Introduction to Electrochemical Impedance Spectroscopy (EIS: Maths and Theory)

okay so good afternoon my name's sam and i'm a senior lecturer in the dyson school of design engineering uh and part of the electrochemical science and engineering group as well as having my own group in the dyson school the tldr group tools for learning design and research and i'm going to talk to you today about eis electrochemical impedance spectroscopy uh and my aim is to give you a an introduction to it in a way that you've not seen before so hopefully it's somewhat distinct from the textbooks so if you like the textbooks then you're already set but if you found them a bit confusing then hopefully i can pitch at you in a slightly different way uh that that you will be able to hang on to so we're going to start from the very beginning and i have to try and remember and if i don't someone please interrupt me to we should take a break at some point at least one so that we can all refresh our brains okay so this is just for my sake just to remind me of all the things i want to mention today so don't worry about that what are we doing okay there's there's electrochemical impedance spectroscopy but actually there's just impedance spectroscopy in general and this can apply to mechanical systems or audio systems a range of systems show impedance-like behaviors and the general idea is we are applying a small periodic stimulation signal and we're going to measure the response of that signal so usually therefore in electrochemical systems we apply a small stimulation voltage signal and it's periodic here and we might measure a small current signal as the response now you can also do the inverse of that you could you could stimulate it with the current and measure the voltage okay but more often than not you stimulate the voltage and measure a few questions before we look into the analysis because i i suspect for the audience many of you have either done impedance or seen some impedance or seen a bunch of nyquist bots all these bits and i'm really keen to start from the very beginning of what we're doing and why to make sure that we're all on the same page so first question from perspective of linearity you may have heard this expression in the context of impedance why do we use a small signal okay and the answer is because electrochemical systems aren't linear i.e if you've got a system where for a particular voltage it gives a particular current if you double the voltage you wouldn't usually expect to get double the current okay they are often very non-linear but if you give it a very small signal i.e the amplitude of your sine wave is nice and small so it's only traveling a short sort of distance in the voltage range then you can say roughly approximately the response will be linear so analysis of linear systems as you might imagine is much more much easier than the analysis of nonlinear systems and so that's why we take this approach a nice small signal so that you can approximate the response as a linear response to that signal okay and so we often would call this a pseudo linear response but not too small okay and this is a classic sort of engineering scenario you've got confiding constraints one of your constraints says don't make it too big otherwise it'll be non-linear and on the other side don't make it too small otherwise you'll have a signal-to-noise problem you won't be able to generate reliably that kind of small signal and you won't be able to measure reliably the response from the system so you've got to compromise there and that's why you pay good money for high quality impedance equipment because it can generate small signals and measure them accurately across a wide range the big model is this when i've spoken to lots of people about impedance whether it's uh you know the beginning of their first year of their cdt or during their phd vive the big model seems to come from the fact that impedance is not the ratio of voltage and current in in time it's voltage and current in frequency and we're going to come back to that about halfway through and hopefully that's going to be a bit of a helpful revelation to some of you so we've got a set of idealized components okay they are idealized they don't exist in reality no things actually behave quite like this okay but they're very helpful models that we can put in our system and they do often represent reality quite well in many scenarios so resistors and you can see the little symbol on the left a little squiggly line resistors and they have resistance r measured in ohms capacitors with capacitance c measured in farads and in each case on the right hand side i'm giving their relationship to voltage so you all have heard of v equals ir and you also should be equally familiar with the voltage relationship to a capacitor which is that the voltage is one over capacitance multiplied by the integral of the current ie the amount of charge that's been moved or stored and lastly inductors with inductance l which is has the units of henry's and the relationship to voltage here is the inductance multiplied by the derivative of the curve so the capacitors are the integral of the current and the inductors are the derivative of the current and here's a sort of very conventional representation sort of bringing together all the bits i've just shown here's a simple series circuit with all three of those components represented you've got some kind of voltage signal and it's generating a current in the circuit in the loop and the result of those three components is that when you put in a known voltage you get a measured current out and it might have a change in amplitude and a change in uh phase depending on the values of r l and c and phase as you can see from that diagram is the degree to which the peaks line up with the peaks and the troughs lined up with the troughs and so you can be in phase or you can be an anti-phase where the peaks line up with the troughs between the voltage and the current and anything in between as i say all of the components are idealizations including the wires so we don't often don't talk about the wires and impedance diagrams they're the bits that are just the black lines that connect all the components together and we assume those to have uh perfect electronic conductivity or zero resistance and certainly no uh reactants reactances capacitance or inductance but that's not even true okay so i just want to like keep sewing in your mind this idea that these are models these are simplified representations that do not describe reality exactly but they describe reality conveniently which is a very useful thing to do so even wires actually have resistance as a function of frequency due to something called the skin effect where you get sort of eddy currents in your flows of electrons inside the wire which i blew my mind when i found that out and so here you've got example of a copper wire and depending on the frequency it changes you know material property resistivity it changes as a function of frequency and of course also these things will change as a function of temperature wires as any of you have done impedance also often have a degree of inductance and if you have long wires because you've got a big rig then you can measure quite significant inductance which can be a pain so all of the components are simplified idealizations but they're very helpful the key framing that i'd like you to to make use of today are relating impedance in electrochemical systems to other physical analogies which i consider to be easier to visualize and i'm sure many of you have heard them before so the first one is the the spring mass damper trolley system you've got a trolley it's on wheels frictionless wheels and it's attached to a wall on one side with a spring of stiffness k a damper of damping constant c and the trolley itself has a mass m and it moves back and forth in direction x right it's constrained by that spring and so this thing will have a sort of oscillation frequency so it'll move back and forth and if it's got damping that energy will gradually leave the system a bit like air resistance and the thing will gradually oscillate back towards the middle position and you may remember from your undergraduate mathematics that this has this second order ordinary differential equation where we say the mass times the acceleration where acceleration is x double dash or the second derivative of position mass times acceleration plus damping times speed which is the first derivative of position plus spring stiffness times position equals the forcing term which is a function of time so x x prime and x double prime are all also functions of time and it's a bit messy to put them all there so all of the terms in this equation end up with the units of force so as you know f equals m a so mass times acceleration is a force damping times speed is a force stiffness times position is a force and so this is a second order ode heterogeneous because it's got a function on the right hand side and it describes this trolley and i and hopefully some of you can look at that system and have a bit of a feel for how it's going to behave so if i move the trolley to the right and let me turn on my laser pointer if i drag this trolley to the right over here and let it go you should all sort of have a feeling that it's gonna get dragged back to the left again by the spring but this damper is going to sort of slow down the rate at which it gets dragged back and for those of you who remember this in a bit more detail depending on the ratio of these three components so depending on the balance of these three parameters will determine whether this thing oscillates forever or oscillates a bit and gradually slows down or really gently moves back to the middle or really slowly creeps back to the middle as if it's moving through honey let me show you a second system this is water in a pipe okay and you've got water and it's moving through a constriction which is sort of a narrowing in the pipe which maybe causes a bit of turbulence or something and it can cause the pipe pipe to sort of heat up in that region and lose energy you've got a water wheel which as the water flows past it this water wheel gains momentum or sort of it has rotational inertia so it likes to keep going in the same direction and then lastly you've got a little membrane that is totally across the whole pipe and it sort of blows up like a like a balloon so as you flow