Harvard University Division of Engineering and Applied Sciences. Fall Lecture 3: The Systems Approach - Electrical Systems

Transcription

1 Harvard Unversty Dvson of Engneerng and Appled Scences ES 45/25 - INTRODUCTION TO SYSTEMS ANALYSIS WITH PHYSIOLOGICAL APPLICATIONS Fall 2000 Lecture 3: The Systems Approach - Electrcal Systems In the last lecture, we dscussed mechancal systems. The basc dea was that although most systems are not comprsed of dscrete sprngs, masses, and dampers, ther behavor can be modeled as such. The fundamental quanttes that we were dealng wth were effort, n the form of force, and flow, n the form of velocty. We saw that there was a verson of ohm s law, n whch the force was proportonal to the velocty through a pure damper (F damper = bẋ). Energy storage was n the form of sprng elements, where the force was proportonal to the ntegrated velocty (F sprng = k R ẋdt = kx). Fnally, the nerta was n the form of Newton s 2nd law, where the force was proportonal to the rate of change of the velocty (F = mẍ). Gven physcal constrants of rgdly connected elements, and the constrants of the force balance, we were able to descrbe the dynamcs of many dfferent types of systems. The mportance of the mechancal elements n descrbng physologcal systems was demonstrated wth a dscusson on skeletal muscle mechancs. What we ll focus on today s the analogous development for electrcal systems, where the relatonshps are nearly dentcal to those we dscussed n mechancal systems. Just as we found n our connecton of mechancal engneerng systems to models of muscle mechancs, the electrcal analog has a smlar relevance n the descrpton of electrcal actvty n the body, whch we wll use to descrbe the passve electrcal propertes of electrcally exctable cell membranes. Electrcal Systems Electrcal systems, as wth mechancal systems, can be dscussed n terms of effort and flow. Perhaps you mght even fnd t more natural to dscuss the dynamcs of electrcal systems n these terms as compared to mechancal systems. For electrcal systems, the effort takes the form of electrcal potental, ψ = E, and the flow takes the form of current, ζ =. Agan, we have elements that are purely dsspatve n nature (resstors), storage elements (capactors), and nertal elements (nductors). The basc elements of electrcal systems are shown n Fgure. Fgure : The Isolated Electrcal Elements e e e R C L e 2 e 2 e 2 An solated resstance element (left), capactance (center), and an solated nductance element (rght).

2 ES45/25 - Lecture 3 2 The resstor s the purely dsspate element, so the potental dfference across the resstor s proportonal to the current, or flow: e R = e e 2 = R Note that f e s greater than e 2, then the current wll be postve, and the flow s the same as the drecton shown. If, on the other hand, e s less than e 2, then the current wll be negatve, and the flow wll be n the opposte drecton shown. Ths s obvously Ohm s law, and s analogous to dampers n mechancal systems. In electrcal crcuts, the capactor s the storage element, and therefore the voltage drop across the capactor s proportonal to the ntegral of the current: e C = e e 2 = Z t dt + e C (0) C 0 The capactor s analogous to the sprng n mechancal systems. The nductor s the nertal term of the electrcal crcut, and therefore the voltage drop s proportonal to the dervatve of the current: e L = e e 2 = L d dt The nductor s analogous to the nertal propertes of the mass n mechancal systems. Smlarly, we can wrte equatons for the current gong through each of these elements: R = e e 2 R C = C d(e e 2 ) dt Z t (e e 2 )dt + L (0) L = L 0 Notce that we haven t really pad any attenton to whether e s greater than e 2, or vce-versa. The correct way to attack these problems s to assume a drecton of current, as we have done, and the rest comes out n the wash. For example, assume that the current s flowng through the resstor as we have labelled t n the fgure. Suppose further that e 2 turns out to be greater than e. Ths results n a negatve current, whch mples that t s n the opposte drecton from what we ve labelled. Electrcal Elements n Seres and Parallel Just as a bref revew, remnd yourself that:

3 ES45/25 - Lecture 3 3 ffl Resstors n seres: R total = R + R 2 +::: ffl Resstors n parallel: R total = R + R 2 +::: ffl Capactors n seres: C total = C + C 2 +::: ffl Capactors n parallel: C total = C +C 2 +::: ffl Inductors n seres: L total = L + L 2 +::: ffl Inductors n parallel: L total = L + L 2 +::: These relatonshps are essental n reducng complex crcuts to relatvely smple ones, whch can easly be descrbed mathematcally. Krchoff s Voltage and Current Laws To determne the dfferental equatons governng the behavor of an electrcal crcut, we can smply use the fact that the sum of voltages around a loop has to be zero (Krchoff s voltage law), or the fact that the current comng nto a node must equal that leavng a node (Krchoff s current law), as shown n Fgure 2. Fgure 2: Krchoff s Voltage and Current Laws E E E 3 Left: Krchoff s current law. The sum of the current enterng a node has to equal the sum of the current leavng the node, = 0. Rght: Krchoff s voltage law. The total voltage drop around a closed loop s zero, E + E 2 + E 3 = 0. Krchoff s current law says that the net current enterng a node n the crcut s zero. Explctly for Fgure 2: n out = = 0

