The Root Locus Method - Introduction

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The Root Locus method is a fantastic way of visualizing how the poles of a system move through the S-plane when a single system parameter is varied from 0 to infinity. I show how to interpret the locations of poles in the S-plane, and how they are tied back to system requirements.
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If you have any questions on it leave them in the comment section below or on Twitter and I'll try my best to answer them.
I will be loading a new video each week and welcome suggestions for new topics. Please leave a comment or question below and I will do my best to address it. Thanks for watching!

[@Paddydothrock]

I never thought I'd see the day when someone could produce a youtube tutorial video which progresses in a logical way, makes sense throughout, doesn't have a highly irritating voice, uses a decent microphone and doesn't constantly fumble with it, stays on topic and has not just legible but actually impressively neat writing.. Thankyou so much, yours are the best tutorial videos I've seen. Please consider giving some lessons to Kahn Academy.

[@hyderali4351]

can't agree more !

[@vuongngo2181]

You are very detailed and thorough. Keep it up!!

[@mrsblackviper]

well it take a lot of effort to achive that.

[@neroskywalker9183]

I've learned more of the basis of root locus in this video than in my whole course in the university.

[@keval761]

NeroSkywalker which uni? xD

[@batman_1st]

National university of singapore

[@TheTeladras]

Could it be that u did not listen during the courses? :d

[@Nazirm7]

Yeah probably didn't because his professor couldn't explain for shit like mine or didn't do any examples

[@julius8631]

Same here

[@ianhanley1213]

I am a fifth year Aeronautical engineering student, currently using root locus to design aircraft flight control systems. This fifteen minute video is a more complete, effective and absorbable explanation than I have been given throughout my degree.

[@fac7orcosplay]

Currently studying Electrical Engineering, I have to pass the final exam of Automatic Control, this videos were much more helpful than the entire course

[@BrianBDouglas]

Hello, I briefly cover that in my video called Stability of Closed Loop Control Systems starting at around 6 minutes. The even faster answer though is that the time response of a pole is e^st, where s is the location of the pole in the s-domain. If s is negative (left half plane) then as time get's larger e^st gets smaller (stable). If s is positive, then e^st gets larger as time increases. No matter how many poles you have, it only takes 1 unstable pole to drive the system to infinity.

[@psp_online]

Pls make a video on designing controllers (PI or PID- is it possible?) for non minimum phase systems.

[@dinmukhamedzardykhan5815]

I am subscribed to you forever.. A lecture have never been so clear..

[@Lolwutfordawin]

my uni has done an awful job at online classes, these series of yours are a life saver! so wonderfully explained!

[@aminijooni]

you're my hero Brian. well explained and super easy to understand. thank you

[@anantchopra1663]

One of THE best videos I've ever seen on Control Systems. I learnt far more from your video than my control systems course at my university. Amazing explanation with wonderful insights! Hats off to you for producing such great work!

[@davoodseify3624]

Man your explanation was extraordinary. I’ve learned so much here than anywhere else. It helps me so much. Thank you🙂🙏

[@ahadajaz674]

You are literally the GOAT. You broke down a concept and explained it so well with amazing visualizations. This video deserves an award!

[@dorsaghaemi]

Thanks Brian. After 12 years in Control Engineering, This is the first time I have this level of understanding, which you provide.

[@santiagoricoy1313]

I met an old coworker of yours! She heard I was doing controls and immediately recommended your videos; I smiled and said, yes, I've watched every single one. If I taught the class, the lectures would just be practice problems, and the homework would be to watch your lectures. 😂 Also, at about 7:29, I thought the imaginary portion was the damped frequency, not the natural; I thought the requirement forms a circle. Anyone able to help me understand?

[@abdulazizsalman6197]

i got tears on my eyes when i watched your explanation, it is the greatest lesson i have heard in entire my life even though i am in a university with among of useless lecturers , thank you so much

[@BrianBDouglas]

viraj shelke and Suraj more, It's hard to explain in writing but I'll try. Start with the root locus form of the transfer function: G(s)/(1+K*G(s)) and then separate G(s) into numerator and denominator. N(s)/(D(s) + K*N(s)). Now if K is 0 then the transfer function is N(s)/D(s) which is G(s), so the roots start at the poles of the original transfer function. When K is increased the denominator D(s) + K*N(s) starts to behave more and more like just K*N(s) which are the zeros of the original transfer function. Does that help?

