nm0724: that you know what's going down this morning er we got some guy where 
are you from anyway 
om0725: Centre for Applied Language Studies 
nm0724: there you are so he's local we er he so he's not that er oh you weren't 
here we had the Polish T-V lecture in er er lecture or videoing a lecture the 
other day but er that wasn't you lot have you lot been er some of you have been 
videoed haven't you the oh yeah when er some of you were in the lab i know 
others of you missed it but for er the N-B-C the American one er that went out 
on the er er N-B-C Today programme s-, oh no not B-B-C N-B-C so it was er i 
think something like thirteen-million Americans would have watched you er and 
there's been feedback saying how good you looked and things like that er and 
and at least one of you appeared on Sky T-V er or er filmed for Sky yesterday 
but oh no names mentioned the celebrities in the audience it's amazing er i got 
a favour to ask in fact as well has anybody got i know all of you'll be skiving 
off this afternoon and 
putting your feet up and things like that as typical students do on a Friday 
afternoon has anybody got a video er machine or equipment or anything or have 
access to it you do well it's n-, s-, n-, n-, n-, noth-, nothing naughty i'm 
asking [laughter] for no er i i rushed out the house this morning and i forgot 
the Newsround tonight would it be possible for you 
sm0726: yeah sure 
nm0724: brill excellent 'cause 
sm0726: 
nm0724: er i think it starts four- fifty-five but i might but it's Beeb one 
four something like that if it's poss just to video it i'll pay you handsomely 
you know er while this is being recorded i don't know what shall we say sort of 
ninety per cent in the exam something like that or [laughter] that all right 
okay i mean er al-, all right ninety-five okay that's the final offer just so 
this is this is me being er very generous so and i-, and if ever you say you 
said ninety-five and you don't get it then it's all been recorded so right what 
it what it is on Newsround is oh don't panic don't panic 
you've had all this before er what it is on Newsround is something to do with 
the Sony dog and they came here and we of course we're the most knowledgeable 
people on earth about the Sony dog Sony excepted and er they wanted a bit of 
waffle on it i think that's what it's about anyway it could be something 
completely different er let me let me remind you oh it's all right step this 
way step this way you need a crash helmet for this lecture it's er i know it's 
a bit of a a bit of a crash course the the s-, stuff we were doing if you 
remember we have a system that we're dealing with and we're shoving a step 
input into it step input is switching it on bing step input the pra-, this is 
practical er [laughter] experimentation and er we're seeing how the system 
responds and we looked at the first type of system where it's overdamped so the 
system responds the output morning whatever this is it it it eventually gets up 
to its final value but it doesn't overshoot it doesn't do anything 
funny really it just gets there eventually and that's it we then had a look thi-
, this is hopefully how i how i left you last time we were on er was looking at 
the critically damped case and how far did we go fine so this this you've had a 
look at this bit so far so if i go over this you won't or shouldn't be saying 
what the hell is that it should be yeah i remember that way back beyond the 
days gone by the in the critically damped case just to remind you the 
critically damped case is really this sort of mathematical ideal on the cusp 
probably never really happens in practice type it it's a it's an ideal type of 
thing to get to it's sort of er in a political sense it's the pure communist 
state you know it's an ol-, i-, ideal but it's never er going to actually occur 
in practice er here we got a system output the the system is G input is U which 
is a step input and where we'd got to was simply at the point of looking at 
what the output response is to a step input and er er again 
what we'd done before well what what i'd done and some of you had listened in 
was to look at the sort of extreme cases when thi-, this is what it looks like 
the output looks like in terms of time physically what the output looks like 
and there are three values to it or three parts to it one is the just the gain 
K is just a number alpha apart from being a Greek symbol just to make it look 
more difficult is also just a number and and this would just be as it is 
squared so this is just going to be a positive value here that's all a gain 
value we've then got one now this this effectively will be the steady state 
value when things have settled down and all we've got are two exponential terms 
which we can look at at the extreme cases and the extreme cases being when we 
