Systematic Nodal and Mesh Analysis

Introduction

Systematic circuit analysis converts Kirchhoff’s laws and element relations into simultaneous algebraic equations. Unlike ad hoc series–parallel reduction, nodal and mesh methods remain effective for networks containing multiple independent sources, dependent sources, and bridge connections.

Nodal analysis treats node voltages as the primary unknowns. Modified nodal analysis (MNA) extends it to ideal voltage sources and source-controlled elements. Mesh analysis uses loop currents in planar circuits, while supernodes and supermeshes handle source configurations that prevent direct equation writing.

These methods underpin computer-aided circuit simulation. Professional tools such as SPICE primarily use MNA because it supports systematic matrix assembly for large networks.

Learning Objectives

After completing this chapter, students should be able to:

• select an appropriate reference node and define node voltages;
• formulate nodal equations containing independent and dependent sources;
• construct and solve supernode equations;
• formulate MNA systems for circuits containing ideal voltage sources;
• define mesh currents and write mesh equations with correct signs;
• construct and solve supermesh equations;
• determine whether a circuit equation system has a unique solution;
• check units, source constraints, current directions, and power balance; and
• apply systematic analysis to practical electrical networks.

1. Foundations of Systematic Circuit Analysis

A node is a set of points connected by ideal conductors and therefore at one potential. A branch is a two-terminal element or series group carrying one branch current. A loop is any closed path, while a mesh is a loop containing no smaller loop within it.

The methods in this chapter rest on three relationships:

1. Kirchhoff’s current law (KCL): the algebraic sum of currents at a node is zero.
2. Kirchhoff’s voltage law (KVL): the algebraic sum of voltages around a closed path is zero.
3. Element laws: for a resistor, v = Ri or i = Gv, where G = 1/R.

KCL and KVL apply to lumped circuits when electromagnetic propagation and field coupling can be represented adequately by concentrated circuit elements.

Choosing a method

The method with fewer unknowns is often—but not always—the easiest. MNA is generally preferred for automated implementation.

2. Nodal Analysis

2.1 Node-voltage variables

Choose one node as the reference and assign it voltage zero. Every other node voltage is measured relative to that reference. A circuit with n total nodes normally has n-1 unknown node voltages.

The reference node should preferably be connected to several elements or voltage sources. The choice does not affect physical branch voltages, but it can simplify the equations.

2.2 General nodal equation

Consider nonreference node k. If currents leaving the node are taken as positive, KCL gives

Here:

• v_k and v_j are node voltages in volts;
• G_kj is branch conductance in siemens;
• i_k,source^out is the algebraic current in amperes contributed by current sources leaving node k.

Equivalently, independent source currents entering the node may be placed on the right-hand side:

For a passive resistor network:

• each diagonal entry G_kk is the sum of conductances incident on node k;
• each off-diagonal entry G_kj is the negative of the conductance directly connecting nodes k and j;
• i contains net independent current injected into each node.

The entries of G have units of siemens, so Gv has units of amperes. Dependent sources can add nonsymmetric terms to the matrix.

2.3 Basic numerical example

Node v₁ is connected to ground by a 6 Ω resistor, to node v₂ by a 3 Ω resistor, and receives 2 A from a source directed from ground to the node. Node v₂ is connected to ground by 2 Ω.

KCL gives

Multiplying both equations by 6 produces

The second equation gives v₁=2.5v₂. Substitution into the first gives

Checking node 2, v₂/2=1.091 A and (v₁-v₂)/3=1.091 A, so KCL is satisfied.

3. Supernodes

An ideal voltage source between two nonreference nodes does not directly provide its current as a function of node voltage. The two nodes and the voltage source are therefore enclosed conceptually as a supernode.

The procedure is:

1. Write one KCL equation for the boundary of the entire supernode.
2. Omit the internal current through the voltage source from that KCL equation.
3. Add the voltage-source constraint.
4. Retain any dependent-source control relation.

Worked Example: Complete supernode solution

Nodes a and b are connected by an ideal 6 V source whose positive terminal is at node a. Node a is connected to ground through 4 Ω and receives 3 A from a current source directed from ground to a. Node b is connected to ground through 2 Ω.

The supernode KCL equation is

The voltage-source constraint is

Multiplying the KCL equation by 4 Ω gives

Using v_a=v_b+6:

Therefore,

Let i_s be the current through the voltage source from a to b. KCL at node a gives

so i_s=1 A.

A power check confirms the result. The 4 Ω and 2 Ω resistors absorb 16 W and 2 W, respectively. The voltage source absorbs 6 V × 1 A=6 W, while the 3 A source delivers 24 W. Thus the algebraic sum of absorbed powers is

4. Dependent Sources in Nodal Analysis

A dependent source must be retained as an active circuit element. Its controlling voltage or current becomes part of the equation set.

