Nodal Analysis

• A procedure for analyzing circuits using node voltages as variables
• It’s better to use node voltages instead of circuit elements because it reduces the number of equations
• Goal is to determine the node voltages using KCL, which are defined to be positive wrt ground
• Steps:
  1. Assign a ground node with $v = 0$ and $v_{i}$ for the remaining $n-1$ nodes
  2. Apply KCL to each $n-1$ nonreference node. Express the currents in terms of the node voltages
  3. Solve the resultant system
• Since current flows from a higher to lower potential in a resistor, the current is

Example 03.1.

(Practice Problem 3.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 109])

Example 03.2.

(Practice Problem 3.2 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 112])

Nodal Analysis with Voltage Sources

• When dealing with voltage sources, we must examine two cases:
  1. The voltage source is between ground and a nonreference node. Then, the voltage at the nonreference node is equal to the voltage of the voltage source
  2. The voltage source is between two nonreference nodes. Then, the two nodes form a supernode, and we must apply both KCL and KVL to determine the node voltages

Example 03.4.

(Practice Problem 3.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 114])

Example 03.5.

(Practice Problem 3.4 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 117])

Mesh Analysis

• Use KVL to find unknown currents using mesh currents as circuit variables
• A mesh is a loop that does not contain any other loop within it
• Only applicable to planar circuits (no branches crossing one another)
• Steps:
  1. Assign mesh currents $i_{i}$ to the $n$ meshes
  2. Apply KVL to each mesh. Express the voltages in terms of the mesh currents
  3. Solve the resultant system
• For each KVL equation:
  • The coefficient of the mesh current being solved is positive and equal to the sum of all resistances in the mesh
  • The coefficient of any adjacent mesh current is the negative of the sum of resistances shared between the mesh and the shared mesh
  • All voltage rises may be summed and written on the RHS (i.e., voltage sources)
• Use capital $I$ for branch currents and lowercase $i$ for mesh currents

Example 03.6.

(Practice Problem 3.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 120])

Example 03.7.

(Practice Problem 3.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 122])

Mesh Analysis with Current Sources

• When dealing with current sources, we must examine two cases:
  1. The current source is exclusive to one mesh. Then, the mesh current is equal to the current source
  2. The current source is shared between two meshes. Then, the two meshes form a supermesh by excluding the branch with the current node and any elements in series with it. Perform KVL on the supermesh and apply KCL to a node where the two meshes intersect

Example 03.9.

(Practice Problem 3.7 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 124])

Analysis by Inspection

Nodal Analysis by Inspection

• KCL not needed if all the sources are independent current sources; the resulting system is $$\mathbf{Gv = i}$$with $\mathbf{G}$ being the conductance matrix, $\mathbf{v}$ being the output vector, and $\mathbf{i}$ being the current matrix/input vector
• For the conductance matrix:
  • The $i$th diagonal term is the sum of the conductances in branches connected to the $i$th node
  • The off-diagonal terms are the negatives of the sum of the conductance in the branches connected between two nodes $i$ and $j$
• For the current matrix:
  • The $n$th term is the sum of the currents entering the $n$th node
• If a circuit with independent current sources has $N$ nonreference nodes, the system in terms of the conductances is:

• Shortcut: there is symmetry along the diagonal, so all the off-diagonals across from one another should be equal

Example 03.10.

(Practice Problem 3.8 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 127])

Mesh Analysis by Inspection

• Matrix is of the form $$\mathbf{Ri = v}$$where:
  • $R_{ii}$ = Sum of resistances in mesh $i$
  • $R_{ij} = R_{ji} =$ Negative of sum of resistances in common with meshes $i$ and $j$
  • $i_{N}$ = Unknown mesh current up to the $N$th mesh
  • $v_{N}$ = Sum of all independent voltage sources up to the $N$th mesh
• Thus, the generalized matrix is:

$$ \begin{bmatrix} R_{11} & R_{12} & \dots & R_{1j} \ R_{21} & R_{22} & \dots & R_{2j} \ \vdots & \vdots & \ddots & \vdots \ R_{i1} & R_{i2} & \dots & R_{ij} \end{bmatrix} \begin{bmatrix} i_1 \ i_2 \ \vdots \ i_N \end{bmatrix}

\begin{bmatrix} v_1 \ v_2 \ \vdots \ v_N \end{bmatrix} $$

Example 03.11.

(Practice Problem 3.9 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 128])