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I teach physics at high school level, and one of the topics I cover involves instructing students on how to calculate the total resistance of two resistors, whether in parallel, in series or simple networks of resistors.

I am currently seeking authentic examples that are accessible at high school level, yet effectively demonstrate the practical application of the formulas for total resistance in parallel or in series with resistors.

While textbooks provide numerous exercises for this topic, many of the examples fail to show students the practical significance of these calculations.

Specifically, I am interested in examples that can address potential student queries such as: "Why not simply purchase one resistor with the exact resistance needed for my application, rather than combining multiple resistors in parallel or in series?"

Note that I am only considering ohmic resistors for this question.

Given my background in physics rather than electrical engineering, I am reaching out here in the hope of receiving practical examples that effectively illustrate various reasons why these formulas are relevant.

Edit Please note that I am looking for concrete (and accessible) examples. For example saying "Maybe you need a resistor with a non standard (or non available value) you can combine..." helps, but is too abstract to really answer my question. So it would be great to get a concrete example that illustrates this relevant, but abstract point.

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    \$\begingroup\$ Please explain the reason for the close vote such that I can improve my question. \$\endgroup\$
    – Julia
    Jan 25 at 13:04
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    \$\begingroup\$ For example on "math educators" there are many non closed questions about "real life" examples (matheducators.stackexchange.com/search?q=real+life+examples) of mathematical concepts. Those questions have a comparable logical structure as mine. I think giving examples that satisfy certain requirements is not opinion based. \$\endgroup\$
    – Julia
    Jan 25 at 13:41
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    \$\begingroup\$ @Julia don't worry about the close vote, not everyone will agree with it. I don't agree with it, in this case, because there are some very objective reasons for mixing resistances, and the question has attracted some factual, objective answers. Your question should be safe. \$\endgroup\$ Jan 25 at 13:53
  • \$\begingroup\$ This isn't opinion-based. Using parallel resistors is common practice in all manner of practical scenarios. \$\endgroup\$
    – Lundin
    Jan 25 at 14:09
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    \$\begingroup\$ @Julia Aside from all the answers already provided, and some do touch on this, there's one huge reason to know about how to combine resistors in parallel and series -- actual circuit analysis requires knowing about Thevenin and Norton equivalents and, for example, a simple CE transistor amplifier stage will use a base biasing pair of resistors to set the base voltage, where the two resistors are treated in parallel for Thevenin resistance reasons. So that's one place I often see the need. \$\endgroup\$ Jan 25 at 20:34

22 Answers 22

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Examples from the real world:

  • Sticking to the "E series".

    The calculated component value you need might not be available in one of the standard "E series", which in turn means that it might be more expensive and hard to purchase. Using a combination of several standard E series resistors in parallel could result in the same value.

    (You could show students briefly what "E series" are here.)

  • Bill of material optimization.

    A single small metal film resistor, as used in most modern electronics, has a fixed power limit. If the specified power isn't enough for your needs, then using 2x 0402 resistors might be cheaper than using 1x 0603 with higher power rating, particularly if you are already using that very same resistor value elsewhere on the board. PCB assembly contractors will charge you a rigging fee for the pick & place and the more different components there are, the more expensive this fee.

    (Here you could actually show some youtube video with the basics of how a placement machine works, with the tape & reel rigging.)

  • Heat dissipation.

    In some applications, notably explosive environments (EX/ATEX) ones, you want heat to be evenly spread across the board rather than focused at a single component. Two somewhat warm resistors might not trigger an EX event, but one hot resistor might. The EX/ATEX standards name a temperature limit which you are not allowed to go across. So it's not uncommon for these applications to have lots of parallel resistors in sensitive areas where current rushes can be expected or might happen during failures.

