Network Equation & Solution Methods - Network Theory (Electric Circuits)

Basic Concepts

• Kirchhoff's Voltage Law (KVL): The algebraic sum of the voltages around any closed loop in a circuit is zero.
• Kirchhoff's Current Law (KCL): The algebraic sum of currents entering a node (or a closed boundary) is zero. Equivalently, ∑I_IN = ∑I_OUT.
• Purpose of circuit equations: Use KCL, KVL and constitutive relations (Ohm's law, element V-I relations) to obtain algebraic equations whose solution yields node voltages and branch currents.

Nodal Analysis

Overview: Nodal analysis (node-voltage method) is a systematic way to determine the voltage at circuit nodes with respect to a chosen reference node (ground) by applying KCL at the nodes and expressing the currents in terms of node voltages.

Procedure (ordered)

1. Identify all principal nodes and choose one as the reference node (ground).
2. Assign a variable for the voltage at each non-reference node with respect to ground.
3. Apply KCL at each non-reference node: the sum of currents leaving (or entering) the node equals zero.
4. Express each branch current using Ohm's law in terms of node voltages (use conductances if convenient, where G = 1/R).
5. Solve the resulting simultaneous linear equations to obtain the node voltages.

Once node voltages are known, voltage across any element and current through any element can be found by direct substitution into Ohm's law or the element V-I relation.

Remarks on choosing the reference node

• A convenient reference node has many connections; this often reduces the number of node equations.
• If a node is connected to a known voltage source referenced to ground, selecting ground to make that node the reference simplifies the algebra.

Example: Find nodal voltages and loop currents using the nodal method

Problem statement: Find the nodal voltage at each node and current in each loop by nodal analysis.

Solution: We label nodes and circuit elements and then follow the nodal method.

The chosen reference node is the one labelled as Ground. There are three principal nodes in this circuit; node n1 is connected to a voltage source so its node voltage is known.

The KCL equation at node n2 equates currents through the branches connected to n2.

Known relation: v₁ = V_s. (This is the node-voltage at n1 because the voltage source directly sets it.)

KCL at node n2: i₁ = i₂ + i₃.

Using the expressions above and the known node-voltage v₁, combine the relations to obtain the nodal equations.

 ……………  (4) 

Rewrite the expressions to form linear functions of the unknown node voltages v₂ and v₃.

KCL at node n3 gives the second equation.

Write both equations together in matrix form for unknowns v₂ and v₃.

Good practice when forming the matrix equations:

• Place all independent sources (current or voltage) on the right-hand side as inhomogeneous terms.
• Ensure diagonal terms (sum of conductances connected to the node) have consistent sign convention.
• If arranged correctly, the conductance matrix is symmetric: A_ij = A_ji. If it is not symmetric, check the equations for errors.

If the determinant of the coefficient matrix is non-zero, the set of simultaneous equations has a unique solution and node voltages can be obtained (Cramer's rule is referred to here as an illustration).

In Cramer's-rule notation, the unknown voltage V_K is given by the ratio of the determinant with k-th column replaced by the source vector to the determinant of the coefficient matrix.

For this particular circuit the node voltages v₂ and v₃ are:

Nodal Analysis with Floating Voltage Sources (The Supernode)

Floating voltage source: A voltage source that does not connect directly to the reference node is called a floating voltage source. It imposes a constraint between the two node voltages it connects.

The portion of the circuit that encloses the floating voltage source and the nodes it connects is treated as a supernode. Apply KCL to the supernode as you would for any node because charge conservation still holds for the closed surface enclosing the supernode.

Applying KCL to the supernode yields a current balance equation such as I₁ = i₂ + I₃.

The voltage source provides a constraint between the node voltages forming the supernode. For example:

Constraint: V₂ = v₃ - v₂.

Use the KCL equation for the supernode together with the voltage-source constraint to obtain a complete set of equations.

Example - Nodal analysis with a supernode

Problem statement: Determine the node voltages v₁, v₂ and v₃ of the circuit shown.

Solution: The circuit contains a current source I_S which provides a direct constraint on a branch current and, through Ohm's law, may set a related node voltage.

Begin by applying KCL to the relevant nodes and using Ohm's law where required.

The current source gives a direct expression for v₃ in terms of I_S and R₃:

v₃ = I_S R₃

Combine the KCL expression and the constraint provided by the current source to obtain the node-voltage equations.

Mesh Analysis

Definition: A mesh is a loop in a planar circuit that does not contain any other loops within it. Mesh analysis (loop-current method) assigns mesh currents to independent meshes and applies KVL to each mesh to form simultaneous equations in the mesh currents.

Procedure (ordered)

1. Label all circuit elements and identify unknowns.
2. Identify all meshes and assign a mesh current to each (choose a direction, commonly clockwise).
3. Apply KVL to each mesh, expressing voltages in terms of mesh currents and element impedances.
4. Solve the resulting simultaneous equations for the mesh currents.
5. Compute desired branch currents and voltages from the mesh currents using algebraic relations.

Example - Find mesh currents I₁ and I₂

Problem statement: Determine the mesh currents i₁ and i₂ for mesh 1 and mesh 2 in the circuit below.

Solution: The circuit contains three loops but only two meshes (inner loops). Assign mesh currents I₁ and I₂ clockwise.

The branch with R₂ is shared by the two meshes, therefore its branch current equals I₁ - I₂ (depending on the chosen reference directions).

Apply KVL around mesh 1 (traverse in the direction of I₁):

I₁ R₁ + (I₁ - I₂) R₂ - V_S = 0.

Apply KVL around mesh 2:

I₂ (R₃ + R₄) + (I₂ - I₁) R₂ = 0.

Rearrange to the standard simultaneous equation form:

I₁ (R₁ + R₂) - I₂ R₂ = V_S

-I₁ R₂ + I₂ (R₂ + R₃ + R₄) = 0

Solve the matrix equation for I₁ and I₂. Once mesh currents are known, branch currents follow from linear combinations of mesh currents.

For this circuit:

I₁ = i₁, i₂ = I₁ - I₂, i₃ = I₂.

Mesh Analysis with Current Sources

There are two common situations when current sources appear in mesh analysis:

• If a current source lies only in one mesh, the mesh current equals the current source value; this reduces the number of unknown mesh currents.
• If a current source is between two meshes, form a supermesh by excluding the branch with the current source and apply KVL around the supermesh. Use the current-source relation as an additional constraint between the two mesh currents.

Example - Mesh analysis when a mesh contains a current source

Problem statement: Find the unknown mesh current I₁ for the circuit shown.

Solution:

I₂ is the mesh current through the mesh that contains the current source, therefore

I₂ = I_S.

Apply KVL around mesh 1, taking into account the shared branch currents:

I₁ R₁ + (I₁ + I_S) R₂ = V_S.

Solve the above algebraic equation for I₁:

Practice Problems with Answers

Q.1. Determine the currents in the given circuits with reference to the indicated direction?

Q.2. Determine the currents in the given circuits with reference to the indicated direction?

Answer: i1 = 3.31A, i2 = 1.68A, i3 = 1.63A, i4 = 0.627A, v2 = 8.39V, v3 = 6.51V