A philosophy of circuits and systems in biomedical engineering, or, Love in the time of COVID-19

Love is what will keep you going. Sometimes the other reasons you have for doing something – money, obligation, a sense of purpose – will all dry up, and you will have only the fount of love from which to draw to quench that thirst, that need for your next heartbeat, that next breath, that next wish to see the new day. Love is that wellspring from which you truly drink. When considering why that next cup of water, sure consider the hemodynamic (in)stability of dehydration sure, but consider also love. 
 
To speak on the topic of love is perhaps superfluous in an engineering course. We came here to study biomedical circuitry, systems capable of measurement in clinical settings, to learn the many jagged borders of the ever-ragged realms teamwork, the units we shall ever assemble in. Excepting those extraordinary circumstances, and even then, we did this with aplomb. We started with a humble resistance to a potential and we have ended with the measurement of biological potentials – the foundations of many modern medical devices. And we did this best when we gave ourselves to the knowing of and the sharing of what we had. Study buddies. In class teams. That shared glance of those of those who have been there. Be it charity, be it comradery, it must at times be love that compels us.
 
I write these philosophies to express a love for the subject. I suspect many of you sit through them for a similar love, whether it be only in the quiet of your heart, the stillness of your mind. I suspect many at this point sit through the lectures because in a sense we must, we are compelled. If not strictly by the subject matter then by at least by one thing whispering in our ear, keep going. Nurture that – that love – it will ever replenish you. Without it, we can’t go on.
 
People, as it turns out, are necessary for that love. No one loves anyone who only loves themself. All loves are held by many hands. And now those hands must be sanitized and really we aren’t doing the whole hand shaking thing anymore and we probably shouldn’t even be leaving our homes unless you swear adamantly it’s absolutely, crucially vital to the operations of the critical infrastructure of this very nation. And it turns out when you can’t reach out and hold others or the thoughts of others, those feelings of love can dissipate. My mother weeps because she can’t hold the hands of elementary school children she teaches. The many little moments we can’t have right now.
 
Distance causes heartbreak, separation causes heartache, containment causes heartnumbness. It’s all too easy to stop caring and stop caring that you stopped caring when there is a pane of glass between you and everything you interact with. I have heard descriptions of depression describe in similar veins. Love lost – a sealed bottle tossed by wide seas – to wash ashore? To be received by whom? Found? Sought?
 
The drop-off in attendance and attention was precipitous. We lost half of ourselves when we lost each other. We all feel that missing half. Whether you’ve said it aloud to yourself, you probably miss some of us. And whether we’ve said it aloud to ourselves, we’ve probably missed you too. We are missing a significant portion of who we are when we are not out with others. And as of late we have not been out. And we have not been with others. And we haven’t been to any places we mustn’t. And that’s one of the worst ways to go about learning.
 
Learning is about exploring those places we mustn’t (“out with others”). Earth on its axis, axis about the sun. Ancestral descent, a whole lot of history. Energy in various forms, thoughts on how to use it. Once never dreamt, now soundly known. Knowledge acquired step by step in a few right directions, a whole lot of wrong directions, and the ability to discern the right from the wrong. Learning requires some stumbling and bumbling in the dark sometimes. It’s okay. It’s part of the process. When you find a good source of light, use it.
 
Knowing by finding is a one way to define learning. And it requires exploration. And exploration requires freedom. Freedom of movement, of communication, of self-in-environment. It is what is allowed on these hallowed academic grounds so well. If you aren’t free to consider all thoughts out there, in here, where can you? 
 
Hence, the beginning of this class made frequent and heavy use of group-based knowledge acquisition. We were working on worksheets in small groups, presenting our results, getting into lab groups. The point was to show you the people around you can aid in your learning and you can aid in theirs. It was meant to foster your sense of community as a biomedical engineering student here at the University of Michigan. To help you through our program. To build you up into a good engineer who will do this every day, excepting those very extraordinary circumstances. And even then… 
 
And while the format of the class changed halfway through – just as we were about to dive headlong into a whole community-based other aspect of this course – I think it is fair to say that we learned a great deal in this class. To summarize
 