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    \$\begingroup\$ RE: Bill of material optimization - yep, there is a cost associated with having more types of surface-mount components needed on the board. It's a bit of a derived example, but if you can fit 2x 2K resistors in parallel to completely remove 1K resistors from your board, that means one less reel of components to stock and to fit on the pick-and-place machine. \$\endgroup\$ Jan 26 at 16:28
  • \$\begingroup\$ RE: Power dissipation - the best practical example is a dummy load. High power resistors are sometimes wire wound, which is bad for RF. A combination of lower power, metal film resistors, solves this problem. \$\endgroup\$ Jan 27 at 2:29
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    \$\begingroup\$ If you are (like I was) curious about what ATEX means: en.wikipedia.org/wiki/ATEX_directive \$\endgroup\$
    – kebs
    Jan 27 at 18:46
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  1. Resistors aren't made in every value. They are made in E-series values. The E-series numbers are selected to match the tolerance of the resistors and to make it possible to get nearly any needed value with a minimum of parts. Can you imagine having to keep a stock of every value of resistor from 1 to 1 million ohms? The E-series lets the manufacturer and sellers produce and sell enough values to cover most needs and requirements.
  2. Power ratings. When current flows through a resistor, the resistor can get warm or hot. That's not a problem when you put 5V on a 10000 ohm resistor - that's only 0.0025 watts. 5V on a 1 ohm resistor, however, will cause the resistor to convert 25 watts of power to heat. Resistors made for high power are expensive. It is often cheaper to use multiple smaller (lower power rating) resistors in a series and parallel combination than it is to buy a single resistor rated for enough power.
  3. Voltage ratings. Resistors have a maximum voltage rating - the ratings are given in the resistor's datasheet. You can put multiple resistors in series so that each is only exposed to a voltage lower than its rated maximum voltage.

As an example, it used to be common for radio amateurs to make their own dummy loads.

A dummy load is a big resistor that has the same resistance as an antenna has impedance. They are used to test transmitters. You don't want to transmit a bad signal to the world while you are adjusting the transmitter, and you can't operate the transmitter without an antenna, so you use a big resistor.

Since the transmitters may put out tens to hundred of watts of power, you need a resistor rated for the needed voltage and power.

High power and high voltage resistors are expensive and hard to get, so amateurs would build their own dummy loads out of multiple resistors. This project describes building a 100 watt dummy load from twenty 3 watt resistors. There are twenty resistors in parallel, each of 1000 ohms. Strictly speaking, each resistor has to handle 5 watts - that project goes over the rated power of the resistors by using an oil bath to keep them from getting hot. Still, bringing it down to 5 watts per resistor is still better than trying to push 100 watts through a part only made for 3 watts.

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    \$\begingroup\$ There's also high voltage failure. For something like a resistive divider to reduce voltage to read into an amplifier: if just one resistor in the fails short in the upper leg the divider is suddenly a short and you don't want the full 1000V-2000V to blow its way through the rest of the circuit. But if you use four or five series resistors to do the same job as one resistor in the upper leg if the divider, if one fails, the division ratio is a bit less but the circuit won't instantly fry. \$\endgroup\$
    – DKNguyen
    Jan 25 at 21:37
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    \$\begingroup\$ @DKNguyen: On the flip side, resistor dividers may use parallel resistors on the low side to ensure that a failure of one resistor won't cause extreme voltage excursions. This sort of issue is, btw, why volume controls in tube amplifier circuits would often have the wiper contact tied to one end of the resistance--so having the wiper be momentarily open might cause gain to unexpectedly increase to maximum, but not increase to infinity. \$\endgroup\$
    – supercat
    Jan 25 at 23:34
  • \$\begingroup\$ Right. Asking why we need a standard series is like asking why we have a positional system for numbers: couldn't we just make up a single digit for whatever exact number we need every time? Of course we could and of course we can't. \$\endgroup\$
    – Rad80
    Jan 26 at 12:11
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    \$\begingroup\$ In the plasma lab we had various things with as many as as 100 resistors in series encased in a plastic tube and mounted at the end of a 3-4 foot plastic pipe for making absolutely, positively sure that the capacitor banks were discharged before anyone touched them. Mostly about the voltage, but they were typically 2 watt and up. \$\endgroup\$
    – Ecnerwal
    Jan 28 at 2:41
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A somewhat complex example where precision resistors are needed is digital-to-analog converters, where a network of resistors are needed to weight digital bits of a number to their power-of-2 equivalents.
You have your choice of two types of networks...R-2R ladder, and the more simple binary-weighted ladder. There are variations of different approaches to binary-weighting the resistor values in the R-2R network.

