# BioMEMS Module 2D – Scaling Laws and Analysis in Micro and Nanosystems

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### BioMEMS Module 2D – Scaling Laws and Analysis in Micro and Nanosystems alright see where we left off here so like I said we’re gonna finish off this chapter today so last time we ended off talking about the thermal domain we’re basically going through the different domains it’s been a week and a half so it’s our last lecture so just to remind you we’re looking at the different energy domains we’re looking at electronic the electronic domain the fluid flow domain the thermal domain we talked a little bit about diffusion and now we’re going to talk about finish up with a mechanic’s domain today what I want you to be able to do in this module is to be able to analyze these different domains with equivalent circuit models let me show you an example of what we did okay in the in the fluid model okay you can model a microfluidic Channel by using an electrical circuit where a current source represents a flow and these different resistors represent hydraulic resistances okay so that’s one way of modeling a fluid system using an electrical circuit in the you know moving ahead I’m not going to go over all this stuff again in the thermal domain we talked about circuits to model the thermal domain okay in this case you have a current source which represents a heat heat being generated in the system if you recall with the micro bolometer you had a membrane and when infrared light hits this membrane it gets it gets hot so that’s represented by this current source and you have a thermal capacitance here and two thermal resistances here just as a reminder what is this thermal capacitance what does it represent physically stored energy that’s correct so if stored thermal energy is the fact that the object is hot okay it’s going to take some energy to increase the temperature of the object and also the energy stored the heat stored in the object can’t go away instantaneously so the temperature of the object can’t attain change instantaneously just like the voltage across a capacitor cannot change instantaneously so it’s modeled by this capacitance these thermal resistances represent the heat flow paths where this pixel is connected to the substrate through these legs heat can flow through the legs okay so from a purely thermal standpoint this is heating up okay and when that when that membrane is heated up heat flows between the hot object to the cold object the substrate is cold so it flows from the membrane into the cold substrate here like this that heat flow can be represented by currents in this electrical circuit model currents flowing through these two resistors basically model heat flow going from the membrane into the substrate this thermal capacitance models the energy storage of the heat storage in this memory the advantages remember of thermal systems is that these thermal capacitances can be very small so basically objects can be heated up and cooled very quickly at the micro scale all right we finished off with the thermal domain here today we’ll talk about the mechanical domain mm-hmm okay and again these concepts I’m showing you are you know they’re very basic concepts that you’ve probably probably learned in in undergraduate mechanics course even in an advanced high school course but it’s it’s useful for us to understand some of these basic concepts especially those coming from different different domains so what we’re going to get at here is that I also want you to be able to analyze a mechanical system using an equivalent circuit model and understand the scaling benefits of mechanical systems just as we’ve understood the scaling benefits of fluid systems thermal systems and so on alright so in the mechanical domain we can just review a few essential concepts mechanical systems with rigid body mechanics were looking at the displacement and velocity and acceleration of some type of object a very basic system can be modeled by a spring and dashpot we’ll get to that in a second but we’re looking at the movement of a mass all right now the different forces that a mass might experience one is acceleration so let’s now look at some of the details of this let’s say that you have let’s say that you have a mass here we’re gonna draw a Western accelerometer system looks like by the way does anyone know where accelerometers are the most popular uses of accelerometers right now what are accelerometers what do they measure acceleration yeah and do you know where accelerometers are used a lot the biggest in in the 90s the biggest application for accelerometers were automotive airbags for detecting crashes so when the accelerometer detected a large acceleration acceleration that would trigger the airbag to deploy all right great technology for crash safety but does anyone know what the big use of accelerometers are right now every one of you is using one cell phones yeah exactly they’re used in cell phones for when you pick it up when you rotate when you rotate your cell phone your screens turn right their accelerometers built into certain phones for detecting like how much I’m walking like my my Samsung Galaxy has you know that the app that detects how many steps I’ve taken per day accelerometers are and things like fitbit’s you know that also detect motion they’re everywhere so the basic I’m using accelerometers as an example because it a nice example of a miniaturised system which is very quite commonly used the way that an accelerometer would be modeled this is pretty ugly let me redraw this you have a mass okay and that mass is attached to a cantilever the cantilever has a spring constant given by K all right now this whole thing is in some sort of housing okay and this cantilever beam is the one half of it is attached to the housing itself we’re drawing a very simple simplified diagram of this the the mass is actually made of a material that’s that’s either conductive or it has an electrode on the bottom and like this most of the time the mass itself is just conductive all right so let’s say we have a mass that has an