Mesh-current analysis
For a planar circuit, choose one mesh current per elementary loop, express shared-branch currents as differences of mesh currents, and let Kirchhoff's voltage law generate the coupled equations.
- Distinguish a mesh from a general loop and state the planarity requirement for direct mesh analysis.
- Choose mesh-current reference directions without claiming they are physical branch currents everywhere.
- Express the current of a shared resistor as the algebraic difference of adjacent mesh currents.
- Construct mesh equations systematically from Kirchhoff's voltage law and Ohm's law.
- Solve a two-mesh resistive network and recover all physical branch currents and node voltages.
- Recognize the resistance-matrix structure and verify the result against nodal analysis and power balance.
Can the same physical circuit be solved systematically without choosing node voltages as the primary unknowns?
Yes—when the circuit is planar, mesh-current analysis chooses one current per elementary mesh and organizes Kirchhoff's voltage law (KVL). The method is not a competing physical theory. It is a different coordinate system for the same network. We will solve exactly the circuit from chapter 02 so the two methods can be checked against each other.
04.1A mesh is more specific than a loop
Figure 4.1 — Two elementary meshes in the same planar network
The left mesh contains the source, R1 and R2. The right mesh contains R2, R3 and R4. R2 belongs to both meshes and is therefore the shared branch.
A loop is any closed path through the circuit graph. A mesh is an elementary loop of a planar representation: it contains no other loop inside it. Figure 4.1 has two meshes, left and right.
Mesh-current analysis in this direct form is a method for planar circuits—circuits that can be drawn on a plane without branch crossings that are not electrical connections. General loop-current methods exist beyond this setting, but the clean “one current per mesh” construction relies on planarity.
For this network the two mesh currents will be the only unknowns. Once they are known, every resistor branch current follows.
04.2Mesh currents are chosen mathematical references
Figure 4.2 — Positive references for the two mesh currents
Both mesh currents are chosen clockwise. On each mesh's unique upper branch, the mesh-current direction is the physical branch-current direction.
Choose I1 clockwise in the left mesh and I2 clockwise in the right mesh. Clockwise is convenient, not physically privileged. Choosing either mesh current counter-clockwise would also work if every voltage drop were written consistently.
A branch belonging to only one mesh has branch current equal to that mesh current. Therefore, with the chosen references:
The shared branch R2 is different because both mesh currents pass through it in opposite directions.
04.3The shared resistor carries a difference of mesh currents
Figure 4.3 — The two mesh currents oppose each other in the shared branch
The arrow defines the physical R2 current from node A down toward ground. With both mesh currents clockwise, I1 crosses R2 downward while I2 crosses it upward.
Define the physical current through R2 as positive downward from node A to ground. The left clockwise mesh current crosses R2 downward; the right clockwise mesh current crosses R2 upward. Therefore
If the solved value is negative, the actual R2 current simply flows upward. Again, the reference direction is not a prediction.
This difference is the mesh analogue of the mutual voltage-difference term in nodal analysis: coupling appears because one physical element belongs to two chosen coordinates.
04.4Write Kirchhoff's voltage law around the left mesh
Traverse the left mesh clockwise. The 10 V source is a rise, while the resistor terms are voltage drops in the mesh-current direction. KVL gives
Rearrange by mesh currents:
For R1=R2=1 kΩ and VS=10 V:
The negative coupling term came from the shared resistor current I1−I2; it was not inserted by a memorized matrix rule.
04.5Write Kirchhoff's voltage law around the right mesh
Traverse the right mesh clockwise. R3 and R4 carry I2. Through R2, the right-mesh traversal is upward, so the branch current in that traversal direction is I2−I1:
Collect terms:
For all resistors equal to 1 kΩ:
04.6Solve the two mesh equations
The compact numerical form above uses kΩ·mA=V. From the second equation, I1=3I2. Substitute into the first:
Recover the shared branch:
Thus the physical resistor currents are 6 mA through R1, 4 mA downward through R2, and 2 mA through R3 and R4.
04.7Recover node voltages and compare with nodal analysis
Ground is the reference. The current through R2 is 4 mA downward through 1 kΩ, so node A is
The current through R4 is 2 mA downward through 1 kΩ, so node B is
Chapter 02 obtained exactly VA=4 V and VB=2 V by nodal analysis. Two different coordinate choices have reconstructed the same branch currents and node voltages.
04.8The mesh equations also form a resistance matrix
The diagonal term for a mesh is the sum of resistances around that mesh. The mutual term is minus the resistance shared by the two meshes when both mesh currents have been assigned the same rotational sense.
In compact notation one may write
Here R denotes the mesh-resistance matrix, i the column vector of mesh-current unknowns, and v the column vector of known voltage-source excitations. The dimensional check is
As in nodal analysis, the matrix is organized Kirchhoff bookkeeping, not a new physical principle.
04.9Power balance and when mesh analysis is attractive
Using the recovered physical branch currents:
The resistors absorb 60 mW. The source delivers
Mesh analysis is especially attractive when a planar circuit has relatively few meshes, many voltage sources, or when the desired outputs are branch currents. Nodal analysis is often more natural when the circuit has relatively few essential nodes or many current-source connections.
This is a method-selection heuristic, not a law. Chapter 05 will compare the two systematically.
04.10Current sources, common errors and mastery practice
An ideal current source on a branch belonging to only one mesh can directly fix that mesh current, subject to the source arrow. An ideal current source shared by two meshes is more subtle because its voltage is unknown, so an ordinary KVL equation through that branch is inconvenient. Chapter 05 introduces the supermesh.
- For the same all-1 kΩ network, change the source to 15 V. Find both mesh currents and the shared-branch current.
- Keep the 10 V source but make R2=2 kΩ while R1=R3=R4=1 kΩ. Write and solve the two mesh equations.
- For the original network, choose the right mesh current counter-clockwise instead of clockwise. Rewrite the shared-branch relation carefully.
- Write the resistance matrix for R1=2 kΩ, R2=3 kΩ, R3=1 kΩ, R4=4 kΩ without solving it.
- Why does a current source shared by two meshes motivate a supermesh?
- Give one reason to prefer nodal analysis and one reason to prefer mesh analysis for a particular circuit.
Solutions
1. Linearity scales every current by 1.5: I1=9 mA, I2=3 mA and the downward R2 current is 6 mA.
2. The equations are 3I1−2I2=10 mA and −2I1+4I2=0. The second gives I1=2I2; substitution gives 4I2=10 mA, so I2=2.5 mA and I1=5 mA. The downward R2 current is 2.5 mA.
3. If the right mesh is counter-clockwise, both selected mesh currents cross R2 downward. The downward branch current becomes I1+I2, but several KVL signs change accordingly.
4. The matrix is [[5,−3],[−3,8]] kΩ for clockwise currents in both meshes.
5. The current source voltage is not prescribed, so traversing it in a direct KVL equation introduces an extra unknown voltage. Combining the adjacent meshes avoids writing that internal voltage while a separate current constraint relates the mesh currents.
6. Nodal analysis is often attractive with few unknown nodes or many current-source branches; mesh analysis is often attractive for planar circuits with few meshes or many voltage sources. Count unknowns and inspect source placement rather than following a universal rule.
The final chapter will treat supermeshes, then put nodal and mesh analysis side by side with an explicit method-selection and verification workflow.