Problem 10-6    Source:   Problem 2.18 - page 64 -   IRWIN, J. David - Book: Análise de Circuitos em Engenharia - 4ª edição - Ed. Pearson Education do Brasil - 2013.

In the circuit show in Figure 10-06.1, determine:

a) The current i_o

b) The voltage V_o

Solution of the Problem 10-6

Resistors associations should be resolved of the circuits highlighted in green and yellow. For the circuit highlighted in yellow, it can be seen that the resistors of 2 and 10 ohms are in series. Therefore, they can be replaced by a single resistor of 12 ohms. And this resulting resistor lies in parallel with the 4 ohms. Then the whole circuit highlighted in yellow can be replaced by a single resistor value equal to 3 ohms. And for the circuit highlighted in green is simply add the two resistors ( 2 + 4 = 6 ohms ) because they are in series. The figure below shows how the circuit was.

In the Figura 10-06.2, three currents of interest were used, that is, i₁, i₂ and i₃. Note that the resistors of 9 and 3 ohms are in series, resulting in an equivalent resistance of 12 ohms. But this resistor is in parallel with that of 6 ohms, resulting in an equivalent resistance of 4 ohms. Therefore, all resistors that are interconnecting the e₁ to ground, can be replaced by a single resistor of 4 ohms, as shown in the Figure 10-06.3.

Note that in the circuit the voltage source is with the positive pole connected to ground. So the e₁ and e₂ voltages will have negative values. As a consequence, the currents, as they were represented, will also have negative values. This means that its direction is opposite to that indicated in the figures. Once this reservation has been made, the Ohm's law can be apply and it is easily calculate the current i₁, that is:

From this moment, there are two different ways to calculate the values of i₂ and i₃.

Way 1

With the value of i₁ the value of e₁ can be calculated, and from this result applying the Ohm's law, we compute i₂ and i₃. Like this:

Note that the sum of i₂ with i₃ results in the value of i₁.

Way 2

The other possible way is to apply a current divider. Since the total current is known that circulates through the two resistors that are in parallel, that is, i₁ and we know the equivalent resistance value of the parallel, which is 4 ohms, then:

Well, so far we have calculated the values of currents and voltages at points of interest. Now looking for the circuit show in Figure 10-06.4, where we indicate the known values, we can calculate what the problem is asking.

Item a

To calculates i_o, the value of i₃ is known and the equivalent resistance at point e₂. Therefore, by applying a current divider we have:

Item b

To calculates V_o is much easier, because by applying the Ohm's law:

Pay attention to the fact that all calculated values are negative values. This means that by looking at the last circuit the indicated direction of the currents must be inverted.