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Mesh analysis

Mesh analysis, also known as the mesh current method, is a fundamental technique in electrical circuit theory for analyzing linear planar circuits by applying Kirchhoff's voltage law (KVL) around closed loops called meshes.^[1]^[2] In this method, arbitrary directions are assigned to circulating currents in each mesh, and a system of linear equations is derived from KVL to solve for unknown mesh currents, from which branch currents and voltages can be determined using Ohm's law.^[3]^[1] The process begins with identifying the meshes in a circuit diagram—defined as the smallest loops that do not enclose other loops—and labeling each with a mesh current, typically in a consistent direction such as clockwise.^[2] Voltage drops across resistors are expressed in terms of these mesh currents; for shared branches between adjacent meshes, the net current is the algebraic sum (e.g., I_1 - I_2) if currents oppose each other.^[3] KVL is then applied to each mesh by summing the voltages around the loop—treating voltage sources as positive or negative based on direction relative to the current, and resistor drops as I \times R—resulting in equations set equal to zero.^[1] These simultaneous equations can be solved algebraically for simple circuits or using matrix methods like [V] = [R][I] for more complex networks, where [R] is the resistance matrix incorporating self and mutual resistances.^[3] Mesh analysis is particularly advantageous for circuits with fewer meshes than nodes, as it generates fewer equations compared to nodal analysis, which relies on Kirchhoff's current law (KCL) at nodes.^[2] It is well-suited for planar circuits—those drawable on a plane without crossing conductors—and extends to supermesh cases involving current sources or dependent sources by adjusting the equations accordingly.^[4]^[5] This method simplifies calculations in multi-loop networks, making it a staple in engineering education and practical circuit design for determining power dissipation, voltage regulation, and component stresses.^[3]

Fundamentals

Meshes and Loops

In electrical circuit theory, a mesh is defined as a closed path in a planar circuit that does not enclose any other closed paths within it.^[6] This topological element forms the basis for mesh analysis, where such paths allow for the systematic application of circuit laws. In contrast, a loop is any closed path formed by circuit branches, which may enclose one or more meshes.^[7] Mesh analysis primarily applies to planar circuits, which are those that can be drawn on a single plane such that no branches cross except at connection nodes.^[8] This planarity ensures that meshes can be identified without ambiguity, avoiding the complications of overlapping paths in non-planar configurations. The method assumes circuits composed of basic elements, including resistors, independent voltage sources, and independent current sources.^[9] To illustrate mesh identification, consider a simple planar circuit with three branches forming a single triangular loop: one branch with a resistor and voltage source in series, another with a resistor, and the third connecting them with another resistor. This entire loop constitutes a single mesh, as no smaller closed paths exist inside it.^[10] In a more complex example, two such triangular meshes sharing a common branch create two distinct meshes, each defined by its perimeter without enclosing the other.

Essential Meshes and Currents

In mesh analysis, essential meshes, also known as fundamental meshes, represent the minimal set of independent loops in a planar circuit that collectively span all branches without redundancy or enclosure of one loop within another. These meshes form the foundational structure for applying the mesh-current method, enabling the reduction of complex circuits to a set of independent current variables.^[11]^[12] The number of essential meshes in a connected circuit equals b - n + 1, where b is the total number of branches and n is the total number of nodes. This formula arises from basic graph theory applied to planar circuits: the circuit can be modeled as a connected planar graph, where the number of independent loops (meshes) follows from Euler's formula for planar graphs, v - e + f = 2, with v = n (vertices as nodes), e = b (edges as branches), and f as the number of faces (including the outer infinite face); solving for the internal faces (meshes) yields f - 1 = b - n + 1.^[11]^[12] To identify essential meshes, begin with the leftmost closed loop in the circuit diagram and proceed rightward, selecting each subsequent loop such that it shares at most one branch with previously chosen meshes while avoiding any loops that retrace paths or enclose other selected loops. This systematic approach ensures independence and complete coverage of all branches, typically resulting in the exact number given by b - n + 1.^[11]^[12] Mesh currents are assigned to each essential mesh as circulating currents, denoted sequentially as I_1, I_2, \dots, I_m where m = b - n + 1, with an arbitrary but consistent direction—conventionally clockwise—to simplify the application of Kirchhoff's laws. These currents represent the direct flow through elements unique to a single mesh and contribute to the net flow in shared branches.^[11]^[12] The current in any branch is the algebraic sum of the mesh currents from all adjacent essential meshes, taking into account their directions: for a branch shared between two meshes, the net current is I_k - I_l if the currents oppose each other, or I_k + I_l if they align. This superposition principle allows all branch currents to be expressed in terms of the mesh currents once the latter are solved.^[11]^[12]