4 ES45/25 - Lecture 3 4 Krchoff s voltage law says that the sum of the voltage drops around any loop s equal to zero. Explctly for Fgure 2, we have: E + E 2 + E 3 = 0 Often, the use of both of these laws requres the assumpton of current drectons, but as long as you re consstent, t does not matter whch drecton you assume. Gven the ablty to add crcut elements n seres and parallel, and the ablty to effectvely mplement Krchoff s current and voltage laws, you now have a set of tools to determne the dynamcs of a varety of dfferent crcuts. The Resstor-Inductor Crcut Suppose we consder the smple resstor-nductor crcut shown n Fgure 3. Fgure 3: A Smple Crcut wth an Inductor and Resstor s R E L A smple crcut, wth a battery, E, a resstance, R, and an nductance, L. The current s denoted as. The crcut s opened and closed by the swtch, s. The crcut conssts of a battery, n seres wth a resstor and an nductor. The swtch s s open untl tme t = 0, at whch pont t closes. Frst, let s use Krchoff s voltage law: E L d dt R = 0 () One way to work ths problem s to let x = E R. We then have: where the soluton has the form: L dx dt + Rx = 0 x(t) = C e R L t We then resubsttute for x: (t) = C e R L t + E R

5 ES45/25 - Lecture 3 5 Snce the swtch, s, s open untl tme 0, we have: so: (0) = C + E R = 0 C = E R and the resultng soluton s: (t) = E R ( e R L t ) Another way to work the problem s to apply the more general technque of fndng homogeneous and partcular solutons. The homogeneous part nvolves fndng the soluton to: for whch we have: L d dt + R = 0 homog = C e R L t The partcular part nvolves fndng the soluton to: L d dt + R = E whch yelds: part = C 2 Substtutng nto the orgnal dfferental equaton gves: So the soluton becomes: C 2 = E R = homog + part = C e R L t + E R Usng the ntal condton of (0)=0, we fnd that C = E R, gvng: (t) = E R ( e R L t ) Fgure 4 shows the response of the crcut for a varety of dfferent values of E;R; and L. The matlab functon used to generate the plots s RL.m, and s avalable on the course webste.

6 ES45/25 - Lecture Fgure 4: Responses of the Resstor-nductor Crcut (t) (t) Tme Tme The left panel shows the response wth E =, R =, and L = (slower dynamcs) and L = 2 (faster dynamcs). The rght panel shows the response wth R =, L =, and E = and E = 2. Suppose we have another stuaton where agan the swtch s open for t < 0, closes at t = 0, and then reopens at t = t. What would the response be? You should convnce yourself that the same soluton as above holds here for t < t. At tme t = t, the swtch opens, and we have: The soluton to ths s: L d dt + R = 0 (t) = C 3 e R L (t t ) To solve for C 3, we can wrte: (t ) = C 3 e R L (t t ) = C 3 where: (t ) = E R ( e R L t ) from the fnal condton pror to openng the swtch. So for t > t we have: (t) = E R ( e R L t )e R L (t t ) Fgure 5 shows the resultng dynamcs for partcular values of the parameters. Fgure 5: Responses of the Double Swtch Resstor-nductor Crcut.2.2 (t) (t) Tme Tme Both panels have parameters E =, R =, and L = 2. For the panel on the left, the tme of the swtch reopenng, t =. For the panel on the rght, t = 3.

7 ES45/25 - Lecture 3 7 The two examples have exactly the same crcut parameters, but have dfferent tmes n whch the swtch s reopened. The example n the left panel shows an earler swtch tme as compared to the example on the rght. The results were generated wth RL doubleswtch.m, avalable on the course webste. An Example So we have learned to deal wth relatvely smple crcuts, but n fact these same technques can be used to analyze much more complcated scenaros. Consder the more complex crcut, shown n Fgure 6. Fgure 6: A More Complex Crcut L C 2 R C 2 2 R In ths crcut, we have two loops, each labelled wth a current, and 2. To attack ths crcut, we analyze one branch at a tme. Frst, however, we can note that the current travelng through capactor, C 2, wll be a functon of the two branch currents. Explctly, we can use Krchoff s current law at the juncton above C 2, as shown n Fgure 7. Fgure 7: Krchoff s Current Law 2-2 The current comng n from the left ( ) mnus the current gong out to the left ( 2 ) has to equal to current gong down the branch of C 2. We can then apply Krchoff s voltage law to each loop of the crcut. Startng wth the left sde we have: L d Z dt ( 2 )dt R Z C 2 C dt = 0 Workng on the rght loop, we have: R C 2 Z ( 2 )dt = 0

