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.
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 hereto 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.
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.
Rseries = Ra + Rb
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.
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!)
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.
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.
Parallel Resistors - A Point to Remember
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.
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 hereto 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.
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.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.
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
- Vs= Va + Vb
- Vs= Is Ra + Is Rb
- Vs= Is (Ra + Rb)
- Vs= Is Rseries
- 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.
- 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.
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.
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.
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
- Ip= Ia + Ib
- Ip= Vp /Ra + Vp /Rb
- Ip= Vp[ 1/Ra + 1/Rb]
- Ip= Vp/Rparallel
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.
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