[@serialglobetrotter]

Brian Douglas sir , i have few problems , 1st ) there was a question where i was asked to plot the damping ratio line with just OS% given as 15% how do i do that? how to plot damping ratio line ? 2nd) how to calculate natural frequency & damping ratio in case of a 3rd order transfer function ? kindly please refer back...thank you!!

[@ph7ryan]

ID Jeet 1) There is an equation that relates %OS directly to zeta, %OS = exp[(-z*pi)/sqrt(1-z^2)]*100, where z is zeta (the damping ratio). There's a few ballpark, pre-calculated OS's that are good to remember, 5%, z=.7, 15%, z=.5, 35%, z=.3. So you would plot the z of .5 when given a OS of 15%. 2) 3rd order+ functions can always be broken into 1st and second order systems. You would simply factor out a first order to create a first order and a second order, and your natural frequency would be the frequency of the second order response.

[@serialglobetrotter]

ph7ryan thank you very much... that was helpful ..

[@bobseagull7600]

It's interesting to see just how much people appreciate good quality videos. Nearly a million views for the nerdiest content on KZbin. Congrats and thank you for your work!

[@fawazmohsin2287]

This series is the best way to learn about root loci

[@Bahloola17]

amazing is the way you explained it!!! I read about root locus 100 times but never understood the logic behind it... now it makes sense :)

[@Plenty_for_Twenty]

KZbin University prevails again. I was not entirely grasping RL from my professor and the textbook. Just kind of head down and mathing into the unknown. Your video placed a ton of perspective into my study. Thank you Brian Douglas 🍻

[@ChrisjayH1]

I think my entire class is watching your videos. Thank you so much for these.

[@darrenp83]

Yup! We are all watching these videos!

[@nemesis3255]

Darren P And I had to find out all by myself...

[@Rmkrh]

Thank you Brian for these videos. I am feeling like I am reborn after watching this. 10 mins of your video has taught me more than what 6 months of studies in my university have.

[@sudmudmud357]

Not everyone got the skills to teach and nail it right there, I feel like I can obtain a degree just attending KZbin videos, your videos are one of the reasons. Keep up the good work .

[@myonlynick]

0:00 up to 6:30 is PACKED with a load of insight info!

[@DhirenDave-v4g]

The way you have explained the concept with very illustrative examples is fantastic. You are doing a great service to the engineering community by this effort. Please keep it up. Thank you.

[@sowasvonkeinplan]

It's wierd... I've been attending 3 lectures on system's theory and control theory during my university studies several years ago and I have the impression that the first 5 minutes of your video tought me a lot more about one of the most useful methods frequently used in control theory than all of the lectures together. Maybe my point of view is a little bit biased as there's a lot of previous knowledge so everything seems total intuitive. Anyway, I wished my professors had been as talented as you explaining and presenting stuff like that. Great video.

[@shovon9412]

This is a gold mine for any control engineer who is still in the learning phase or even for those who is just revising man

[@BrianBDouglas]

Hi Echo, is your question why I didn't set b such that no oscillations occur at all (poles only on the real axis)? You could set b to be anything along that blue line that I drew. The reason I didn't choose a larger value is because when you're buying an actuator (like a damper) typically the larger the value (or larger damping) the more expensive the unit, the larger the unit, and the heavier the unit. Normally I try to select the cheapest smallest part I can that still meets requirements.

[@imafirinmalaser]

Brian, you are amazing. Honestly. As an undergrad control system engineer, I am indebted to you!

[@chrisp7255]

I am using this to study for a final right now, and I am so mad I didn't discover this earlier. This would have cleared up an entire semester's worth of confusion. Thanks for the explanation Brian, this is really well done.

[@Carrierski]

This is literally the best explanation i have EVER seen in my life. Very clear, the speed is perfect and answers all important questions. Instant sub.