switched the thing on and it's like er time is zero it's the big bang time well 
in a physical sense that is not in a er well any other sense and i've got to 
watch what i say here haven't i 'cause i mean th-, this is 
this is with language studies you get sort of er mystical foreign students er 
listening in on this or yeah 
om0725: possibly 
nm0724: okay for the so for the the Latin American students that might listen 
to this transcript the big bang i'm talking purely in a sort of Stephen Hawking 
sense not in er er other other connotations and er anyway so actually when when 
t-, [laugh] oh God let's go for it when T is zero then exponential of zero is 
sm0727: one 
nm0724: one great and when this lot is zero because we've got a T in there that 
going to zero the whole lot of that goes to zero so all we've got when we 
switch the thing on this is when we apply the step input at that time we go 
zing then the output will be this value one-minus-one and one-minus-one is 
sm0728: zero 
nm0724: one student knows anyway that er th-, yeah one-minus-one is zero which 
means that the output is zero which is pretty obvious we switch the thing on 
and the output's er well i was going to say i was going to swear but i won't er 
the output is zero er but why not but er now let's let's take the other case 
when time goes off to a big number such as infinity or the end of the lecture 
whichever is first and in this case the exponential minus infinity is 
sm0729: must be zero 
nm0724: must be zero it sounds good doesn't it it sounds good so that must be 
zero and here we've got er time is off to infinity or a big number and this bit 
is down to to zero so what's er anything multiplied by infinity is 
sm0730: infinity 
nm0724: infinity except for one specific case because anything multiplied by 
zero is 
sm0731: zero 
nm0724: zero and er we're on one of these sort of parad-, what is it a p-, a 
paradigm or something i don't know what or is that some exotic drink 
sm0732: 
nm0724: no all right anyway it's zero multiplied by any old thing and T is a 
big number so the whole lot's going to go to zero there so it's because that's 
zip all so we've got one-minus-zero which is 
sm0733: one 
nm0724: one thanks very much well we ge-, we're getting there even if it's the 
odd sort of person there so in a steady state this would be K-over-alpha-
squared whatever the number is multiplied by one which turns out to be 
K-over-alpha-squared so that's the critically damped case that's the steady 
state value and the two extremes er are the ones we've just looked at the 
steady state when time goes off to infinity or a or a big number whichever you 
want er is this number K-over-alpha-squared which is the the inherent gain of 
the system again they're just numbers K and alpha are just numbers i mean Kay 
is also a girl's name i know that but for the purposes of this thing just treat 
her as a number if anybody knows anybody called Kay tonight say to her Kay 
you're just a number if if she gets upset about that you know especially if you 
say you're number fifty-eight or something like that she gets upset then say no 
the prof says you're just a number say how does he know er and then in the 
initial does anybody know anybody called Kay just to get you into trouble here 
just as well just as well 
sm0734: bloke 
nm0724: don't pardon 
sm0734: there's a bloke called Kay 
nm0724: is there er okay well i'll [laughter] 
sm0735: 
nm0724: right yeah well [laughter] the er 
sm0736: these Japanese 
nm0724: really okay let's move on [laughter] er the in fact not that i'm going 
to go through 'cause you know we got language students who don't know what all 
this derivative stuff is but in fact the derivative er of the output when T is 
zero also turns out to be zero you know we had that we had that little trinket 
cropping up before and here it comes again it's not particularly important no 
big deal all it means is when you er er er it's it's very much like er er a 
member of the opposite sex when you actually suggest something to them it takes 
them a bit of time to get going you can't suggest something to them and they're 
off straight away and the same with this system so in fact the the other person 
could be a critically damped system er and in this case so the derivative is 
zero the takes a bit of time to burst into life essentially that's all it means 
what this critically damped system looks like there's no overshoot 
the dotted this this dotted thing here is the steady state value er this is the 
output here's time along here we switch the switch and up it goes but it 
doesn't actually overshoot in theory it doesn't overshoot the final value that 
it settles down to 
in fact it is the most rapid response the critically damped thing what it means 
is it's the most rapid response that you can possibly get on the face of this 
earth without any overshoot for the overdamped case you remember it it didn't 
