Root Mean Square

The first time I was in school for electrical engineering (long story), I had a professor who had never worked in the industry. I was in her class and the topic of the day was measuring AC waveforms. We got to see some sine waves centered on zero volts and were taught that the peak voltage was the magnitude of the voltage above zero. The peak to peak was the voltage from–surprise–the top peak to the bottom peak, which was double the peak voltage. Then there was root-mean-square (RMS) voltage. For those nice sine waves, you took the peak voltage and divided by the square root of two, 1.414 or so.

You know that kid in the front of the class? They were in your class, too. Always raising their hand with some question. That kid raised his hand and asked the simple question: why do we care about RMS voltage? I was stunned when I heard the professor answer, “I think it is because it is so easy to divide by the square root of two.”

So What’s the Right Answer?

This made me really angry. I was paying good money to be there and that was the answer? Even at that young age I knew better. There are two things wrong with the professor’s answer. First, the dividing by the square root of two is only valid on the pretty sine waves we were studying. Any more complex waveform required calculus to get the right answer. For example, a triangle wave’s RMS voltage is the peak value divided by the square root of three.

However, the biggest problem is that the answer is nonsense. It is even easier to divide by ten, but that’s of no value. The reason you want to measure RMS voltage is simple: 1 V RMS does the same amount of work in a load as 1 V DC. So a resistor subjected to 1 V DC and 1 V RMS will generate the same amount of heat, for example.


Not only is this practical, but it makes Ohm’s law continue to work. We know that power is I times E. Since I is also equal to E/R, you can deduce that power is E (the voltage) squared over the resistance. But in an AC circuit, what is that voltage? It isn’t peak or peak-to-peak–that would give a wrong answer. It is, instead, the RMS voltage.

Take another example. If you look up common RMS values for different waveforms, you’ll see the RMS voltage of a square wave with peak voltage V is just V. That’s because a real square wave goes in equal amounts positive and negative. So a 5 V square wave, for example, is always at either 5 V or -5 V and, either way, the same amount of work gets done.

But what about a pulse train from 0 V to 5 V with a duty cycle of 50%? Now the RMS value is the peak voltage (5 V) time the square root of the duty cycle (0.5) or about 3.5 V. The calculus can get hairy, but if you have a set of discrete measurements (as you probably do in any real-life situation) you simply apply the name backward. Square each sample. Find the mean (average). Then take the square root.

Consider the pulse train example. With eight samples taken at twice the frequency of the PWM, you’d expect to get four 5 V readings and four 0 V readings. If you square these samples, you get four values of 25 and four that are still 0. So the average will be 100/8 or 12.5 and the square root of 12.5 is about 3.5. That matches the answer from the table (that is, five times the square root of  0.5).


You can compute RMS voltage using an oscilloscope. However, with a meter, it can be tricky depending on what the meter measures. Most meters that don’t claim to measure RMS, read the average value of the voltage. Some meters measure RMS but only for a sine wave. Older true-RMS meters used thermal or electrodynamic methods to measure the RMS value. However, modern meters are usually adept at measuring RMS, at least for pure AC signals. A very few meters will have an option to measure RMS voltage for signals with a DC component, like an offset.

According to Fluke, an average-reading meter will read 10% high on a square wave, 40% low on the output of a single-phase diode rectifier, and anywhere from 5% to 30% low on the output of a three-phase diode rectifier. Big difference.

A Mad Professor (as in Angry)

In those days, I wasn’t smart enough to hold my tongue, so I raised my hand from the back of the class and explained the above (perhaps a little more succinctly). The response from the professor: “Oh yeah. That too.” I won’t mention the name of the school or the professor, but it did prompt me to find a new school.

With the increased use of digital instrumentation and calculators, there is a propensity for spewing off numbers without thinking about what they mean. In this case, saying that an AC voltage is 20 volts isn’t really a complete answer. We need to know what kind of voltage measurement the 20 represents and we also need to understand what that means and how it fits for the question at hand.

If you want to get more into the calculus, you might enjoy [Darryl Morrell’s] video, below.