[@magyarandrastamas2485]

If only 1/10 of the teachers had such a huge enthusiasm to make the students got the curicullum then people would love to go to school. Good teacher equals knowledge + fedback enthusiastic repetition till the understood lecture equals the determined level. :) Well done Brian!

[@gang57]

i must post this, i just came back from my exam "control systems" today. during my preparations i used all your videos about the root locus and the nyquist. I must say, thank you very very much Brain, because of you i managed to get the proper insight i needed. your techniek is awesome, good animated and keeps the interest locked on target! thanks many times bro

[@nikosips]

I have no words.. thumbs up from an undergrad of electrical and computer engineering! The intuition you give on this subject is amazing and really helpful . Keep the videos coming !!

[@akankshapassi8962]

+brian douglas you are love. Your videos are gonna save my exam tomorrow. Thanks.

[@nalostta]

man,you explained not just the technique ,but also the right questions that should go through ones mind while doing this...amazing!!!

[@RobBabu]

Great Video Brian! You're a huge help to all of us struggling with control systems (aka all of us) :)

[@MrRawnik]

brian you are awesome. like your methodology.

[@greersimpson]

Thank you so much you I just watched all of your lectures in one sitting and it literally just clicked. Your the man, keep it up!

[@johnhan75]

oh my god.. Your video is so much helpful, I can't belive it,,, I am an university engineer student, and I actually think you teach way better than our proffessor, eventhough school I am attending is quite a good school.. thank you so much for lecture, have a good day!

[@seang5476]

Thank you sir! Helping students 10 years later

[@Pyrohawk]

Your tutorials are amazing. You've given a gift to humanity.

[@deekshas3936]

This is such a great video oh my god. Our teacher didn't even tell us what k is supposed to be. We just solve problems mindlessly. But you helped me to understand the meaning and the real use of the root locus. It was so intuitive that something just clicked in my brain. Also along the way, your explanation of the s plane helped me to fill in some gaps in understanding. The video was very engaging throughout. Thank you so so much!!!

[@JD-jl4yy]

It's embarrassing how much better this explanation is compared to my professor's...

[@deanw8206]

You need to grow your channel and cover more topics. You just explained the 2.5 hours of confusing controls lectures I sat through in under 15 minutes.

[@jorgerive7335]

Your are a talented instructor!! Excellent job on these tutorials --they move quickly, they're thorough, and your explanations are clear and presented in a very easy-to-understand fashion. Kudos and thank you.

[@余坤奇]

we dind`t learn any thing about the loucs root method but our teacher ask us to do a course work with this. thank for u help , your video helped me a lot !

[@matthewoliver88]

I have a test tomorrow and this is the first video that i clicked on. You're my hero

[@サメ頭のくま]

This is a REALLY good explanation of this. I came for later R.Locus stuff but I decided to watch this intro video (so I understood 99% of it prior to watching). Your explanations are very thorough but concise. I have a friend taking Control Systems next term and I'm going to recommend your videos!

[@dr.ganeshbhokare6547]

Very simple and conceptual explaination of root locus...really very impressive due to to simplicity of explaination

[@shengzhu9716]

Great teaching. You really help me understand control much better!

[@xxanimangaloverxx]

I wish I could print out what write out in your videos. Best notes ever.

[@kdmq]

My professor never explained everything up to 6:30 and just explained root locus and threw me off for half the semester. I should have skipped more classes!

[@rubenayla]

That's the conclusion I come to everytime I go to class.

[@MrDoz01]

the amount of planning that must go into this ...so good ...thank you so much

[@elzarqua]

wow, learnt more in this tutorial than i have all semester! love how you link everything together! cheers!

[@jaydesai7386]

very well and logical video . Understanding level is very high . Well done.

[@sullivan3503]

At 7:30, you mention that natural frequency requirements dictate that the pole must lie inside a horizontal band. However, this would correspond to limiting the imaginary part of the mode. The natural frequency corresponds to the norm of the pole, which would mean the pole must lie inside or outside of a certain circle.

[@EdgarPoe_Raven]

4 minutes to get to know what I NEEDED. Dude,you have no idea how grateful I am. Perfectly explained. I should donate you to your Patreon or PayPal or whatever, for the coffee at least, you fucking deserve it.

