While Kirchhoff’s Laws provide the fundamental approach for analyzing complex electrical circuits, alternative methods such as Mesh Current Analysis or Nodal Voltage Analysis offer enhancements by reducing the mathematical complexity involved. This reduction in mathematical computations can prove advantageous, particularly in managing large networks.
Mesh Current Analysis Circuit
A straightforward approach to minimize mathematical complexity is to utilize Kirchhoff’s Current Law equations for analyzing the circuit, thereby determining the currents I1 and I2 flowing through the two resistors. Consequently, there’s no necessity to compute the current I3, as it’s merely the summation of I1 and I2.
Thus, Kirchhoff’s second voltage law can be simplified accordingly:
- Equation No 1 : 10 = 50I1 + 40I2
- Equation No 2 : 20 = 40I1 + 60I2
therefore, one line of math calculation has been saved.
Mesh Current Analysis
A simpler approach to solving the given circuit involves utilizing Mesh Current Analysis or Loop Analysis, also known as Maxwell’s Circulating Currents method at times. Instead of assigning branch currents, each “closed loop” is labeled with a circulating current.
As a guiding principle, circulating currents within loops are labeled clockwise to ensure comprehensive coverage of all circuit elements at least once. Any required branch current can then be deduced from the appropriate loop or mesh currents using Kirchhoff’s method, as done previously.
For instance: i1=I1, i2=−I2, and I3=I1−I2.
Subsequently, Kirchhoff’s voltage law equation is formulated similarly to previous approaches for solving them. However, this method offers the advantage of ensuring that the circuit equations yield the minimum necessary information for solving, as the information is more general and readily translatable into matrix form.
For example, consider the circuit from the above section.
To efficiently solve these equations, we can employ a single mesh impedance matrix �Z. On the principal diagonal, each element is “positive” and denotes the total impedance of each mesh. Conversely, elements of the principal diagonal are either “zero” or “negative,” signifying the circuit element connecting the relevant meshes.
Initially, it’s essential to comprehend that when dealing with matrices, dividing two matrices is equivalent to multiplying one matrix by the inverse of the other, as demonstrated:
Once the inverse of R is determined, V/R becomes equivalent to V×R−1, enabling us to calculate the two circulating currents.
Here’s the breakdown:
- [ V ] represents the total battery voltage for loop 1 and loop 2.
- [ I ] denotes the names of the loop currents that we aim to determine.
- [ R ] stands for the resistance matrix.
- [ R-1 ] signifies the inverse of the [ R ] matrix.
Subsequently, the calculation yields 1 as -0.143 Amps and I2 as -0.429 Amps.
Given that I3=I1−I2, the combined current I3 = −0.143−(−0.429)=0.286 Amps.
This matches the previously obtained value of 0.286 amps current from Kirchhoff’s circuit law tutorial.
Mesh Current Analysis Summary
In summary, the “look-see” method of circuit analysis stands out as one of the most effective techniques among various circuit analysis methods. The fundamental steps for solving Mesh Current Analysis equations are outlined below:
- Assign circulating currents (I1, I2, …IL, etc.) to all internal loops.
- Compile the [L x 1] column matrix [V], representing the sum of all voltage sources within each loop.
- Construct the [L x L] matrix [R] to encompass all resistances in the circuit:
- R11 signifies the total resistance in the first loop.
- Rnn denotes the total resistance in the Nth loop.
- RJK represents the resistance directly linking loop J to loop K.
- Formulate the matrix or vector equation [V] = [R] × [I], where [I] comprises the currents to be determined.
In addition to Mesh Current Analysis, node analysis serves as another method for calculating loop voltages, thereby further simplifying mathematical computations through Kirchhoff’s laws alone. In the subsequent tutorial focusing on DC circuit theory, we’ll delve into Nodal Voltage Analysis to achieve this goal.
| Read more Tutorials on DC Circuits | |
|---|---|
| 1. | DC Circuit Theory |
| 2. | Ohms Law and Power |
| 3. | Electrical Units of Measure |
| 4. | Kirchhoffs Circuit Law |
| 5. | Kirchhoffs Current Law |
| 6. | Kirchhoffs Voltage Law |
| 7. | Mesh Current Analysis |