It is complex because students must first understand how digital binary bits are arranged to represent a numeric value. Many scientific calculators can assist in decimal-to-binary conversion. Microsoft's calculator app in WINDOWS does radix conversion as well.

schematic

simulate this circuit – Schematic created using CircuitLab


Another type of network requiring precision resistors is a bridge-type network. A simple network requires four resistors and a source of voltage or current. The power source can be AC or DC (a 9V battery is shown):

schematic

simulate this circuit

With this network, use your imagination to select resistor elements. For example, two of the resistors (R1 & R4) could be thermistors. You ask students to balance the bridge so that Vm=0 by varying R2 and/or R3. Then you can compare the "hotness" of one student with another - each trying to unbalance the bridge in his/her favour: one student on R1, other student on R4.

R1 & R4 could be photoresistors whose resistance varies with how much light they see.


Kudos to you in your attempts to apply network principles with practical examples. This is a very dry and difficult part of electrical engineering that has put countless classes of students to sleep.

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Just a hobbyist but I've used combinations of resistors in many different situations, just a few I can think of right now:

  • I needed to terminate a video signal with 75 ohm, but that is only available in the E24 series of resistors. I don't stock all those at home, but 150 ohm is readily available, and can be connected in parallel to give 75 ohms exactly.

  • Generating different supply voltages with the venerable LM317 regulator requires a voltage divider with a ratio that can sometimes be difficult to achieve with only two resistors. Replacing one of the sides with a parallel or series combination of two resistors gives you a lot more flexibility.

  • Generalizing from the above, any voltage divider (for things like connecting analog signals to an ADC, other interfaces) are more easily implemented with a combination of standard components.

  • I've had a situation where I needed more power handling but lacked the physical space for a larger resistor. A combination of several thin SMD resistors solved the problem.

  • I have a large bank of hefty 50 Watt resistors mounted on a heat sink for passive load tests. Every time I use it I calculate which combination of parallel and series brings me into the range of power I want to test.

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The formulas are relevant for many reasons. The question such as "Why not buy the exact resistance" is a good one and very simple.

You can't buy any resistance you want, because it either does not exist as stock item, or you place a custom order for a resistor manufacturer to make you one, which will be both expensive and take time.

There are only standard resistance values available. For example, if after calculations you end up needing a 4.9 kilo-ohm resistor, the value does not exist in any standard. So you have to buy 4700 ohms and 200 ohms which do exist, or 4700 ohms and two 100 ohms, or any other combination ending up with 4900 ohms.

Same applies with resistors in parallel. If you need a 4545 ohm resistor, it does not exist, but if you put two resistors of 9090 ohms in parallel, you end up with the value you need. Same for any combination of resistances in parallel that end up with a value you need.

In the examples I used very precise values available in E196 series which has very precise tolerances and as per the name, there are 196 different values per decade. Usually you don't need such high precision and to ease up your inventory of resistors accessible to your students in a lab, you might only use E12 series resistors, which have 12 values per decade. So instead of stocking 196 resistors per a single decade, you can limit to 120 resistors of E12 resistors spanning 10 decades. Which should already be larger range than you need. You rarely need resistors smaller than 0.1 or 0.01 ohms, or larger than 10 or 100 megaohms.

So even if you need non-standard values, or even a standard value if the lab is out of stock for some popular resistor, you can calculate a substitute that is close enough for the required value. Say you need about 5k resistance, but both 4k7 and 5k6 are out of stock in the lab and you must finish your electronics project for a deadline and can't wait, you can just use two 10k resistors in parallel to get the 5k needed.

Also it allows to do some slight adjustments. For example if a voltage divider is used to set the output voltage of an adjustable regulator, it may be that with the resistor values you have available you cannot hit the target voltage exactly, but slightly below or above say 5V or 3.3V that is needed. You can always put a third resistor (in parallel or series) to fine-tune the value of one of the resistors to fine-tune the voltage to be closer or nominally exact 3.3V or 5V, if it matters.

So understanding the applications of what you can do with the theory is useful.

Another example is to calculate input and outptut impedances of voltage dividers. For example, if I am making a product with coaxial SPDIF digital audio output, it must have 1Vpp (unterminated) voltage output and 75 ohms output impedance driving the coaxial cable that has 75 ohm characteristic impedance. If the digital signal is on a 3.3V buffer, I can calculate the voltage divider values needed because I know the 3.3Vpp must be brought down to 1Vpp (unterminated) and 75 ohm impedance.

