Nodal analysis is a powerful technique for solving complex circuits. It simplifies the process by focusing on node voltages instead of individual branch currents. This method leverages Kirchhoff's Current Law and Ohm's Law to create a system of equations.

By applying nodal analysis, we can determine voltages at key points in a circuit. This approach is especially useful for circuits with multiple components and interconnections, making it a fundamental tool in electrical engineering problem-solving.

Node Analysis Fundamentals

Node and Reference Node

• A node is a point in a circuit where two or more circuit elements connect
• Nodes are used to analyze voltages and currents in a circuit
• A reference node, also known as the ground node, serves as a common reference point for measuring voltages in a circuit
  • The reference node is typically assigned a voltage of 0 V
  • All other node voltages are measured with respect to the reference node

Node Voltage and Branch Current

• Node voltage represents the electric potential at a specific node in a circuit
  • Node voltages are always measured relative to the reference node
  • The voltage between any two nodes is the difference in their respective node voltages
• Branch current refers to the current flowing through a specific branch or element in a circuit
  • Branch currents are determined by applying Ohm's law and Kirchhoff's current law (KCL)
  • The direction of branch currents is typically assigned based on the reference direction chosen for the analysis

Kirchhoff's Current Law

Kirchhoff's Current Law (KCL)

• Kirchhoff's current law (KCL) states that the algebraic sum of currents entering and leaving a node must equal zero
  • Mathematically: $\sum_{k=1}^{n} I_k = 0$, where $I_k$ represents the current in branch $k$ connected to the node
  • KCL is based on the conservation of charge principle, ensuring that charge does not accumulate at any node
• To apply KCL, assign a reference direction for each current entering or leaving the node
  • Currents entering the node are considered positive, while currents leaving the node are considered negative
  • Write an equation expressing the sum of currents equal to zero

Conductance

• Conductance is the reciprocal of resistance and represents the ease with which current flows through a circuit element
  • Conductance is denoted by the symbol $G$ and is measured in siemens (S)
  • The relationship between conductance and resistance is given by: $G = \frac{1}{R}$, where $R$ is the resistance in ohms ($\Omega$)
• Conductance is often used in nodal analysis to simplify equations and express currents in terms of node voltages
  • Ohm's law can be rewritten in terms of conductance as: $I = GV$, where $I$ is the current and $V$ is the voltage across the element

Solving Nodal Equations

System of Equations

• Nodal analysis involves setting up a system of equations based on KCL and Ohm's law to solve for node voltages
  • Each node in the circuit (except the reference node) will have a corresponding equation
  • The number of equations will be equal to the number of unknown node voltages
• To set up the system of equations:
  1. Identify the nodes in the circuit and assign a reference node (usually ground)
  2. Label the node voltages as unknown variables (e.g., $V_1$, $V_2$, etc.)
  3. Apply KCL at each node, expressing branch currents in terms of node voltages and conductances
  4. Solve the resulting system of equations using linear algebra techniques (e.g., Gaussian elimination, matrix inversion)

Nodal Equation

• A nodal equation is an equation obtained by applying KCL at a specific node in the circuit
  • The nodal equation relates the node voltage to the voltages of adjacent nodes and the conductances of connected branches
• The general form of a nodal equation for a node $i$ is: $\sum_{j=1}^{n} G_{ij}(V_i - V_j) + I_i = 0$
  • $G_{ij}$ is the conductance between nodes $i$ and $j$
  • $V_i$ and $V_j$ are the voltages at nodes $i$ and $j$, respectively
  • $I_i$ represents any current source directly connected to node $i$
• The nodal equation for each node is included in the system of equations to be solved for the unknown node voltages
  • Once the node voltages are determined, branch currents can be calculated using Ohm's law