overshoot it er came in from once i eventually got there and i've said here 
it's desirable this particular goal this particular type critically damped 
response if you're looking at machine tools er or robots in fact or er a lot of 
production machinery welding machines is a a prime example i know tonight 
you'll all go home and play with a welding machine as you do and with that 
welding machine the aim the target is to make the thing critically damped so if 
if you can imagine for a moment that i am a welding machine and this the top of 
the projector is the thing like what i am supposed 
to be welding then here's my little torch or stick or whatever the welding 
thing is and i the design of it i want this arm to come in as quickly as 
possible do do you need a v-, a vocal commentary he moved his arm [laughter] in 
the direction of [laughter] so i want the arm to come in as quick as possible 
to to that point zing like that do the do the welding and move away again now 
if it's overdamped the case we saw before it will not come in zing it will come 
in zing and eventually get there if it's as the case we're going to move on to 
it goes past it overshoots then of course it would go zing and we messed the 
and you can imagine whatever it is we've gone too far and know what going too 
far gets you er so this critically damped case is the the the perfect the ideal 
case if we had a perfect welder see knows all about perfect welders he can 
afford to miss the first ten minutes if we had the perfect welder it would be 
critically damped however i this this thing here is important that 
that's really in theory this is the theoretical for Christ's sake if you get a 
job at the end of this course as a welding machine setter upper or whatever it 
happens to be or someone who's in charge of people who set welding machines up 
or or whatever er then don't do it quite this way aim for this as a perfect 
thing but remember probably you're going to get a bit of noise floating around 
probably you're going to get a bit of variability one robot's welding stick 
might be a little bit longer than another robot's welding stick you know in 
this world robots have welding sticks of different lengths we have to be er 
aware of that and that might mean if you've set the thing up to come in just 
from one side the the critically damped case it it might bang the so you might 
break it it might you you could go too far er the other thing with noise really 
you may not want this to overshoot and er with a bit of noise on the top the si-
, the signal wouldn't look like that it would come in 
and it it might go like which means you would in fact be overshooting a little 
bit so in practice what happens is er you would slug it down a bit so in a a 
lot of manufacturing cases in fact it's really slugged down a hell of a lot er 
typical production line down at say Ford who are about to close the production 
line down i don't know whether that's any coincidence but so at Dagenham the 
robots there they have some big KUKA robots that do welding and parts 
manipulation and so on and they're slowed down to about a third sort of three 
times if you like slower than they could actually operate at for safety reasons 
for that reason that if they operated faster than that then they could the the 
to achieve more production because if the things are operating faster they get 
more things going through more welds three welds a m-, a minute instead of one 
weld or whatever it happens to be but as you get towards that optimum this 
critically damped case which is what we're looking at you've got 
all sorts of potential problems because of the variability in what you're 
dealing with so this is a an ideal a theoretical case to aim for here's the 
theory now let's detune slow things down let's come back to the overdamped case 
a bit in a practical way in order that things don't blow up and we don't break 
things and so on so forth so that's what we're about so there er critically 
damped is a sort of theoretical ideal er it's a bit like my wage packet it's 
completely theoretical but the it's not ideal either it seems seem that the 
language they get paid loads of dosh over there so they get p-, get paid it all 
in foreign money and er [laughter] right er we've done the overdamped case 
we've done the critical damped case and er we're going to have a look at the 
this is this is in fact this is a really sad moment because this underdamped 
response this is it this is the final thing we're going to be covering on the 
course so i know i know it's a time for holding back the tears for er you know 
stifling not a yawn but stifling something or other er so the o-, the the 
overdamped it didn't didn't overshoot critically damped's right on the cusp 
underdamped is the the final thing we're going to look at and it's called 