Another, maybe even obvious example, is that if you need to make a 2000W heater, but have only certain type of heating wire used to make many types and different power heaters, you can calculate how many parallel wires you need to make an European 240V model and American 120V model with the same heating wire, just arranging the heaitng wire pieces in parallel and series combination as required.

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You will (nearly) always be able to find a manufacturer who can create the exact specification you need, but at a price, and a delay. If you can get the voltage handling, or the tolerance, or the area, or the value, or the low inductance, or the reliability, or the components now, by putting cheap components you already have access to into combinations, then you do. It's too real a reason to dismiss as 'abstract'.

For the professional, the reason is usually standardising on a small range of resistors, to keep stock holding, and all the associated expenses of that, down. For the amateur, much the same, except now you are rifling through your spare parts box looking for any components that can be combined right now into the value you want.

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I have a real-world example from a circuit board in my company.

Signals coming into a circuit board are normally amplified to a level which the rest of the circuit board can deal with; and similarly signals going out are amplified from the level the rest of the circuit uses to the level the receiving equipment expects. This is typically done with operational amplifiers, using inverting or non-inverting amplifier configurations. In both of those configurations, the gain of the amplifier circuit is set using resistors - so if you want a specific gain, you need to use a specific combination of resistor values.

Like most manufacturers, we use resistor values from the E24 series, because they are cheaper and more readily available, so that limits the combination of resistors we can get off the shelf. Even then though, if we only need a simple gain (for example a gain of 2), we can often choose specific values to get this.

But just recently (with component shortages over the COVID period), we found that one of the ADCs we use in the circuit was not available. A very similar part did exist - but the original part translated a voltage level of +/-10V to full-range measurements, whilst the potential replacement took a voltage level of +/-10.24V as being its full-range measurement. In order to drop in the replacement ADC without requiring changes everywhere else, we needed to put an extra gain of 1.024 on the input amplifier. It started with a gain of 1.5, but the new gain required was 1.536.

With a non-inverting amplifier circuit, a gain of 1.5 can easily be achieved with R1=2K and R2=1K, both standard values from the E24 resistor range. A gain of 1.536, not so much! Keeping R1 at 2K, we needed R2=1.072K. The best match for this was a 1.2K resistor in parallel with a 10K resistor (combined resistance of 1.0714K).

Hopefully this will give your students an example of where resistors are used, why values are important, and the kind of calculations an engineer has to do with them.

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First of all, stop saying "ohmic resistors" — it's simply redundant and sounds a bit silly. All resistors are "ohmic".

It isn't so much that you'd use these formulas to create a particular value of resistance per se, except when you also have some other constraint. For example, it is often necessary to bias a transistor or opamp with a particular combination of voltage and resistance. This is called a "Thévenin source". The most convenient way to create the voltage is with a voltage divider from the power supply, but you also select the resistor values so that their parallel combination is the required Thévenin resistance.

The series and parallel formulas are also used in circuit analysis all the time. For example, computing the effective input or output resistance of a circuit so that you can predict how it will interact with other circuits around it.

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  • \$\begingroup\$ 0 ohm resistors aren't ohmic :) \$\endgroup\$
    – Lundin
    Jan 25 at 12:56
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    \$\begingroup\$ Oh, yes they are! ;-) \$\endgroup\$
    – Dave Tweed
    Jan 25 at 12:56
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    \$\begingroup\$ Depends on tolerance perhaps. Though I'd give a bonus cookie to someone who can explain to me how tolerance is calculated on 0 ohm resistors, because the math doesn't add up. \$\endgroup\$
    – Lundin
    Jan 25 at 13:00
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    \$\begingroup\$ @Lundin It's an additive tolerance for 0 ohm 'resistors', not a fractional one. \$\endgroup\$
    – Neil_UK
    Jan 25 at 13:29
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    \$\begingroup\$ In my circles, "ohmic" refers to a linear I-V curve. Commercial resistors are mostly designed in this way, but it's not a major spec. And certainly not everything that can be used as a resistor is ohmic, not even all metal resistors. \$\endgroup\$
    – tobalt
    Jan 26 at 7:56
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A strand of decorative holiday lights has 24 bulbs in series. The strand draws 0.5 A at 120VAC when it is lit. What is the resistance of each bulb when lit?