Common source relations are:

A control parameter must have the correct dimensions. For example, multiplying a voltage by a dimensionless number cannot directly produce a current.

When superposition is used, dependent sources remain active in every partial calculation because they describe the circuit itself. Only the selected independent sources are deactivated. An independent voltage source is replaced by a short circuit and an independent current source by an open circuit, provided the source has zero internal impedance or infinite internal impedance, respectively.

5. Modified Nodal Analysis

5.1 Motivation

A supernode is convenient for manual work, but MNA provides a more systematic treatment of ideal voltage sources. In addition to node voltages, MNA introduces the currents through selected voltage-defined elements as unknowns.

For n nonreference node voltages and m ideal voltage sources, the standard block form is

where:

• G contains conductance and controlled-source contributions;
• B connects voltage-source currents to node KCL equations;
• C imposes voltage-source constraints;
• D is usually zero for independent ideal voltage sources but may contain terms for more general elements;
• v is the node-voltage vector in volts;
• i_v is the voltage-source-current vector in amperes;
• i is the current-injection vector in amperes;
• e contains prescribed source voltages in volts.

For independent ideal voltage sources, C=B^T under consistent orientation. The complete matrix contains rows with different physical dimensions: upper rows are KCL equations, while lower rows are voltage constraints. This is valid because each row is dimensionally consistent on its own.

5.2 Correctly assembled MNA example

Node 1 is held at 2 V by an ideal voltage source connected from ground to node 1, with node 1 positive. A 1 Ω resistor joins nodes 1 and 2. Node 2 is connected to ground by 2 Ω. A dependent current source injects current from ground into node 2 with value g_m v₁, where g_m=1 S. Let i_s be the current through the 2 V source from node 1 to ground.

Node 1 KCL is

Node 2 KCL, with currents leaving node 2 positive, is

Because g_m=1 S, this becomes

The voltage constraint is v₁=2 V. The MNA system is therefore

From the third row, v₁=2 V. The second row gives

The first row gives

A direct node-2 check gives a resistor current from node 2 to node 1 of 2/3 A, a ground-resistor current of 4/3 A, and a dependent-source injection of 2 A. Thus total leaving current is 2/3+4/3-2=0 A.

6. Mesh-Current Analysis

6.1 Applicability

Mesh analysis is directly applicable only to planar circuits—circuits drawable on a plane without branch crossings other than at nodes. It assigns a current to every mesh, commonly clockwise.

For a resistor belonging to only mesh k, its voltage drop is Ri_k. For a resistor shared by meshes k and j, the drop in the direction of mesh k is

The general resistive mesh system can be written

For a DC resistor network, diagonal coefficient Z_kk is the sum of resistances around mesh k, while off-diagonal coefficient Z_kj is the negative of the resistance shared by meshes k and j. Matrix entries have units of ohms, and Zi has units of volts.

6.2 Mesh example containing a dependent source

Two clockwise meshes share a 2 Ω resistor. The left outer branch contains a 4 Ω resistor and a 12 V rise. The right outer branch contains a 3 Ω resistor and a current-controlled voltage source producing a clockwise voltage drop v_d=r_m i₁, with r_m=1 Ω.

Left-mesh KVL is

or

Right-mesh KVL is

Using r_m=1 Ω:

Thus i₁=5i₂. Substitution gives

Therefore,

The shared-branch current in the direction of mesh 1 is i₁-i₂=12/7 A.

7. Supermeshes

An ideal current source shared by two meshes prevents writing separate KVL equations through that branch because the source voltage is unknown. The two meshes form a supermesh.

The procedure is:

1. Write KVL around the outer boundary, excluding the current-source branch.
2. Write the current-source constraint relating the mesh currents.
3. Add the control equation if the source is dependent.

Worked Example: Complete supermesh solution

Two clockwise meshes share a 2 A current source. Its orientation and the chosen mesh directions impose

The outer boundary contains a 2 Ω resistor in mesh 1, a 4 Ω resistor in mesh 2, and a net 12 V source rise. Supermesh KVL is

From the constraint, i₁=i₂+2. Therefore,

Hence,

and

Verification gives i₁-i₂=2 A and 2i₁+4i₂=12 V. Both the source constraint and KVL are satisfied.

If a current source lies on the exterior boundary of only one mesh, that mesh current is known directly, subject to sign orientation; a supermesh is unnecessary.

8. Solvability and Physical Consistency

8.1 Rank conditions

For a linear system

a unique solution exists when A is square and nonsingular, equivalently when rank(A)=n. If

the equations are inconsistent. If

infinitely many solutions exist.

8.2 Common causes of difficulty

• Floating subcircuit: no reference path defines its absolute potential.
• Conflicting ideal voltage sources: parallel ideal sources demand different voltages.
• Conflicting ideal current constraints: series or cut-set sources demand incompatible currents.
• Redundant constraints: identical source equations create dependent matrix rows.
• Missing control relation: a dependent source is specified without defining its controlling variable.
• Incorrect topology: an element is assigned to the wrong node or mesh.