in one direction this membrane gradually and it's got stiffness pulling it in one direction and then it can ping back in the other direction this system has an equation like this where i've given v is my little volume of displaced water so we can say that alpha is a bit like m and m is the inertia of the trolley and alpha can be the inertia of our little water wheel beta is a bit like c c was our damping depending on how fast you go is how much energy you lose or how much force you require to move through that space so beta relates to our constriction in our pipe and lastly gamma is a bit like k ie the stiffness and this one's a very obvious one the stiffness in our sort of blocking elastic membrane is a bit like the stiffness in your spring okay and now instead of a force because you're in a pipe it's a force per unit area so it's a pressure so all of these have units of pressure now we're back to the one that we're really interested in today the inductor resistor capacitor series circuit and we can look at that and say okay what do we learn by knowing that this is the second order heterogeneous ode that describes it what do we learn by drawing an analogy to these other two systems well let's start with the middle term r multiplied by q prime equals v well you should be able to sort of work out by deduction that q prime is clearly i is to the current so this is v equals ir is the middle term okay and so what is q q is displaced charge okay so initially imagine all the electrons in your circuit are in some initial position and if i move the current a little bit if i let the current flow a little bit you can compare where they are now to where they were that's the sort of amount of distance your charge has been displaced by so q is displaced charged and it's very much like displaced volume of water or displaced position of trolley and so it's the first time derivative of the displacement of charge so it's just the current so resistance times current is voltage what about the other two terms here let's look at the first one uh l q double prime so this is l times the second derivative of displaced charge which is the first derivative of the current so l times the first derivative of current equals v and this one is like m the mass of the trolley so you can think of an inductor as being like momentum in your circuit right so on what's momentum like well when you've got a current flowing and it's flowing along in one direction and you suddenly give it a signal to do something else it wants to keep going how it was okay it takes a while to slow down and change course that's momentum conversely this term over here c to the minus 1 times q notice there's something a little bit annoying here i've given you k for spring stiffness gamma for membrane stiffness but c to the minus one for our capacitor i'm afraid as with a few things that we're going to talk about today that's just convention so we could have defined a different parameter that was you know not to the minus one but just you know just a straight up coefficient but by convention we've defined the parameter b one over that convention c in the other two systems so you've got c to the minus one so c is like the inverse of stiffness okay so when c is very very large it's like having a very very soft spring attached to your mass i.e if your spring is very soft you don't even see it there at all whereas when c is very very small it's like having a very stiff spring and therefore it really blocks the current from going through at all so i hope you can keep these analogies in your mind between these idealized circuit components l r and c and the components of either the spring mass damper or the waterfall plug membrane system uh because as i say i deal with this maps quite a lot and i find this super useful to keep in mind so let's stick with the one that we're most interested in and let's deliberately reframe it back to something slightly more familiar let's forget about displaced charge which is q and think about current okay so all i'm doing is i've replaced q prime with i q double prime with d i by dt and q with the integral of i with respect to t okay and that is something that's going to turn out to be quite important in in making a convenient description of our model so let's try and extract the impedance okay so i've talked about resistance but i haven't really mentioned impedance and i want you to walk through this analysis with me so i i've you know i'm teaching the maths course in my department and i've found as much as i enjoy writing on an ipad that actually the number of mistakes and the clarity of my handwriting are not good enough so i've i've painstakingly made this animation by animation so hopefully that works a bit better there's a lot of notation here here we've got the same equation again okay but what if i want to just have a resistor well a system a circuit with just a resistor is a circuit where essentially we can imagine that l was zero okay and c to the minus one was zero ie c has gone off to infinity right that would make this term here disappear and it would make this term here disappear and it would give us r times i equals v which you should all be familiar with so here it is i've got this displaced charge notation which we're going to disappear in a minute and what we're going to do as i showed in the early slide is replace my mystery v as a function of t function with a specified sinusoidal stimulation so it's going to have amplitude v naught some specified voltage amplitude it's going to have frequency omega some rotational frequency angular frequency and it's a sine wave and so it's a function of t i'm just going to substitute that in and rearrange okay so i now have an expression for the current in my system as a function of time q prime which is just i equals v naught over r uh times sine omega t okay very obviously this is the simplest one so all seems very fine and what you know what you've seen is that z equals v over i which in my case is v over q prime uh and so we just sub in all the things we know v was v naught sine omega t and q prime as we've just found was v naught over r sine omega t and quite clearly almost everything just cancels out and we end up with one over one over r which is just r so it looks like we've progressed through a sort of visual understanding to a mathematical ode type description and we can reduce this to the thing that we're after which is the impedance what is the impedance of a system obviously this one is an easy one so let's look at another scenario scenario two just a capacitor so we'll start with the full equation and once again say okay if we want just a capacitor then we'll have we'll set l to zero we'll set r to zero and we'll keep c to the minus one as c to the minus one okay it's some unknown value of c it's just a capacitor if we do that our equation reduces to this c to the minus one q equals v as a function of t uh and we're going to choose uh our same stimulation function v naught sine omega t and solve well if i just rearrange the equation above i get q on its own and that says to me the displaced charge equals c v naught sine omega t but if i want the impedance then i want the derivative of displaced charge i want the current so i have to differentiate this thing with respect to time and that's not hard at all of course sine omega t differentiates to omega cos negative and now i can do the same thing again z equals v over i which equals v over q prime which equals the things we just found as a ratio which equals something wrong right z equals 1 over c omega multiplied by tan omega t as picard is saying here that is not how this works that is not how any of this works okay z is not the ratio of v of t over i of t okay that's not what it is clearly you know by pursuing this method what we've got is some very strange thing that says well this is the our impedance of our system what we've been calling zed is a function of time i don't think that's correct right so it's saying that our you know and also it's it's a tan function very strange functions with all these sort of discontinuities in it okay so something's gone wrong here and i think it's the thing that's caused a lot of people confusion in this topic so let's undo that confusion model is this z does not equal v as a function of t over i as a function t z is this v tilde as a function of omega divided by i tilled as a function of omega it's the voltage as a function of frequency divided by the current as a function of frequency why have i got these little tildes there the little wiggles over the top of these variables and the answer is it's because i've done something to them uh and what i've done to them is i have furrier transformed them okay i'm no longer looking in the time domain i'm looking at them in the frequency domain what does that mean i suspect almost all of you would have at least heard of freya transforms before most of you would have done some for your transforms before and your undergrad and probably almost all of you would have forgotten how to do a free transform and that's fine we don't need to remember how to do them but we need to not be intimidated by them so this is the fourier transform equation and it does look quite intimidating do not panic we're not going to use it i'm just going to give you four results from it that will make our lives spectacularly easier and bring all of these elements together so the first result is that actually just to look at that first line for a second just from a notational perspective you can see that i've got squiggly f or curly f okay and that's me just saying what is the fourier transform of x of t some function x of time okay and to find that i would do the integral between plus or minus infinity of that function multiplied by this interesting thing e to the minus i omega t all with respect to dt and one way of writing the output you know i haven't worked it out but one way of writing down that i have applied a fourier transform to x of t is to write x tilde of omega okay that's just me saying to myself that x tilde is the free transform of x so it's nice to use the same symbol and you put a little tilt some people use capital letters instead so they use capital x it doesn't matter which one you do i quite like the tilt what is the fourier transform of the first time derivative of x okay the fourier transform of the first time derivative of x but it turns out if you sub it in to that