underdamped because it's underdamped er the the damping is not enough to stop 
overshoots essentially and this this type of response the output overshoots its 
steady state value so i'll just flick back to this previous overhead he put the 
previous overhead on the projector flick back to that then in this case we was 
the the sort of ideal in the n-, furthest we could go quickest we could go 
without actually overshooting here we're actually going to go past this final 
value or er for language students from er the southern English past the final 
value er the transfer function roots in this case you remember we had the first 
case the overdamped case the roots of the denominator were were like two 
different roots as it were but real roots and in the 
critically damped case the roots were in fact exactly the same and in this case 
for the underdamped case we've got complex conjugates we've got er er a real 
part and an imaginary part and and in this case for the underdamped response 
the output oscillates following a step input oscillates meaning it it goes 
backwards and forwards and just just to see what i'm getting at here what we're 
talking about here an an experiment for tonight is trying to get the key into 
the door of your room or house wherever you're living er for a an overdamped 
case just just go normally just go and you should be all right to go in fact 
that's probably critically damped isn't it just go straight to the keyhole turn 
the door and so on just for the purposes of experimentation have a few drinks 
and then try and get the key in the door and you'll find more than like well i 
find anyway i don-, is is that the key you'll have to sort of j-, j-, jiggle it 
around a bit like that and eventually you get 
the key in the door the more drinks the have you have the more jiggling around 
that you have to do until if you have too many i'm not suggesting you have too 
many but if you have too many probably goes completely unstable and you'll fall 
over sideways while you're trying to [laughter] get the key in the door but 
this one the oscillations are they're trying to to get r-, right on to the then 
you over you go past and you come you know the you know the feeling you go it's 
a it's a bit like my wife's driving so [laughter] [cough] don't laugh it's true 
unfort-, [laugh] that's another another rear reflector i shall be mending 
tomorrow er the amount of damping dictates how quickly these oscillations die 
down a lot of damping the oscillations don't hang around for very long not much 
damping and they stick around for quite a while and in the limits if there was 
no damping then it's purely it's just oscillating off it goes just oscillates 
and for the foreign language students very much like a Citroen Diane suspension 
it's getting towards that or le Diane de Citroen or whatever it [laughter] 
whatever it is am i supposed to interact with this or should this be a 
om0725: 
nm0724: [laughter] just that no i mean normally this is deadly serious is these 
lectures are they not [laughter] and er 
sm0737: compared with your other ones 
nm0724: compared with other ones yeah 
sm0738: 
nm0724: that's right so this is i mean there have been no Skoda jokes today at 
all they can't remember any 
sm0739: opportunities for one 
nm0724: oh i don't need opportunities 
sm0740: 
nm0724: do you want a Skoda joke 
sm0741: 
nm0724: yeah i don't know have i had any Skoda this is ah this is for the Czech 
listeners er my my wife is Czech this is a true story actually and er w-, years 
ago when she first came to England er the university photographer caught us at 
the degree ceremony and said aha you're Czech i've got a Skoda joke as people 
do tend to do and is i-, you may have heard this bit but stick with it and he 
said er why why do Skodas have a heated rear windscreen and my wife er and he 
said so you can keep your hands 
warm when you're pushing it [laughter] and we we sort of well me and another 
guy we were sort of laughing a little bit 'cause we haven't heard it before see 
it was many years ago but my wife didn't laugh at all i thought well maybe he's 
offended her wh-, wh-, why aren't you laughing it's quite funny and she said 
well Skodas don't have a heated rear windscreen [laughter] [laughter] right way 
to go [laughter] the true jokes are the funniest er as the damping is in-, 
sorry that's got nothing to do with the lecture but what the hell as the 
damping is increased so we get to the critically damped case the case like what 
we were at before so no damping it's just purely oscillatory increase the 
damping increase the damping and we will get the critically damped case where 
you don't get any overshoots no oscillations at all i just put the what a shame 
'cause that's what i felt like that's an optional extra as far as your notes 
are concerned if anybody in the exam puts something like what a shame then well 