electrode on it and then on this side we also have an electrode and this is connected to some type of circuit and the circuit measures capacitance what’s going to happen to the capacitance here fastened in sizzle to epsilon a divided by D this is the distance D the distance between the place right now let’s say we take this whole master so we’re talking about this entire thing here and we subject that to an acceleration what’s going to happen to this mess mohammet it’ll swing that’s correct it’ll swing it’ll swing up and down that’s correct absolutely exactly right when this thing is exposed to acceleration this mass is going to end up either moving closer to this electrode or farther away that changes this distance D okay and that distance D changes the capacitance and you can have a circuit that measures like practice so you can electrode electrically read out the acceleration because the force on this mass is force is equal to MA all right we’ll go back to the equations here but this is this is what the system looks like now accelerometers are used like I said you have accelerometers in your cell phones you have accelerometers and automotive airbags are there everywhere so in this this is a good example because it demonstrates rigid body mechanics and you have a rigid mass in there that’s attached to something that’s holding on to it that that rigid mass is going to be subjected to acceleration F equals MA right when you subject it to acceleration this is a mechanical model okay so when let’s say this mass was connected attached to the housing here and this whole device was accelerated in in the upwards direction okay so in the direction of X the acceleration is happening in the positive x-direction so the first force we’re going to talk about is that the force of the uncertain on this mass is equal to mass times acceleration whatever acceleration was being applied to the entire system that’s the first force on this mess the second force well let’s let’s do this one as the second force the second force is that the the mass is attached to a spring okay in this case you had the mass that was connected to some sort of beam element okay these beam elements can be microfabricated all right this beam element is going to resist the mass removing it acts like a spring now with Springs there’s a very simple way that we can model Springs us by using Hookes law force is equal to KX K is a spring constant of the cantilever and X is the displacement okay you imagine what the spring the more you displace it the more force the spring tries to pull back so that’s a simple law Hookes law K is a spring constant so the more rigid the spring is that yeah the more rigid the spring is the higher the spring constant is going to be all right so you have this force that you have the mass here and then you have this resistor element or this is a spring element and that’s represented by K that represents a force exerted on this mass that’s pulling back on it it’s it’s being accelerated in this direction but the force is pulling back in the negative x-direction the third force is the friction force okay now the friction force in this case in the case of a simple cantilever beam is suppose the mass is being accelerated in one direction now let’s say it’s a it’s going in the upwards directions since that’s the example that we showed now there may be air molecules in here okay so if those air molecules are there the mass has to push aside the air molecules right so the air molecules will actually damp the system that’s called a damping coefficient all right so that’s where this force comes in a frictional force or a damping force that force is equal to be V it turns out that that it scales with the velocity the faster the object is moving through the air the more force is going to be exerted on it it’s like a drag force right you imagine a car moving on moving through a highway faster it’s going to more drag forces there’ll be so that’s your frictional force drag force force is equal to B times B so if we look at all these three forces together we have this three system this simple system here the the frictional forces represented by something called a dashpot that’s what this see is just to clarify this C and this B are the same thing so in this mechanical system we have three forces at work one forces acceleration dependent force equals MA another force is velocity dependent and then another force is displacement dependent you can add all those three things up so the force in the upper X direction the acceleration force ma is equal to negative B V minus KX okay this is the force from the the frictional force and then this is the force from the spring all right and you can express that as a differential equation M times d squared X DT X’s X is the displacement plus B DX DT minus KX equal to 0 DX DT is velocity and then you have this this term here so reminding ourselves from our undergraduate mechanics class that this is just a basic differential equation it’s a second-order equation and it can be modeled by this system here very easy to solve these types of differential equations this particular system will give you if you recall it will give you a resonant frequency so the mass can actually vibrate the resonant frequency is a square root of K divided by M and the damping coefficient is given by B divided by 2 times square root K M now we’ll talk a little bit more about this in a second when you solve this system basically you’re looking you’re either looking for the displacement of the the  that’s a displacement in an under damped system from the moment you tap the beam the beam will start to resonate at its resonant frequency so it’ll just basically vibrate okay and in a perfectly under damped system where there’s no this is if there’s no damping at all right that damping coefficient B that we looked at previously if that’s if that doesn’t exist this thing will vibrate indefinitely it’ll continue forever so this is an example of an under damped system and an overdamped system you know it might just look like this where you displace it at time zero but then in Ischl e goes