  • In the first lecture,
    • E. C. said we “walked […] through the syllabus and [got] important suggestions and notes on how to succeed this semester” and
    • B. M. said “we will learn circuits and linear systems”, “will cover all fundamental circuit components and how to derive and characterize functions from them”, “a lot of this will be motivated by biomedical examples, “electrical circuits are everywhere”.
  • In the second lecture,
    • J. R. noted this “section of class focuses on describing the basic elements to be used in circuit analysis along with equivalent impedance”.
  • In lecture 3
    • I. D. relayed the fact that “impedance is the opposition of current, Z = R + jX. Equivalent impedance of circuit elements can be derived from Ohm’s law” and 
    • J. S. in his immaculately LaTeX set notes said: “We began our study of nodal analysis by learning the fundamental theorem of network topology and applying the definitions established there to the calculation of equivalent impedances. We then learned Kirchhoff’s current and voltage laws, analogs of the laws of conservation of mass and energy for circuits.”
  • In lecture 4
    • K. B. said “We discussed applications of Ohm’s Law and Kirchoff’s Voltage Law to evaluate circuit configurations and analyze nodes”.
  • In lecture 5
    • Y. L. gave procedures for nodal analysis, “1. Select a node as a ref, assign nodal voltages everywhere else, 2. Apply KCL at each node, and 3. Solve simultaneous equations” and mesh analysis “for each mesh, assign a mesh current, 2. Apply KVL to each mesh, 3. Solve the resulting equations”.
    • C. D. showed us examples of everything from “nodal analysis w/ controlled sources”, to “mesh analysis w/ independent sources”, even supermeshes.
  • In lecture 6
    • We as a class came together to solve Homework I more correctly and more efficiently than any one of us could alone. Thus providing yet another example of one of the central tenets of this philosophy: many hands make for quick work.
  • In lecture 7
    • H. K. told us how “We applied the Thevenin and Norton theorems [to transform] circuits. The Thevenin theorem took a complex circuit between two terminals and represented it as a resistor in series with a voltage source. Norton theorem took a complex section of a circuits between two terminals and represented it as a current source in parallel with a current source.”
    • J. O. outlined “a step-by-step procedure: 1. Perform two of these – determine the open circuit voltage, determine the short circuit voltage, zero the independent sources and find the resistance – 2. Use the equation Voc = isc*R to compute remaining value. 3. Thevein, 4. Norton”.
    • Rory M. told us how  “the voltage (or current) through an element (in a linear circuit) is the algebraic sum of the voltages across (or the current through) that element due to each independent source”.
  • In lecture 8
    • P. R. provided “the answers to Worksheet Lecture 8, Theorems and Transformations II, discussed in class on February 5th, 2020. The first three problems served as review, and the last three are more difficult problems that could appear on an exam. Questions were retyped to make the answers more clear.”
  • In lecture 9
    • B. R. said “We learned about the basic circuit within an operational amplifier as well as rules and conveniences associated with op-amps. We then learned about several different kinds of amplifiers” and
    • S. S. said “we went over some basic examples of operational amplifiers (comparators, voltage followers, inverting amplifiers, summing inverting amplifiers, noninverting amplifiers, differential amplifiers, instrumentation amplifiers, differentiators, and integrators)”.
  • In lecture 10
    • K. W. said “Today was a worksheet day, where we just did the worksheet in class” – O, how we long for such days.
  • In lecture 11
    • A. B. presented “a long derivation of a simple model” starting from images of the system and seamlessly flowing into math describing it
    • J. T. said “these notes are dedicated to exploring and examin[in]g bioimpedance”, “graphs were also made to convey our results graphically”
  • In lecture 12 we worked in small groups to reexamine our exams and understood more thereby Of course, little did we know then, this was also one of the last times we would ever meet. One of the last times we’d get to help each other. 
  • In lecture 13
    • E. F. said “We analyzed source free RC and RL circuits. Conditions can be analyzed in order to solve for responses of the system”
    • Y. C. had a handwriting and organizational manner that made it clear how complete response is the sums of natural and forced responses and transient and steady state responses.
  • In lecture 14
    • D. M. said “Initially, we finished up last lecture’s topic concerning first-order ODE’s in an op-amp circuits. Then we got into second-order ODE’s”
    • E. D. claimed “It started w/ an introduction to second order systems. Then, we talked about the extreme cases. Finally, we discussed source-free series RLC circuits and the 3 important cases (overdamped, critically damped, and underdamped).” – subtle use of red to highlight important results
  • In lecture 15
    • E. P. getting straight to the point said in this lecture there was an “Introduction to the Laplace transform and s domain here is a table of useful equations” getting a jump of this last homework of ours
  • In lecture 16