Because the holiday lights are in series, a single bulb burning out causes the whole strand to fail. To work around this, each bulb has a shunt that is designed to short-circuit when its filament fails. After one of the bulbs burns out, how much power do each of the remaining bulbs dissipate?

Consider a strand with 50 or 100 bulbs. How can these strands be designed to have the same brightness as the original strand?

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  • \$\begingroup\$ Thanks, but bulbs are not ohmic. \$\endgroup\$
    – Julia
    Jan 26 at 8:41
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    \$\begingroup\$ True, but they do approximate it once lit. \$\endgroup\$ Jan 26 at 14:01
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You don't just have resistances where you intentionally place resistors, you have it everywhere, and calculating the effect of that is important. An example from experimental physics:

I have a stack of electrodes for a Penning trap (image c/o Dhdpla on wikipedia):

Image of a Penning trap consisting of a central grounded electrode, with two biased endcap electrodes

I want to bias the two endcaps to +1 kV, but I measure that the resistance from each endcap to the central grounded electrode is 1 megaohm. My cheap high voltage power supply can only output 1 mA of current at 1 kV. Can I use this single supply to supply the bias voltage for both endcaps? This situation can be represented by this circuit:

schematic

simulate this circuit – Schematic created using CircuitLab

Which is equivalent to a 500 kOhm resistor to ground, and would require 2 mA from the HV supply, so the answer is no. That probably requires some more explanation of simplification to make it accessible, but I'll leave that up to your expertise.

For resistor networks, bridges (like the Wheatstone) are critical for high-precision electrical measurements, like in resistive thermometers (and many other situations).

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I am currently seeking authentic examples [of] the practical application of the formulas for total resistance in parallel or in series with resistors. […] Why not simply purchase one resistor with the exact resistance needed for my application, rather than combining multiple resistors in parallel or in series?

One of the reasons why we need to know those formulas is so that we can simply purchase one resistor with the exact resistance needed.

Let's say that I've built a circuit that includes a 2 kΩ resistor in series with an LED, but then I find that the LED is too dim. In order to make the LED a little brighter, I try putting a 3 kΩ resistor in parallel with that one, and I find that that results in a good brightness. If I didn't know the formula for parallel resistance, I would have to put a 2 kΩ resistor and a 3 kΩ resistor in parallel in my next revision of the circuit board. Fortunately, I do know that formula, so I can simply use one resistor.

It's also important to realize that often, we a study a thing not because that exact thing is useful, but because that thing is a simpler, easier version of a different thing which is useful. For example, it's not usually useful to simply have two resistors in series, but one bit of circuitry which is extremely useful is a voltage divider. If you want to calculate the amount of current that a voltage divider consumes, the easiest way to do that is to make the circuit simpler and easier by pretending that nothing is connected to the output node. When you do that, the voltage divider becomes... two resistors in series! We previously learned how to understand the simple-but-not-so-useful thing, and now we can use that knowledge to help us understand the complicated-and-useful thing.

Here are a few practical examples of things that are "more complicated versions" of resistors in series or in parallel. Once we understand how series and parallel resistors work, that helps us to understand all of these things.

  • A voltage divider – used almost every time we do anything with an analog signal
  • A thermistor in series with a resistor – used along with an analog-to-digital converter for measuring temperature
  • Two transmission lines in parallel, with a resistor in series – used for splitting a high-frequency signal into two high-frequency signals without producing unwanted reflections
  • Two resistors in series, except that a voltage clamp is connected to the center node – used for protecting against overvoltage events when there are two directions that they could come from
  • Several heating elements in series and/or in parallel – used for heating a large area
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Safety reason. Some safety standards use phrase "single fault condition". Eg. Use 2 resistors in series. Circuit must be considered safe if one resistor will fail into short condition.

