7.1 Mesh Analysis
LEARNING OBJECTIVES
- Write simultaneous linear equations in matrix form
- Apply techniques of linear algebra to solve for current vector
- Solve linear circuits by advanced mesh analysis methods
Kirchhoff’s rules of circuit analysis—at any junction, the sum of incoming currents equals the sum of outgoing currents; and, the potential difference around a closed loop is zero—are the necessary and sufficient conditions for determining the current through each component of any linear circuit—that is, any circuit composed only of elements that respond linearly to changes in current or voltage (such as Resistors, Capacitors and Inductors). Thus, the problem-solving strategy you learned in Kirchhoff’s Rules is sufficient to analyse any such circuit. However, that approach is inefficient and its steps are prone to human error—problems we’d like to mitigate in particular when applying the basic principles of circuit analysis to more complex circuits.
The mesh analysis techniques you learn in this section will not necessarily advance your knowledge of the physics of electrical circuits. Rather, the objective of this section is to provide a simpler and more straightforward approach to the analysis of planar circuits (also known as “meshes”). By the end of this section, you will see that it is possible to determine the current in all branches of a circuit in two easy steps: read off the components of a vector and a matrix simply by observation; solve for the vector of mesh currents . But before this useful result can be realised, we must first explore a few nuanced points to explain why the technique works.
We begin by modifying the problem-solving strategy from Kirchhoff’s Rules, adding a few particular requirements and removing some unneeded steps.
Problem-Solving Strategy: Mesh Analysis
- Draw mesh current loops, ensuring:
- each loop is unique; and
- all circuit elements—voltage sources, resistors, capacitors, inductors, etc. and short circuits—are covered by at least one loop.
- Apply loop rule as described in Kirchhoff’s Rules (particularly with reference to Figure 6.3.5) and solve simultaneous equations.
- Add or subtract mesh currents in branches that are covered by multiple loops, depending on the direction of each loop and the sign of each current calculated in step 2.
Referring back to the problem-solving strategy described in Kirchhoff’s Rules, you should see that we have removed three steps and replaced them with step 3 here. Rather than defining arbitrary currents at each junction, then using the loop rule to eliminate some of the current variables, we now skip straight to defining current loops, apply the loop rule, and solve directly for each mesh current. Then, as these may turn out to be positive or negative depending on the direction of the loops chosen arbitrarily in step 1, we add or subtract values as appropriate in branches covered by multiple mesh currents to find the actual current through each.
The tradeoff results