result in a decrease in mathematical complexity, especially when working with complicated and big network. This ease of use can be very beneficial for effective circuit analysis.
Circuit Analysis Using Mesh Currents
Let's take for example the electrical circuit which was discussed in our previous article:
One innovative technique to reduce the number of calculations is to determine the resistors I1 and I2 currents using Kirchhoff's Current Law. We dont need to do a separate step to solve for the total current (I3) because it is simply the sum of I1 and I2. Kirchhoff's voltage law may thus be reduced to the following two equations:
- Eq. No 1 : 9 = 45I1 + 36I2
- Eq. No 2 : 18 = 36I1 + 54I2
In this manner, we spare ourselves one whole line of mathematical calculation.
Using Mesh Current Analysis
Mesh Current Analysis, sometimes referred to as Loop Analysis, is a less complicated technique that can be used in place of measuring the current in each branch of the circuit individually.
In this case, branch currents are disregarded in favor of closed loops inside the circuit. For every loop, we designate a "circulating current" that usually flows in a clockwise manner.
It is important to ensure that every component in the circuit has at least one loop covering it. We can later locate any specific branch current we require once we have these loop currents.
It's similar to figuring out the individual currents by using the loop currents as building blocks. Although we use Kirchhoff's Laws with loop currents rather than branch currents, they are still applicable.
For example, branch i1's current would just be the loop current I1, while branch i2's current would be the loop current I2's negative (i2 = -I2). Likewise, branch I3 current is determined by deducting loop current I2 from loop current I1 (I3 = I1 - I2).
Like previously, this approach makes use of Kirchhoffs voltage law, but it has a significant advantage. The bare minimum of information is required to solve the circuit.
This is a result of the more universal loop currents that we employ, which are simple to arrange into a tidy table (matrix) for problem-solving.
As an illustration, consider the circuit from our earlier section.
An effective method of solving these equations is to use a unique table known as a "mesh impedance matrix" (Z). Essential circuit information is contained in this table.
Positive elements may be found along the primary diagonal, which runs from top left to bottom right. These show the overall impedance of every circuit loop, or mesh.
Element outside the primary diagonal may be negative or zero. They serve as a representation of the impedance and connections between various loops.
It is noteworthy that while working with matrices, dividing by one matrix is equivalent to multiplying by its "inverse." In essence, the actions carried out by the original matrix are reversed by its inverse.
[ V ] = [ I ] * [ R ] or [ R ] * [ I ] = [ V ]
We may utilize the unique form of R, denoted as R-1, that we have discovered to find the values of the two circulating currents. Recall that V multiplied by something else equals V / R. That "something else" is represented by R-1, and it quickly helps in identifying the currents.
In the above calculations:
- [ V]: This indicates the combined voltage of all the batteries in every loop, such as loops 1 and 2.
- [I]: The currents flowing around each loop (the ones we are attempting to solve for) are named here.
- [ R]: This is identical to a table that contains data for each resistance in the circuit.
- [ R-1 ]: This is a modified version of the resistance table ([ R ]) that facilitates the easier calculation of the loop currents. It functions somewhat like a shortcut.
This allows us to determine the currents around each loop (I1 and I2). With a value of -0.143 Amps, the calculations indicate that I1 is flowing in the other direction from what we first thought. With a value of -0.429 Amps, I2 is likewise flowing in the opposite direction.
Recall how we were interested in the total current (I3). Well, we can locate it with ease using this strategy. To put it simply, I3 equals I1 minus I2. After entering the figures, we obtain 0.286 amps, which is equivalent to: -0.143 – (-0.429) = 0.286 Amps
This is the exact same 0.286 amp current that Kirchhoff's Current Law allowed us to find previously! It demonstrates that while both approaches get the same result, this one could be a little quicker.
Conclusions:
Mesh Current Analysis is frequently regarded as one of the most effective techniques for circuit analysis. The main stages involved are broken out as follows:
Determine Loops and Give Currents to Them: Give a distinct current variable (I1, I2, etc.) to each independent loop in the circuit. The current flowing through each of these loops is represented by these currents.
Sources of Voltage in Loops: Make a column matrix of L rows (referred to as [V]), where L is the number of loops. Within each loop, each member in this matrix indicates the total of all the voltage sources.
Constructing the Resistance Grid: Create a square matrix with dimensions of L x L, designated as [R]. The resistance data of the circuit is captured by this matrix:
- The total resistance inside each loop is represented by the diagonal elements (R11, R22, etc.).
- Loops J and K directly share resistance, which is represented by off-diagonal elements (Rjk).
Finding Loop Current Solutions: Write the matrix equation [V] = [R] x [I], where the unknown loop currents are represented by the column matrix [I]. You may solve for each loop current in the circuit using this equation.
Mesh Current Analysis deals with finding loop currents quickly, whereas Nodal Voltage Analysis, which will be discussed in the upcoming tutorial, deals with figuring out voltages at certain circuit locations. When compared to depending only on Kirchhoff's Laws, both methods provide methods for circuit analysis using less complicated arithmetic.
References: Mesh analysis