equation there it is i omega times x tilt so something there's something quite profound there and i don't want to dwell too much on the furry a bit you know i think that'll be an individual preferences type thing where you might go away on your own and remind yourself about exactly what's going on but something quite profound that is going on there is that calculus has turned into multiplying by a coefficient so so differentiation in this case has become multiply your transformed function by i omega where omega is the frequency and i is the square root of minus one okay i is the the complex coefficient the complex unit so the fourier transform of the first time derivative of some function x is just i omega x tilde therefore unsurprisingly oh hold on there should be a little minus sign there just to keep everyone on their toes therefore the second time derivative of this function is minus omega squared x tilde and if you're struggling to see why that is well just differentiate the first derivative again i e just multiply by i omega again if you multiply by omega again i times i is just minus one and omega times omega is just omega squared so this is minus omega squared x tilt again very handy calculus traditionally you know a bit of a pain in the furrier world in the fourier domain very straightforward the third identity to keep in mind is that integration is exactly as easy as you might hope it would be it's just divide by i omega okay so differentiation is multiplied by omega integration is dividing by i omega and the last one is just a demonstration of the fact that the fourier transform is a linear operation so the fourier transform of coefficient a times function x plus coefficient b times function y is a times x tilde plus b times y tilt ie you can break up a you can break up that addition so it's just the fourier of ax plus the fourier b y and also you can take the coefficients out so it's a times the fourier of x plus b times the fourier of y so that linearity property is going to come in really handy as well it means you can break up your big ode into all its separate bits so we just need to keep those in mind those are going to come up those are going to be the explanation that links together the representations of impedance that you would have seen in papers to this mechanical physical ode type model that i showed you a few slides ago so let's have another go at our capacitor now that we've realized that the method that we showed first that we managed to succeed with the resistor on didn't work with the capacitor can we have a go at the capacitor with our new fourier knowledge so let's furious transform our system using our knowledge of linearity we just take all the three coefficients out and fourier transform each term separately and what we get is this okay uh we can see that i have kept q and turned it into q till to free a transformed version of q the coefficients all stay the same but i've now got i omega where q prime was and minus omega squared where q double prime was and my v has also turned into a v tilde so let's try again with our capacitor if i want a capacitor then i set l to zero i set r to zero and i set c minus one to c to the minus one okay plug it in and delete the terms that aren't relevant i just get c to the minus one multiplied by q tilde is v so once again i can say that my function v of t i'm going to choose to be v naught sine omega t but it doesn't actually matter at this point i don't need to put in the actual function i can just put in my v tilde and say well that is the function that i'm going to use okay and uh all i've done is move that c to the minus one i've multiplied both sides by c okay and i've you know you can see i've turned the fourier transform of v into just v tilt as you know to get the impedance i don't want the ratio of v and i i want firstly i don't want the ratio of v and q i want the ratio of v and i or more specifically v tilde over i tilt so if i want not q tilled but q prime tilled then as you know all i do is multiply by i omega okay all i do is multiply by i omega and that is the same as differentiation so my impedance is v tilde over i tilt or v tilde over q prime till and that's just the ratio of these things we've just found and what you end up with is one over c apologies a little typo there as well get rid of that c v the stimulating frequency was just uh voltage was just v but okay now hopefully lots of you will recognize the final right hand side of the final equation the impedance this notion this thing z of just a capacitor is one over i omega c okay one over i omega c now you're making progress now you've finally got like a piece of the puzzle that you couldn't access you you really can't access this without fourier transforms unless there are some ways to sort of fudge it that aren't really what's going on but now you are moving forward so you've got the sort of agreement of picard here actually we can make life even easier again okay this is just a capacitor take three okay really really hammering it to death now i find that well i think you can all agree that we'd rather work with currents than whatever a displaced charge is and so that's a very easy substitution right q prime becomes i q double prime is just i prime and q is the integral of i so not a very complicated thing to have done i'm just slightly reframing the discussion and once again i can do the fourier transforms of those that one derivative and that one integral and i end up with this uh it's the same equation but now i've fourier transformed it and once again if i want to solve for the capacitor so the same problem as i did last time then i just set the coefficients to what they need to be and quite straightforwardly most of the terms disappear the first two terms disappear and now all i've got to do to get z is just move that i over to the other side right divide both sides by i tilled so we could once again talk about the fact that our stimulating signal is v naught sine omega t but again why bother it doesn't matter we we know it's going to cancel out so it doesn't make any difference so that's an even faster way to get your uh impedance only w if you're working with i to begin with or the uh the frequent the fourier transform of um the current then you don't even need to do that differential or derivative stack again now you're really cooking now we've got to the point where actually we are on the cusp of being able to analyze absolutely any circuit i need to give you about three more pieces of information and you have all the maths needed for almost all circuits okay um it's possibly a slight exaggeration but not far away right all the classics so so far we've just analyzed in fact uh are there any questions so i just want to check right that's like a key little moment just there um so if anyone's got any questions they're welcome to ask out loud or put it in the chat i think one of the challenges will be that that as ever with with a network you've got a very diverse group of uh people watching so some will have lots of experience of impedance and some will have almost none but you're welcome to ask absolutely anything okay i've got the chat in front of me so i can see that either people are with me or too far away to ask a question so we press on okay so you've done a resistor and you've done a capacitor and you recognize the significance of being in the frequency domain i didn't mention the fact that why does the resistor work well the resistor is not a functional frequency anyway so you get away with it without moving to the frequency domain with a resistor but it would still work in the frequency domain of course what about two components in series well in series you just add them up okay so if i want r and c in series then here's my nice fourier transformed current equation and once again i'm going to pick my coefficients to solve the system energy then so put l to zero r to r c to the minus one to t to the minus one and here we go so i've just thrown away the first term because i set l to zero and at this point solving basically is just factorizing so you know i've just taken all the coefficients in front of the uh the fourier current right the i tilt um and then i just divide both sides by itild and i get that and there's always endless sort of algebraic rearrangement that you can do once you've got that so my impedance of a resistor and a capacitor in series is r plus 1 over i omega c that's it it is often quite helpful to rearrange things to be a single fraction that turns out to be quite helpful for certain scenarios so uh we can do that so uh you know i don't need to walk through every step of the algebra but that's not too hard to do and also it's often good practice to realize your denominator i e don't have any complex numbers on the bottom of your fraction and all i do there is multiply top and bottom by i and that that that solves that problem and then multiply by minus one as well to clean up okay so i now have an expression for the impedance of a series resistor and capacitor and what you can see is it's a function of omega that's one thing it's also a it's it's a it's got a real component and a complex component okay and as i'm sure many of you recognize already you know when we come to analyzing these things we know that we're dealing with complex numbers and we're going to want to plot them which we're going to talk about later today and so but it's not a function of time time has disappeared when we do our fourier transform time sort of evaporates out of it through the integration process so it's just a function of frequency it is a value parameterized in terms of frequency and it's a function of your various circuit components coefficients so this is like a little micro recap of all the steps we just walked through i hopefully convinced you of why we need to furrier transform our nice second order displaced charge ode but actually life gets a bit easier if instead of thinking about displaced charge you think about current so i just did a little substitution i equals q prime and because of linearity we can split it up into all these separate bits and because of the way that calculus works in fourier space we can actually just factorize it out so we can factorize i tild out of each of those bits and say that our new description of that of the first line if you will is that we've got the sum of three series components in this case multiplied by the fourier