Resistor Combinations
Series and Parallel Resistor Combinations
Series Combinations of Resistors
Parallel Combinations of Resistors
What If You Have A More Complex Circuit?
Problems
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Combinations of Resistors

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.

Goals

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.

Combinations of 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.

Series Combinations of Resistors

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.

Questions

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?

You might wonder just how often you actually find resistors in series.  The answer is that you find resistors in series all the time.

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 (Va and Vb) and current that flows through both resistors (Is) and a voltage variable, Vs, for the voltage that appears across the series combination.

Let's list what we know.

• The current through the two resistors is the same.
• The voltage across the series combination is given by:
• Vs= Va + Vb
• The voltages across the two resistors are given by Ohm's Law:
• Va = Is Ra
• Vb = Is Rb
We can combine all of these relations, and when we do that we find the following.
• Vs= Va + Vb
• Vs= Is Ra  + Is Rb
• Vs= Is (Ra  + Rb)
• Vs= Is Rseries
Here, we take Rseries to be the series equivalent of the two resistors in series, and the expression for Rseries is:
Rseries = Ra  + Rb

What do we mean by series equivalent?  Here are some points to observe.

• If current and voltage are proportional, then the device is a resistor.
• We have shown thatVs= Is Rseries, so that voltage is proportional to current, and the constant of proportionality is a resistance.
• We will call that the equivalent series resistance.
There is also a mental picture to use when considering equivalent series resistance.  Imagine that you have two globs of black plastic.  Each of the globs of black plasic has two wires coming out.  Inside these two black plastic globs you have the following.
• In the first glob you have two resistors in series.  Only the leads of the series combination are available for measurement externally.  You have no way to penetrate the box and measure things at the interior node.
• In the second box you have a single resistor that is equal to the series equivalent.  Only the leads of this resistor are available for measurement externally.
Then, if you measured the resistance using the two available leads in the two different cases you would not be able to tell which black plastic glob had the single resistor and which one had the series combination.

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.

After you have grown the blobs around the resistors there is no electrical measurement you can make that will allow you to tell which one has two resistors and which one has one resistor.  They are electrically indistinguishable!  (Or, in other words, they are equivalent!)

Question

Q4.  Is the series equivalent resistor larger than either resistor, or is it smaller?

Problems

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?

Parallel Resistors

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.

Note that we have defined the voltage across both resistor (Vp) and the current that flows through each resistor (Ia and Ib) and a voltage variable, Vp, for the voltage that appears across the parallel combination.

Let's list what we know.

• The voltage across the two resistors is the same.
• The current through the parallel combination is given by:
• Ip= Ia + Ib
• The currents through the two resistors are given by Ohm's Law:
• Ia = Vp /Ra
• Ib = Vp /Rb
We can combine all of these relations, and when we do that we find the following.
• Ip= Ia + Ib
• Ip= Vp /Ra + Vp /Rb
• Ip= Vp[ 1/Ra + 1/Rb]
• Ip= Vp/Rparallel
Here, we take Rparallel to be the parallel equivalent of the two resistors in parallel, and the expression for Rparallel is:
1/Rparallel = 1/Ra  + 1/Rb

There may be times when it is better to rearrange the expression for Rparallel.  The expression can be rearranged to get:

Rparallel = (Ra*Rb)/(Ra + Rb)

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.

Question

Q5        Is the parallel equivalent resistor larger than either resistor, or is it smaller?

Problems

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.

Parallel Resistors - A Point to Remember
• It is important to note that the equivalent resistance of two resistors in parallel is always smaller than either of the two resistors.

Problems

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, R3 and R4 in series?

What If You Have A More Complex Circuit

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.

• Ra = 1500 W
• Rb = 3000 W
• Rc = 2000 W
• Rd = 1000 W
• Vs = 12 v
We need to figure out where we can start.  We can start by trying to find any of the combinations we've learned about.  So let's think about whether there are any series or parallel combinations and if there are let's see if we can identify them.  Then we can apply what we know about series and parallel combinations.  There's no guarantee that approach will work, but it is worth a try.  Let's look at two resistors at a time.

The first question is are there any series or parallel combinations?  Click the red button below to see two resistors in series.

Question

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, we should be able to replace the two resistors in series with their series equivalent.  If we do that, there's a node in the middle with a voltage, and we'll lose information about that voltage.  Right now, we're not interested in that voltage, and we'll willing to lose that information.  Let's just replace the two resistors with their series equivalent.  Click the red button to make that replacement.  Depressing the button will remove the two resistors in series, and releasing the button will insert the replacement.

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?

Question

Q7  What replacement can be made?

Ok, you see how it goes.  Let's take a numerical example using the values mentioned above.
• Ra = 1500 W
• Rb = 3000 W
• Rc = 2000 W
• Rd = 1000 W
• Vs = 12 v
Here is the circuit.

Problems

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?