[@19hashan93]

THANK YOU SO VERY MUCH DR. BRIAN!! In a little over 10 minutes, you managed to teach a topic that I couldnt understand in 2 hours at class. You are extremely awesome for uploading these. We are eternally grateful! Liked and Subscribed! Its also amazing how you linked this to a real world engineering problem so well. It reminds one that engineering is not just numbers and maths but directly mirrors what goes on in reality. (btw, the "we messed up" cartoon was a brilliant idea!)

[@condedennisful]

excellent explanation. i like how you explain very quickly and don't waste time like other videos.

[@kevincorrales9774]

Left in the middle of my lecture for this topic. That’s how good this video is lmao

[@tomasmartini105]

Hi Brian. Really good video! I think you should clarify that you are talking about natural damped frequency at 7:25. Otherwise it could be mistaken with the natural undamped frequency. This one would introduce a constraint in the form of a circumference.

[@q1q1q1q1q1q1q1q11]

Thanks for your clear teaching style, it is much appreciated.

[@jacquesnicolay9221]

Thanks for making KZbin useful. Thank you sir.

[@PunmasterSTP]

I'm so glad I came across your channel, because your videos are so superb. I'm sure other commenters have already given much praise, and so long ago, in more eloquent ways, so I'll just say "Thank you!"

[@prateekmuthiahnitap8447]

Hi samter from Neso academy comment section ( ͡° ͜ʖ ͡°)

[@PunmasterSTP]

@@prateekmuthiahnitap8447 Hello back! How are you doing? Also, are you studying for the JEE or another test? As for myself, I tutor in electrical engineering a bit, so I try to stay sharp...

[@prateekmuthiahnitap8447]

@@PunmasterSTP I'm doing great! Well atleast for now... Wish me luck for my currently ongoing college exams ^^

[@PunmasterSTP]

@@prateekmuthiahnitap8447 I'm glad to hear it, and best of luck on your exams!

[@ShahzadHassanBangash]

great work sir keep up the good work our society need people like you .

[@zahidfaheem3233]

love it when you find a video that gives more than you expected! thanks a lot bro and hope the book becomes a bestseller :D

[@elmerosorto1680]

Amazing explanation, i am reviewing all the content i've learn in these university years, because this is my last semester, thanks for the good work.

[@abdohajar908]

Sir, you don't know how helpful these videos are!!! Thank you :)

[@khamisi__6378]

thx a lot i was going down a rabbit hole reading the chapter.

[@MrAskOnce]

Very well put together... I'm about to have to develop the root locus for a project I have in my controls class and this is so much better of an explanation in ~13 minutes than my professor has given in 3 lectures. Woo. Will continue watching the others -- thank you for posting this. xD

[@ishakaimen1576]

Your videos are so good and useful.Thank you so much. May I suggest? if you show how to solve one or two examples on the concept in hand, that will really help the viewer understand the concept. Thank you again for the helpful videos

[@LuigidBlackhouse]

Thank you, you saved my live (or at least my grade). This video (and the other two) helped me catch up some lectures I missed due to an illness.

[@BrianBDouglas]

Hi Ahmed, there are a lot of nuances when using general rules of thumb and they don't work in ever situation. If you have a 2nd order system and it has a damping ratio between 0 and 1 then you will have a system with imaginary roots and the cosine of phi works (or with zeta = 1 you have two repeating real roots). However, if you have a damping ratio greater than 1 then you have two real roots (which can be thought of as two 1st order systems) and you can't say zeta is the cosine of phi anymore.

[@pedroklepa]

This is a skilled explanation on the subject and thus super helpful. Thank you!

[@MANGOBAPRODUCTIONS]

Thanks for your passion of sharing knowledge to the world, keep the good work. You saved my control course and made me enjoy it.

[@protiiooi]

This is awesome! I finally understand what my teacher has been trying to teach for the past few weeks

[@Berilium2]

Good man Brian! You're saving my degree one video at a time!

[@guruprasad_1984]

Super Brian...you have covered beautifully the significance and notion of root locus in a 13 minute video....

