Week 5: Methods of Analysis(Nodal Analysis Based on KCL and KVL)

                           Nodal Analysis Based on KCL

The steps in the nodal analysis method are:

• Count the number of principal nodes or junctions in the circuit. Call this number n.
• Number the nodes N₁, N₂, . . . , N_n and draw them on the circuit diagram. Call the voltages at these nodes V₁, V₂, . . . , V_n, respectively.
• Choose one of the nodes to be the reference node or ground and assign it a voltage of zero.
• For each node except the reference node write down Kirchoff's Current Law in the form "the algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a current flowing into a node is to be considered a negative current flowing out of the node.)
  
  Example:

                Use nodal analysis to find the voltage at each node of this circuit.
Solution:

• Note that the "pair of nodes" at the bottom is actually 1 extended node. Thus the number of nodes is 3.
  
  
• We will number the nodes as shown to the right.
• We will choose node 2 as the reference node and assign it a voltage of zero.
• Write down Kirchoff's Current Law for each node. Call V₁ the voltage at node 1, V₃ the voltage at node 3, and remember that V₂ = 0. The result is the following system of equations:

  

  The first equation results from KCL applied at node 1 and the second equation results from KCL applied at node 3. Collecting terms this becomes:

  

  This form for the system of equations could have been gotten immediately by using the inspection method.
• Solving the system of equations using Gaussian elimination or some other method gives the following voltages:

  V₁=68.2 volts and V₃=27.3 volts

Learnings:
We learn first where the  nodes are in the circuit and finds a super node to solve the nodal analysis based on KCL and KVL law.After that, we can cumpute what are the blank in the circuit.