Photo credit: sine wave graph by [AlanM1] (CC BY-SA 3.0 )]

66 thoughts on “Root Mean Square

    1. Those who can, do. Those who can’t do, teach.

      And those who can’t teach, teach gym.

      Maybe Al’s school had given up on its athletics department and needed to find alternative jobs for the instructors.

      1. That saying really rubs me the wrong way!

        Or rather, it makes me re-count my lucky stars to have had so many great teachers throughout my life. It’s a bummer that some people actually believe this is a “truth” when it’s just a reflection on the bad teachers they’ve (personally) had.

        Great teachers very often “can”, but instead they choose to teach, because teaching others is a force multiplier. It makes the world full of more people who “can” than if they had just done it themselves.

        1. It may not sit well with you, but there is also a grain of truth there.

          Here are some examples why professors are crazy. Lets stick to college type jobs, since everyone knows high school teachers only do that job for the summer vacation.
          1) Good jobs in academia are incredibly hard to get. Why would a capable person put up with that crap? Getting a tenured position at a university is a serious ordeal with absolutely no certain outcome. Imagine spending 8+ years “training” for your profession, and in many cases paying for that training, and then having to wait for someone to die to actually get a steady job.
          2) They don’t pay well. Sure a professor makes more than a grocery bagger, but typically less than the equivalent job in industry.
          3) Teaching is not the main responsibility. Research is. So professors have to split their focus on what they really want to be doing (research) with what pays the bills (teaching).
          4) There is a HUGE amount of red-tape associated with everything. Especially for public funded schools. So even the research aspects of the job are messed up.

          It’s crazy talk for a capable person to pick a job in academia if they can get a job in industry. Professors are typically professional students that only know how to be students and only know how the academic system works because that’s the only thing they have experienced.
          The case could be made for retired industry veterans coming back and doing some teaching. But even that doesn’t guarantee good professors since the academic way of thinking of problems isn’t even close to the real world problem solving. What the professor thinks is important and what the industry veteran knows is important are often different.

          Oh, and just because you had a “great teacher” doesn’t mean they would have been capable at a job in industry where they would be actually required to make things. It’s two very different skill sets.

          1. A professor would make around what an engineer project manager would make back in the dot com boom era. They do have a good benefit package and job security once tenured vs in the industry. Some would also prefer the research part of the work. Eventually as an engineer, you’ll get more and more management side of things instead of getting your hands dirty.

            My manager showed me a newspaper article with a startup in the area I was working on. I recognized the name of the CTO as I worked with the guy when he was a PHD student across the other side of the lab. He published a few papers. I also know someone else that worked with him in a different company before. He eventually went back to the university as an associated professor.

        2. Back in school day, the student bod had to partition a department to reverse their decision of denying tenure to one of the good professors. We spent a lot of time talking to him outside class. He suffered because in his field, it is difficult to publish papers. He tell us that the ones who do care about students already go for additional training course. Our CS department has probably the best professors because it is a new science and the ones that teach still remember when they were students. [Being a new science also means that they are still learning too.]

          It was funny to see of one of the other professors referenced in a SiFi book (just 2 words with his last name and his field) as he is a recognized expert. I found a paper that say how his work was great, but not practical in real life because of the complexity except the most critical systems. That kind of matched what I felt.

    2. Or sometimes a teacher has to teach a topic they are not an expert in. For example I had a VoIP class where the professor literally read everything out of the book, and still needed corrected often. At the end of the class someone asked what was his background experience? The guy had worked for Panasonic, and had helped design DVD decoding chips. We spent the last day learning about MPEG2, and I learned more in that day than the whole VoIP course. Sometimes professors are great, but staffing issues put them in the wrong class.

    1. Seriously? Another? This sentiment is vile. Of course there are poor teachers; that can be said of, literally, any field. Why disparage an entire profession, particularly one as honorable as teaching? Were all of your teachers truly this bad? Seems statistically improbable. More likely that the problem lies with the common factor — you.

      1. +1

        Teaching is not as easy as everyone seems to think. It takes knowing your subject way beyond what you would be able to get by with in the industry. You have to explain things, not just apply what you know. You have to be able to explain things in many different ways, because that one explanation that worked with someone, does not necessarily work with someone else. You have to be ready for any kind of question and be ready to research what you don’t know. Teaching takes commitment. It is one of the most difficult jobs out there, and also one of the least recognized.