[@drsandeepvm5622]

Been teaching control engineering since many years, but understood today 😇. Thank you Sir 😊

[@BrianBDouglas]

Hey James, no problem. The transfer function is 1/(s^2 + 1.5s + k). For such a simple transfer this is the easiest way. Put the den. in standard form which would be s^2+1.5s+k = s^2+2*z*w*s+w^2. When k = 1.1 then it's easy to solve for the damping ratio. w = sqrt(1.1) =1.049. Then z = 1.5/(2*1.049) = 0.715. Of course the root locus is a good way to visualize how the damping ratio changes with k. On the diagram zeta = sin(angle between pole to origin and the vertical axis).

[@sizmostudent]

Finally, I understand it, I have been trying to understand it for 2 months, thanks a lot for this amazing video!

[@Postermaestro]

Love the "side notes" in your videos. It seems to me like these are things that have confused you in the past, and most likely confuse students (like myself) taking these courses.

[@jessstuart7495]

If you write the transfer function in a different form, you can directly relate b and k to the damping ratio. TF = 1/(s^2 + b*s + k) => 1/(s^2 + 2*ζ*ω0*s + ω0^2) b = 2*ζ*ω0, and k=ω0^2 b = 2*ζ*sqrt(k) rearrange to ζ = b/(2*sqrt(k)) k is multiplied by factor p, sqrt(k) will multiply by a factor (sqrt(p)). To maintain at-least a 0.75 damping ratio, you'll have to multiply b by a factor of sqrt(1.1) = 1.049. b_new = 1.5*1.049 = 1.6 (only keeping significant digits)

[@ECOMMUSK]

these are amazing examples. I did not know to think of it like that. It makes more sense when you say the marketing department says a system with a mass of 1 will sell the best, and the spring department wants to use a constant of 1 etc... thanks!

[@medetauyenur1817]

Thank you. Easy and intuitive explanation

[@frl-less9282]

Really appreciate your videos! You made me understand the whole control system thingy in such a short time. I tried to study myself but it took me days to understand the basic. You are awesome!!!

[@debasismt]

In my college time I thought I will never be able to understand this. Now its crystal clear.

[@jcapper2]

I'm honestly shocked how well you've explained this. Thank you so much!

[@BrianBDouglas]

Hi Anthony, I will be posting on Nyquist, but not for another month or so. I only put out videos once a week and I have a few other topics in work already. Hopefully you'll still find it useful even after your class is over :)

[@re.liable]

I hope you won't ever take down these videos. It's still a little bit over my head but I expect to be coming back to these over and over again lol EDIT: I just noticed that I'm already subscribed. I must've seen another video series from you before. You're the best, thank you very much

[@gabrieljalil1562]

Great tutorial! Not that many tutorials of this quality discussing engineering subjects. Keep up the good work!

[@TheBirdMan]

Thanks very much. Great video. I understood every single point and got the whole point of importance of Root Locus

[@shanilperera]

Thanks a lot , a great effort Mr.Brian (Y) being supported me in several difficulties.

[@ainagalofreballbe8175]

I love you so much for this video right now, I had totally forgotten about this concept because I hadn't developed an intuition for it. Now I believe I can actually use it for something else than a passing grade in an exam. Thanks!

[@ryanvickramasinghe1971]

give this guy tenure already.

[@mohamededbey]

Great content as usual Control engineers are very proud to have you

[@StevenMcconnon]

Thank you. I'm an engineering student who is trying to teach himself control systems and this helps a lot.

[@SolidSnakePlissken108]

i am racing against time to study for an exam which is about 8 hours away (it is my last ever exam at uni i might add), and 3 hours ago i was in utter panic that i will fail at the Root Locus problems that await me. i cant happily say: not anymore. why? because of your Root Locus lectures. Thank you.

[@thinquin]

Brilliant Lecture... I am so much less confused... THANK YOU!!!

[@codesrhodes3757]

Just amazing sir. Don't have the words to describe how awesome you teach It is fabulous.

[@xDR1TeK]

I loved it. Very smooth. I'm into oscillators and this is exactly what I do to find when my system will remain astable.