### 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

mass divided by time or you’re looking for the velocity of the mass divided by time now in practical circumstances what we’d be interested in with the 6lr ometer is one is how quickly can it respond all right let’s say you’re and let’s say you’re using your this accelerometer sensor for an airbag deployment you need that thing to respond very quickly right so to deploy the airbag in time so these types of mechanical systems will have a frequency response just like an electrical circuit will have a frequency response so when we’re analyzing these things we’re analyzing these equations here one of the things we’re interested in is how fast does the object move how quickly does it get to its equilibrium position does it vibrate does it resonate okay if if the if a sudden acceleration causes the device to vibrate at which frequency will have vibrate it turns out a lot of accelerometer sensors take advantage of vibrations they run the system at the resonant frequency because that’s where you get the the best energy characteristics or the least energy loss all right so this is the basic string a spring mass mass dashpot system of course you can have much more complicated things we’re just going over the basics right now another important and basic concept is the beam bending of a cantilever beam since this comes up all the time in in MEMS type devices so this cantilever beam is going to exert a force that right and the dimensions of this beam and the rigidity of the beam is described by its spring constant K right and the spring constant K is given by this equation here K is equal to three EEI over l cubed and this e is the Youngs modulus so that’s a property and mechanical property of the material used to make the beam it’s a rigidity of the material I is the moment of inertia that depends on the geometry of the beam it’s equal to wh cube divided by 12 in the case of a rectangular beam for non rectangular beams like circular beams and other cross-sections you can look this up in a book there are I think there’s a book called Rourke’s fan formula for stress and strain that has all this information in there it’s all been solved analytically their exact equations that’s the moment of inertia you notice that it’s W times H cubed divided by 12 H is the thickness of the beam so you can imagine what’s going to happen if you have a very thin beams as is the case when we micro fabricate devices we we make we deposit thin layers of material and then we pattern those layers so devices the thicknesses of the beams will be very very thin that equates to small spring constants so things become flexible things that you would consider to be rigid like silicon is a traditionally rigid material but if you make a very thin beam out of silicon that that silicon won’t Bend and then we have L cubed in the denominator okay the longer the beam is the shorter the beam is the the higher the spring constant so another thing that happens at the micro scale is if you’re if you’re scaling down the length by the same dimension and then your spring constants will become larger I’m sorry if your scale that is scaling down the the thickness of the beam along with the length and your spring constants will remain the same if you only scale down the length while keeping the other parameters constant of course your spring constants going to jump way up right it turns out that the we’ll talk more about the scaling of these devices we’ll get to that in a second the damping factor here so this system is described by this differential equation if you recall from your differential equations class the solution you may have a system that’s under damped you may have a system that’s over damped this should read over down by the way now or you could have a system that’s critically damped okay now just to remind you of what the response of that is if you were to take your system so if you were to take this mass for example and just like tap it here to tap the mass and then you were to plot the velocity V of T versus time if you have an under damped system what will happen is that the sorry let’s make this X of T

