Combining Resistors

Part 1: Fundamental Concepts: Combining Resistors

It is possible to combine multiple resistors together so that they behave as a single resistor.  The resistors can be connected in series, which causes the same current flows through each resistor, or they can be connected in parallel which causes the current to divide as it flows through the resistors.  The combined resistors, whether connected in series or parallel, have two connection points after they are combined.  As a consequence, they have a single value and act as a single resistor with a new value.

In a previous section, the use of series connected resistors was explored in terms of the voltages that appeared across them when in a circuit. When resistors are connected in series, we saw that the total, or equivalent, resistance is found by adding the values of the individual resistor.

Figure 1. Series resistor equivalent value.

As shown in Figure 1, Req = R1 + R2 ... +Rn where Req is the equivalent resistance of the series combination and Rn signifies the last resistor in the chain.  

Resistors can also be connected in parallel.  When the current enters the parallel combination it has more than one path to follow and the current flow splits to flow through all of the resistances.  The effect of this is to reduce the resistance to current flow when compared to a single resistor in the group.

Figure 2. Parallel resistor equivalent value.

Figure 2 shows the method for finding the equivalent value of parallel connected resistors.  The easy way to remember how to find the value of resistor in parallel is by remembering the phrase “the reciprocal, of the sum of, the reciprocals”.  As a rule of thumb, the value of parallel connected resistors is always going to be lower than the value of the smallest resistor in the group.  

Let’s try an example to see how this works. In the following figure, what is the total resistance measured between the indicated points in the following figure?

Figure 3. Series resistor example.

In this case the three resistors are connected in series.  We know that Req = R1 + R2 ... +Rn so, to find the answer we simply add the values.

Req = 1,000Ω + 2,000Ω + 3,000Ω   therefore   Req = 6,000Ω

That was easy. What is the value of the resistors connected as shown in the following figure?

Figure 4. Parallel resistor example.

In this case the resistors are connected in parallel so the math is still easy,  albeit a bit messier.  The parallel combination is found by:

Req = 1/(1/1,000 + 1/2,000 + 1/3,000)  therefore Req = 545.45Ω

There is really no end to the way in which we can combine resistor values.  For example, how would you find the resistance value between the two points indicated in the following figure?

Figure 5. Parallel and series resistor example.

If we look at the resistor connections carefully, we can see this is really the two previous problems combined into one.  Remember that when combining resistors in parallel or series, the resulting value acts like a single resistor.  This allows us to simplify things by combining the parallel resistors into a single resistor as shown in the following figure.

Figure 6. Simplifying by making the parallel resistors a single value.

All that happened in Figure 6, was the replacement of the three parallel connected resistors with a single one whose value is equal to the parallel combination.  This makes the problem simple to solve because we now have four resistors connected in series.  The final value is then found by:

Req = 1,000Ω + 2,000Ω + 3,000Ω + 545.45Ω   therefore   Req = 6,545.45Ω