Series and Parallel Resistor CombinationsYou are at: Elements - Resistors - Resistor CombinationsSeries Combinations of ResistorsParallel Combinations of ResistorsWhat If You Have A More Complex Circuit?Problems
Resistors do not occur in isolation. They are almost always part of a larger circuit, and frequently that larger circuit contains many resistors. It is often the case that resistors occur in combinations that repeat.
What are our goals for this lesson? Here are some.
Given a circuit with a number of resistors,Be able to determine resistor combinations within the circuit where two or more resistors can be combined.
Be able to replace series and parallel resistor combinations with the correct equivalent resistors.
In this lesson we will look at two recurring resistor combinations, series combinations and parallel combinations. Those are common combinations, not only for resistors but other elements as well. (For example, we can speak of "a resistor in series with a capacitor".)
We'll start by examining series and parallel combinations and then move on to identifying those combinations when they are "buried" within a larger circuit. What we're doing is learning how to recognize small portions of larger circuits. Experts do that. You can click here to see how experts are able to recognize larger combinations in many situations. It is a part of the basic "tool box" that an expert in an area acquires as s/he becomes an expert.
Two elements are said to be in series whenever the same current physically flows through both of the elements. The critical point is that the same current flows through both resistors when two are in series. The particular configuration does not matter. The only thing that matters is that exactly the same current flows through both resistors. Current flows into one element, through the element, out of the element into the other element, through the second element and out of the second element. No part of the current that flows through one resistor "escapes" and none is added. This figure shows several different ways that two resistors in series might appear as part of a larger circuit diagram.
Here is a circuit you may have seen before. Answer the questions below for this circuit.
Q1. Are elements #3 and #4 in series?
Q2. Are elements #1 and #2 in series?
Q3. Is the battery in series with any element?
An example of series resistors is in house wiring. The leads from the service entrance enter a distribution box, and then wires are strung throughout the house. The current flows out of the distribution box, through one of the wires, then perhaps through a light bulb, back through the other wire. We might model that situation with the circuit diagram shown below.
In many electronic circuits series resistors are used to get a different voltage across one of the resistors. We'll look at those circuits, called voltage dividers, in a short while. Here's the circuit diagram for a voltage divider.
Besides resistors in series, we can also have other elements in series - capacitors, inductors, diodes. These elements can be in series with other elements. For example, the simplest form of filter, for filtering low frequency noise out of a signal, can be built just by putting a resistor in series with a capacitor, and taking the output as the capacitor voltage.
As we go along you'll have lots of opportunity to use and to expand what you learn about series combinations as you study resistors in series.
Let's look at the model again. We see that the wires are actually small resistors (small value of resistance, not necessarily physically small) in series with the light bulb, which is also a resistor. We have three resistors in series although two of the resistors are small. We know that the resistors are in series because all of the current that flows out of the distribution box through the first wire also flows through the light bulb and back through the second wire, thus meeting our condition for a series connection. Trace that out in the circuit diagram and the pictorial representation above.
Let us consider the simplest case of a series resistor connection, the case of just two resistors in series. We can perform a thought experiment on these two resistors. Here is the circuit diagram for the situation we're interested in.
Imagine that they are embedded in an opaque piece of plastic, so that we only have access to the two nodes at the ends of the series connection, and the middle node is inaccessible. If we measured the resistance of the combination, what would we find? To answer that question we need to define voltage and current variables for the resistors. If we take advantage of the fact that the current through them is the same (Apply KCL at the interior node if you are unconvinced!) then we have the situation below.
Note that we have defined a voltage across each resistor (V_{a} and V_{b}) and current that flows through both resistors (I_{s}) and a voltage variable, V_{s}, for the voltage that appears across the series combination.
Let's list what we know.
What do we mean by series equivalent? Here are some points to observe.
Here are two resistors. At the top are two 2000W resistors. At the bottom is single 4000W resistors. (Note, these are not exactly standard sizes so it took a lot of hunting to find a supply store that sold them!). You can click the green button to grow blobs around them.
Q4. Is the series equivalent resistor larger than either resistor, or is it smaller?
P1. What
is the series equivalent of two 1000 W
resistors in series?
P2. What
is the series equivalent of a 1000 W
resistor and a 2700 W
resistor in series?
P3. What is the series equivalent of three 1000 Wresistors in series? You may want to do this problem in two steps.
P4. Imagine that you have a 100 W resistor. You want to add a resistor in series with this 100 W resistor in order to limit the current to 0.5 amps when 110 volts is placed across the two resistors in series. How much resistance should you use?
The other common connection is two elements in parallel. Two resistors or any two devices are said to be in parallel when the same voltage physically appears across the two resistors. Schematically, the situation is as shown below.
Let's list what we know.
There may be times when it is better to rearrange the expression for R_{parallel}. The expression can be rearranged to get:
Either of these expressions could be used to compute a parallel equivalent
resistance. The first has a certain symmetry with the expression
for a series equivalent resistance.
Q5 Is the parallel equivalent resistor larger than either resistor, or is it smaller?
P5. What
is the parallel equivalent of two 1000 W
resistors in parallel?
P6. What
is the parallel equivalent of a 1000 W
resistor and a 1500 Wresistor
in parallel?
P7. What is the equivalent of three 1000 Wresistors in parallel? You may want to do this problem in two steps.
P8. What is the equivalent resistance of this resistance combination?
P9. What is the equivalent resistance of this resistance combination?
P10. What is the equivalent resistance of this resistance combination? Here all three resistors are 33 kW. Remember to input your answer in ohms.
P11. Here is an operational amplifier circut, a Wien-bridge oscillator. The circuit is taken from Wojslaw and Moustakas' book Operational Amplifiers (John Wiley & Sons, 1986, p100). Assuming that the amplifiers take no current at the "+" and "-" terminals are resistors, R_{3} and R_{4} in series?
Here's a circuit with resistors that has them connected in a different way. For a short while we're going to work on the question of how to analyze this circuit. For a start we're going to assume that this is a resistor. It has two leads at the left (marked here with red dots) and we'll assume that we want to find the equivalent resistance you would have at those leads.
We will use the following numerical values for the resistors in this example, and we will work through using these values.
The first question is are there any series or parallel combinations? Click the red button below to see two resistors in series.
Q6 Would the two resistors above (highlighted when the button is clicked) be in series if any current were drawn from the circuit by attaching a load?
Now you should have the circuit with the two resistors in series replaced by their series equivalent. Now, we can see that there is another replacement we can make. What's that replacement?
Q7 What replacement can be made?
P12. What is the equivalent resistance of the two resistors in series - 1000W and 2000W?
P13. Next you should have two resistors in parallel. What is the parallel equivalent?
P14. Now you should have two resistors in series attached to the source. What is the value of the series equivalent?
P15. With a 12v source - as shown in the figure - what is the current that is drawn from the source? Give your answer in amperes here.
Give your answer in milliamperes here, if that's what you want.