transform of the current equals the fourier transform of the voltage okay and clearly to get the impedance is just the thing in the bracket because i just would divide both sides by i tilt to get it now you probably recognize each of those terms in the bracket and that's because they are the sort of well uh well well-used or well-known um impedance formulations or sort of simplified forms of impedance formulations for those three classic idealized components are c and l okay so we've moved through that process and you you now have just some like lego bricks basically you now have three little bits of maths that plug together to give you a very wide range of circuits okay if you've got a resistor and it's in series with an inductor i can tell you straight off the the in the impedance of that circuit is going to be r plus j omega l i've switched from j to i or i to j apologies for that but if you see a little j it's the imaginary number as well it's the square root of minus one okay so and if i've got all those three things in series i just add all three together okay and i can put in any values of r c and l that i want so you now have all of series circuits in the bag done it's just adding those things together what about parallel circuits okay well actually not very difficult at all you need a few pieces of information one keep in mind ohm's law v equals ir just keep that in mind the next law you need is kirchhoff's law i.e uh currents in a closed loop sum to zero so you don't accumulate currents in a wire uh is another way to think about it what that tells you and you know is sort of move through this fairly quickly but what that tells you is that uh the voltage or the voltage drop across resistor one uh and across resistor 2 must be the same so i1 r1 equals i2 r2 and what that means is we can construct or what that allows us to do is use that information to construct an equivalent resistor req okay and what we want to know is if we know r1 and we know r2 and there are a pair of parallel resistors what would the value of req be well we know the value of i in the second circuit we know the value of i through this circuit which is the same as the value of i through once this thing comes back to being a single wire again before it branches it must be the sum of i1 and i2 okay because that's kirchhoff's law so we know i is the same here as it is here sort of definitionally and once you know that actually you just do a little bit of algebra um and what it turns out to what it results in is and many of you'll be familiar with this that it's this inverse sum of the inverse impedances or resistances so if you had the resistances one and two as your two resistors then your impedance of your system would be one over one plus one over two which is three over two all to the power of minus one which is two over three so two thirds okay um and that would make sense if you've got a one ohm and a two ohm your equivalent resistance is two-thirds of an ohm and the reason for that is that yes one ohm and two ohms are both quite big obstacles in your path but you've now got two paths so having those two paths even though they've got big obstacles is easier than just having one path on its own so there's a sort of intuitive sense to why it's it's an inverse sum of inverses and crucially this isn't just about resistance you notice that i didn't use the squiggly line for the resistor i used a little box i tricked you uh these are for any component okay and so this rule would still apply so it doesn't matter whether it's resistance capacitors inductors or whatever so now if you want an equivalent impedance for parallel uh arbitrary impeding components it's just this uh parallel resistance some people often call it but it's the inverse sum of inverses or the harmonic mean of the two so let's try and let's try and use that approach uh in our framing of the equivalent circuit for in our framing of a resistor and a capacitor in parallel with each other so i just drop straight in using that z equivalent the one over r plus one over i omega c to the power of minus one so it's a sort of a one over a one over and then the whole thing to the power of minus one and i multiply that by i tild and that equals v tilde and you know now now that we've got this nice uh fourier transform of current formulation of our impedance you know actually that was the impedance right you you only have to divide both sides by r to recover the the term you want so the thing that's to the power of -1 that whole object is the impedance of the system if you like rearranging algebra then fill your boots you can endlessly rearrange this thing okay here's a bunch of them so you know sometimes it's helpful and sometimes it's it's not um so if you want to make them one fraction then have both of them be over r so multiply the right hand side bit by r over r and then you've got that and then you flip it over and then again you want to realize your fractions if you've got this bit over here on the bottom it's got a full complex number 1 plus i omega rc and if you want to make that into a real number you can multiply top and bottom by its complex conjugate which would be 1 minus i omega rc and that would result in this thing over here and there is one more step that you might choose to do this so you know everything here each one of these equalities because i've written the equal sign they are exactly the same thing they're just me writing the same equation or writing the same expression in in five different ways you know and and hopefully not just to be a boar but also because it is quite helpful this last formulation here i've separated up in a very particular way okay notice they both got the same denominator of the fraction okay here but i've chopped the top into into two separate fractions one just gets this r and the second one i've taken also the minus i out and put it down here and then left the omega r squared c on top and what this formulation does is has a real component this is just real numbers and an imaginary component so all of this is imaginary and for those of you who've you know looked at nyquist spots before those plots are of the complex plane with a real axis and an imaginary axis so splitting it up like this is very handy and so that brings us to nyquist plots um and i think now is probably a good moment to have a pause as we've been going for 45 minutes um so has anyone got any questions you're welcome to shout out loud or or type it into the chat um before and we're going to take a sort of maybe even i don't know what or deal what would you recommend a sort of five or ten minute break at this point but really as you wish maybe we can do ten minutes yeah ten minutes okay i don't know yeah i think that that's that's what the lecture would be like normally so if anyone's got a question you're very welcome to ask at this point yeah about any of the steps we've gone through so far i really hope that i've managed to keep most of you with me to this point uh because that's actually most of the hard stuff done you've got these nice uh simplified expressions for each component each of the classic components and uh you know if you want them in series you just add them and if you want them in parallel you do the uh inverse sum of the inverses and and that's it um and that gets you you know such a wide range of circuits and now uh tractable to you okay so we're gonna we'll that that means i think we should probably reconvene uh i'll start again at one so let's let's reconvene again just before let's reconvene again just before one okay right i'm going to have a go at some drawing but before i do oh we had a question from sauron uh and i'm like how does the potential stat actually measure the real number of parts of impedance and but it looks like we've got an answer from apostolos i imagine the potential that is just measuring the impedance real part and reacting to measuring part in that case but would be happy to hear more from the lovely speaker that's very sweet excellent question um so full disclosure i am very much a theoretician and and have never actually touched a potential stat um but but i do have some idea of what's going on and so i've come back to this first slide excuse me and you have clearly your potential stat and this is obviously sort of diving way down the path a bit but your potential spat is measuring or taking measurements at a variety of frequencies and it does them usually not always actually but usually sequentially so it'll do let's measure the impedance at this frequency let's measure the impedance at this frequency okay so at any given moment in a conventional potential step it is injecting a sinusoidal voltage of known frequency and known amplitude okay now something crucial to bear in mind is that um the system starts from rest if you want right the system uh starts from not having a current at all and you go bang here's a here's a voltage off you go and it takes actually a while for uh this response current to settle down okay imagine that you are sort of what's a nice example i don't know like sort of wiggling wiggling a rope initially it's a bit chaotic it takes some time before it starts to fall into a fixed relationship with the motions of your driving force of your hand okay initially it sort of flaps all over the place and so your potential stat when it applies this sinusoidal voltage initially the current that comes out is all over the place so it has to apply enough cycles such that it believes that the current has settled down into its steady state in the frequency domain steady-state response okay and this is of course a sort of limiting step on in terms of how quickly you can measure the response um and you know as you know measuring a full impedance spectrum can take a long time and that's part of the reason why you have to do many cycles at any particular frequency in order to have uh sort of to minimize the um unsteady component of your um exponentially decaying ringing when you start the measurement that being said how does it measure it so so i'm fairly confident in saying that it doesn't do what you might imagine it does in terms of like a speaker so a speaker to measure you know frequencies in the air takes lots of little points dot dot of the pressure signal and plots out a um oh sorry a microphone plots out a sinusoid and very has very rapid fitting of that to extract but i think rather than doing that digitally with the signal