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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.

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:
       o Vs= Va + Vb
     The voltages across the two resistors are given by Ohm's Law:
       o Va = Is Ra
       o 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      resistors in series?

Enter your answer in the box below, then click the button to submit your
answer. You will get a grade on a 0 (completely wrong) to 100 (perfectly
accurate answer) scale.


                          Your grade is:

P2. What is the series equivalent of a 1000     resistor and a
2700 resistor in series?
Your grade is:

P3. What is the series equivalent of three 1000     resistors in series?
You may want to do this problem in two steps.




                          Your grade is:

P4. Imagine that you have a 100 resistor. You want to add a resistor
in series with this 100 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?



                          Your grade is:


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:
        o Ip= Ia + Ib
      The currents through the two resistors are given by Ohm's Law:
        o Ia = Vp /Ra
        o 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     resistors in parallel?

Enter your answer in the box below, then click the button to submit your
answer. You will get a grade on a 0 (completely wrong) to 100 (perfectly
accurate answer) scale.


                         Your grade is:

P6. What is the parallel equivalent of a 1000     resistor and a
1500 resistor in parallel?




                         Your grade is:

P7. What is the equivalent of three 1000    resistors in parallel? You
may want to do this problem in two steps.




                         Your grade is:


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?
Your grade is:

P9. What is the equivalent resistance of this resistance combination?




                          Your grade is:

P10. What is the equivalent resistance of this resistance combination?
Here all three resistors are 33 k . Remember to input your answer in
ohms.




                          Your grade is:

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
     Rb = 3000
Rc = 2000
     Rd = 1000
     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
     Rb = 3000
     Rc = 2000
     Rd = 1000
     Vs = 12 v

Here is the circuit.
Problems

P12. What is the equivalent resistance of the two resistors in series -
1000 and 2000 ?




                         Your grade is:

P13. Next you should have two resistors in parallel. What is the parallel
equivalent?




                         Your grade is:

P14. Now you should have two resistors in series attached to the source.
What is the value of the series equivalent?




                         Your grade is:

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.
Your grade is:

Give your answer in milliamperes here, if that's what you want.


                          Your grade is:


Problems

     Problem   Resist2P01 - Resistor Combinations - 1
     Problem   Resist2P02 - Resistor Combinations - 2
     Problem   Resist2P03 - Resistor Combinations - 3
     Problem   Resist2P04 - Resistor Combinations - 4
     Problem   Resist2P05 - Resistor Combinations - 5


Links to Other Lessons on Resistors

     Resistors
     RCombinations
     VoltageDividers
     Bridge Circuits
     Resistance Experiments

Send your comments on these lessons.

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Combinations of resistors

  • 1. 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. 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.
  • 2. 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?
  • 3. 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.
  • 4. 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
  • 5. 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: o Vs= Va + Vb The voltages across the two resistors are given by Ohm's Law: o Va = Is Ra o 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.
  • 6. 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.
  • 7. 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 resistors in series? Enter your answer in the box below, then click the button to submit your answer. You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale. Your grade is: P2. What is the series equivalent of a 1000 resistor and a 2700 resistor in series?
  • 8. Your grade is: P3. What is the series equivalent of three 1000 resistors in series? You may want to do this problem in two steps. Your grade is: P4. Imagine that you have a 100 resistor. You want to add a resistor in series with this 100 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? Your grade is: 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.
  • 9. Let's list what we know. The voltage across the two resistors is the same. The current through the parallel combination is given by: o Ip= Ia + Ib The currents through the two resistors are given by Ohm's Law: o Ia = Vp /Ra o 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?
  • 10. Problems P5. What is the parallel equivalent of two 1000 resistors in parallel? Enter your answer in the box below, then click the button to submit your answer. You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale. Your grade is: P6. What is the parallel equivalent of a 1000 resistor and a 1500 resistor in parallel? Your grade is: P7. What is the equivalent of three 1000 resistors in parallel? You may want to do this problem in two steps. Your grade is: 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?
  • 11. Your grade is: P9. What is the equivalent resistance of this resistance combination? Your grade is: P10. What is the equivalent resistance of this resistance combination? Here all three resistors are 33 k . Remember to input your answer in ohms. Your grade is: 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
  • 12. 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 Rb = 3000
  • 13. Rc = 2000 Rd = 1000 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
  • 14. 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 Rb = 3000 Rc = 2000 Rd = 1000 Vs = 12 v Here is the circuit.
  • 15. Problems P12. What is the equivalent resistance of the two resistors in series - 1000 and 2000 ? Your grade is: P13. Next you should have two resistors in parallel. What is the parallel equivalent? Your grade is: P14. Now you should have two resistors in series attached to the source. What is the value of the series equivalent? Your grade is: 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.
  • 16. Your grade is: Give your answer in milliamperes here, if that's what you want. Your grade is: Problems Problem Resist2P01 - Resistor Combinations - 1 Problem Resist2P02 - Resistor Combinations - 2 Problem Resist2P03 - Resistor Combinations - 3 Problem Resist2P04 - Resistor Combinations - 4 Problem Resist2P05 - Resistor Combinations - 5 Links to Other Lessons on Resistors Resistors RCombinations VoltageDividers Bridge Circuits Resistance Experiments Send your comments on these lessons.