it's up 
to you isn't it the decision is yours [sneeze] bless you has anybody seen 
Bicentennial Man yet no okay so y-, note we are with science students 
engineering students we're not into arty things like Robin Williams films and 
stuff so has every-, everybody finished with that 
sm0743: 
nm0724: okay we wait while the slow student catches up [laughter] well 
apparently it's about some i i only know this 'cause some Argentinian 
journalist contacted me to say what do you think of it as Argentinian 
journalists do 
sm0744: based on the Asimov story 
nm0724: i think it's yeah it this robot 
sm0745: about the robot that wanted to be human 
nm0724: th-, that's right it couldn't be human and wants to have human feelings 
and stuff like that so the journalist asked me can robots have feelings and so 
on that's why i've put what a shame that was the link putting what a shame 
'cause the system doesn't have oscillations [laughter] that's it let's go for 
the next one now unfortunately you're not going to get away with this lightly i 
mean normally because we don't do stuff like maths and 
things like that i mean but we have to because we've got er language student 
listening in we have to do some tough stuff okay so er purely for this purpose 
we'll throw some equations and maths on the board just so it looks as though we 
actually do some work and then they'll go away thinking cor i'm glad i didn't 
do that degree that's blooming hard that one wh-, how the hell do they know all 
that stuff and then once once we're not recording then we'll get back to the 
easy life that we normally do [laughter] like 
Skoda jokes and things like that right just to remind you we're still we're 
still looking at the what case are we looking at the underdamped case that's 
the one thanks for that and in this instance we have got complex conjugate 
roots so the the transfer function G that relates input to output has this gain 
K there she is again on the top line or he again or he again and on the and on 
the bottom line we have S-plus-alpha-minus-J-beta bring in another Greek symbol 
make it more complicated for the Greek listeners this is probably all right 
though and and multiply by S-plus-alpha- plus- J-beta and you can see here why 
they use the term J and not some word like egg because if they called J egg 
then we'd have egg beta and that would be a bit silly [laughter] see there's 
there's logic in these symbols you thought why they use J now you know now you 
know er so we've got complex conjugate routes i'll just just to make it look 
again for the language people listening in 
make it multiply things out and so on and we can also get if we multiply this 
stuff out to get rid of the J or egg if you like whichever the alternative er 
just i thought i'd scramb-, scramble it up oh God [laughter] no no no that's 
enough that was almost a joke there so scramble it up and this is just looking 
at the denominator forget K's forgotten for the moment poor girl er and if we 
can also if we multiply it out because this this little bit this little Frankie 
here is going to crop up again he's going to we're going to go through a few 
and er and whoo in it comes just to surprise us in a little bit so it's 
important to know where we're going S- squared plus two-alpha-S and then alpha-
squared plus beta-squared so multiplying the bit alpha-plus-J-beta alpha-minus-
J-beta we're just getting rid of that er complex conjugate thing or lo and 
behold we can even look at it in one yet one other way which is this is in fact 
also S-plus-alpha- all-squared plus beta- all-squared A-K-A this form and 
now those astute members of you that are wide awake unlike myself will note 
that this S-plus-alpha- all-squared is exactly what we had for the previous 
case the underdamped case we had didn't we didn't we have something like that 
the critically damped critically damped case for the critically damped case we 
had two roots exactly equal we did somebody's somebody's nodding yeah we had 
something like that well it might not have been alpha it might have been A or 
something but but we did did we have alpha we did the lectures actually tie 
together loosely so S-plus-alpha-all-squared was what we had before and now 
we've got this extra term the beta that's the only difference really is this 
extra bit which when we're factorizing it above er we can only do it in terms 
of the complex conjugates so that's where we are apply a unit step input this u-
, here's U unit i've just to remind you it's just unit single value if it was 
five units then we'd just multiply the whole thing by five 
just to see what happens and the output then is just the multiple of G and U or 
G operating on U so we've got Y here's here's G K over S-plus-alpha-all-squared 
plus beta-squared and then multiplied by one-over-S the step input thing 
business and now this is the bit when you close your eyes and just write and 
