For any voltage sources, identify them as Case I or Case II.  Which node voltages are known right away as a result of Case I

            [1]  For any voltage sources, identify them as Case I or Case II.  Which node voltages are known right away as a result of Case I voltage sources?

            [2]  Label the resistor currents.  You can use the same label for the currents between resistors R ₁ and R ₂.   Generate the KCL equations for nodes 1 and 2 in terms of the resistor currents you labeled.  Convert the resistor currents in the KCL equations to node voltages and resistor values using Ohm's Law.

Using nodal analysis we can write the KCL equation for node one, V1 :

I0 + I5 – I1 = 0

Converting this in terms of node voltages and resistor values:

(VA – V1)/R5 + (V2 – V1)/R7 – (V1-VB)/(R1+R2) = 0

Using nodal analysis we can write the  KCL equation for node two, V2 :

            [3]  Substitute  V_A and V_B  for the known voltages in the KCL equations and manipulate the equations to show that they can be written in the matrix form Ax = b as shown below.

voltage sources