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Examples Using Formal Circuit Methods

Module by: Don Johnson

Summary: Two examples of using the node method to solve for a circuit.

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Example 1

Figure 1
Node
Node (circuit22b.png)

The presence of a current source does not affect the node method greatly; just include it in writing KCL equations as a current leaving the node. The circuit has three nodes, requiring us to define two node voltages. The node equations are

e 1 R 1 + e 1 − e 2 R 2 − i in =0      (Node 1) e 1 R 1 e 1 e 2 R 2 i in 0      (Node 1) (1)
e 2 − e 1 R 2 + e 2 R 3 =0     (Node 2) e 2 e 1 R 2 e 2 R 3 0     (Node 2) (2)

Note that the node voltage corresponding to the node that we are writing KCL for enters with a positive sign, the others with a negative sign, and that the units of each term is given in amperes. Rewrite these equations in the standard set-of-linear-equations form.

e 1 1 R 1 +1 R 2 − e 2 1 R 2 = i in e 1 1 R 1 1 R 2 e 2 1 R 2 i in (3)
- e 1 1 R 2 + e 2 1 R 2 +1 R 3 =0 e 1 1 R 2 e 2 1 R 2 1 R 3 0 (4)
Solving these equations gives
e 1 = R 2 + R 3 R 3 e 2 e 1 R 2 R 3 R 3 e 2 (5)
e 2 = R 1 R 3 R 1 + R 2 + R 3 i in e 2 R 1 R 3 R 1 R 2 R 3 i in (6)
To find the indicated current, we simply use i= e 2 R 3 i e 2 R 3 .

Example 2

Figure 2
Nodes Example
Nodes Example (circuit22c.png)

In this circuit, we cannot use the series/parallel combination rules: The vertical resistor at node 1 keeps the two 1 Ω resistors from being in series, and the 2 Ω resistor prevents the two 1 Ω resistors at node 2 from being in series. We really do need the node method to solve this circuit! Despite having six elements, we need only define two node voltages. The node equations are

e 1 − v in 1+ e 1 1+ e 1 − e 2 1=0     (Node 1) e 1 v in 1 e 1 1 e 1 e 2 1 0     (Node 1) (7)
e 2 − v in 2+ e 1 1+ e 2 − e 1 1=0     (Node 2) e 2 v in 2 e 1 1 e 2 e 1 1 0     (Node 2) (8)
Solving these equations yields e 1 =25 v in e 1 2 5 v in and e 2 =513 v in e 2 5 13 v in . The output current equals e 2 1=513 v in e 2 1 5 13 v in . One unfortunate consequence of using the element's numeric values from the outset is that it becomes impossible to check units while setting up and solving equations.

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