you can instead do that with power electronics so you actually just directly infer the impedance from the power electronics setup in the box but i've never built one uh and so uh that could all be out the window by now maybe that's all old news and there's some clever way of doing it that i'm missing out on but hopefully that gives you some idea that it's like there's there's a lot to take into account here and actually the uncertainty in your impedance measurement i.e the expected error that you're going to find is a function of the frequency that you're measuring and the resistance of your system okay uh and so at different frequencies your potential that would prefer you to have different real components in order to accurately measure the the imaginary component so you really shouldn't just measure an impedance you really need to look at the machine and look at the handbook and it gives you a plot of the uncertainty as a function of uh the frequency of your measurements usually you report one appropriate to your discipline but if you're measuring something unusual it may be that the potential staff that you're using is simply not able to take that measurement certainly they all have very high um they all have cutoffs at certain high frequencies where they just are no longer able to record meaningful measurements okay so we're now on the ipad so apologies this is a bit more chaotic right so now we've got uh the knowledge to do components of the series and components in parallel we understand that the impedance is the ratio of the frequency domain uh voltage and current uh and i've corrected my typos and so now let's take that knowledge and get to the the point that you'll see in papers which is nyquist plots okay these are parametric plots what does it mean well often you'll be familiar with a plot that might be speed as a function of time so you put in a time and you get a speed these plots aren't quite like that these plots are parametrized in terms of omega and both of the axes are a function of omega okay so you could even imagine and that's actually some nice papers have this a three-dimensional graph where the third axis is omega and your impedance is a path through that 3d space which is nice so it's like a bow two different bow plots and a nyquist plot all stuck together so it's a parametric plot so every point you say okay this is omega therefore what is my x-axis coordinate what is my y-axis coordinate they are conventionally drawn with the negative axis inverted which is very irritating so so sorry the vertical axis negative um so it's flipped over vertically and a notational simplification or convention that the real component of the impedance z is often written as z prime that's not the derivative that's just saying just the real bit and the imaginary component of the impedance z is often written as z double prime and both of those things have the units of ohms although it's sometimes slightly uncomfortable to think of an imaginary number having a unit but they very much do but their interpretation is a bit subtle and so this graph has axis minus a double prime versus z prime okay and that is almost all impedance plots you'll ever see why is the vertical axis negative because most of the time the things that you're interested in are capacitances and they would have negative imaginary components so it saves us from always having to draw the axis at the top and go down instead we make it negative and draw up okay so it's just a useful convention so now it's time for us to have a go at some plotting and i'm sure any of you have been doing lectures or meetings during the pandemic have seen that this could be a bit of a pain but we will overcome so uh i want to draw the impedance spectrum of a resistor okay and i think it's important to do the obvious ones just so we don't skip over them i want to draw a graph this parametric plot that shows me all of the different values of z i e z prime and minus z double prime as we change omega the frequency for a circuit that just has a resistor can you please all just make sure that you've got a vision in your head for what this graph is going to look like what is this graph going to look like what is the impedance of just a resistor okay and now i will ruin the surprise and the graph looks like this all of the i don't know why that's a p all of the different frequencies all have the same value it doesn't doesn't change anything as you can see z subscript r the impedance of a resistor is just r it's not a function of omega so all the points sit on top of each other in this graph doesn't matter what your value of frequency is z has the same value and that is it's all real there are no i's in that formulation either so it just sits on the real axis it's got no imaginary component it's just r okay that's the easy one obviously what about just a capacitor okay what about just a capacitor well often we don't want to have imaginary numbers on the bottom of a fraction it can be a bit of a pain so instead we might rewrite this as if we multiply top and bottom by i we get i on the top and on the bottom we get minus omega c and so that's probably more conveniently written as minus i times brackets 1 over omega c okay and now you can immediately see what i'm talking about why do we want the vertical axis to be the negative of the imaginary component well because all of our capacitive bits are going to be negatively bits so how do we plot this what happens as the value of omega ah well here's another question oh doesn't matter what values of omega and c and we're going to assume that omega and c are both all positive real numbers your frequencies and your capacitances are all positive real numbers okay they can be zero as well okay so that's that's the first assumption which is absolutely how the universe seems to work that being true all values of this are going to be imaginary only there will be no real component of this so we can immediately say that all the impedance spectra will sit on the vertical axis it won't take any forays into the positive um real space okay as omega goes towards infinity so limb as omega goes towards infinity of z c is going to be well as omega gets very large that fraction is going to get very small so it's going to go towards zero and the limit as omega goes towards zero of z c is that this thing is gonna is gonna head off to uh minus i times infinity okay so i'm gonna draw a bunch of points on my graph here and it's often useful to annotate and say well when the impedance is zero that is when omega is going to infinity and when the impedance goes off up here that is when omega is going to zero so it's just a vertical line okay okay we haven't given inductors much love so let's do that now uh what's the graph of this thing look like try and talk yourself through it just now what happens when omega goes to um zero when omega equals zero z l equals zero okay fine we're here and that's omega equals zero and when omega goes to infinity this thing is going to go to i infinity complex infinity okay and that's positive so this thing is a another straight line on the vertical axis only but this time it goes down from the bottom and you can see that that for my capacitor zero starts at the top and goes oh sorry omega goes to zero at the top and to infinity at the bottom and the same thing is true with uh an inductor but they just start from different places if you know i mean so so still the the highest value if you want is is near the top of the graph but then it goes off as it gets to infinity as zombie goes infinity it goes down and that's useful to keep in mind okay so that's the sort of three classics the three trivial ones what about for a series combination of a resistor and a capacitor notice i've drawn a capacitor and then a resistor and this is something that's going to be really crucial we'll talk about a bit later but the spatial ordering of a series of components is irrelevant right components in series all experience the same current at the same time so the spatial ordering is in the relevance okay well we solved this problem already using our fourier series just before the break uh for a uh transform just before the break and we even rearranged it in a nice way such that it had a real bit and um imagine a bit okay so now how do we draw this one right well this thing just to write out even more explicitly is r minus i times one over omega oh see yeah that's an omega for the for anyone who's not sure it's r minus it's r minus i times one over omega c okay that means that the real component for for any value of omega or any value of c is always just r summaries at my door two seconds think about that graph get that graph going in your head what does it look like oh if it's one of you that just sent me flowers that was very quick very sweet okay right you've got the real bit is just r in all scenarios and the complex bit is this slightly more interesting function of omega 1 over omega c so okay what happens when omega goes to infinity it's this it's the same analysis that we're doing last time so as omega goes to infinity we just say well this thing it goes to r minus i multiplied by one over it's going to infinity it's going to go to zero so it's going to just go to r z equals r and as omega goes to zero z okay you this thing what's going to happen when it's going to be equals r minus sort of i infinity and you realize i'm using sort of slightly sloppy notation around limits but you'll you get my point okay uh well there's a question coming up okay fine it's an answer it's an answer from ed thanks ed okay and so we can we can now plot this already okay so here is r and as we increase our value of omega this negative imaginary component is just showing up so now we've got a vertical line over here and it's good to put on your frequency so this is as r goes to is so omega goes to infinity and this is as omega goes to zero okay so it's just a vertical line but it's been displaced okay and keep that in mind we've seen on these other two examples that r was just a point on the real axis c was a vertical line on the imaginary axis and the series combination of those two is this displaced vertical line so that actually gives us a clue for some future ones that we're going to do where if you add an r in series to something you just shift it along the horizontal axis you just shift it along the real axis by the value of r okay what about a parallel rc circuit okay