Parallel DC Circuits

Two network configurations, series and parallel, form the framework for some of the most complex network structures. A clear understanding of each will pay enormous dividends as more complex methods and networks are examined. The series connection was discussed in detail in the last chapter. We will now examine the parallel circuit and all the methods and laws associated with this important configuration. The term parallel is used so often to describe a physical arrangement between two elements that most individuals are aware of its general characteristics. In general,

• Fig. 1(a): the two resistors are in parallel because they are connected at points a and b.
• Fig. 1(b): Resistors $R_1$ and $R_2$ are in parallel because they again have points a and b in common. $R_1$ is not in parallel with $R_3$ because they are connected at only one point (b). Further, $R_1$ and $R_3$ are not in series because a third connection appears at point b. The same can be said for resistors $R_2$ and $R_3$.
• Fig. 1(c): Resistors $R_1$ and $R_2$ are in series because they have only one point in common that is not connected elsewhere in the network. while ($R_1 + R_2$) are in parallel with $R_3$, because their combined effect goes parallel with $R_3$. Resistors $R_1$and $R_3$ are not in parallel because they have only point a in common. In addition, they are not in series because of the third connection to point a.

Furthermore, even though the discussion above was only for resistors, it can be applied to any two-terminal elements such as voltage sources and meters.

Parallel Resistors in a circuit

For resistors in parallel as shown in [Fig. 2], the total resistance is determined from the following equation: Lets go for a deep calculation of the total resistance obtained from the parallel combination of resistors as given in eq.(1): Looking to Fig. 2(a), applied voltage source $E$ is connected directly to all of the resisters in parallel. So voltage drops across $R_1$, $R_2$, and $R_2$ are same. But the currents in the parallel resistors are different due to different values of resistance. $I_1$ is the current passing through $R_1$, $I_2$ through $R_2$, and $I_3$ through $R_3$. Current $I$ is the overall current extracted from the battery, which divides further into three currents $I_1$, $I_2$, and $I_3$. In equation form, we can get: where $I_1 = E/R_1$, $I_2 = E/R_2$, and $I_3 = E/R_3$. By inserting these values in the above equation, we can get: But Fig. 2(b), which is the resulted form of parallel resistors, can gives us the value of current $I = E/R_T$. Now we get; Since conductance G = 1/R, the equation can also be written in terms of conductance levels as follows: If you review the examples above, you will find that the total resistance is less than the smallest parallel resistor. That is, in Example 1, 2 Ω is less than 3 Ω or 6 Ω. In general, therefore, This is particularly important when you want a quick estimate of the total resistance of a parallel combination. Simply find the smallest value, and you know that the total resistance will be less than that value. It is also a great check on your calculations. In addition, you will find that

Special Case: Equal Parallel Resistors

For equal resistors in parallel, the equation for the total resistance becomes significantly easier to apply. For N equal resistors in parallel,