Node Voltage Method

The node voltage method of analysis solves for unknown voltages at circuit nodes in terms of a system of KCL equations. This analysis looks strange because it involves replacing voltage sources with equivalent current sources. Also, resistor values in ohms are replaced by equivalent conductances in siemens, G = 1/R. The siemens (S) is the unit of conductance, having replaced the mho unit. In any event S = Ω-1. And S = mho (obsolete).

We start with a circuit having conventional voltage sources. A common node E0 is chosen as a reference point. The node voltages E1 and E2 are calculated with respect to this point.

A voltage source in series with a resistance must be replaced by an equivalent current source in parallel with the resistance. We will write KCL equations for each node. The right hand side of the equation is the value of the current source feeding the node.

Replacing voltage sources and associated series resistors with equivalent current sources and parallel resistors yields the modified circuit. Substitute resistor conductances in siemens for resistance in ohms.

           I1 = E1/R1 = 10/2 = 5 A
           I2 = E2/R5 = 4/1  = 4 A
           G1 = 1/R1 = 1/2 Ω   = 0.5 S
           G2 = 1/R2 = 1/4 Ω   = 0.25 S
           G3 = 1/R3 = 1/2.5 Ω = 0.4 S
           G4 = 1/R4 = 1/5 Ω   = 0.2 S
           G5 = 1/R5 = 1/1 Ω   = 1.0 S

The Parallel conductances (resistors) may be combined by addition of the conductances. Though, we will not redraw the circuit. The circuit is ready for application of the node voltage method.

           GA = G1 + G2 = 0.5 S + 0.25 S = 0.75 S
           GB = G4 + G5 = 0.2 S + 1 S = 1.2 S

Deriving a general node voltage method, we write a pair of KCL equations in terms of unknown node voltages V1 and V2 this one time. We do this to illustrate a pattern for writing equations by inspection.

           GAE1 + G3(E1 - E2) = I1             (1)
           GBE2 - G3(E1 - E2) = I2             (2)


           (GA + G3 )E1         -G3E2 = I1     (1)
                  -G3E1 + (GB + G3)E2 = I2     (2)

The coefficients of the last pair of equations above have been rearranged to show a pattern. The sum of conductances connected to the first node is the positive coefficient of the first voltage in equation

(1). The sum of conductances connected to the second node is the positive coefficient of the second voltage in equation

(2). The other coefficients are negative, representing conductances between nodes. For both equations, the right hand side is equal to the respective current source connected to the node. This pattern allows us to quickly write the equations by inspection. This leads to a set of rules for the node voltage method of analysis.

  Node voltage rules:

•    Convert voltage sources in series with a resistor to an equivalent current source with the resistor in parallel.

•    Change resistor values to conductances.

•    Select a reference node(E0)

•    Assign unknown voltages (E1)(E2) ... (EN)to remaining nodes.

•    Write a KCL equation for each node 1,2, ... N. The positive coefficient of the first voltage in the first equation is the sum of conductances connected to the node. The coefficient for the second voltage in the second equation is the sum of conductances connected to that node. Repeat for coefficient of third voltage, third equation, and other equations. These coefficients fall on a diagonal.

•    All other coefficients for all equations are negative, representing conductances between nodes. The first equation, second coefficient is the conductance from node 1 to node 2, the third coefficient is the conductance from node 1 to node 3. Fill in negative coefficients for other equations.

•    The right hand side of the equations is the current source connected to the respective nodes.

•    Solve system of equations for unknown node voltages.