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A very basic practical example on how the knowledge of parallel resistance can be applied is during acceptance testing of video distribution amplifiers (VDAs), also used for NRZ-L telemetry data distribution. You may have a Chassis loaded with 15 VDAs; each VDA has One Input and 6 ouputs, the Input and Output impedance is 75-Ohms. The Chassis has dual/redundat power supplies rated at +/- 12VDC/100W. All the inputs are fed a 1Vpp and the ouputs are adjusted to a maximum of 5Vpp/terminated with 1/2W 75-Ohm resistors. The fully loaded Chassis is usually run for 6-8 hours continously at room temperature 77F.The question is can the individual Power Supplies handle the fully loaded Chassis? Sometimes new power supplies would fail under full load due to power supply defects. The student can calculate the required full load current by adding each VDA required current (6-ouputs/75-ohms) and then multiplying by 15.The Chassis usually have two exhaust cooling fans rated at 12VDC/80mA.

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  • \$\begingroup\$ In certain cases involving two resitors in parallel, it is useful to find an unknown resistor by using the formula Rx = RtR/R-Rt \$\endgroup\$
    – A.M.
    Jan 26 at 3:51
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In certain cases involving two resitors in parallel, it is useful to find an unknown resistor by using the formula Rx = RtR/R-Rt. For example, you may need a 1/2 watt, 75-ohm resistor and you only have 1/4 watt resistors. If you choose a 100-ohm resistor what value resitor do you need to make it +/- 5% 75-ohms? you can do the simple math (75x100)/100-75 = 300-Ohms. If you use the formula for Rt = R1 x R2/R1 + R2 it will give you (300x100)/300 +100 = 75-Ohms. So the two 1/4 watt resitors in parallel will dissipate 1/2 Watt together.

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I'm not an electrical engineer, but a programmer who has led a team developing software for electrical engineers. Everything has resistance (OK, superconductors, but let's not go there): the wires in your house, the wire from the power pole to your house, the wire from the substation to the power pole. That resistance dissipates power, as @A.M. pointed out), which drops the voltage that your appliances run at, and costs you money. Moreover wiring codes normally restrict the amount of voltage drop that is allowed, so you need to be able to do calculations with resistances.

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resistive heaters like a in a hair dryer

here high heat uses a second resistor in parallel with the first.

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It's not only about solely resistors. A device might be modeled as a resistor under certain situations, so the series and parallel resistor concepts would be useful in determing currents or voltages across these devices in a circuit.

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    \$\begingroup\$ Do you have a concrete example? \$\endgroup\$
    – Julia
    Jan 25 at 12:57
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Sometimes, as in a resistor divider, you care about how well resistors are matched to each other instead of their absolute value. The sheet resistance tolerance of resistors on an integrated circuit may be +/-40%, but you can still get a 0.1% output accuracy from these inaccurate components as long as they match each other pretty well.

The way to do this is to have a standard physical size of resistance, with a fixed length and width, and then make a lot of them right next together. If you make them different width or length, they will no longer match each other as well.

If you have a typically 10kΩ link size, a reference voltage of 1V, and you need 200mV with 25kΩ total typical resistance for the entire divider, you would use series 2 links on the top and parallel 2 links on the bottom.

They also sell packaged, matched resistors for this purpose, and you could do the same thing with them outside of an IC.

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Using multiple resistors makes it easier to tune the exact value later.

YouTube channel BigClive likes to change how bright an LED is (less bright takes way longer to burn out).

Normally, he does this by replacing only one of a pair of resistors.

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In addition to achieving specific values, power handling, safety, and bill of materials optimization, you can achieve a lower tolerance resistor by combining several high tolerance resistors in series.

For example, assuming normally distributed component variance, you can get a 3 % 10 kΩ resistor by putting ten 10 % 1 kΩ resistors in series.

This could be a good classroom lesson on sums of uncorrelated random variables.

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The primary reason I use parallel or series resistor circuits is to reduce part variety. If I use 1K resistors almost everywhere on the PCA, and need one 500ohm and one 2K resistor, then series and parallel circuits allow me to buy one resistor type as opposed to three different resistors.

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Fixing something after it's been produced with a wrong part. Dave loaded the wrong part into the assembly machine and now there's a 2kΩ resistor where there should be a 100Ω one. This causes the part to not work. It'd be a royal pain to desolder that resistor, but really simple to solder another one in top. What value should we use?

Most speakers are 8Ω, most amplifiers operate best at 4-16Ω, sometimes as low as 2Ω. How do you hook up all your speakers? With 4 speakers you can do 2 parallel sets of 2 in series to still have 8Ω load on the amp. Plenty of opportunity for questions like does this arrangement of speakers fit in the specs for this amplifier.

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