  1. Another reason RMS is important is that the nameplate voltages on everything that is utility AC (or intended to be run on it) is expressed in RMS voltage. For example, 125VAC is RMS. The peak voltage is actually closer to 176-180 depending on the crumminess of the sinewave you’re actually getting. To prove this, measure the AC-to-DC stage on an ATX power supply. The AC line power is rectified to DC straight into some big caps, thus capturing the peaks. It is greater than the line voltage by a factor of 1.414, roughly.

    Another reason RMS is important is that multimeters with an AC scale express the value as RMS. It’s important to know that.

    1. Um. Ok. Techincally you are right, those are good reasons that it is important to understand RMS. However, those things are only true because RMS is important. If RMS were not important for the reasons the author mentioned then no doubt multimeters wold not measure in RMS nor would power lines be labeled in RMS.

    2. “Another reason RMS is important is that multimeters with an AC scale express the value as RMS. It’s important to know that.”

      Absolutely not true in all cases. MANY multimeters read AC as an average value. I have a Fluke 8050A (true RMS) and a Fluke 73 III (averaging) on my bench and they read exactly the same for sine waves but different for any other waveform. It’s important to know that ;-)

  2. I had a similar experience my first and last semester at a community college. 2 of my 3 Electronics classes were taught by a math teacher that clearly would need help replacing batteries in a TV remote. I vaguely remember something about how he was trying to explain the two different theories of how current moves (Electrons moving vs ‘holes’ moving), and he described it as two different KINDS of current, as if we would be encountering both kinds in the real world. I quickly learned the nerdy ham class helper in the lab area knew a LOT more, so I instead spent time with the helper, and wound up flunking the classes.Still managed to get a good education, despite the teacher.

      1. You come up with a lot of simplifications over the years. What is the resistance of a 47k and 470k resistor in parallel? 47k is equal to ten 470k in parallel, so 470k/11. But … 1/11 is 0.90909, so that’s somewhere between 42k and 43k.

        Calculated, it is 42.727272k. But for 99.99% of my purposes, 42.5k is close enough.

  3. It is equally important to note that while most AC devices are specified by their RMS voltages, where you are using components, you still need to design based on the peak voltages, and even there, you need to be aware of utility variations. In the US for example, most consumer electric devices are called 110VAC or 120VAC devices, but if you have good line quality and few other demands on your mains transformer, the utility may be delivering 125VAC (RMS), and your design needs to accommodate that peak RMS * 1.414 (or dividing by 0.707, which is another number you may have seen bantered about). So, if you’re using a MOV to protect your circuit from transients, it needs to be rated for 176.75V or so (commonly, you will encounter 175V and 185V MOVs which are typical component values for 125VRMS and 130VRMS), or the rest of your circuit design needs to be tolerant of the surges and other power anomalies.

    1. Usually your MOV would list a working voltage rating. These parts are already worked out for the common input voltages as that’s what their customers would be buying.

      Essentially, it is made up of a series of micro spark gaps inside the material. The tolerance of a MOV is so loose that it is not unusual to begin clamping at 2-3X the working voltage. It can only clamp the peak part of the spike while your electronics still have to deal with the over voltage.

  4. Area under the curve, averaged over a cycle. The squaring is to make everything positive, but you then need to un-square to get the correct value. It’s simple, once you understand why each step is there.

    I always tried to understand “why” things were done, rather tna memorizing “how” to do them. Once you understand why, the how is easy (and you understand how to handle the case that wasn’t discussed, like triangle, sawtooth, etc).

    1. Not area under the curve. The squaring isn’t just to make the numbers positive. The mean absolute value is a very different thing.

      RMS, at the most common level, is an indicator of power delivered, which, for a resistive load, will be proportional to the square of voltage (as well as the square of the current) at each instant. RMS can be thought of as a measure of the constant voltage (or current) equivalent that will deliver the same power to a resistive load as the non-constant potential (or current) measured.