back to zero displacement and so this would be over damped and in the case of a critically damped system you might have some oscillations that eventually die down and until it rests if you want to look more at physics concepts this is another nice page from hyper physics any questions on this stuff this is really hopefully just a reminder of things that you’ve looked at earlier good question so the question is is a goal to have an under damped system if you want the system to recover from whatever perturbation that you gave it the fastest response the system that returns to its initial state the quickest would be the would be the critically damped system see the over damped system it resists the over damped system would be if if there’s a lot of let’s say if there are a lot of air molecules next to the mass if it was a high pressure system those air molecules would resist that thing from moving at all that the problem with over damped systems is that there’s a lot of energy loss it’s actually resisting those air molecules would resist the mass from moving at all okay now a critically damped system is an under damped system is where there were no air molecules at all in which case we were just Asli indefinitely there are certain situations where you would like the system to oscillate indefinitely in the case of resonators you actually want the system to be resonating with without losses and with at least energy input as possible the critically damped system is important when you want to when you want the system to recover from oscillations as quickly as possible so initially there’s some oscillation with those oscillations died down relatively quickly we’ll see a few examples of the resonant system so here’s an example of a micromechanical system the accelerometer that we just talked about so you assume that in as accelerometer as a proof mass that’s attached to a cantilever beam the beam bends on the proof mass experiences acceleration just like I drew earlier in this case the arrows are sort of opposite here the the acceleration is going down in the spring the spring force and the damping force is going in the optimum going up now these this is a practical device as I said these accelerometers are in your accelerometers are in your mobile phones you know you can see these types of devices here you can also see these STMicroelectronics accelerometers in for airbag deployment as I mentioned earlier some of the systems you know these are examples of what the systems look like I drew a very simple diagram in in the notes here all right but sometimes these devices can actually be a little bit more complicated than that the device that I drew here this only detects acceleration along one axis all right the accelerometers that are in your cell phones and some of the sensors even in the automobiles are multi axis sensors so they’re like sense   models because when you’re dealing with mechanics you’re not really thinking about something flowing from one place to another if we think about this rigid body mechanics problem we were thinking about just the movement of the mass period there’s no flow of one thing to another okay but it turns out you can model mechanics using circuits and it’s it’s done all the time what you do is that you consider that the potential variable so the voltage at a specific node in the circuit you consider that the velocity the flow variable you consider that force okay now this is all mathematically you can you can justify it I’ll show you in just a second the resistor becomes lubricity right 1 over B so this is represents a friction element now resistors always in in circuits resistors are the energy dissipative element and if there’s some type of energy loss going on if you have a resistor in the circuit in the mechanical domain the way you lose energy is by friction so whenever you have friction in the system that could be be modeled by every mr. element now in the electrical system we have a capacitance that’s an energy storage element remember in the thermal domain we were talking about the heat capacity the fact that an object retains you know that it takes energy to heat up an object so that represents energy storage now in mechanic’s the energy storage element is mass because when a mass is moving at a specific velocity it has inertia it has kinetic energy it has momentum right that momentum itself is a form of energy storage that’s why this can be represented as a capacitor inductor is also an energy storage element we didn’t really talk too much about inductors in the fluid flow in the thermal domain but we all know what inductance is in the electronics domain it’s energy storage in the form of a magnetic field in the mechanical domain the spring is represented by an inductor in the force current modeling mechanism whatever the spring constant is the compliance is one over K and K is the spring constant and this is represented by a energy storage element the idea behind the spring is that if you were to take a spring and you wind up the spring and you push the spring down it’s storing energy when you let it go it Springs the mass Springs away so that’s also a form of energy storage all right so we can model the mechanic’s system in terms of these two energy storage elements so let’s go back up to here and we’ll see how this how this system can be Bob so if we look at this electrical circuit here actually have a let me write this down the mechanical system here we have I’m gonna redraw this here’s just so you can see it mass-spring alright so this is the dashpot this is the spring and this is the mass so this mass is subjected to an acceleration okay so we’re going to assume this mass is being accelerated upwards and the spring is going to resist that force the dashpot is also going to resist that force so if we were to write a an equation describing this let me say that the force the acceleration force minus the spring force minus the drag force is equal to zero so the acceleration force is equal to MA the spring force is KX and the drag force is B times V so this is equal to zero so this becomes M DV DT minus K