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# Combinations of resistors

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### Combinations of resistors

1. 1. Combinations of Resistors Resistors do not occur in isolation. They are almost always part of alarger 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 commoncombinations, not only for resistors but other elements as well. (Forexample, we can speak of "a resistor in series with a capacitor".) Well start by examining series and parallel combinations and thenmove on to identifying those combinations when they are "buried" within alarger circuit. What were doing is learning how to recognize smallportions of larger circuits. Experts do that. You can click here to seehow 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 ass/he becomes an expert.Series Combinations of Resistors Two elements are said to be in series whenever the same currentphysically flows through both of the elements. The critical point is thatthe same current flows through both resistors when two are in series.The particular configuration does not matter. The only thing thatmatters is that exactly the same current flows through both resistors.Current flows into one element, through the element, out of the elementinto the other element, through the second element and out of the secondelement. No part of the current that flows through one resistor"escapes" and none is added. This figure shows several different waysthat two resistors in series might appear as part of a larger circuitdiagram.
2. 2. Questions Here is a circuit you may have seen before. Answer the questionsbelow 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. 3. You might wonder just how often you actually find resistors inseries. 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 thatsituation with the circuit diagram shown below. In many electronic circuits series resistors are used to get adifferent voltage across one of the resistors. Well look at thosecircuits, called voltage dividers, in a short while. Heres the circuitdiagram for a voltage divider.
4. 4. Besides resistors in series, we can also have other elements in series- capacitors, inductors, diodes. These elements can be in series withother elements. For example, the simplest form of filter, for filteringlow frequency noise out of a signal, can be built just by putting a resistorin series with a capacitor, and taking the output as the capacitor voltage. As we go along youll have lots of opportunity to use and to expandwhat you learn about series combinations as you study resistors in series. Lets look at the modelagain. We see that the wiresare actually small resistors(small value of resistance,not necessarily physicallysmall) in series with the lightbulb, which is also a resistor. We have three resistors in series althoughtwo of the resistors are small. We know that the resistors are in seriesbecause all of the current that flows out of the distribution box throughthe first wire also flows through the light bulb and back through thesecond wire, thus meeting our condition for a series connection. Tracethat 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 thoughtexperiment on these two resistors. Here is the circuit diagram for thesituation were interested in. Imagine that they are embedded in an opaque piece of plastic, sothat we only have access to the two nodes at the ends of the seriesconnection, and the middle node is inaccessible. If we measured theresistance of the combination, what would we find? To answer thatquestion we need to define voltage and current variables for the
5. 5. resistors. If we take advantage of the fact that the current throughthem 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) andcurrent that flows through both resistors (Is) and a voltage variable, Vs,for the voltage that appears across the series combination. Lets 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 Ohms Law: o Va = Is Ra o Vb = Is Rb We can combine all of these relations, and when we do that we findthe following. Vs= Va + Vb Vs= Is Ra + Is Rb Vs= Is (Ra + Rb) Vs= Is RseriesHere, we take Rseries to be the series equivalent of the two resistors inseries, and the expression for Rseries is: Rseries = Ra + Rb What do we mean by series equivalent? Here are some points toobserve.
6. 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 seriesresistance. Imagine that you have two globs of black plastic. Each of theglobs of black plasic has two wires coming out. Inside these two blackplastic 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 thetwo different cases you would not be able to tell which black plastic globhad the single resistor and which one had the series combination. Here are two resistors. At the top are two 2000W resistors. Atthe bottom is single 4000W resistors. (Note, these are not exactlystandard sizes so it took a lot of hunting to find a supply store that soldthem!). You can click the green button to grow blobs around them.
7. 7. After you have grown the blobs around the resistors there is noelectrical measurement you can make that will allow you to tell which onehas two resistors and which one has one resistor. They are electricallyindistinguishable! (Or, in other words, they are equivalent!)QuestionQ4. Is the series equivalent resistor larger than either resistor, or is itsmaller?ProblemsP1. What is the series equivalent of two 1000 resistors in series?Enter your answer in the box below, then click the button to submit youranswer. You will get a grade on a 0 (completely wrong) to 100 (perfectlyaccurate answer) scale. Your grade is:P2. What is the series equivalent of a 1000 resistor and a2700 resistor in series?
8. 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 resistorin series with this 100 resistor in order to limit the current to 0.5 ampswhen 110 volts is placed across the two resistors in series. How muchresistance should you use? Your grade is:Parallel Resistors The other common connection is two elements in parallel. Tworesistors or any two devices are said to be in parallel when the samevoltage physically appears across the two resistors. Schematically, thesituation is as shown below.Note that we have defined the voltage across both resistor (Vp) and thecurrent that flows through each resistor (Ia and Ib) and a voltagevariable, Vp, for the voltage that appears across the parallel combination.
9. 9. Lets 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 Ohms Law: o Ia = Vp /Ra o Ib = Vp /Rb We can combine all of these relations, and when we do that we findthe following. Ip= Ia + Ib Ip= Vp /Ra + Vp /Rb Ip= Vp[ 1/Ra + 1/Rb] Ip= Vp/RparallelHere, we take Rparallel to be the parallel equivalent of the two resistors inparallel, and the expression for Rparallel is: 1/Rparallel = 1/Ra + 1/Rb There may be times when it is better to rearrange the expressionfor Rparallel. The expression can be rearranged to get: Rparallel = (Ra*Rb)/(Ra + Rb) Either of these expressions could be used to compute a parallelequivalent resistance. The first has a certain symmetry with theexpression for a series equivalent resistance.QuestionQ5 Is the parallel equivalent resistor larger than either resistor, oris it smaller?