hope for the best and then we'll get back to the the easier stuff you know what 
we did before with partial fractions and expanded out and so on and so forth 
you ready for this say yes namex we're ready for it 
sm0746: yes namex we're ready for it 
nm0724: thank you very much here we go there we go i've i spent all morning 
working this one out before i came in 
sm0747: 
nm0724: so that's right this is this is the one earlier and er you just need a 
squeezy bottle and three matchsticks and it all works out so the output is he 
said describing the difficult equation the output is K over alpha-squared-plus-
beta-squared now all of these K alpha beta are just numbers a typical number is 
three so they might all be three in fact 
and in fact if K was three she might feel a bit happier than if she was fifty-
eight and then inside the parentheses or brackets or whatever you like to call 
them we have one-upon-S which in fac-, as er you'll note is something to do 
with the steady state value and it turned up up there for the unit step input 
and then we have two terms which look absolutely awful the fir-, both of them 
have the same denominator the same S-plus-alpha-all-squared-plus-beta-squared 
so that that's the denominator on both of those the first one is S-plus-alpha 
and the second one is alpha why the hell have i done it that way because i've 
sort of cheated a little bit and looked ahead a step er what i'm trying to do 
with this is to put them in partial fractions into a form that you can just go 
to a book and look up how they are and it tells you what that actually means in 
terms of time signal representations so really you can go and look up what sort 
of forms you can get to and then try and 
chop chop this thing up er into a form that actually gives you something that 
you can just look up so you don't have to do any fancy stuff er don't worry 
when it comes to examsville this i'm not going to ask you shove this into 
partial fractions and look it up in a book and tell me what the answer is 
'cause i'm assuming you'll all be able to do that sort of stuff if you really 
wanted to eventually er so that is not going to crop up so here it is but don't 
worry don't think blimey how the hell am i going to learn that stuff er don't 
panic or anything it's not going to you're not going to be asked that sort of 
thing you're going to be asked lots more difficult questions than that but not 
this particular thing what does this which is more important what does this 
actually look like in terms of a time signal which is the important thing time 
is of the essence what the hell does that mean i thought essence was sort of 
some spice or something like that 
what's time got to do with being spicy 
sm0748: 
nm0724: no perhaps it's something to do with Geri Halliwell i don't know right 
or maybe that's just wishful thinking again this is just to confuse er all of 
you never mind those listening in here's the gain so this is the output in 
terms of time this is what it what it actually looks like what the syste-, for 
a an underdamped system what it would actually look like the response that you 
get would would appear like if it is the Citroen Diane that i spoke of earlier 
it would be the position of the shock absorbers after a a bump or something 
like that after a hump or hump's not a good word to use [laughter] not in a 
Citroen Diane anyway [laughter] i don't know might be possible this probably if 
this tape gets sent to the Vice Chancellor this will be the last lecture that 
i'll be giving [laughter] there'll be women's rights campaigners men's rights 
campaigners Skoda rights [laughter] campaigners there'll be all sorts 
[laughter] er the anti-, anti-Slovak lobby or something probably the 
probably some refugees will crop up somewhere [laughter] in there as well er K 
all complaining K over alpha-squared-plus-B and then there'll be people 
complaining that they weren't included K over alpha-squared-plus-beta-squared 
is the same gain that's just come through the whole thing just like a dose of 
prunes there it is again [laughter] nothing's affected it whatsoever er and 
then inside the bracket here's that one which was the one-over-S there it is 
again and two terms er exponential- minus-alpha-T multiplied by the cosine of 
beta-T and the same sort of thing again exponential-minus-alpha-T multiplied by 
the sine of beta-T and all we got is that that's just m-, little gain A over 
alpha-over-beta which are just numbers remember alpha beta and K are just 
numbers and the nice little thing here you'll see the alpha and the beta i find 
these little things nice the exponential has the alpha bit in it so there's no 
beta in the exponential and the beta turns up with the sines and 
cosines the frequency type of the the changing element and there's no alphas 
