Problem + Hard 13-3.    Source:   Problem worked out by the website author.

For the circuit shown in Figure 13-03.1, determine:

a) the values ​​of I₁, I₂ and < f4>I₃;

b) the values ​​of the currents flowing through the voltage sources and the powers supplied or received by them.

Solution of the Problem + Hard 13.3

Item a

Between points c and d, there are four resistors connected, two resistors in parallel. The parallel of 36 Ω and 18 Ω, results in an equivalent resistor of 12 Ω. So, summing the values ​​of the resistors an equivalent resistance is found between points c and d, of 40 Ω. Applying Ohm's law, one can easily find the value of I₁, or:

For the I₂ loop:

And finally, for I₃, we have:

Therefore, we have a system of two equations with two unknowns that can be solved by any method. We found the following values:

Item b

To find the current flowing through the 80 V voltage source, you must do the equation of the nodes for the node d, as it is easily verified that the current flowing through by the voltage source is the sum I₁ + I₂, that is:

For the 9 V source, the current flowing through it is the current I₃. Note that for the case of the two voltage sources, the currents are leaving through the positive pole. Therefore, both sources supply power to the circuit. Then:

And for the 9 V source, we have:

Therefore, the total power supplied to the circuit is the sum of the powers calculated above, that is, - 689.25 W. On the other hand, it is known that this power must be equal, in module, to the power dissipated by all the resistors that make up the circuit. It is left as an exercise for the student to prove this statement.