SOLUTIONS Homework #1

Transcription

1 SOLUTIONS Homework #1 INTRODUCTION TO ENGINEERING IN MEDICINE AND BIOLOGY ECEN Fall 2003 R. Mihran Due Tuesday, 9/9/03 Question 1: a) Chapter 1 of the text discusses the fact that a significant fraction of the fluid in your body exists in the spaces around the exterior of your cells, i.e. the extracellular fluid. This fluid represents the immediate "environment" that each cell exists in, and since this fluid is constantly in motion through blood circulation and tissue diffusion, essentially all cells in your body live in the same "internal environment". In the same sense that the house that you live in is designed to maintain a comfortable environment by controlling temperature, ventilation, etc. within it, your body goes to great lengths to maintain its cellular "internal environment" at a constant, near-optimal condition. Things like body temperature, blood oxygen and carbon dioxide levels, blood ph, blood pressure, etc., are thus constantly monitored by the body and carefully maintained at as close to optimal values as possible under differing conditions. This internal maintenance of a constant, optimal set of conditions is what physiologists refer to as "homeostasis". As an example of class 1 functions which contribute to homeostasis, when you begin to exercise, a number of critical physiological parameters are affected. The increased motion and work performed by your muscles will generate additional heat, consume additional oxygen, and generate additional carbon dioxide as compared to resting. Special sensory cells called "chemoreceptors" detect the change in blood gasses, as well as the effect that rising CO2 has on the ph of your blood. Other special sensory cells in your brain detect the rising blood temperature. In addition, the blood vessels which supply the muscle(s) being exercised will enlarge (this is called vasodilation) to try to channel more of your blood to that part of your body. This will in turn cause the average blood pressure in the rest of your body to drop, since the total volume of your circulatory system is thus increased. This drop in blood pressure is detected by special sensory cells in major arteries, and this information too is sent to the brain for the proper response for homeostasis. As an example of class 2 functions, your body will respond by executing certain familiar responses, most notably, increasing your heart rate and respiration. Increased respiration clearly helps to increase the rate at which oxygen and CO2 can be exchanged in the blood, as well as the rate that additional heat can be dissipated. The increased heart rate and other changes to the heart muscle also contribute to bringing your blood pressure back to normal. b) A feedback system, also known as a "closed loop" system, refers to a system in which one or more parameters describing the state of that system at any given moment has an effect on, or is "fed back to", the system to modify it in some manner. In the context of a system like a machine which has a simple input and output, feedback would exist if some portion of the output was returned or fed back to its input. This would be in contrast to an "open loop system" in which the output of the machine was essentially "pre-set".

2 To help you conceptualize the difference between a negative and positive feedback system, try, as you read this, to perfectly balance a pen or pencil on its point on your desk, and think of that balanced (or near balanced!) state as the starting point of the system. Even though it is theoretically possible to locate the pencil in a perfectly balanced position, in practice, you can never do so successfully for any length of time. Why not? It is because when even the slightest perturbation on the pencil occurs - as from an imperceptible air current, or vibration - the gravitational forces on the pencil are no longer symmetrically balanced, and it will experience a net acceleration in the direction of the initial disturbance. This acceleration results in further displacement of the top of the pencil, which in turn upsets the balance further. This process continues as the pencil falls. Thus, the displacement of the pencil from its starting position causes feedback to the system which creates yet more displacement. This type of feedback is known as positive feedback, and clearly leads to instability in a system. Now, instead of attempting to balance the pencil on the desk, try to balance it for as long as you can on the tip of your finger. With a little practice (or a longer pencil) you can make enough adjustments to keep it balanced for an extended (or indefinite) period of time. Those adjustments represent a calculated response by you to the tendency for the pencil to fall in any given direction, and are intended to counteract any displacement of the top of the pencil. Thus, any displacement of the system (pencil) which you detect initiates a response from you to reduce further displacement and bring the pencil back to its starting point. Your response is thus negative to the displacement - this type of feedback is known as negative feedback. In general, therefore, negative feedback leads to stability in a system, and is used extensively in both nature and in engineered systems to control or regulate system behavior. c) Refer back to the examples you gave in your answer to part a): for each of these, would you say that these represent examples of negative or positive feedback? Each of the examples I described in part a) represent examples of negative feedback used by your body in response to the disturbance of exercise: The increased respiration will increase the rate of oxygenation of blood (in response to falling O2 levels) and the removal of CO2 (in response to rising levels), as well as the rate of heat dissipation (in response to rising temperatures). The increased heart rate will increase systemic blood pressure (in response to falling levels) as well as increase the rate of blood gas exchange and heat dissipation. By far, most of the control systems in your body represent negative feedback systems designed to maintain stability of physiological states. In a few rare cases, where a very rapid or even explosive response is required, the body may exploit the natural instability of positive feedback systems. Two examples where positive feedback is used in a limited manner are the sequence of chemical events leading to the formation of a blood clot, and the generation of the electrical impulses in nerve cells, known action potentials. We will talk more about these later this semester. d). Try to identify at least two examples of negative and positive feedback in some familiar systems other than your body (such as a thermostat, etc.) Positive Feedback: Other than the pencil case, another familiar example of positive feedback occurs when a microphone used in a public address system is pointed directly at a speaker, or when the volume (amplification) is turned too high. In this case, any small sound emerging from the speaker is picked up by the microphone and amplified, and again emerges from the speaker - but this time, a little louder. This is picked up again by the microphone, and re-amplified. This process continues to build very quickly into that very unpleasant screeching sound that we have all heard when the PA systems are turned up too high.

