6. Branch + Mesh Node Analysis

• Branch Mesh and Node Analysis


  Method to solve for the current in each branch of a circuit using Kirchhoff's voltage and current laws
• Branch Current Method


  Use Kirchhoff's voltage and current laws to solve for the current in each branch of the circuit. Once the branch currents are known the voltages can be determined
• Loops and Nodes


  • Loops are complete current paths, Nodes are junctions where two or more current paths meet
• The Branch Current Method has 2 voltage sources and 3 branch currents

  
• General Steps in Branch Current Method

  1. Assign a current in each circuit branch in an arbitrary direction

  2. Show the polarities of the resistor voltages according to the assigned branch current directions

  3. Apply Kirchhoff's voltage law around each closed loop (sum of voltages = 0)

  4. Apply Kirchhoff's current law at the minimum number of nodes so that all the branch currents are included (sum of currents at node = 0)

  5. Solve the equations resulting from steps 3 and 4 for the branch current values

  
• The general circuit for branch current analysis has Kirchhoff's Voltage Law equations for two loops and a Kirchhoff's Current Law equation at one node

  
• Solving the Branch Current Method equations

  1. Substitute the given values into the Kirchhoff's Voltage Law and Kirchhoff's Current Law equations

  2. Solve the 3 simultaneous equations to find the branch currents I1, I2 and I3
• Mesh Current Method


  Use loop currents instead of branch currents. Branch current is the actual current through a branch, loop current is a mathematical quantity used to make analysis easier.
• Steps in Mesh Analysis

  1. Assign a current in a clockwise direction around each closed loop

  2. Indicate the voltage drop polarities in each loop based on the assigned current directions

  3. Apply Kirchhoff's voltage law around each closed loop, including voltage drops for components shared by multiple loops

  4. Solve the resulting equations for the loop currents using substitution
• The general circuit for mesh analysis has two Kirchhoff's Voltage Law equations, one for each loop
• Solving the Mesh Analysis equations

  1. Substitute the given values into the two Kirchhoff's Voltage Law equations

  2. Solve the two equations by substitution to find the loop currents I1 and I2

  3. Calculate the branch currents from the loop currents
• Circuits with more than two loops require three Kirchhoff's Voltage Law equations to solve for the three loop currents I1, I2 and I3
• Node Voltage Method


  Alternate method of analysis for multiple-loop systems, uses node voltages instead of loop or branch currents
• Steps in Node Voltage Analysis

  1. Determine the number of nodes

  2. Select one node as a reference, assign voltages relative to this node

  3. Assign currents at each node where the voltage is unknown, except the reference node

  4. Apply Kirchhoff's current law to each node where currents are assigned

  5. Express the current equations in terms of voltages and solve for the unknown node voltages

  
• The circuit for node voltage analysis has one unknown node voltage (VA) which is solved for using Kirchhoff's current law