A mathematically singular formulation does not always mean that every physical quantity is indeterminate. For example, absolute floating-node voltages may be undefined while their voltage difference remains fixed.

8.3 Numerical conditioning

A nonsingular matrix can still be ill-conditioned. Very large ratios of resistance, conductance, or controlled-source gain can amplify rounding errors. Practical simulators use scaling, pivoting, sparse matrix ordering, and convergence controls. Reporting excessive digits from an ill-conditioned model creates false precision.

9. Verification and Best Practice

A defensible solution should pass several checks.

Units

• Nodal KCL terms must all be currents.
• Mesh KVL terms must all be voltages.
• Conductance is measured in siemens; resistance and transresistance are measured in ohms.
• Transconductance is measured in siemens.

Signs and orientations

Define every voltage polarity and current direction before forming equations. A negative computed value indicates that the actual direction or polarity is opposite to the assumed reference; it does not by itself indicate an error.

Independent equation count

The number of independent equations must match the number of unknowns. Equation count alone is insufficient: the equations must also be linearly independent and mutually consistent.

Local and global checks

Substitute the result into at least one original KCL equation and one source constraint. For the entire circuit, apply the passive sign convention and verify

where positive power denotes absorption. Small nonzero sums can result from rounding; large mismatches indicate an equation or sign error.

Recommended workflow

1. Redraw and label the topology clearly.
2. Select the reference node or mesh-current directions.
3. Mark all source polarities and control variables.
4. Write equations symbolically before inserting numbers.
5. Check dimensions row by row.
6. Solve using elimination, matrix methods, or verified software.
7. Substitute into the original equations.
8. Check source constraints and power balance.
9. Report units and an appropriate number of significant figures.

10. Illustrative Applied Case: Sensor-Bias Network

Background

This fictional illustrative case concerns a battery-powered industrial sensor. A 12 V supply feeds a bias node through a 4 kΩ resistor. The node is connected to ground through 6 kΩ. The sensor electronics draw a voltage-dependent current g_m v, where g_m=0.1 mS. A signal-conditioning stage also injects 1 mA from ground into the node. The node must remain below 8 V.

Problem

Determine the bias voltage and verify the current balance.

Analysis and solution

Taking currents leaving the node as positive gives

Rearranging,

Thus

The current entering from the 12 V supply is

The currents leaving through the 6 kΩ resistor and dependent load are

and

Total entering current is 1.0645+1=2.0645 mA, equal to the total leaving current 1.2903+0.7742=2.0645 mA.

Outcome

The predicted node voltage is approximately 7.74 V, below the 8 V requirement. However, component tolerances and temperature-dependent transconductance must be included before production approval.

Learning points

• Dependent-source models can represent voltage-dependent electronic loads.
• Unit conversion between kΩ, mA, and mS must be consistent.
• Nominal compliance does not replace worst-case tolerance analysis.
• A direct KCL check can expose modeling and arithmetic errors.

11. Practical Limitations and Contemporary Practice

Linear nodal and mesh equations apply directly to linear resistors and linear controlled sources. Nonlinear devices such as diodes and transistors require operating-point iteration and local linearization. In sinusoidal steady state, resistance is replaced by complex impedance or admittance; the topology and equation-writing principles remain unchanged.

Modern simulation programs use sparse MNA matrices because practical systems may contain millions of variables. Automated schematic capture and AI-assisted equation generation can reduce clerical work, but they cannot guarantee that source orientation, grounding, device models, or safety assumptions are correct. Engineers remain responsible for independent verification.

In safety-critical products, analysis records should preserve model versions, component tolerances, assumptions, and verification results. Relevant electrical safety, insulation, and electromagnetic-compatibility standards must be addressed separately; a correct nodal solution does not establish regulatory compliance.

Chapter Summary

Nodal analysis applies KCL to node-voltage unknowns, while mesh analysis applies KVL to mesh-current unknowns in planar circuits. Supernodes incorporate ideal voltage sources between nonreference nodes, and supermeshes incorporate current sources shared by meshes. Modified nodal analysis systematically introduces voltage-source currents as additional unknowns and forms the basis of most circuit simulators.

Dependent sources must remain active and require explicit control relations with dimensionally correct gains. Reliable analysis also requires attention to source polarity, current direction, matrix rank, numerical conditioning, and physical verification. Substitution into original equations, source-constraint checks, and conservation of power provide strong safeguards against modeling and algebraic errors.

Key Terms

Node: A set of ideal-conductor-connected points at one electrical potential.

Reference node: The node assigned zero potential.

Branch: A two-terminal path carrying one branch current.