knocking around with the betas so the alphas and the betas are sort of er 
separated er and they they turned up in a different form in the the denominator 
and here they're having different effects in the output but can we can we see 
that in another way well we can this this will hopefully make it a little bit 
clearer the only thing i mean it may be first instance look a little bit more 
complicated but in fact it simplifies when we're looking at what happened this 
can be written by playing around with so-, sine and cosine what do they call 
trig identities or something like that is it sounds like something out of Only 
Fools and Horses whatever and we've got the gain that's still there it's sort 
of lingering in there it is K over alpha-squared-plus-beta-squared no change 
there and then inside the one is still the one and all they've done is play 
around with these two 
terms messing around with the exponentials and the sine er i've cheated a 
little bit because there should be some sort of phase shift that appears in the 
ins-, it's not really just beta there should be a phase shift but i so when you 
look at in in years to come when you're ruminating on it with your cocoa one 
night and you're you're looking at this think blimey this should be a phase 
shift he's got it all wrong well i i'm just excluded it or execu-, got rid of 
it just for the purposes of going through this there is a bit of a phase shift 
in there but it's not really important in terms of what how we're looking at 
this er later on it may become important but not at this stage so we've got the 
gain er that that one is still there which multiplies by the gain and then 
we've got this time dependent thing that in here is the square root of alpha-
squared-plus-beta-squared on the top divided by beta that's that which again is 
just a gain component just a number 
and then we've got the two terms exponential-minus-alpha-T multiplied by sine-
beta-T and again the exponential has the alpha and the sine has the beta and 
the two things are separate so what does this all look like er well couldn't be 
bothered to work it out myself so i thought let you do a bit of work i thought 
make this an interactive lecture the steady state value er this is as time goes 
off to infinity again when time is B well what is it can we work it out can we 
see 
sm0749: 
nm0724: oh you've given it away yeah if if you set T to be zero in that lot 
then the overa-, it doesn't really matter what the sine is doing that's going 
up and down as sines do but this this thing exponential of minus-infinity will 
be zero and that wipes the floor with anything else so er as time goes off to 
infinity we've got this gain multiplied by one-minus-zero and one-minus-zero is 
sm0750: one 
nm0724: one who said zero one yeah so it's just K over alpha-squared-plus-beta-
squared is the steady state value in fact in fact if we 
went back just out of interest if we went back here and had a look at at this 
thing if er we remember with the step input [sneeze] if we just bless you if we 
just set S to zero and multiply this thing out then which is alpha-squared-plus-
beta-squared K over alpha-squared-plus-beta-squared so we we can just look at 
the thing in the first place and if you remember the steady state value for a 
step input we just set S to zero and we've got our answer and that's the same 
doing the same thing there 
and that's what we've got K over alpha-squared-plus-beta-squared but let us 
this this denominator i threw in er a few minutes ago just to to add a bit of 
spice and ele-, some-, something there 'cause it's it's something we can sort 
of ask exam questions on sometimes and i want to look at it a little bit more 
closely so let us let us return to the denominator the denominator is the bit 
on the bottom you should know the numerator's the bit on the top and in the 
middle well keep your eyes closed you've got to draw the line somewhere there 
ha ha ha ha [laughter] so the numerator is commonly thi-, this is the bit on 
the right hand side see i writ-, i wrote the right hand side first just so you 
what the hell's he doing and on the right hand side this is what we had before 
S-squared plus two-alpha-S plus alpha-squared-plus-beta-squared that's what we 
had before for the denominator er 
well about ten lines ago was it not was it not and this is just to say that 
this this left hand side is how it is often written and we'll see why in a 
second so the S-squared has not changed it's S-squared and then we've got plus 
two- zeta- omega- baby-N or subscript-N- S plus omega-N-squared and again zeta 
we're trying to bring in people half the names that are nearly married to 
American film stars things like that so it sort of adds a bit of artistic 
licence to what we're doing so two-zeta-omega-N-S plus omega-N-squared and we 
can see that i mean straight even i could do this this morning with before my 