      This leads to a nice puzzle that highlights this: Use the 50% duty waveform to deliver power to a resister. Lets say it is 1ohm. 50% of the time, the power is 25W, 50% of the time, and 0W the other 50%, for 12.5W. If you look at the average voltage, you get 50% of 5V, which is 2.5V. The power delivered by a constant 2.5V across a 1ohm resistor is 6.25W. No the same.

      The real puzzle comes in when you look at the power delivered by a PWM line, like the PWM output of a microcontroller. Does that pin deliver 12.5W or 6.25W to the 1ohm resistor? We think of the output voltage as being 2.5V in this case, but that would indicate power delivered is 6.25W. The math above comes to 12.5W. What is going on? (hint: are the conditions here the same throughout? Or, are there changed assumptions….)

  5. As a former EE prof myself, I can only guess you misheard the question or the answer. Or mayber the professor did. Or maybe she was having a joke at the asker’s expense. I can’t imagine she got through her PhD without knowing something basic.

    1. a) may not have been a PhD, depending on the school, and b) I can quite well imagine someone getting a PhD without knowing a lot of the basics, having known, and worked with, a number of them. For example, the EE that held hard to the ‘only electrons carry current’ and ‘Ben Franklin messed up all up’ mode. Then again, he also didn’t believe in quantum mechanics (‘tunneling? No such thing’). That said, I’d love to meet someone that never, ever hangs up on a basic item. I hope that I generally catch it and rectify more smoothly than that when I do.

      1. One of the first things you learn in graduate school is how much you _don’t_ know. (And how much of it you never will.)

        Humility and avoiding making statements about things you don’t know go a long way to not sounding like a fool in public. The rest, of course, is actually knowing the stuff that you’re responsible for knowing.

        1. What I learned in graduate school (Columbia and, no, that was not the school in question for this post) was that it is better to have an employer pay for it than to pay out of your own pocket ;-)

          I am always amazed when I conduct job interviews how many people will make up an answer to a question instead of saying “I don’t know.”

          The other thing I’ve found–not related to professors–is that you have to be careful if you are an expert in something. Because people often don’t differentiate between you being an expert on, say, computers and not being an expert on cars, or plumbing, or everything else. The Professor on Gilligan’s Island wasn’t realistic–he knew about everything (except how to build a boat). So you always have to be careful when dealing with people who ask you about things to make sure they understand if your answer is authoritative or just your opinion.

          1. I’ve heard campfire stories about employers paying for education. Figured it was before the WW’s. My last four employers wouldn’t even pay for minor continuing education. They bought Altium (second hand), wouldn’t pay for even Fedeval’s courses and expected a six layer BGA design in three days.

            I also received orders from on-high to only hire engineers with the extremely narrow and wildly unrelated skill required for the next half dozen projects lined up. Considering my last twenty job interviews, these penny smart, pound foolish management decisions are the norm.

            Maybe I should become a professor…..

    2. Well, that wasn’t her only thing (and, yes, she was a PhD). She used to say MUL-timeter (like you’d say potentiometer) instead of MULTI-METER. There were a lot of very good instructors at that school, too, but no I can promise you I didn’t misunderstand anything about the situation nor did she. I also had another EE professor who happened to be up on an antenna tower with me assert that we could tell how high we were by timing something dropping but we would need to know the weight of what we dropped. No kidding. There are dumb people in every profession and academia is no different, I’m sorry to say.

      1. Most university Professors have little hands-on experience. I’ve known University profs who couldn’t use breadboards. Partly this is because most Universities value a Professors ability to churn out research papers much more than his/her ability to actually educate/teach students…especially in a hands-on manner..

        1. That was another pet peeve. The grad students that ran the labs would have people build stuff on breadboards and then if it didn’t work, tear it up and build it again instead of troubleshooting. Those of us who “knew” would do troubleshooting and help others which was not always popular with the grad students.

          As someone who has spent a lot of time in front of a classroom, I’ve often thought that maybe you shouldn’t expect profs to be great at managing a classroom. And for what a big uni charges, maybe they should consider having two people in charge of every class: an instructor and a classroom/learning manager.

