**Series and Parallel Resistor Circuits**

The previous examples show, as practice does, that circuits usually have more than one component. The question is, can we find a way to simplify calculations using Kirchoff's laws? The answer is yes, and the elements can be coupled together into two types of connections: series and parallel connections. We will now investigate the series and parallel connections for electrical resistances.

A *series connection* has all resistors connected in a line and the same amount of current passes through each resistor as shown in Figure 14.11.

If the three resistors are of different resistances, the voltage drop on each of them is different (they oppose differently charge motion). According to Ohm's law, the voltage is:

*V*_{1} = *R*_{1} · *I*, *V*_{2} = *R*_{2} · *I*, *V*_{3} = *R*_{3} · *I*

The voltage drop on the entire line of resistors has to be the same as the sum of all the voltages in the previous equation:

*V* = *V*_{1} + *V*_{2} + *V*_{3}

*V* = *R*_{1} · *I* + *R*_{2} · *I* + *R*_{3} · *I*

*V* = (*R*_{1} + *R*_{2} + *R*_{3}) · *I*

If we imagine the entire circuit represented by just one resistor that has the same effect on the current as do the three, we can consider the previous equation to be written as:

*V* = *R _{s}* ·

*I*

If we put together the last equations, we have:

(*R*_{1} + *R*_{2} + *R*_{3}) · *I* = *R _{s}* ·

*I*

*R*_{1} + *R*_{2} + *R*_{3} = *R _{s}*

The resistance of an equivalent circuit of resistors connected in series is equal to the sum of the resistances. More generally, the previous equation can be written as:

where *i* is the number of resistors connected in series.

A *parallel connection* has multiple resistors hooked to the same junction (see Figure 14.12), and the current splits, taking the *path of least resistance*, literally! The same equations of currents and voltages in junctions and loops can be set to determine the equivalent parallel connection resistance.

The current in the resistors is different and dependent on the value of the resistors, but the voltage drop between the ends of each resistance is the same, as the voltage on the parallel connection, *V*.

The resistance of an equivalent circuit of resistors connected in parallel is equal to the sum of the inverse of the resistances. More generally, the previous equation can be written as:

where *i* is the number of resistors connected in parallel.

**Example**

Two resistors, *R*_{1} = 100 mΩ and *R*_{2} = 50 mΩ are connected first in series and then in parallel to the same battery. When is the equivalent resistance larger: in the series or in the parallel connection?

**Solution**

To answer this problem, set up the equations for each of the series and parallel connections for the two resistors.

*R*_{1} + *R*_{2} = *R _{s}*

100 + 50 = *R _{s}*

*R _{s}* = 150 mΩ

*R _{p}* = 33 mΩ

*R _{s}* >

*R*

_{p}Actually, the above result is true for all resistors, and the parallel connection will always yield a smaller value of the equivalent resistance than does a series connection made up of the same resistors as in the parallel one.

Practice problems of this concept can be found at: Electrical Current Practice Questions

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