remember acceleration is dv/dt right displacement is the is the integral of the velocity minus B times B is equal to zero all right so we have this equation here in terms of one variable the velocity V DV M DV DT minus K integral of V minus B V is equal to zero so that’s an equation describing the system now in the electrical equivalent model just draw this out here next to it hmm the way that we describe this system is through this electrical equivalent circuit that looks like this all right on the left here you have a spring that represents the energy dissipative element now the resistance is equal to 1 over B well let’s just let’s just say this is a resistance R this is L this is L sorry and this is C what I want to show you here is why the electrical circuit can model this mechanical domain now in this in the circuit the way that we would analyze it is that we would use Kirchhoff’s current law ok kirchoff’s current law states that all the all the currents in the system have to add up to zero so let’s say we do kirchoff’s current law at this at this node right we if we see so this is i2 i3 I won so i2 equals i1 + i3 or sorry listen I – – I 1 minus I 3 equals 0 so this is Kirchhoff’s current law at this node I want to show you how this how this matches up ok if we’re you know for those of you who are electrical engineers you’ll see this find this find this analysis relatively straightforward this I – for a capacitor i is equal to c dv/dt alright and this is V is the voltage at this node you redraw that looking messy alright so this I 2 here is going to be C DV DT I won the resistant that the current through that resistor is going to be for a resistor I is equal to V over R so it’s going to be just 1 over R times V and I 3 here you recall the current and inductor there’s this rule V equals L di DT in a circuit and so the current if you solve for it it’s equal to the integral of V divided by L so this was 1 over L times the integral of V this is equal to 0 right let me just switch these two around just so you can see the let’s make this a three and make this I 1 1 over R times V this is just so you can see the analogies between the two the two equations here alright so if you look at the left here right look m dv/dt minus K integral of V – V V is equal to zero you notice that this equation is pretty much the same c dv/dt minus 1 over l times the integral of v minus 1 over r times V is equal to 0 these two equations basically have the same form this is the reason we use equivalent circuit modeling ok if if you solve one if you can solve one system and the equations are the same as the other system those results can be used in the other system right we use circus because it’s circuits circuit models help develop intuition you know after you’ve been working with circuit models for a while you develop an intuition that if you if you change this resistance here then it’ll change this performance of the circuit and so on the whole point is to develop intuition but just so you see the equivalence here all right we can say that number one we can say that this voltage represents velocity so this capacitance here is equal to M C is equal to M that’s this thing here now the inductance the L is going to be equal to 1 over K so that it matches up with this K integral of V over here and the resistance in order to make this match up with negative B times V the resistance will be equal to oops this sense is equal to one over these and B is the damping coefficient right so this circuit really does model the same thing as a mechanical system so this is the diagram of the circuit here now why do we go to all that trouble well first of all just to show you that how you can model these types of systems with circuits but also you know from this standpoint once we have a circuit we can analyze that circuit to determine the the behavior of the system so in a circuit element you know that we have these things called resonances right in the mechanical system we also have resonances with any RLC circuit the resonant frequency is equal to this is the given in terms of radians per second is a square root of 1 over LC and if we plug in the values for L and C we find that in the mechanical system the resonant frequency is the square root of K divided by M all right so just looking at this what do you think is going to happen in a mechanical system as we miniaturize it what will happen to the resonant frequency if we have a vibrating object we make that vibrating object smaller what’s going to happen to the resonant frequency it’s going to go up it’s going to go up quite significantly look we have the mass on the bottom right mass is proportional to length cubed we have the square root of that but still it’s in the denominator so this is going to be resonant frequencies go way up at the micro scale this is one of the scaling advantages another thing here we have the quality factor now quality factor represents this idea of how much energy losses in the system if we were to tap that if we were to tap that mechanical system and it starts vibrating if it’s if there’s energy losses if there’s damping that those vibrations are eventually going to disappear right but in a in a perfect system where there’s no energy loss those that thing will vibrate forever now something that has very low damping very low energy loss that’s called a high quality system this is governed by or it’s quantified by this quality factor Q the higher the Q is – less damping the less the less energy losses there are in the system when we make resonators for sensors we want there we want them to have a very high quality factor we don’t want energy losses in the system right so when we talk about the quality factor the quality factor scales a square root of M times K divided by B so you see this square root of M here at the top however you also have the B at the bottom when you make the device as small the damping coefficients get smaller because there’s less air resisting the movement of the of the mass so you can still get very high quality factors at the micro scale so just to sum up this slide before we   