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

acceleration the x axis the y axis and the z axis call those triaxis accelerometers this is an example of the accelerometer where you have some moveable plates and you have a moveable microstructure here this is what an example of what a system might look like in the diagram that I showed you earlier you had a single mass and just a single electrode that’s a good way to describe the system but the systems that are used commercially in order to increase the sensitivity they try to increase the amount of capacitance in the system all right so that you have more change let’s look at what I mean here you have a moveable plate here let’s look at this one for example when you have motion in the X direction this moveable plate has fingers on it and located right next to the fingers are electrodes now if you look at this this structure here you notice that you have one capacitance here all right this is an electrode and this I think this is what’s moving you have an electrode up here so there you have a capacitance form there and then you also form a capacitance here as well so this is what’s called a differential sensor when the plate is moving up what’s going to happen to this capacitance that’s right will it increase or decrease it’ll increase that’s great it’ll increase because it’s capacitance is epsilon a over D all right so smaller distance means that the capacitance will increase at the same time this the second capacitance is going to decrease right so you have one capacitance that gets larger another capacitance that gets smaller this is a differential way of sensing it’s a very accurate way of doing things it’s is very good at rejecting noise all right so they have some differential sensing element here and that the other length of the capacitor works works the same way this is an example of I believe this is a multi axis accelerometer notice there are spring elements that would allow the proof mass to move in this direction in this direction and even the Z Direction okay these spring elements here for example will allow the device to resonate in in this direction these two spring elements here will allow the device the proof mass to move in this direction and these elements here this is kind of interesting these elements will actually allow the device to pop up in the Z direction parallel are orthogonal to the surface okay and the nice thing about some of these initial devices that were made is that the electronics were actually integrated right below the sensor so the electrode was under there all the detection electronics were there as well now these types of devices you know that this sort of system is both a blessing and a curse because when you put the electronics and the sensor on the same chip there are certain advantages in the sense that you you get your whole device done in in one in one device right you don’t need a separate separate chip for the moving element and a separate chip for the electronic element however this process is very expensive so when they’re doing when they’re trying to adapt a design and they improve it there’s a lot of work that has to go in between different design runs alright but this is you know the main point here is just an example of a commercial micros accelerometer system the accelerometers oh yeah the accelerometers I mean if you were to there’s so many accelerometers just there’s a bunch of them in your cell phone the accelerometers you can buy now for only $1 to $5 and the reason why is because they’re so small they take up very little real estate on it on a wafer you know uh the the thing that you do micro fabrication on is you start off with a silicon wafer there yeah big and then you put all you make all your devices each wafer will ultimately end up containing you know it could get to name thousands of devices the fact that the devices are small as what makes them cheap ultimately definitely yeah the question is can you can you buy some of these chips and put integrate them into whatever system that you’re working on your lap yeah yeah of

course of course it’s there’s so many tests and measurement systems that are integrating the accelerometers in there just because they’re cheap yeah and there’s entire industries that have popped up because of this sensor like for example the the whole like Fitbit industry monitoring monitoring your boat your you know it’s like essentially a pedometer that whole industry has come up because of this accelerometer device they’re so cheap yeah they’re cheap and they’re super small which you talking about this here oh yes yes yes so what there’s two aspects that I should clarify one is a differential sensing aspect of it right so when this is moving up this capacitance decreasing this one’s increasing I’m sorry this one’s increasing this one’s decreasing that’s a differential part of it the reason they have multiple legs on here is to increase the effective capacitance so these types of sensors will become more sensitive the more capacitance you have so by putting another leg here they have a second capacitor here they’re getting a larger capacitance change when this plate moves and it’s making the sensing yep making the system more sensitive so this is an example of an accelerometer that’s only moving in a single axis and this one here below you can imagine how this type of idea can be extended to and triple axis sensors so so this is the example now what we’re interested in is how can we model these devices and what are the scaling benefits of these devices all right so this is the equivalent circuit modeling of an accelerometer and I think clarity sake okay so let’s assume that this entire accelerometer package was subjected to acceleration so you took the thing and you pushed it down alright so this mass experiences an acceleration force going down this the spring and the damping force resists that force so there’s a-there force balance we saw that in the previous slide this is the mechanical model over here so we have the mass and then we have the spring constant and then we have the dashpot here right proof mass experiences of force when the device experiences acceleration force is equal to MA there’s two opposing forces the the cantilever serves as a spring so the spring force is equal to KX and the spring constant we can calculate and the velocity of the proof mass in the air creates drag forces and this is called damping FD is equal to B times B the damping coefficient so the question is how can we model this using a circuit this is an equivalent circuit model just so we can give a little bit of context here remember we were in this slide here we were showing the flow analogies of charge fluid and heat and we’re showing that you can model these different systems with electrical circuits whether you’re talking about electricity the flow of fluid from one place to another the MU the flow of heat from one to another in my opinion the fluid flow and the heat transfer are pretty straightforward they’re straightforward analogies because you imagine current the same thing as fluid flow and the same thing as heat flow okay so the the different energy domains when we were looking at our flow variables in electrical systems it’s current in fluid flow is volumetric flow rate in thermal systems it’s heat flow these are all types low type things is very intuitive to understand and when we talked about the different resistor elements in here you’re talking about the electrical resistance here you’re talking about hydrodynamic resistance and here you’re talking about thermal resistance again in my opinion quite intuitive to understand in the mechanics domain when we make equivalent circuit models there are two different analogies that we can use we can use the and the one that I’m going to talk about in this class which in my opinion is more intuitive is the force current model it’s less I have to say it’s less intuitive than the electronics of fluid flow in the thermal