parallel rc circuit um should be appearing on your screen but it's not appearing on mine okay here we go okay parallel rc second so we've done the maths already and i even just before the break finished by saying let's arrange it in this very particular way to have all the real bits here and all the negative complex bits here and i hope now you can see why we're bothering why are we bothered to do that okay so just like the other times let's start by analyzing key points so the key point we're going to analyze first is as omega goes to zero what's going on when omega's going to zero what have we got well the real bit just becomes r over one plus zero right r over one plus zero so it's just r and what happens to the imaginary bit minus i times zero over one plus zero zero over one zero so it just equals r so as your frequency goes towards zero your impedance of this quite complicated little circuit is just r just a real value part so here is r so that's one of the points and this can be omega goes to zero what about as omega goes to infinity our next most sort of obvious point to analyze to scrutinize what happens there okay and again apologies for the robust mathematicians amongst you i'm going to use quite informal approach to limits well the real component is r over one plus infinity squared so r divided by a very large number that the real part is going to go to zero okay minus i times what's going to happen to this imaginary part so as we go to infinity we get infinity times some stuff plus oh sorry divided by 1 plus infinity squared so you know again avoiding robustness for a second the denominator wins this battle it becomes one over omega and omega is going to infinity it becomes uh zero so actually the impedance of this thing that as frequency goes to infinity is zero okay so we've got two points so far and they're both just on the real axis okay what do we do next now there is different answers to that question depending on what you do but if you if you are you know not fussed about why then what you do is just write that equation directly into your favorite coding language and it will plot you the result okay but we can do a bit better than that today we're going to analyze one particular [Music] value of frequency and it's called the characteristic frequency in fact it's not goes to at all it's an equals there's one frequency here that is different from the others okay and that frequency is 1 over rc rc to the minus 1. 1 over rc why is that different from the others well if you sub in omega equals one over rc if you look at the equations most of the time omega is being multiplied by rc so rc multiplied by one over rc is just one okay so if we try and do this z would equal r over one plus one over rc times rc which is one squared one squared is one so one plus one is two so r over 2 is the real component minus i times and again same analysis here so omega is 1 over rc so this is r squared c over rc so that's just r on the top what about the bottom well it's the same as the other one two okay so we've now got this particular point it's r over 2 which is here and it's also r over 2 here r over two okay so we've got this point here and we can do this and shall we well you can investigate as many points as you wish right but what you will see is that this thing is actually a beautiful little slightly skewy semicircle okay and uh that's very satisfying and so actually you know if you crush down out of the impedance maths and just turn this into sort of regular graphing this is just the parametric equation for a circle displaced from the origin okay that's where these sort of squared bits come from and so if you had negative frequencies you could complete the other side of the circle but you don't so this is as good goes to infinity and this is when omega equals r c to the minus one and we call that point you know you can see it's like it's it's the most unreal it's the highest value of the imaginary component it's the characteristic frequency this is omega subscript c the character frequency or you could also use tau which is the time constant the characteristic time constant and frequency and time have this inverse relationship so tau is just equal to rc and if you look at rc the units of resistance ohms and units of capacitance farads you multiply those together you do just get seconds i promise you do so it's quite that's quite a fun little exercise to work through okay so so by working through this like second order heterogeneous od analysis we fourier transform it we frame it in terms of currents we analyze it we understand kirchhoff's law so that we can do parallel systems we work all that together and we end up with a formulation that if we spit it into real and imaginary actually drawing the graphs is quite straightforward um and often there's a bit of a trick you know not so much a trick but you have to remember to look for the um characteristic frequencies but even just by looking at the equations now that you've seen one hopefully they'll seem relatively obvious to you you know quite nice to see that omega times rc all right how do we get how do we simplify that make omega 1 over rc what could be easier okay so if you measure the impedance spectrum of this thing you'd get points like this all along your semicircle okay any any questions at that point characteristic frequencies why frequencies should always be included on the plot that is how you see capacitance and that's that i need to stress that point let's go back to this graph there's such an important point here and i hope it would like exactly what the electrochemical network is for hopefully is part of this which is you must report frequencies on your nyquist plots okay otherwise you've got no idea what your capacitances are when you look at them for example with this plot okay if we look at this plot here it doesn't matter what my value of c is you will always have the graph that is a semi-circle between 0 and r for any value of c okay only by showing the frequencies do you show your value of c you know when you look at the way this is formulated well it's just r and and r over two right it's only by reporting omega c or tau c or just reporting your log space frequencies on your uh diagram can the reader have any insight into what the capacities capacitances are and that i think that's so important and so often you'll look at a paper especially as a reviewer and be like i don't know what's going on here because i've got no insight into what kind of processes you'd be measuring because you haven't reported your frequencies okay um yes right okay so we're sort of almost at the home street now in fact we are this is the home street welcome you've made it to the home stream i've made i i love this topic as you can probably gather so i've made some animations that show the circuit that you can see in this image and so this is the keyboard i really want to draw dwell on and i really want you guys to have like a predictive feel for what's about to happen look at the circuit i've shown it's a resistor followed by two parallel rcs okay i've given a value for each component half an ohm two ohms one ohm one farad and uh two thousandth of a pound okay i've drawn the nyquist plot for that circuit and you can see the legend in the top okay my green line is the actual uh response of this circuit and the distance between each little cross in my green line is is sort of root two um uh root two log spaces so root two omega okay so this is it's a it's a geometric series it's multiplied by root two every time okay i've also drawn a green circle around one particular point and that point is one hertz okay and in addition i've drawn the semicircles for the first rc and the second rc rc1 and rc2 now a bunch of things to remember one is i told you that the order of components in series is irrelevant to the plotting of impedance spectrum so i could put the shift the blue parallel pair all the way to the left have the resistor alone in the middle and then have the red ones on the right we wouldn't change the diagram really important to remember that okay some people might look at this and say hang on why is the red one first then the blue one in the circuit but the blue one first and then the red one in the nyquist plot i hope you could answer that question now okay what's going on here so if we look at and the answer that question is characteristic frequencies or characteristic time constants so notice i've also reported the characteristic frequencies of those two parallel rc pairs um that are in series with each other and i've converted those into towels so parallel times so you know the time constant of the red one is two times one two seconds and the time constant of the blue one is one times point zero zero zero five so point zero five seconds okay so there's a lot of information taken there and the last thing to really make sure that you're very comfortable with is look at the frequencies omega goes to infinity on the left and only goes to zero on the right okay we've got two semi-circles and and look at the point where they meet right and the first thing i suppose the other thing to know is this point here it should be no surprise to you that this point here is at 0.5 because that is squarely the result of that resistor there okay so it's just shifted everything to the to the right another piece of analogy whether it's springs or water wheels or whatever i like the water wheel one personally okay if you've got a very big capacitance if you've got a really big but it's like a very soft elastic membrane in the way you don't even you don't even notice it right okay and if you've got a very small capacitance it's like a very stiff membrane in the way that's one thing to think about the next thing i appreciate i'm really laboring the point here but i'm about to make this come to life so i'm about to animate this diagram within a very particular way and i really would love you to preempt what it's about to do i want you to guess correctly what an earth is about to happen in this diagram the next thing to consider is that um why is the blue on the left and the red on the right well the red has a higher time constant or a lower frequency constant characteristic frequency than the blue that's that's what's going on here so what's going on as we have when we have a uh very high frequency resist sorry capacitors are like short circuits at a very high frequency they're just like a