Equation Formulation

Applying Kirchhoff's Voltage Law

Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all voltages around any closed loop in a circuit is zero.^[6] In mesh analysis, this law is applied to each essential mesh, where the sum of voltage drops across resistors and the effects of voltage sources within the mesh must equal zero.^[13] Essential meshes are independent loops defined by the circuit's topology, and mesh currents are assigned to these loops, typically in a clockwise direction for consistency.^[14] For each mesh, the voltage drop across a resistor is determined by the resistance multiplied by the net current flowing through it, which is the algebraic sum of the mesh currents passing through that resistor.^[6] If a resistor is shared between two meshes, the net current is the difference between the two mesh currents (e.g., I_m - I_n) if they flow in opposite directions through the resistor.^[14] Independent voltage sources within the mesh are incorporated directly into the equation, added as positive if the source's voltage rise aligns with the direction of the mesh current traversal, or negative if it opposes it.^[13] The sign convention in mesh analysis treats voltage drops across elements in the direction of the mesh current as positive and voltage rises (from sources opposing the current) as negative.^[6] This ensures that traversing the loop in the direction of the mesh current yields the correct polarity for each term. For the m-th mesh, the general KVL equation takes the form:

\sum_{k=1}^{M} Z_{mk} I_k + \sum V_{s,m} = 0

where M is the number of meshes, Z_{mk} represents the mutual impedance between meshes m and k (with Z_{mm} as the self-impedance of mesh m), I_k is the current in mesh k, and \sum V_{s,m} is the net voltage contribution from independent sources in mesh m, signed according to the convention.^[14] This formulation captures the voltage balance without explicitly solving for currents at this stage.^[13]

Calculating Mesh Impedance Coefficients

In mesh analysis, the impedance coefficients form the left-hand side of the system of equations derived from Kirchhoff's voltage law applied to each mesh. These coefficients represent the voltage drops due to currents in the meshes and are categorized as self-impedance and mutual impedance terms. The self-impedance Z_{mm} for mesh m is the sum of all impedances within that mesh, accounting for the total ohmic drop caused by I_m alone.^[3]^[15] The mutual impedance Z_{mk} (where m \neq k) arises from shared branches between meshes m and k, equal to the impedance of the shared branch, with the sign determined by the relative directions of I_m and I_k through that branch: negative if the currents flow in opposite directions (resulting in a subtractive effect in the voltage drop) and positive if in the same direction (additive effect).^[3]^[15] This convention ensures the equations correctly capture the interaction between adjacent meshes. To compute these coefficients systematically, traverse each branch in mesh m and sum the voltage drops Z \times (net current through the branch from all contributing meshes). For branches unique to mesh m, the term is Z \cdot I_m; for shared branches with mesh k, it includes \pm Z \cdot I_k based on the direction convention. Voltage sources within a mesh do not contribute to the impedance coefficients; instead, they are transferred to the constant term on the right-hand side of the equation, with the sign depending on the assumed current direction (positive if opposing the current, negative if aiding it).^[3]^[15] Consider a two-mesh circuit where mesh 1 has resistors R_1 (unique) and R_2 (shared), and mesh 2 has R_2 (shared) and R_3 (unique), with mesh currents I_1 and I_2 both clockwise. The self-impedance for mesh 1 is Z_{11} = R_1 + R_2. The mutual impedance term from mesh 2 in the equation for mesh 1 is Z_{12} = -R_2, since the currents flow in opposite directions through the shared R_2. Similarly, Z_{22} = R_2 + R_3 and Z_{21} = -R_2. If a voltage source V is in mesh 1 opposing I_1, it appears as +V in the constant term for that equation.^[3]^[15]

Solution Approaches

Simultaneous Linear Equations

Once the mesh equations have been formulated by applying Kirchhoff's voltage law around each essential mesh, as described in prior sections, they result in a system of simultaneous linear equations of the form R \mathbf{I} = \mathbf{V}, where R is the resistance matrix containing self- and mutual-resistance coefficients, \mathbf{I} is the vector of unknown mesh currents, and \mathbf{V} is the vector of voltage sources driving the meshes.^[6]^[16] For small systems, such as 2x2 or 3x3 matrices typical in basic circuits, these equations can be solved using standard algebraic techniques like substitution, elimination (e.g., Gaussian elimination), or Cramer's rule, which leverages determinants to find each unknown directly.^[16] Consider a two-mesh resistive circuit yielding the equations:

R_{11} I_1 + R_{12} I_2 = V_1

R_{21} I_1 + R_{22} I_2 = V_2

To solve via substitution, isolate one variable from the first equation, say I_2 = \frac{V_1 - R_{11} I_1}{R_{12}}, and substitute into the second equation to obtain a single equation in I_1, which is then solved explicitly; back-substitution yields I_2.^[16] Alternatively, using elimination, multiply the first equation by R_{21} and the second by R_{11}, then subtract to eliminate I_1, solving the resulting equation for I_2 before back-substituting for I_1.^[16] For Cramer's rule, compute the determinant of the coefficient matrix \Delta = R_{11} R_{22} - R_{12} R_{21}; then I_1 = \frac{\Delta_1}{\Delta} where \Delta_1 replaces the first column with [V_1, V_2]^T, and similarly for I_2.^[16] These methods assume the circuit is linear and time-invariant, with elements obeying Ohm's law, and for DC analysis, no initial conditions or transient effects are considered.^[17] To verify the solutions, substitute the computed mesh currents back into the original KVL equations and confirm that each balances to zero (or equals the source voltage as appropriate).^[16]^[6]

Matrix Representation

Mesh analysis lends itself to a compact matrix representation that systematically organizes the Kirchhoff's voltage law equations for multiple meshes. The resulting linear system is expressed as \mathbf{R} \mathbf{I} = \mathbf{V}, where \mathbf{R} is the resistance matrix, \mathbf{I} is the vector of mesh currents, and \mathbf{V} is the vector of equivalent voltage sources. This formulation is particularly advantageous for computational implementation, as it transforms the problem into solving a single matrix equation.^[9] The resistance matrix \mathbf{R} is square, with dimensions equal to the number of meshes. Its diagonal elements R_{ii} represent the self-resistance of mesh i, which is the sum of all resistances within that mesh. The off-diagonal elements R_{ij} (for i \neq j) denote the mutual resistance between meshes i and j, typically the negative sum of resistances shared between them, assuming opposite current directions in the shared branches. For resistive circuits, \mathbf{R} is symmetric, satisfying R_{ij} = R_{ji}, due to the reciprocal nature of resistive elements.^[9]^[18] The source vector \mathbf{V} has entries V_k that equal the algebraic sum of all independent voltage sources in mesh k, with signs determined by their polarity relative to the assumed direction of the mesh current. If no voltage sources are present in a mesh, the corresponding entry is zero.^[18] The mesh currents are obtained by solving \mathbf{I} = \mathbf{R}^{-1} \mathbf{V}. For practical computation, especially in larger circuits, methods such as Gaussian elimination or LU decomposition are employed to invert \mathbf{R} or directly solve the system. This matrix approach enhances computational efficiency and scalability, facilitating integration with software tools like MATLAB for numerical solutions or SPICE for circuit simulation. Furthermore, the determinant of \mathbf{R} relates to the circuit's input resistance in specific configurations, such as two-port networks where it influences parameters like r_{11}.^[9]^[19]

Special Cases

Supermeshes for Current Sources

In standard mesh analysis, currents are assumed to circulate freely within each defined mesh, allowing independent assignment of mesh current values. However, an ideal current source shared between two adjacent meshes imposes a fixed current through the branch connecting them, violating this assumption and complicating the application of Kirchhoff's voltage law (KVL) due to the unknown voltage across the source.^[20]^[21] To address this, a supermesh is formed by combining the two adjacent meshes that share the current source into a single larger loop, effectively treating them as one entity for KVL application while excluding the current source branch itself.^[22]^[23] This approach reduces the total number of independent meshes by one, as the supermesh replaces the two original loops. The key properties of a supermesh include the absence of its own dedicated current (it relies on the mesh currents of the combined loops) and the need for both KVL and Kirchhoff's current law (KCL) to fully define the system.^[21]^[20] The procedure for incorporating a supermesh begins by identifying all meshes in the planar circuit and labeling clockwise mesh currents for consistency. For meshes sharing an independent current source, draw the supermesh contour around the outer perimeter of the combined loops, omitting the current source branch. Apply KVL to this supermesh by summing the voltage drops across all elements in the path, expressing them in terms of the relevant mesh currents and known sources. Additionally, derive a constraint equation using KCL at the nodes adjacent to the current source branch, which relates the difference between the two mesh currents to the source value.^[23]^[22]^[20] If additional non-supermesh loops exist, write standard KVL equations for them. The resulting set of equations—one fewer KVL than the original number of meshes, plus the constraint—is solved simultaneously for all mesh currents.^[21]^[22] The constraint equation typically takes the form I_m - I_n = I_s, where I_m and I_n are the mesh currents on either side of the current source, and I_s is the source current (with sign depending on direction relative to the mesh currents).^[21]^[20] For the supermesh KVL, the equation excludes any term for the current source voltage, as it is indeterminate, focusing instead on resistive and other voltage drops:

\sum V = 0

where the sum traverses the supermesh path.^[23]^[22]

Dependent Sources

Dependent sources in electrical circuits are active elements whose output is controlled by a voltage or current elsewhere in the circuit, complicating mesh analysis by introducing interdependencies among mesh currents. The primary types encountered are the voltage-controlled voltage source (VCVS), where the output voltage is proportional to a controlling voltage; the current-controlled current source (CCCS), where the output current is proportional to a controlling current; the voltage-controlled current source (VCCS); and the current-controlled voltage source (CCVS). In mesh analysis, these sources require expressing the controlling variable in terms of the mesh currents before substituting into the Kirchhoff's voltage law (KVL) equations, ensuring the system remains solvable as a set of linear equations for linear dependencies.^[24]^[5]^[25] For a VCVS, the dependent voltage appears as an additional term in the relevant mesh equation. Specifically, if the VCVS with gain \mu is in mesh k and controlled by a sensing voltage V_s across elements in other meshes, V_s is first written as a linear combination of mesh currents (e.g., V_s = R (I_m - I_n) for a resistor shared between meshes m and n), then the term -\mu V_s is added to the right-hand side of the KVL equation for mesh k, modifying the voltage vector in the matrix formulation. This approach maintains the structure of the impedance matrix Z while accounting for the control through the source vector. In contrast, for a CCCS with gain \beta, the dependent current must be incorporated by adjusting the branch currents in voltage drops. If the controlling current is a mesh current I_p, the CCCS current \beta I_p is directly substituted, altering the self-impedance and mutual-impedance coefficients in Z for the affected meshes; if controlled by a branch current (e.g., the difference between two mesh currents), it is expressed accordingly before updating the coefficients, potentially leading to a non-symmetric Z matrix.^[24]^[25]^[5] These modifications introduce challenges, as the resulting impedance matrix may lack symmetry, requiring general linear solvers rather than simplified methods assuming reciprocity, and if the dependency is nonlinear, iterative techniques like Newton-Raphson must be applied to converge on the mesh currents. In the matrix representation, dependent source terms are placed in the source vector for voltage-controlled types or embedded in Z for current-controlled types, preserving the form Z \mathbf{I} = \mathbf{V}.

Examples and Applications

Basic Resistive Circuit

To illustrate the standard application of mesh analysis, consider a basic resistive circuit with two essential meshes. The circuit features a 10 V DC voltage source in the left mesh, connected in series with a 2 Ω resistor. This left mesh shares a 4 Ω resistor with the right mesh, which also includes a 3 Ω resistor closing the loop. All components are purely resistive, with no current sources or dependent elements.^[1] Identify the two meshes and assign clockwise mesh currents: I_1 for the left mesh and I_2 for the right mesh. This assignment simplifies the analysis by ensuring Kirchhoff's current law is inherently satisfied at shared branches.^[6] Apply Kirchhoff's voltage law to each mesh, summing the voltage drops around the loop equal to the supplied voltage (or zero for the source-free mesh). For the left mesh, the drops across the 2 Ω resistor is $2I_1 and across the shared 4 Ω resistor is $4(I_1 - I_2), equaling the 10 V source:

$2I_1 + 4(I_1 - I_2) = 10

For the right mesh, the drops across the shared 4 Ω resistor is $4(I_2 - I_1) and across the 3 Ω resistor is $3I_2:

$4(I_2 - I_1) + 3I_2 = 0

Solve the resulting simultaneous linear equations. The first equation simplifies to $6I_1 - 4I_2 = 10. The second simplifies to -4I_1 + 7I_2 = 0, so I_1 = \frac{7}{4}I_2. Substitute into the first: $6\left(\frac{7}{4}I_2\right) - 4I_2 = 10, yielding \frac{42}{4}I_2 - 4I_2 = 10, or \frac{26}{4}I_2 = 10, so I_2 = \frac{20}{13} A (approximately 1.54 A). Then, I_1 = \frac{7}{4} \times \frac{20}{13} = \frac{35}{13} A (approximately 2.69 A). The branch currents are I_1 = \frac{35}{13} A through the 2 Ω resistor, I_1 - I_2 = \frac{15}{13} A through the 4 Ω resistor, and I_2 = \frac{20}{13} A through the 3 Ω resistor.^[1] To verify, recompute the KVL for each mesh using these currents. For the left mesh: $2\left(\frac{35}{13}\right) + 4\left(\frac{15}{13}\right) = \frac{70}{13} + \frac{60}{13} = \frac{130}{13} = 10 V. For the right mesh: $4\left(\frac{20}{13} - \frac{35}{13}\right) + 3\left(\frac{20}{13}\right) = 4\left(-\frac{15}{13}\right) + \frac{60}{13} = -\frac{60}{13} + \frac{60}{13} = 0 V. Additionally, power balance confirms accuracy: the source supplies $10 \times \frac{35}{13} = \frac{350}{13} W, while the resistors dissipate $2\left(\frac{35}{13}\right)^2 + 4\left(\frac{15}{13}\right)^2 + 3\left(\frac{20}{13}\right)^2 = \frac{2450 + 900 + 1200}{169} = \frac{4550}{169} = \frac{350}{13} W.^[6] This worked example highlights the efficiency of mesh analysis for planar circuits with voltage sources and resistors, reducing the solution to a compact set of equations compared to enumerating all branch currents via more laborious techniques.^[1]

Circuit with Voltage and Current Sources

Consider a planar circuit with three essential meshes that incorporates both an independent voltage source and an independent current source. Mesh 1 contains a 5 V voltage source in series with resistors, while a 2 A current source is positioned on the shared branch between meshes 2 and 3. The circuit includes resistors valued at 1 Ω in mesh 1, 2 Ω shared between meshes 1 and 2, 3 Ω in mesh 2, and 4 Ω in mesh 3, forming a typical configuration for demonstrating combined sources.^[1] To solve using mesh analysis, first identify the essential meshes and recognize that the current source between meshes 2 and 3 necessitates a supermesh, combining those two meshes into one loop for KVL application while reducing the number of independent equations.^[4] Apply Kirchhoff's voltage law (KVL) to mesh 1 independently, accounting for the voltage source and resistor drops based on mesh currents I_1 and the adjacent I_2:

$5 = 1 \cdot I_1 + 2 \cdot (I_1 - I_2)

For the supermesh encompassing meshes 2 and 3, write the KVL equation around the combined path, summing voltage drops across the resistors while excluding the current source branch (which imposes no voltage drop in the equation):

$3 \cdot I_2 + 2 \cdot (I_2 - I_1) + 4 \cdot I_3 = 0

The current source provides the additional constraint equation:

I_2 - I_3 = 2

Solving this system of three equations (two KVL and one constraint) simultaneously yields the mesh currents I_1 = \frac{61}{23} A (approximately 2.65 A), I_2 = \frac{34}{23} A (approximately 1.48 A), and I_3 = -\frac{12}{23} A (approximately -0.52 A; negative value indicates current direction opposite to the assumed clockwise flow).^[4] With these currents, further computations such as output voltage across a specific resistor (e.g., the voltage across the 4 Ω resistor in mesh 3 is V = 4 \cdot I_3 = -\frac{48}{23} V, approximately -2.09 V) or power dissipation (e.g., power in the 2 Ω shared resistor is P = 2 \cdot (I_1 - I_2)^2 = 2 \cdot \left(\frac{27}{23}\right)^2 = \frac{1458}{529} W, approximately 2.76 W) can be determined directly. Mesh analysis proves advantageous in this scenario due to the planar topology and presence of a current source, resulting in only two independent KVL equations despite three meshes, compared to potentially three equations in nodal analysis for the same circuit configuration.^[1]

References

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Below is a merged summary of mesh analysis based on all the provided segments from "Electric Circuits" by Nilsson & Riedel (10th Ed., 2014) and other related content. To retain all information in a dense and organized manner, I will use a table in CSV format for key details, followed by a concise narrative summary that incorporates additional context and variations mentioned across the segments. Since the system has a "no thinking token" limit, I’ll focus on directly synthesizing the data without additional interpretation beyond what’s explicitly provided.
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### Supermesh for Current Sources in Mesh Analysis
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### Supermesh Analysis for Independent Current Sources
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