3 Another example of positive feedback often occurs with combustion - for example, a spark igniting a flammable gas or liquid. The initial heat generated by the spark initiates combustion of the surrounding gas. This in turn generates more heat, which ignites yet more gas, and so on, resulting in a form of instability known, quite simply, as an explosion. Negative Feedback: There are many types of "control systems" which you use regularly which use negative feedback to achieve stability, or control. The thermostat which turns a furnace or oven on or off to maintain a constant temperature is one example, and the "cruise control" used to maintain a constant speed in cars is another. Another example is a sprifng return on a door. A more subtle example from nature is the balance in the predator/prey populations. Question 2: a.)some of the obvious responses include: Sweating: The phase change from liquid water to water vapor occurring on the surface of your skin requires latent heat energy (i.e. is endothermic), thereby cooling the surface of your skin as evaporation occurs. Increased heart rate: This increases the circulation of warm blood from deeper tissues through the surface vessels, where the heat is dissipated. Peripheral vasodilation (enlargement of surface blood vessels): This brings more blood closer to the surface of the body for more rapid cooling through evaporation. This is also the reason why your skin become flushed, or appears redder, when you are very warm. b.) The gain of this negative feedback system for temperature regulation is described by: gain = correction/error In this case, the correction is the difference between the actual temperature (37.5 deg) with the negative feedback temperature regulation system operating, and the temperature which would occur without the feedback response (i.e. 53 deg), or =-15.5 degrees. The error is the deviation from the optimal control temperature, or = 0.5 degrees. In this example then, gain = (-15.5 deg)/(0.5 deg) = -31

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5 c) If you were to generate this chord by playing an A note, an E note, and the next A octave, the actual air pressure variations will be very similar to this form. However, as we discussed in class, the piano will also generate some harmonics, which are might be some fraction of the fundamental frequency (these are called subharmonics) or some integer multiple of the fundamental. These harmonics would slightly modify the form of the pressure signal you would actually measure, and are in fact what allows your ear to distinguish this chord played on a piano from the same chord played on a guitar, or even the same chord played on two different pianos. d) A frequency domain plot shows the relative amplitude (or power, which is proportional to the square of the amplitude) of a signal as a function of frequency, instead of time. For a signal of unknown origin, there are mathematical analyses and algorithms which have been developed to generate the frequency domain representation. These are known as Fourier transforms and spectral estimation techniques. The spectrum analyzer which was used in the class demo performs a fast fourier transform (FFT). We will be talking more about the fourier transform shortly, because as you will see, your inner ear (the cochlea) actually performs a fourier transform on the sound as it travels through your auditory structures. In this particular problem - since you generated the signal - you know that it is composed of three distinct frequencies, and thus it is not necessary to perform additional analysis to draw the frequency domain representation. It is simply composed of three distinct spikes at each of the three individual frequencies, as shown below.