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

go on to more of the these resonant systems in the equivalent circuit model we can sum up the forces acting on the proof mass similar to Kirchhoff’s current law the voltage at the node represents the velocity of the mass so you Kirchhoff’s current law we sum up these currents just as in the mechanical system we always add up the forces in the mechanical system the sum of all forces acting on an object is equal to zero in the electrical circuit all this the sum of all the currents entering a node of the electrical circuit must sum to zero so that’s what we’re using as the analogy here okay it you know if you’re confused about this if you do if you model a few problems using equivalent circuit models you’ll see that it is in fact quite it’s it’s quite useful so let’s talk about more about the scaling advantage higher resonant frequencies means a higher quality factor resonant frequencies scale as 1 over L a times C or the square root of K divided by M so this scales as Eldar the negative 1 actually that’s it’s not quite correct it it really depends on how you’re scaling your spring constant and your mass if you’re just looking at the mass mass skills as is linked to the negative 3 then you have the square root of that which is 1/2 so scales if you’re just looking at the mass and the scales as negative 3 halves length to the negative 3 halves which is still quite significant you also have to look at how the the spring constant scales as well but why why are high resonant frequencies good well if you if you have devices that are operating at high resonant frequencies it turns out that they’re very useful for making resonant sensors now let me give you an example here mathematically we look at this we say the the resonant frequency is a square root of K divided by M now if the mass were to change the resonant frequency is also going to change right if you’re working with very if if your mass is very small to begin with in your resonant frequency is very high to begin with it turns out you can detect very very tiny changes in mess like incredibly small the best mass sensors on the planet right now are these types of resonant mass sensing devices what you have is you have a micro fabricated device that has it could be as simple as a cantilever beam you can see that this is a this is a simple beam element that’s overhanging the side of a substrate and that is being actuated us at its resonant frequency you can have a circuit that drives the system at its resonant frequency this particular device has is using a magnetic force so it’s running a current through here and when those when that current is exposed to a magnet it causes it causes a Lorentz force which pulls the beam up and down this is a more conventional approach where they use electrostatic actuators we saw examples of electrostatic actuators with the with the micro mirror race you have two capacitor plates you apply a voltage between them and that causes the place to go towards each other it’s same idea here except you have a series of beam elements here you apply a voltage between this beam of this element here and this element here that causes the attractive force between the plates and that causes the device to resonate in a resonate in the in plane so it’s sort of in the Y direction here I’m sorry in the X direction this is what an example of the device looks like because it has a small mass it has a high resonant frequency these devices are even smaller these are just single cantilever x’ all right that that are also resonating now I’m the main point I wanted to get at here is when the masses are small the resonant frequencies are very high we actually have the ability to detect very small changes in frequency one of the things that we can measure very well using the instrumentation that we have today is that we can measure very small changes in frequency because we have we have circuits that can that are very good timing circuits that can detect small changes in frequency these elements what happens is that even even if emptor gram change in mass can cause a significant change in the resonant frequency and we can check that so these types of resonant biosensors rely on that idea you have this cantilever that has a certain resonant frequency then you expose that cantilever to some type of biological or chemical agent all right

