Problem 10-5 Source:
Problem elaborated by the author of the site.
On the circuit show in Figure 10-05.1, determine:
a) The currents i1 and i2.
b) What is the power dissipated in the 5 ohm resistor?.
Figure 10-05.1
Solution of the Problem 10-5
Item a
Note that the circuits highlighted in green and yellow are the same, only reversed.
Therefore, the same procedure can be used to calculate the equivalent resistance of each circuit.
For the circuit highlighted in yellow, note that the resistors of 30 and 60 ohms
are in parallel. By multiplying its values and dividing by its sum, we find
the value of the parallel, or 20 ohms. This, in turn, is in series with that
of 4 ohms,
resulting in 24 ohms equivalent resistance for this branch. Similarly,
for the circuit that is highlighted in green, results in 4 ohms for the parallel of
resistors of 5 and 20 ohms. And as 4 ohms is in series with
that of 8 ohms,
results 12 ohms as the end result. Observe in the Figure 10-05.2
how was the circuit simplified.
<Figure 10-05.2
Note that we have slightly changed the topology of the circuit. The current flowing
through the 12 ohms resistor that is connected to the voltage source has been
called it. Note that the 12 and 24 ohms resistors
continue in parallel, where the 12 ohms resistor represents the
highlighted in green and that of 24 ohms represents the circuit highlighted in
yellow.
The 12 ohms resistor continues in series with the voltage source. Note that
the current
i1, that we represent in the figure, is the same current
i1 that cycles through the resistor of 8 ohms in the initial
circuit. To calculate i1 we need to calculate it.
To do so, simply calculate the parallel of 12 and 24 ohms that results
at a equivalent resistance of 8 ohms. And this in turn is in series with that of
12 ohms.
Therefore, the equivalent resistance of the entire circuit is equal to 20 ohms.
Given these values we can compute it, or:
it = 60 / Req = 60 / 20 = 3 A
From this moment we have two different ways to calculate
the value of i1.
Way 1
Notice that the resitor 12 ohms is between the points a and b.
With the value of it we can calculate the value of Vab by multiplying the current it by the equivalent resistance of the parallel of 12 and 24 ohms , that is, 8 ohms. Applying the Ohm's law, the value of Vab is calculated, or:
Vab = 3 8 = 24 Volts
Now just divide the value of Vab by 12 ohms, which is
the value of
resistor where i1 circulates, or:
i1 = Vab / 12 = 24 / 12 = 2 A
Way 2
The other possible way is to apply the current divider. How do you know the total current
that circulates through the circuit, then:
i1 = it ( 8 / 12 ) = 2 A
Therefore, by an alternative method we find the same value.
Item b
As we already know the value ofi1, just apply a current divider
to calculate i2.
i2 = i1 ( 4 / 5 ) = 2 0.8 = 1.6 A
Notice that the 4 value is the result of the parallel of the resistors
of 5 and 20 Ohms. Thus, by knowing the value of the current that
crosses the 5 ohms resistor we can calculate the
power dissipated by this resistor.