    • N. S. gave us the “General Form of the Transfer Function” and the time-dependent behavior of each of the four forms of BIBO stable systems
  • In lecture 17
    • M. D. stated that “Convolution shows how multiple inputs are [transformed] based on the transfer function to give the outputs”
    • G. A. noted that this was “A mathematical operation of two functions that will produce a third function that expresses the shape of one is modified by the other or as shown in the notes below how they affect each other.”
    • D. W. said “We learned about the pulse and impulse equations and how any signal can be represented as an infinite sum of shifted and scaled impulse. […] The main point […] is that of the effect of any linear time (or shift) invariant system on any arbitrary signal is the convolution of the input signal with the system’s impulse response function”
  • In lecture 18
    • We learned about the relationship between inputs and outputs of systems and how they be understood in both the s-domain and the frequency domain.
  • In lecture 19
    • W. C. documents it as being “all about filters. In the beginning we quickly went over passive filtering, focusing on a graphical understanding respective to frequency. We then moved onto a building block of active filtering, inverting amplifiers. We realized that we could write a lot of the other types of op. amps. that we learned initially by thinking about impendance (helpful in the s-domain).”
    • M. O. notes that “High and Low pass filters can be combined”, that “Active Filters can be created by adding circuit elements that give way to a gain (amplification)”, that “Band Pass gain can be generalized for”, and “Transfer functions can be converted to bode plots)”
  • In lecture 20
    • A. R. shows how “we moved from considering open loops to considering closed loops” with “The Key difference is that closed loops attempt to achieve a desired output by comparing its current output to [a] desired value and adjusting based on the difference, which we will call error.”
    • R. E. J. relays that “we ideally want the system transfer function to be equal to one (X = Y). However, at times this situation is not possible and we must consider certain trade-offs or change the system”
  • In lecture 21
    • M. L. said “we explored where electricity in the body comes from and how this electrical activity is measured in various parts of the body”
    • E. B. said “These sources of electricity include the sodium/potassium pump, the capacitor-like properties of the membrane itself, the potassium channel, the sodium channel, and the leakage channel. […] The capacitance of the membrane governs this response. […] Ultimately, the membrane potential is described as [a combination of] conductance and […] Nernst potential […] of each of the ion channels”
    • S. S. said “A generalized biopotential amplifier consists of [a] Pre-amplifier, Two voltage buffers connected to a differential amplifier, [and] Driver amplifier, High pass and low pass filters”
    • E. P. shared “how an ECG measures signal from the heart. We split the wave up into different parts understanding […] each portion represents”
  • In lecture 22
    • P. S. reports of a “demonstrate[ion of[ the capabilities of function generators and oscilloscopes, as well as [a] display how a breadboard can be hooked up to produce a desired output. We started by looking at the Agilent 3320A function generator, learning how to change different aspects of the wave which it outputs. We visualized the output of the function generator using the Tektronix TDS 2012C oscilloscope. Following this, we saw how the SRS Model SR560 could filter input based on parameters like high pass, low pass, cutoffs, and gain. After demonstrations as to how these machines worked, we imitated an output of the function generator and preamplifier using the AD620 and LM 741 operational amplifiers, resistors, capacitors, and a power supply. Finally, we saw that it is possible to hook yourself up to an oscilloscope, and – given the proper equipment – you can visualize your heartbeat.” Something you, yourselves, were all meant to achieve during the lab portion of the class.
  • In lecture 23
    • A. K. remarked that “Today was a review day for the exam […] We did an example involving finding poles and zeros, interpreting dampedness, and sketching s-planes and bode plots. We then did a review on bode plots and dampedness, and sketching s-planes and bodes plots, […] Next we did a biopotential example of designing and bandpass filter given specific constraints.”
    • L. W. tidily concluded “The behavior of a system can be characterized by looking at the equation in the Laplace domain. Setting the numerator equal to zero gives us the zeros, where the function equals zero, while setting the denominator equal to zero lets us solve for the poles, where the function approaches infinity. Negative poles decay away to zero as t \rightarrow \infty, giving us a stable system.”
  • And here in the final lecture
    • We learn it all again.
Please understand, this is a review session infused with a few final thoughts. While I’m happy to have you consider all my deep and profound thoughts of love, what is more important here is to have a general structure of the system we’ve erected in mind. How does all of this fit together? How does it work together? What is its response, its stability, its output to a given input? These things matter. Dare I say to both class and love. Learn to see it with soft eyes. Take it all in.
 