the let’s say the cantilever is exposed to a certain gas and those gas molecules adsorbed to the surface of the cantilever beam that will change the cantilever beams mess right resonant frequency changes we can detect that turns out that we can detect as little as tens of FEM programs web programs is 10 to the negative 10 to the negative 15 grams you can imagine how small that is you can detect very very small mass changes with these devices it’s quite amazing you can also detect biomolecules now proteins DNA these types of biomolecules can also attach to the surface of these cantilevers in fact you can get even you can get fancier and you can functionalize the surface of these cantilevers functionalizing means we put if we want to detect protein a and we don’t want to detect protein B we only want to detect protein a in a solution that might contain protein B and like thousands of other proteins what we can do is we can put an antibody on we can coat the cantilever with an antibody that antibody will only attach protein a the one that we’re interested in these types of surface functionalisation z’ are very very specific because in your body you have antibodies and antigens though the the interaction between antibody and antigen is very very specific in your body so only certain cell types will connect to other certain cell types I I don’t want to get into all the biology of that right now but just suffice to know that you can make these things very specific so what you would do is you could take this type of bio sensor that’s coated with this antibody specific to the protein that you’re looking for and you just dip it in a solution the dip it into your sample right those proteins attach to it and that changes the mass of the cantilever now you take your cantilever out of the solution you re measure its resonant frequency and now you have a different resonant frequency you can measure the amount of protein on that cantilever down to femto grams okay very very sensitive mechanism it relies on the favorable scaling of resonant frequencies in micro scale devices I also mentioned want to mention this thing about quality factor I’m just gonna go over this very briefly we can we may get into more detail on it later some of this may look familiar to some of you so this is time on the x-axis now let’s say we have a system that is over damped and then we we resonate it in time what will happen is that the oscillations will eventually they’ll die down like this so this is over damped in an under damped system when we have very where we have fewer energy losses these oscillations go on forever now in the frequency domain this looks something like this so if we were to look at the frequency and what’s called the power power density in this over damped system we see something that looks like this where this is f naught this is the resonant frequency of the system so these types of plots the power density pops plots of spectral density you have frequency on the x-axis and the amount of power contained at a certain frequency if you have a device that’s resonating like this at a specific frequency then what will happen is that you will have your spectral power power density pop plot will look something like this the highest amount of energy is contained at F 0 the resonant frequency when the device is oscillating at that frequency then you have a peak at F 0 in the case of so let’s say this is what’s called a low low Q system and this is high Q in a

high Q system you’ll have a peak that’s much sharper Center to f 0 this is also the the power density if we have a sharper peak it allows us to measure the the resident frequency much more accurately in a high Q system we can measure the resonant frequencies much more accurately unfortunately I think to give this a proper treatment we could spend an entire lecture talking about equality factors and resonant frequencies and so on I think for now I think we’ll just leave it at this a low Q system will have a relatively wide spectral band so which means that even though it’s the system is resonating at a certain frequency there is some power in the system that’s that’s also present at other frequencies in a high Q system all the energy of the mechanical system is right at the resonant frequency okay this makes it easier to measure the resonant frequency there’s less energy losses in the system and it turns out the system is much more accurate so the quality factor it has to do with the the width of the peak divided and the height of the peak that the larger the height of the peak the smaller the width of the peak the higher the quality factor all right so just to sum up this module then so the the main points here so I hope that you’ve seen some of the benefits of miniaturization it’s not just being able to make things smaller and cheaper and to put a lot of be able to make a lot of devices on a wafer but it’s there’s a lot of physics that improves at the micro scale ok we’ve spent a lot of time on this module but I think it’s I think in terms of the rigor of you know understanding Micro Devices I think this is very important for you to understand regardless of what area of Micro engineering you end up going into whether it’s something that’s very chemically focused or whether it’s in the thermal domain mechanical domain whatever the physics changes at the micro scale because of the relative importance of you know the length dependent phenomenon service area dependent phenomena in volume dependent phenomena and the way that we can understand that is by analyzing the governing equations how the respective phenomena scale with length so that’s an example that we did earlier we just wrote out the equations and find out found out the different factors that were geometrically dependent and we can see how phenomena scales with length that was your assignment in some cases we also saw that you can gain some intuition by looking at dimensionless numbers things like the bond number the reynolds number the peclet number these are typically used in analyzing fluids and heat transfer but there’s no reason that you couldn’t have similar dimensionless numbers to analyze other domains alright these dimensionless numbers allows us it’s a very intuitive way to understand how one phenomena is more significant than the other phenomena and then finally the another way that you can analyze it by is just by doing these equivalent circuit models and these circuit models aren’t limited to micro scale analysis any type of analysis you can do with circuit models are also called lumped element models so you can use those to analyze micro systems in multiple domains so what I hope you’ll get out of this is that you understand that the different domains as different as they are physics there’s actually a lot of similarities between them the way that heat flows is very similar to how fluid flow is very similar to how electricity flows if you can understand some of those analogies then when you start analyzing problems as interdisciplinary engineers people in BioMEMS most most of the folks are have learned about different domains there’s a certain level of inter inter disciplinary expertise you have to have you may not be an expert in all domains but if you understand your own domain very well you can extend that knowledge to other domains that’s the whole point of this exercise all right so let’s let’s end there

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