Loop: Any closed path through a circuit.

Mesh: A loop that contains no smaller loop in a planar circuit.

Node voltage: A node’s potential relative to the reference node.

Mesh current: An assigned current circulating around a mesh.

Nodal analysis: A method based on KCL and node-voltage unknowns.

Mesh analysis: A planar-circuit method based on KVL and mesh-current unknowns.

Supernode: A group of nodes connected by an ideal voltage source and treated as one KCL region.

Supermesh: A combination of adjacent meshes formed around a shared current source.

Modified nodal analysis: A nodal formulation augmented by currents through voltage-defined elements.

Dependent source: A source whose value is controlled by another circuit variable.

Transconductance: The current-to-controlling-voltage ratio, measured in siemens.

Transresistance: The voltage-to-controlling-current ratio, measured in ohms.

Planar circuit: A circuit drawable without branch crossings except at nodes.

Singular matrix: A matrix without a unique inverse, indicating nonunique or inconsistent equations.

Conditioning: The sensitivity of a computed solution to numerical or data perturbations.

Review Questions

Short Answer Questions

1. Why is one node designated as the reference node in nodal analysis?
2. State the two equations required to analyze a basic supernode.
3. Why must dependent sources remain active during superposition?
4. Under what topological condition is conventional mesh analysis applicable?
5. What does a negative calculated branch current mean?

Long Answer Questions

1. Derive the conductance-matrix rules for a passive resistive network from KCL.
2. Explain the construction and interpretation of the MNA block matrix.
3. Compare supernode analysis with MNA for ideal voltage sources.
4. Discuss singularity, inconsistency, and ill-conditioning in circuit equation systems.
5. Develop a verification procedure using KCL, source constraints, units, and power balance.

Analytical / Application Questions

1. Nodes a and b form a supernode with v_a-v_b=5 V. Node a is connected to ground by 10 Ω, node b by 5 Ω, and 2 A is injected into node a. Determine both node voltages and the voltage-source current.
2. A driven two-mesh circuit has supermesh equation 3i₁+6i₂=15 V and current-source constraint i₁-i₂=1 A. Determine both mesh currents and verify both equations.
3. Form an MNA model for a two-node circuit containing a resistor between the nodes, a grounded resistor at each node, an independent voltage source fixing node 1, and a voltage-controlled current source injecting current into node 2. State the units of every matrix block.

Multiple Choice Questions with Answers

Question 1

A circuit has five nodes including the reference node. How many node-voltage unknowns are normally required?

A. 3

B. 4

C. 5

D. 6

Correct Answer: B. 4

Question 2

For the chapter’s supernode example, the equations are v_a/4+v_b/2=3 and v_a-v_b=6. What is v_b?

A. 1 V

B. 2 V

C. 6 V

D. 8 V

Correct Answer: B. 2 V

Question 3

During superposition in a linear circuit, how must dependent sources be treated?

A. Deactivate all dependent sources

B. Replace them by their internal resistances

C. Keep them active in every partial calculation

D. Keep them active only when their controlling source is selected

Correct Answer: C. Keep them active in every partial calculation

Question 4

Conventional mesh-current analysis is directly applicable to which class of circuits?

A. Planar circuits

B. Only series circuits

C. Only source-free circuits

D. All nonplanar circuits

Correct Answer: A. Planar circuits

Question 5

In the MNA example, node 2 satisfies -2v₁+1.5v₂=0 and v₁=2 V. What is v₂?

A. 1.333 V

B. 2.000 V

C. 2.667 V

D. 3.000 V

Correct Answer: C. 2.667 V

Question 6

A supermesh satisfies 2i₁+4i₂=12 and i₁-i₂=2. What is i₂?

A. 0.667 A

B. 1.000 A

C. 1.333 A

D. 3.333 A

Correct Answer: C. 1.333 A

Question 7

What is the SI unit of the gain of a voltage-controlled current source?

A. Volt

B. Ohm

C. Ampere

D. Siemens

Correct Answer: D. Siemens

Question 8

For Ax=b, which condition indicates an inconsistent equation system?

A. rank(A)=n

B. rank(A)<rank([A|b])

C. rank(A)=rank([A|b])<n

D. det(A)≠0

Correct Answer: B. rank(A)<rank([A|b])

Question 9

The dependent-source mesh example gives 6i₁-2i₂=12 and -i₁+5i₂=0. What is i₂?

A. 0.214 A

B. 0.429 A

C. 1.714 A

D. 2.143 A

Correct Answer: B. 0.429 A

Question 10

Which result provides a valid whole-circuit power check under the passive sign convention?

A. The sum of absorbed and delivered powers is zero

B. Every source absorbs positive power

C. Every resistor has zero voltage

D. All branch currents have the assumed positive direction

Correct Answer: A. The sum of absorbed and delivered powers is zero