wire capacitors don't care at all about a very high like a spring if you just want to go backwards and forwards very very quickly the spring doesn't care about that at all neither does the little membrane in the pipe neither does a capacitor if you want to go backwards and forwards very very quickly it's invisible it becomes a wire okay so at our highest frequencies this circuit becomes so as omega goes to infinity this circuit becomes a resistor of 0.5 ohms and that is what's going on at very very low frequencies your spring or your water pipe membrane or your capacitor that they've been stretched to their limits because your sinusoid is changing so slowly that from its perspective it's like oh you've been pushing in the same direction for ages okay so in where as you go to very low frequencies and of course at zero frequency is now just to steady state it's dc not ac as you go to very very low frequencies your capacitor becomes a block it stops you going through that path so it becomes an open circuit so as omega goes to zero your circuit becomes three series resistors okay 0.5 2 and 1. and that's what you see that's this point here 3.5 as you get to zero frequency it's only the frequencies in between and of course everything is in between zero and one half of all the things are in between zero and infinity where something interesting happens okay and that is thing only interesting only interesting things only happen when you are around the characteristic frequencies of the behaviors and that's the whole point of impedance you're investigating the processes by having some insight into their characteristic frequencies okay and so as it stands these two processes have quite significant separation between their two characteristic frequencies right or time constants either way right two and 0.005 there's a big gap between which means that as you start from very very high frequency very very high frequency and you you just you just see one resistor as you slow that frequency down you first start to experience a bit of resistance from the the blue is the blue capacitor the blue capacitor starts to say whoa you're going a bit slow here just a bit slow for my liking here so you start to see some current having to go through the blue resistor and then uh still the red capacitor is like whoa this everything's so fast i don't care i'm just a wire and then as you go slower still the red capacitor starts to say whoa that's a bit slow you're gonna have to send some of that current through the resistor because i you know i'm not going to let everything through and so you start to experience the red resistance okay that's what's going on that's why you've got two separated processes and that's only because they're only separated because the two time constants are very far apart but if the two time constants got a bit closer to each other those two moments when the capacitors say oh i'm now aware of like there's a process going on here they start to overlap okay and they don't overlap the semi-circles but they blur what's going on what they blur which processes is which process is responsible for which response okay so what i'm about to do that was a long buildup but what i'm about to do is um change one value i'm gonna start to slowly increase by a factor of two each time i think every half second i'll multiply by two the parameter c2 okay i'm just going to change c2 everything else stays constant and i want you to try and predict what's going to happen so i'm going to keep increasing c2 eventually i'm going to increase c2 so much that its value will be um two farads and when the value is two fads tau two will also have the value of two seconds just like tau one even though they've got different resistor capacitor combinations the product of them two times one and one times two gives you the same time constant they'll have the same time constant and you won't be able to tell that there are even two processes going on at all you will have no way to know okay so hopefully this animation works and i want you to watch this process only thing i'm changing is c2 do you see that moment there where the time constants were the same okay and now c2 continues to increase and then i'm going to start producing it here we go we're going to go in the opposite area i'm reducing c2 by a factor of 2 each time you see the time constants moving towards each other and then bang they're the same so those semi circles are kind of only to guide your eye as a reader about the fact that there are two processes okay those two semicircles in many electrochemical measurements would be one of these mushy you know two slightly blurred together semicircles the semicircles themselves never overlap but i imagine a kind of like there's a sheet over the top of them and you kind of sort of blow a puff of air under that sheet and it sort of starts to lift the actual response which is the green line off those two semicircles and crucially at the point where they have the same time constant it looks like one big semicircle and so if you've got two processes let's say one's a bit of kinetics and one's a bit of transport and they happen to have the same time constant very difficult to have any insight into that in impedance so you better hope they don't have the same aurinous behavior and then and then you can change the temperature and measure it again okay so i hope you enjoyed that that's that's the sort of the what we've been building to that was the kind of cool bit there is another component which i'll talk about in a second called cpes but basically the logic is all the same in fact yeah let's talk about that there's this component called the cpe a constant phase element and a constant phase element has this formulation so you know imagine this is the thing that would go in front of the current term and it's got two parameters all the other components we've seen so far had one parameter you know r resistance c capacitance l uh um inductance this thing's got two parameters it's got q and alpha okay and it's slightly weird and it gets it's quite a contentious uh it's quite a source of contention in the literature because just by having two parameters it becomes quite flexible and maybe too flexible so let's just think about this as alpha goes to zero when alpha is zero z c p e p e goes to what well it's 1 over q naught uh i omega to the power of 0 is is one anything to the power of zero is one so it's just you know times one so this is just this which you can think so so when alpha is zero when you're your exponent term if your frequency is zero you just have um a resistor it's just got a real value you can think of this as just a resistor okay what about when alpha is one okay z c p e well this would then be one over q zero i omega well that's just like a capacitor okay that's just like one over i omega c so a cpe can recreate two of our favorite components one resistor the other one a capacitor okay but it can also do stuff in between okay and so you could have alpha as anything between these values and let's look at one particular one alpha equals not point 0.5 okay what happens there when it's 1 over q naught i oh well the square root of i omega right if it's not 0.5 okay what's that well what does it look like on a plot and it looks like this it looks like a diagonal line so it's got a constant phase relationship which is why it's called a constant phase because you're this is your phase angle here okay and of course each of the values of alpha between 0 and 1 give you a different angle of that line alpha equals 0.5 one way to think about this fyi and you know you can look at the frequency relationship as frequency gets very big this thing goes small so this is omega goes to infinity and this is omega goes to zero um and okay what else well this thing is quite fun because it's at each different frequency you get some more as your frequency is changing you get a bit more capacity but you also have to pay a resistive cost for that so it's a combination of a resistor and a capacitor in a sense you you do store energy in the system but you pay to store that energy with some resistance um and people say they don't have a physical meaning none of them have a physical meaning all of the components are idealized none of the idealized components can be built well maybe a superconductor has zero resistance but you see my point they're all idealizations so there's no worries about getting particularly stressed about a cpe being made up because they're all made up okay it's just what you do with them that's that's interesting and how you choose to interpret them in terms of mechanisms when alfred equals 0.5 some of you might be aware that a 45 degree slope is often associated with diffusive type processes and there's a there's a lot of maths uh that uh i could talk about another day that explains why that is and so uh this is a warburg diffusion element with a semi-infinite boundary condition okay there are three types of warburg warburgs that are a finite length of material that you've got to diffuse through warburgs that are a fine length material that is blocked at one end and so that you get in but it gets full after a while and warburgs that are semi-infant alfreque was 0.5 for the cpe that is a semi-infinite warburg and that's a bit more involved than we're going to go into today but let's go back to that uh animation so you tend to draw cpes as two little slightly crooked like two arrows although they don't have a directionality but two arrows and here i've just chosen some arbitrary values i've said you know uh i've set q naught to 1 and 0.039 and i've set alpha to 0.8 and 0.4 and i've kept all the r's the same as the last example and you can calculate time constants for for cpes even though they've got a weird uh power next to their frequency you just undo that power so the time constant of a cpe is just r q to the power of one over alpha it's the alpha root of r q okay let's watch it go so all i'm doing is just changing the q coefficient of the second cpe okay still the time constants come back and sit when the time constants are separated it's two separate processes but as the time constants start to move back together again they turn into one weird process look it's sort of inside the red semicircle for a bit very strange and the initial angle so if you look at where the let's go back to the non-animated version the initial angle for example here the tangent of the point