          1. Some people just aren’t cut out for teaching.
            There was one Prof who was a regular Dr. Jekyll & Mister Hyde. In the classroom he would tear people apart about timeliness & not leaving “just because the clock said class was over” (yet wouldn’t hesitate to waste class time himself), didn’t care if you understood the material, basically couldn’t be bothered to be a lecturer.
            If you had him in the lab, he was the best professor on the planet, always making sure you understood what was going on, cracking jokes, agreeing that X protocol was in fact stupid but it kept the safety folks happy, etc.

        2. “most Universities value a Professors ability to churn out research papers much more than his/her ability to actually educate/teach”

          No matter the quantity or quality of publications, professors who don’t bring in grant money don’t get tenure. I suppose there’s some correlation, but it’s all about the money.

          A professor that produces a valuable patent for the university’s foundation will get tenure and probably never *have* to publish again.

      2. Actually your EE professor was right about the dropped object, as terminal velocity also depends on the weight of the object. So actually your thinking here is a bit like: “Let’s assume a spherical skydiver in a vacuum…”.

  6. I had a similiar situation in a class I was taking on transmission line theory. We were going through the equations on antennas and It was asked, “how does the current get from the one line to the other?” The professor replied, “I don’t know, but that’s how the math works.” Brutal.

  7. On the flip side – I can think of many people I’ve worked with that have been great at their job but just don’t try and get them to teach you. There is a big difference between doing and teaching how to do a job. The people who are good at a particular job AND good at teaching I think you will find are the exception and not the rule.

  8. In those days, I wasn’t smart enough to hold my tongue, so I raised my hand from the back of the class and explained the above (perhaps a little more succinctly). The response from the professor: “Oh yeah. That too.” I won’t mention the name of the school or the professor, but it did prompt me to find a new school.

    1. You changed schools because you ran into a person you though you were smarter than?
      Does that mean everyone at the new “smart” school realized they were smarter than you? Did they all change schools then?

    2. A smart person isn’t about what he knows. Anyone one can google answers or remember trivial stuff e.g. sport stats.
      It is how they deal with things beyond their knowledge. They would be the one to ask the right questions and make associations that a normal person can’t see easily.

      Given the IQ normal distribution curve, it is very likely to run into a very smart person. I tend to run into 1-2 at school or work.

      1. I worked most of my career without a degree. When I did get a chance (2 years tuition paid for under the Displaced Worker Retraining program), I determined to get all I could out of it, so I got permission to sit in other classrooms even though I didn’t get credit. I’d mostly stay quiet, unless it looked like there were other students who didn’t get it but didn’t want to ask. Then I’d ask the questions. One day after asking such a question, one of the other students stage-whispered to another, “that’s how he knows so much stuff, he isn’t afraid to ask questions.” If you want to learn, you can’t be afraid to look stupid.  Steve Greenfield AE7HD

  9. Have seen no one mention exactly why RMS is so important. Quite simply it represents what the equivalent DC voltage would be in terms of heat producing potential if applied across a resistive load. 120 VAC RMS is actually 169.7 VPP (sine wave).

    1. RTFA: The reason you want to measure RMS voltage is simple: 1 V RMS does the same amount of work in a load as 1 V DC. So a resistor subjected to 1 V DC and 1 V RMS will generate the same amount of heat, for example.

  10. “Those who can, do. Those who can’t do, teach.”

    Usually when I see somebody throwing out this old line, it comes on the heels of them having just failed an exam in that teacher’s class.

  11. I am saving THOUSANDS of dollars by going to the school of Anonymous and the Jolly Wrencher! I have concluded that with dedicated study, I could be living the neuromancer dream of a purely technical wizards lab and house. I just need 1.1 jigawatts to get the Delorean hooked up to jump start the TARDIS. I left the lights on. It killed the battery. Note to self: the TARDIS, despite being awesome, does have a weak battery system. Other than that, and living a boring mundane life, hack a day keeps my brain functional. I learn something everyday and keep every email I get. I’ve been so inspired by hack a day that I am teaching myself C++. I will learn visual basic as well, and java. But now I want my own logic analyzer, scope, and soldering tools. I want to learn raspberry pi, and I want to build a Linux machine. Hack a Day is the single greatest resource in the world. This website is a school in itself and has inspired me to continue learning.