coffee zeta is if we just look at this we equate S-squared equals S-squared 
there's a profound statement and then if we equate the S terms two-zeta-omega-N 
is two-alpha and the twos cancel which leaves us zeta is alpha-over-omega-N if 
we want and also the omega-N-squared is alpha-squared-plus-beta-squared that's 
just so so what have we got this is where we came from 
the right hand side and all i'm saying here is well you know if you go out in 
the big world the wide boys down Oxford Street they wouldn't talk in terms of 
alphas and betas they would talk in terms of zetas and omega-Ns well they maybe 
wouldn't even talk like that but that that's the the common terminology is that 
one and as we can see let us let us look at extremes let us look at extreme 
cases what for instance happens when beta tends to zero what goes down what 
happens and it's er it's intriguing really isn't it when beta goes to zero 
'cause as beta goes to zero if you're looking at at this side here we're we're 
back to the critically damped case when beta heads off as beta gets to zero 
we're back to the critically damped case er the roots are just S-plus-alpha- 
squared all-squared just like what they were sort of thing before and and now 
for the the foreign listeners sorry maintenant er nous will set zeta to un 
[laughter] and there it is le damping ratio [laughter] so zeta 
zeta is actually known A-K-A the damping ratio and is as we set it to one it's 
it's all the same thing here if you look here it's all magic beta goes to zero 
so omega tends to alpha and as omega tends to alpha so zeta alpha-over-alpha is 
one here so just just from these equations beta to zero omega becomes alpha 
effectively alpha-over-alpha is just is one so zeta tends to one so this thing 
the damped zeta is known as a damping ratio and as beta goes to zero zeta the 
damping ratio goes to to a value of one and the other and and really what beta 
was linked to the sine term the sine term going er really the oscillations so 
what this is saying effectively is as the oscillations die down as beta goes to 
zero the frequency of oscillations dies down and so on the damping ratio tends 
to one and we tend to the critically damped case where we don't have 
oscillation so it all sort of ties together really the er another extreme as 
alpha tends to zero then we can see see from here as 
alpha's zero then omega-N becomes beta or omega-N-squared is beta-squared and 
this omega-N which is why we've got this subscript-N is also known as the 
natural frequency and for this for this reason effectively er as beta goes to 
zero er as alpha goes to zero omega-N tends to to beta becomes beta and er with 
alpha being zero it means zeta is zero so and zeta being zero we've got no 
damping we've we've gone in the direction of the pure oscillations we we switch 
the thing on and it just oscillates it's a pure oscillator it just goes up and 
down up and down up and down never gets exhausted never gets tired never dies 
down just keeps going it's a bit like Bolton Wanderers really just keep going 
up and down never never settle down and er 
sm0752: 
nm0724: pardon oh [laugh] you're not a Bolton fan are you sorry about that i 
didn't as opposed to namex which is just going down and down so and so for the 
underdamped case this is the whole skaboodle this term zeta will go from zero 
in the case 
when there's no damping we've just got pure oscillations to one in the case 
where we were at before the critically damped case and realistically as the 
term zeta tends towards zero so the oscillations get bigger there's more of 
them as zeta gets w-, towards one then the oscillations get smaller so so zeta 
is just a a knob that you can twiddle effectively and between zero and one and 
as you twiddle it so oscillations get bigger or smaller just just to conclude 
this and then we'll have a er a breather it's really bringing things together 
the extremes when zeta is one we've got the case of critical damping that's i 
know you all know that but i thought i'd write it down just for the sake of 
completeness and in fact when zeta is greater than one it's not really 
what we're looking at but when it's greater than one we're off back to the 
overdamped case and again the extremes the other point was when zeta is zero 
that's these there's natural frequency oscillations omega-N and they just go on 
forever and ever and ever and don't die down and i think at this juncture how's 
about we have 'cause er this is a double header this morning for me isn't it i 
think 
sm0753: yeah 
nm0724: hopefully yeah i'm glad you're giving me support in that yeah so er 
maybe er as i fancy making myself a cup of coffee if we said five past is that 
all right five past and we'll get going and we'll get this wrapped up so i'll 
see you all then have a wonderful time in the fifteen minute break or fourteen 
minutes