here that is uh corresponds to this alpha thing here where the flat line will be zero and a vertical line will be one and so naught point four is you know this is 0.5 and so 0.4 is here which is what you see and this one is oops sort of not 0.8 so that's here and so you can often think about it as a people call it a depressed semicircle which i think is a rather disheartening name but it is a semicircle that has been not squished but pushed so that its center is below the horizontal axis and so the tangent to it at the intersection of the horizontal axis is uh corresponds to the value of alpha when you normalize to one and zero okay let's watch that again because i really love these okay so you get this blend of processing there's so much going on there if you want to interpret that spectrum it's very very sort of uh confusing to try and work out what would be going on at one of those intermediate systems it looks like you've got all kinds of things okay [Music] there is a few so that is basically it that's as far as i want to go today but all i want to say in the last few minutes is just a few cool concepts that you might be interested in hearing more about on another day distribution of relaxation times known to its friends as d r t is instead of having a graph of uh i uh z double prime versus z prime you have a graph of gamma versus tau where tau is just time but like a normalized time time constant time and and gamma is well and instead of having um uh this parametric plot you've now got a conventional plot so you're just going along and different processes will occur at different time constants and it's very tempting because it's like unlike a nyquist plot which is a sort of model of semicircles this thing is like ah there is a process here this time there is a process here at this time and there is a process here at this time sounds good doesn't work in practice very often because you need incredible uh frequency resolution in order to extract drt spectrum with any confidence and you have to do all kinds of things like ticking off regularization to smooth out your curves of all the bits that you haven't measured and your noise propagates through terribly so it is lovely in principle and they are lovely if you're a theoretician and have infinite resolution but actually in practice they don't necessarily offer you any more insight what they risk doing is offering you false insight because you have to apply some interpolation to get your eis data into drt data and that can sometimes make you think you've seen a thing that you have not there are often equivalent circuits and so equivalent equivalent circuits and so you can recreate the behavior of that 45 degree cpe element that i showed you with an infinite series of resistors and capacitors in this in this configuration there's almost always another way to think about a circuit um and that applies to drt in fact yeah and that's a really interesting topic where you could you could think about this uh the the framing of drt as and this is going to take too long to draw but um resistor capacitor resistor capacitor or capacitor resistor okay and you just have if you have lots of these and they all stuck together in an infinitely long chain which would take too long to draw what you can imagine on the nyquist plot is this has got little tiny delta r little tiny delta r little tiny delta r this is c one c two c three c four if c one c two three three three four are all really far apart from each other they're very different values of capacitance but are delta r is always the same it's some small amount of r then your impedance response would just be semi-circle same signal same cycle semicircle all the way lots of little delta r's each one as you move through the frequencies experiences the next component not in their order although we often put them in time order but as those values of c if all of the values of c were the same then you would get this spectrum and the concept is that of course depending on those little values of c and depending on very small values of delta r you could create almost any weird thing you wanted in the envelope between many separate circles and one big circle you could almost recreate anything it has to it has to a base certain rules and so one way of formulating your drt analysis is to think of it as an infinite c or a very long series but finite actually of of little delta r parallel rc components which is quite common uh gary sure you often see them and they sort of look like a hedgehog okay and this is uh i've showed you warver which i mentioned was the solution to a diffusion equation garisher is a solution to a reaction diffusion equation so not only are you perhaps moving through an electrode but you're also reacting to the surface and disappearing into it and that's that's that's beyond what we want to talk about today but you'll see these more elaborate components pop up and they are often um physical physically motivated so think about a particular set of equations and some mathematician would have said oh i can solve that set of equations and give you an analytical solution and that is worth mentioning that you know what's the point of eis in general and equivalent circuits is because they all have analytical solutions and so analysis of them and fitting of the data is really straightforward um a more general notion of equivalence that i'd like you to think about although we won't talk about now because we're running out of time a resistor is just a parallel rc pair where the uh c has a value of infinity and a capacitor is just a parallel rc pair where r has gone to infinity every circuit you could think of if you put an r in parallel to that whole circuit and put that r off to infinity you've changed nothing right as r goes to infinity it's like having an open an open circuit okay so that sounds like a silly thing to say and it partly is but it's nice to think of for example here we've got what we saw before so we've got a parallel rc okay and this is the value of r and this is the value of c for some arbitrary value of r we get a semicircle like this if we increase r we get a semicircle like this if we increase r again we get a semicircle like this if we keep increasing r would eventually end up with a giant semi-circle and if we tend send r off to infinity we won't be able to tell whether we're on a vertical straight line which is just what a capacitor is or a sort of tangent to an infinitely wide circle so there's this notion of equivalence that there's always other ways to represent things and make components redundant and you're looking for the sort of the least redundant version of a system like a kind of occam's razor approach where you're not over complicated things in fact you're battling to try and keep it as simple as possible and then lastly in the dying few minutes just thought i'd share some of my own research um so this is a paper called simulated impedance of diffusion in porous media and this is just i wrote a little numerical solver that models diffusive transport in arbitrary spaces and then gives you back the impedance spectra and the sort of the the conclusion of that research is you know in this second graph here the the dense black line with no points on it that's the analytical warburg diffusive finite length behavior each of those colored lines is from one of the little simple 2d systems above it and the point is they look really different from the dense black line and so if you are looking at it porous media maybe the reason your impedance spectrum is a bit uh jaunty is not because you've got some fabulously interesting electrochemical process going on maybe it's just because it's a funny shape okay and you know here we've got two more examples of that and top the inside of that graph is a little distribution of relaxation times analysis of the same data so it's it's the same compliment and this is a porous media that either goes from dense to less dense or vice versa and also whether it's finite length which is blocked or finite oh sorry which is open or finite space which is rot okay all more detail than we've talked about today but might be familiar language to some of you who've been looking at this topic and then a more recent paper um called the electro tortuosity factor why conventional tortuosity factor is not well suited to quantifying pac transport in porous lithium-ion battery electrodes and what to use instead this is another numerical solver in 3d spaces where you try and understand uh migrational transport through a poor network that concludes in um a charging a capacitive process at the surface which is a lot like what you do when you're trying to characterize batteries um and we show that you know it matters whether your dense layer is near the top or near the bottom which is no surprise to anyone but but contradicts a lot of the modelling work that's done a little scary fact just to keep you on your toes that the conductivity of copper for example drops by 30 percent if you introduce point one percent of aluminium into it so you know when someone says i've got a copper wire you need to ask them yes but you know which particular copper one um uh one thing that's come up quite a lot in in fibers and discussions is that if you make a symmetrical cell so you you sell of any kind but you know often a fuel cell you make a symmetrical cell so you put the same electrode on both sides and then you do impedance science at any given moment one of your cells is the anode of one and the cathode right and that keeps reversing which means one of them's doing the processes that it wasn't really designed for doing one of them is doing sort of the opposite of what it would like to be doing it maybe it would like to be reducing oxygen but right now it's produ it's you know producing oxygen um and so if you analyze your impedance of a symmetrical cell you need to ask yourself which side any given moment is my response dominant from and am i over interpreting what i'm seeing here um and that is is the end um so thank you very much for your attention and this is a lovely photo from from the past when we all met up uh of of the extended electrochemical science and engineering group although it's a few years ago now if you've got any questions about impedance i love talking about it so uh please feel free to get in touch you've got my email address there um and thanks to evan cortiles pengnan and anna for inviting me