# T³: Adventures in Science – Series and Parallel Circuits

There are two basic ways to connect electronic components with two terminals: series and parallel. In this episode, we examine those two types of circuits and show how to calculate equivalent resistance.

Most basic electronic components with two terminals, like resistors, can be connected in one of two ways: series or parallel. These connections allow us to mix and match different values of components (or even different components!) to achieve some desired behavior in a circuit.

In the video, we examine how resistors can be connected, and how to calculate the equivalent resistance of each type of circuit.

In a series circuit, components are attached end-to-end and share one common node:

To extend it to more than one resistor, we can calculate the effective resistance (i.e. equivalent resistance) of resistors attached in series by adding up the resistances:

In parallel, components share two common nodes, and current has the option of flowing through one of the components:

From that, we can calculate the equivalent resistance of the resistors attached in parallel by taking the reciprocal of the sum of reciprocal resistances. The following equation has been extended to include any number of resistors:

If you would like a written refresher on what a node is and how to calculate equivalent resistance from series and parallel circuits, we have a written tutorial created by our very own Pete:

That's the fifth and final (for a while) Adventures in Science concept video! Thank you all for the great feedback. I need to work on some other things for the upcoming holiday season, but if we get back around to doing more of these, what other concepts would you like to see covered?

## Comments 12 comments

• Good video and explanation. I thought that Liquid Circuitry 2.0 could be confusing if you didn't understand how the pipe valves worked (Handle parallel to flow = ON, handle perpendicular to flow = OFF).

Initially, the fact that the pipe layout didn't match graphically to the written schematics was a little confusing, too. It took me a minute to figure out what was going on.

Personally, I liked the algebra explaining how to arrive at the final formula, but I can see how that could be intimidating. Maybe a "Mythbusters" style "Warning: Science" graphic would help. But in a Sparkfun kind of way. :- )

• First, in the blog article above (but not in the video), you show two resistors in series and then the general formula for arbitrarily many resistors in series. IM[NS]HO, it would be better to first show the formula for two, then say something like "this can be extended for lots of resistors in series" and show the general formula. This same complaint goes for the parallel resistors, too.

My one complaint about the video is that you ass-u-me that the viewer is conversant with Algebra. I know I was learning about electronics several years before I (formally) studied Algebra. It is an unfortunate fact that many, many people, even talented ones, have numerophobia to some degree. (I'm convinced that it's a "learned phobia", thanks to teachers [and parents] who are numerophobic themselves, but that's beside the point.) I'm just afraid that such an in-your-face algebraic approach might scare off some otherwise very talented kids. I really think that a better approach would be "here's how we calculate it", and then an "oh, by the way, if you don't mind the math, here's how we can show that this is so." Remember, we're trying to give kids some info they can use so that maybe, someday, they'll become Engineers -- we're not trying to train Engineers here. (I know I, and I think you, too, attended good Engineering schools, with very good teachers and pre-reqs that included Algebra, where they spent a couple of weeks covering what you cover in less than 8 minutes.)

Please remember, Shawn, that I mean this as constructive criticism. I really appreciate your efforts in putting together this series. I sure wish I'd had it back in about 1966 or so!

One last thing: I've been meaning to ask about the pen you use: what is it? I've been looking for a pen that has white ink.

• No worries, your feedback has consistently been fantastic and well-received. Good point about the blog post - I can make the correction fairly easily.

I wanted to show the proof for series/parallel calculations, but if it's too much algebra, I can see how it might be a bit scary. Are you suggesting that I show the formula first with an example and then show the proof with the disclaimer that it contains a bit of algebra?

The pen is actually a silver Sharpie, which looks white-ish in the video when used on the blue paper.

• FWIW, I just bought a 2-pack of the silver Sharpies at a local grocery store (Safeway). With my "frequent shopper" for \$3.49, plus \$0.27 of local sales tax. Amazon wants \$5.14, plus tax, plus shipping. Plus I don't have to await delivery!

As soon as I got home, I used one to mark the "pin 1" end of a black socket! (Then wrote this...)

• Uh-oh, the silver Sharpie secret is out. The revolution begins :) They are very useful for writing on dark surfaces.

• I think, Shawn, that it would be less scary if you said "here's how we calculate it" (showing the final formula first), "but if you're interested in where this comes from and not just wiling to take my word for it, here's the derivation". Also, be SURE to say the "two component" case prior to showing the "arbitrarily many component" case.

Given that I'm not an "Amazon prime" member, I'll probably just check local stores. :-) (I suspect that a 50' roll of paper might be a lifetime supply for me, given my age!)

• Resistance is pretty easy. Things get more fun when the mix includes capacitance and/or inductance. Are there some easy ways to analyze some of these cases or does it take Higher Mathematics?

• I just happened to think I should mention this: If you look in appropriate math book (e.g., a Calculus textbook), you'll find literally hundreds of different functions under "integral" and "differential" tables. I got my Engineering degree more than 35 years ago, and have found that most of the time, I only needed to know how about half a dozen or so worked, and even then I only needed the "final Algebra". I've had to use others fewer than a half dozen times, and then it was an easy matter to find them in a book.

To be an Engineer, you really do need to have a "feel" for how the underlying "Higher Mathematics" works so that you can analyze a novel situation, but the vast majority of the time you can "divide and conquer" the problem at hand without having to work out the "Higher Mathematics" every time. The "novel situations" are rare enough that the typical hobbyist should not get scared off by their existence.

• If you're looking to calculate equivalent inductance, the calculations are similar to resistors. For capacitance, it's the opposite. Parallel capacitors add, and you do the summation of reciprocals for series.

Like Member #134773 mentioned, you need some Calculus to analyze how capacitors and inductors behave, as they affect a circuit based on the change in voltage or current.

• Actually, you don't need calculus to derive the equations for capacitors. The trick is to use electric charge, not current, and the definition of capacitance, C = Q/V. When the capacitors are connected in parallel, the charge supplied by the source is distributed between the two capacitors, thus Qs = Q1+Q2. When in series, the source pushes a charge into C1 which pushes the same charge into C2 which finally pushes the same charge back to the source, thus Qs = Q1 = Q2. With just a little algebra, the equations are yours.

Note: The calculus buffs would remind us that charge is just the integral of current, so I guess the calculus is there, just hidden!

• Depends on what you mean by "Higher Mathematics". :-) For a "full analysis", you don't need Tensors or Bessel functions, but you do need some Calculus and Trig.

That said, however, if you're willing to "take the instructor's word for it" and not see the full derivation, they do reduce to simple Algebra functions. As I recall, the Amateur Radio Handbook does a pretty good job of covering it (though I haven't looked at this updated-annually book in a few years). It is an excellent reference, even if you aren't interested in Amateur Radio per se.

Full disclosure: I am an ARRL member, though I've had no hand in writing the manual. I've kept a copy around since more than 20 years before I joined ARRL!

• I should probably have clarified: For inductors and capacitors there are two distinct cases: when you have a DC supply and when you have an AC supply. For the DC case, you need to use an "exponential" function to analyze what happens when you turn the DC on or off, and for the AC you need to involve the frequency of the power source.

OK, that's simplifying it a bit much, but that's the basic situation. I'm assuming the AC source produces a sine wave -- if not (e.g., for a square wave), it gets a lot more messy.

BTW, sorry for the delay. I had to go vote. I went before my "morning" shower -- I figured given the candidates for some of the races, nobody would notice if I stank a little!

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