Combinations of Resistors

Name ______Date ______

As we investigate and analyze electric circuits, we must develop techniques for dealing with combinations of multiple resistors in the same circuit. There are essentially three separate combinations that we will investigate: Series, Parallel, and Complex.

Series

Resistors in series offer no options in terms of the path that the current follows. There is only one choice, only one path: straight through each one. In diagrams, the resistors will resemble the cars in a choo-choo train, with one connected to the next, as below:

In order for the current to move from, say, the left (a higher potential) to the right (a lower potential), it MUST pass through each of the resistors, one right after the other. Furthermore, there is no other way for the current to pass from left to right without passing through each resistor, so ALL of that current must pass through EACH resistor.

Assume that there is a certain total current IT that must pass from the left to the right. Since there are no other options, all of that total current IT must pass through each resistor. In other words, the current through resistor R1 is I1, but that is equal to the total current IT. The same is true for resistors R2 through R5. In other words,

IT = I1 = I2 = I3 = I4 = I5

In moving this charge IT from left to right, there will be a certain difference in potential VT between the left side and the right side. The total voltage drop, then, in passing through these 5 resistors would be that difference in potential VT. But, as we know from Ohm’s Law, this voltage drop must be equal to the product of the current times the resistance such that

VT = ITReq where Req is the equivalent resistance (or combined resistance) for the 5 resistors we have here. Notice that, in order to move from left to right, through an equivalent resistance Req, we had to move through each of the 5 resistors separately. So, we had to overcome each of those resistances separately, and they combine to form the equivalent resistance. So, we can then say that, for our 5 resistors in series:

Req = R1 + R2 + R3 + R4 + R5

Physics Page 1 of 5 Bradshaw 05-06 Combinations of Resistors Name ______Date ______

So, we can now say that

VT = IT[R1 + R2 + R3 + R4 + R5] = ITR1 + ITR2 + ITR3 + ITR4 + ITR5

And finally

VT = V1 + V2 + V3 + V4 + V5

So this leads us to three very important rules for analyzing a collection of resistors (and eventually a collection of equivalent resistances) that are in series. These rules apply for however many resistors we have in Series, from 2 to N.

IT = I1 = I2 = I3 = I4 = I5 Req = R1 + R2 + R3 + R4 + R5 VT = V1 + V2 + V3 + V4 + V5

Parallel

Unlike the case of resistors in series, when resistors are wired in parallel, the current is offered a choice of which path it chooses to follow when moving from high to low potential. In diagrams, resistors in parallel resemble the rungs of a ladder, as seen below:

The two dots on the diagram represent Junction Points, where the current splits (say, on the left) and rejoins again (on the right) after passing through the resistors.

One of the key ideas when looking at resistors in parallel is the fact that each possible branch, each possible path that the current could take (in the diagram, through R1, or through R2, or R3, R4, or R5), is connected to the same pair of junction points. As a result, this guarantees that each of these separate paths MUST have an identical voltage drop; in other words, the voltage for each path (in this case, for each resistor) is exactly the same. Furthermore, each of these voltage drops is equal to the total voltage drop required to pass from junction point to junction point. This indicates that

VT = V1 = V2 = V3 = V4 = V5

At the junction points, the current will split up. Imagine a large group of people trying to enter a building. If there are numerous open doors, some people will follow a path through each doorway. Some people are lazier than others, and so will choose the quickest and easiest pass, while others will attempt to avoid the crush of the crowd and may choose a slightly more difficult path. Regardless, some people will enter the building through each doorway.

Current behaves in a similar manner. If there are 5 separate paths, the current will split up so that some fraction of the total current IT follows each individual path. The larger fractions of the current will follow the paths with lower resistance, and the smaller fractions will follow the paths of higher resistance, but some will go each way. The key is that the product of the current times the resistance for each path is equal; that is, for each path, IR = VT.

Once the current passes along each path, it all meets up at the second junction point so that the amount of current entering the first point is equal to the amount of current leaving the second point. In other words, it makes sure that no current was lost along the way. So, we can express the total current IT as the sum of the amount of current that follows each separate path:

IT = I1 + I2 + I3 + I4 + I5

Now, let’s consider Ohm’s Law for the group of 5 resistors. We can rearrange it in terms of the current:

IT = _VT_ Req

Now, let’s examine this relationship a bit further:

Recall that, for resistors in parallel,

IT = I1 + I2 + I3 + I4 + I5

Also, recall that, according to Ohm’s Law, that

I = V R

So, substituting from Ohm’s Law and knowing that each of the individual voltage drops V1 through V5 must be equal to VT yields:

VT = VT + VT + VT + VT + VT

Req R1 R2 R3 R4 R5

Factoring out VT from the numerators on the right-hand side give us:

VT = VT 1 + 1 + 1 + 1 + 1

Req R1 R2 R3 R4 R5

Which yields the following when we cancel out the VT from both sides:

1 = 1 + 1 + 1 + 1 + 1

Req R1 R2 R3 R4 R5

The expression above is the reciprocal of the equivalent resistance for 5 resistors in parallel. To determine Req, we would mathematically determine the value on the right- hand side of the equation, and then INVERT that number to get the equivalent resistance.

An interesting consequence of this is that the equivalent resistance for resistors in parallel (or, as we will see later, equivalent resistances in parallel) is ALWAYS less than any of the individual values of resistance. This is a mathematical certainty; if you find otherwise, you have made a mistake and need to recheck your work. It is a 100% money- back guarantee that the equivalent resistance is less than any of the individual resistances.

So, we are left again with three rules for resistors (or equivalent resistances) that are in parallel. These rules apply for however many resistors we have in Parallel, from 2 to N.

VT = V1 = V2 = V3 = V4 = V5 IT = I1 + I2 + I3 + I4 + I5 1 = 1 + 1 + 1 + 1 + 1

Req R1 R2 R3 R4 R5

Physics Page 5 of 5 Bradshaw 05-06 Combinations of Resistors