The more fully you see, the more fully you know, the more fully you love. I am sorry we did not learn more, but I am grateful to all efforts of education.
 
In a circuits class you should pop a few capacitors, learn to measure components with a multimeter, learn to troubleshoot with an scope and known probe points. In a systems class you must work in teams to see the larger picture. Both require resources, people, and patience. We have the resources to offer you, whole labs full of oscilloscopes and function generators and circuit components and I would on any given day – excepting those extraordinary circumstances – give you free reign to learn by pushing any button you see before you. We had the people to help you through the thicket of even the trickiest problems because with enough eyes you’ve seen enough things. We need the patience to understand that we could not use them this semester to aid in our learning. Damn those exceptionally extraordinary circumstances.
 
See the effects of solitary confinement. O, how the caged birds sing. O, the Home of the Brave when confined to the Homes of the Brave. It is a shame how things wen, are going. I am not the first to recognize that we do not choose the history we live through it, only how we will live. How we will love, a question to ponder alone for awhile. How to do our best without others, not easy. Without love, not possible. One without the other, non-existent.
 
One of the more distasteful tries at self-promotion during these times was a commercial showing off a video-conferencing app being used to allow family members to say goodbye face-to-face with their loved one as their lungs succumbed to viral load. It was meant to show what technology makes possible (goodbyes at a distance) but it also made clear what technology has made possible (a mass-surveilled, self-isolated quarantine). Who we see in the green-grey dark reflection of our screens is not who others see in person. Who we see across that splay of pixels is not who they really are either. A funhouse trick mirror distorting all who enter. A nightmare scenario you can see each and every hour of the days you currently live. A slim fragile pane between your world and all that. A distasteful metaphor. 
 
And yet to employ it here, I would like to say one last goodbye through this video conferencing app. I would like to say goodbye to each and every one of the students who have participated in this class. Please join me in thunderous applause.
  • G. A.
  • A. B.
  • K. B.
  • E. B.
  • Y. C.
  • E. C.
  • W. C.
  • J. C.
  • C. D.
  • M. D.
  • E. D.
  • I. D.
  • R. E. J.
  • E. F.
  • H. K.
  • A. K.
  • Y. L.
  • M. L.
  • D. M.
  • R. M.
  • B. M.
  • M. O.
  • J. O.
  • R. P.
  • E. P.
  • P. R.
  • J. R.
  • A. R.
  • B. R.
  • P. S.
  • N. S.
  • S. S.
  • J. S.
  • S. S.
  • J. T.
  • L. W.
  • D. W.
  • K. W.
I request especially thunderous applause for our IAs M. A., K. S., and A. T. They have done more for this class than any of us.
 
I say all this to say this much. When this class started many of us were taking it merely because it was the next thing we were supposed to do. It’s required by my major, it fulfils a technical requirement of my major, I’ve really got to get up to speed in this other thing I do – we all had this place to go and a route mostly charted and this was the next step on each of our paths to greatness. Walking it was as easy as showing up.
 
But there came a time when you couldn’t just show up. Where going toe-to-toe as David to each Goliath was a state mandated six-feet apart and then eventually you couldn’t be face-to-face without a mask. Life changed.
 
How well you navigate the ever-changing seas will be how true a bearing your love gives you. Where are you? Where do you want to go? You need only love to resolve the difference. That a well-designed feedback control system.
 
With that, I thank you for the opportunity to dismiss this class one final time under these exceptionally extraordinary times. May you find that which you love and continue on your ever-to-the-horizon journeys. I have done what I can for you here and I trust you can do more. 
 
Good luck.