  12. Something doesn’t match for me about this. School, professor and RMS. Should we replace School with a University or College – fine. Otherwise, I haven’t seen a professor in school ever(it’s like having F1 engineer replacing exhaust in nearby garage for $20). Nor can I recall RMS being part of standard physics classes.

  13. This all seems overly complicated.

    If you have a wave ie triangle draw a rectangle from x axis to peek and from 0 degrees to 180. Now you have two back to back triangles in a box. You can easily calculate the area of the triangles vs the area of the box.

  14. I have a suspicion that concerns engineering and teaching, Those that can do, at Boeing, Those that cant do at Boeing teach, but those who can’t teach Work for Ford, GM, or Chrysler as head designers! Cause face it cars have things so backwards to what makes any sense or logic one must conclude no one designing cars graduated in the upper 2/3rds of their engineering program. For example, just last week a news report caught my attention, a car company is taking action to prevent children being accidentally left in a hot car, so far sounds good, then they describe the car mfg’s planned fix, a buzzer and idiot light that will sound & flash if at a trips start the rear door is opened & shut then at the end of the trip (when car is shut off) the rear door is not opened with in a short time after the driver door is opened and shut with the key off. ?? so the car is going to beep and flash cause I put my gym bag in back or I don’t take my golf clubs in when I pay for my gas?? That’s detroits solution! Maybe cause the folks drawing up car designs don’t drive (or not allowed to) or have no clue how we really use cars, like it’s a rule only thing ever carried in the back seat is a child! MORONS!
    A Better idea would go like so , using sensors already in newer cars, and perhaps a few more similar sensors covering additional areas, and the existing control modules, existing communication and mechanical systems being tied together with a dash of code. to do this : If eternal temp is above XX deg. and seat occupancy sensors (same that turn front passenger air bags off when kid sits in front seat) detects child present and windows closed (or open to less than x%) Engine & A/C are not on no adult in drivers seat (same as other seat sensors) Then Open Windows (all via software & pw ) AND Sound horn(possibly with distinct pattern used by all companies making the pattern universal child needs help alarm sound), and using onstar to dial for ems or the car owners cell phone to alert them about car’s location and child possibly trapped in hot car! that is a method that takes action, not just an buzzer and alight, but opens the windows, sounds
    the car horn and makes calls to get help . That’s how a real engineer thinks. Detroit’s idea doesn’t prevent injury & due to how it senses an alarm, would lead to lots of false alarms. and fall into disuse.

    1. Well, as someone who worked for the first company you named for a number of years I can tell you that like all big companies they have some smart people and some extremely stupid people. One of the things I’ve noticed with them and other large companies is that they are really more amalgams of little small companies, not always connected by any common thread. So site A might have smart management that fosters good teams, but site B might be full of yes men who do anything to look good including throw the engineers under the bus, overpromise and underbid, etc. So you can’t always judge the whole company by one group/product/site.

      One of the things I’ve noticed over my career is that we have become taught that “all men are created equal” means that everyone is the same. It doesn’t. Pro sports is about the only place I know that is left where they got it right. It is perfectly OK to say, “Wow, you are much better than the other people who do this job. Here’s 20X the compensation, plus lots of rock star perks.” In today’s corporate environment, you are supposed to pretend that everyone has equal talent and abilities and, of course, there’s no room to overcompensate stars or punish poor performers. I also see a lot of large companies fail to realize that an electrical engineer (for example) who does FPGAs is probably not the guy to do your new HVAC facilities build out, work purchase orders, or any of a dozen other jobs that are necessary, but not a fit for that person. This leads to bad performance, unhappy employees, and many other maladies.

    2. But that’s a horrible design from the marketing point of view! Who cares if it saves some kids life, if it’s a feature that brings trouble to the owner?
      A futuristic car would never have a feature to automatically e.g. report an accident and incriminate you in the process, but instead have a shovel compartment and be smart enough to help you hide the body.

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