A circuit has two clockwise mesh currents I₁ and I₂ sharing a resistor R. What is the actual current flowing through R from the perspective of mesh 1?
AI₁ + I₂, because both mesh currents pass through the shared resistor
BI₁ − I₂, because the two mesh currents flow in opposite directions through the shared branch
CI₁ alone, because I₂ is a fictitious variable that doesn't affect the branch
DThe average (I₁ + I₂)/2, since the resistor is shared equally
When two clockwise mesh currents share a branch, they flow in opposite directions through it — I₁ flows one way, I₂ flows the other. The actual branch current is the algebraic difference I₁ − I₂ (or I₂ − I₁, depending on the reference direction). This is a key consequence of how mesh currents work: they are not physical currents but mathematical variables. The actual current in any branch is found by summing — with appropriate signs — all mesh currents that pass through it.
Question 2 Multiple Choice
A 3 A current source lies on the boundary between mesh 1 and mesh 2. How should mesh analysis handle this?
AAssign the voltage across the current source as an unknown and include it in the KVL equation for each mesh
BIgnore the current source — ideal current sources have zero resistance and don't appear in KVL
CWrite KVL around the outer perimeter of both meshes combined (skipping the current source branch), then add the constraint I₁ − I₂ = 3 A
DSet I₁ = 3 A and I₂ = 0, since the source fixes the current in one mesh
An ideal current source has an unknown voltage across it, so you cannot write KVL around a loop that includes it without introducing another unknown. The supermesh technique avoids this: merge the two meshes into one by going around the outer perimeter (skipping the branch with the current source) to write one KVL equation. Then add the constraint equation I₁ − I₂ = 3 A (or its negative), which the current source directly specifies. Together these give the same number of equations as unknowns.
Question 3 True / False
Mesh currents are fictitious variables — they are not directly measured anywhere in the physical circuit.
TTrue
FFalse
Answer: True
True. Mesh currents (I₁, I₂, ...) are mathematical abstractions assigned to the enclosed loops of a planar circuit. They have no single physical wire where you could place an ammeter and read off I₁. The physically measurable quantity — actual branch current — is obtained by algebraically summing all mesh currents sharing that branch. Their value as a technique lies in producing a systematic, minimal set of equations, not in having direct physical meaning.
Question 4 True / False
Mesh analysis can be applied to any circuit, including those whose wires is expected to cross when drawn on a flat surface.
TTrue
FFalse
Answer: False
False. Mesh analysis requires the circuit to be planar — drawable on a flat surface without any wires crossing. In a non-planar circuit, the concept of an 'enclosed mesh region' breaks down because the topology does not admit a consistent set of independent loops in the plane. For non-planar circuits, nodal analysis (which works on any circuit) is the correct systematic method.
Question 5 Short Answer
Explain why a current source between two meshes prevents writing a standard KVL equation around either mesh, and describe how the supermesh technique resolves this problem.
Think about your answer, then reveal below.
Model answer: KVL requires summing known or expressible voltage drops around a loop. An ideal current source fixes the current through a branch but leaves the voltage across it as unknown and unconstrained — it adjusts to whatever the circuit requires. Including this unknown voltage in a KVL equation adds a variable without adding a useful equation. The supermesh resolves this by forming a larger loop that goes around the outside of both meshes, bypassing the current source branch entirely. This produces one KVL equation free of the unknown voltage. The missing equation is replaced by the current source's own constraint: I₁ − I₂ = I_source, which directly relates the two mesh currents.
The supermesh is not a workaround — it is the correct application of KVL to a loop that avoids the problematic branch. The number of independent equations is preserved: two meshes normally give two equations; a supermesh gives one combined KVL equation plus one constraint equation, still totaling two equations for two unknowns.