8 ES45/25 - Lecture 3 8 If we defne q such that q =, and q 2 such that q 2 = 2, we then have: and: L q + C 2 (q q 2 )+R q + C q = 0 R 2 q 2 + C 2 (q 2 q ) = 0 Note that the q s represent the charge, where current s the rate change of charge over tme. The result s just a set of coupled ordnary dfferental equatons that descrbes the dynamcs of the crcut. The Connecton Between Electrcal and Mechancal Systems We have ntroduced a generalzed way of descrbng both electrcal and mechancal systems through effort and flow. So t seems plausble that gven a mechancal system, we can fnd the electrcal analog, and vce-versa. Consder the sprng-mass-damper system, shown n Fgure 8. Fgure 8: Sprng-Mass-Damper System b x m f(t) k The classc sprng-mass-damper system. the mass moves on frctonless bearngs. Agan, The sprng s a storage element, whch s equvalent to a capactor. The damper s a dsspatve element, whch s equvalent to a resstor, and the mass s an nertal element, whch s equvalent to an nductance. The elements all have a common velocty, whch s equvalent to the flow (or current) for electrcal systems. The equvalent electrcal crcut s shown n Fgure 9. Fgure 9: Electrcal Crcut Analog to Sprng-Mass-Damper System C R voltage source f(t) L The analogous electrcal crcut for the sprngmass-damper mechancal system.

9 ES45/25 - Lecture 3 9 A Physologcal Example Agan, the dscusson of electrcal systems, as wth mechancal systems, s nterestng and mportant for the number of relevant engneerng applcatons. But how mght t apply to physologcal systems? Consder the electrcally exctable neuron. The neuron s the fundamental unt used by the bran to perceve the world, transmt nformaton, and take acton. Neurons vary wldly n ther functon and form, but they share some very fundamental propertes whch wll be the focus of upcomng lectures. For the moment, we wll consder the passve electrcal propertes of a squd neuron, and make tes to a very basc electrcal crcut that descrbes the behavor qute well. The cell body s the metabolc center of the cell, separated from the outsde world by a layer of lpd molecules. Electrcal actvty s ntated n the cell body, and travel down the fnger-lke projecton called the axon. The projecton termnates at synapses, whch are chemcal or electrcal contacts wth adjacent neurons. The electrcal actvty s carred by dssocated ons, ncludng K + ;Na + ;Cl ;Ca 2+, movng across the cell wall. We wll consder the classcal preparaton of the squd gant axon, not to be confused wth the gant squd axon. The axon s bathed n an onc soluton, and the potental s measured across the cell wall, as shown n Fgure 0. Fgure 0: The Squd Gant Axon V Cell Body Axon The classcal preparaton of the squd gant axon. The normal potental of the cell s around 60mV, measured from nsde the cell to outsde the cell. By electrcally stmulatng the axon surface wth an njected current, we can nduce changes n the potental. When ths s done expermentally, the observed tme course of the potental s shown n Fgure. Fgure : The Passve Response of a Neuron current tme potental The passve response of a squd axon to current njecton.

10 ES45/25 - Lecture 3 0 A model that was proposed for the passve propertes of the cell membrane s shown n Fgure 2. Fgure 2: A Model for Passve Neuron Propertes C R Vm A smple electrcal crcut representng neuron membrane dynamcs. In ths model, the current njecton s modeled as the current source, and the passve propertes of the membrane are modeled as a parallel combnaton of a resstor and capactor. The capactor represents the buld-up of charge across the lpd blayer. Ths relatvely smple model qute ncely predcts the passve response of the membrane to njected current, and wll be elaborated upon n future lectures. The shortcomng of the model, however, s that t does not explan some of the more nterestng phenomena we observe n neural tssue. When stmulated n such a manner, a neuron wll respond as shown. Increasng the sze of the current njecton ncreases the sze of the response n a roughly lnear fashon, consstent wth the model. However, ncreasng the current njecton past a certan pont results n a very non-lnear behavor - the potental of the neuron rushes rapdly upward and the neuron fres what s called an acton potental, the all-or-none event assocated wth neurons as shown n Fgure 3. Ths phenomenon s not characterzed by ths smple model, but nstead requres more elaborate non-lnear models whch wll be dscussed. Fgure 3: A Model for Passve Neuron Propertes current tme potental A smple electrcal crcut representng neuron membrane dynamcs. Readng Requred readng: Sectons n Khoo [] Addtonal readng: Chapter 3, Secton n Ogata (handout) [2]

11 ES45/25 - Lecture 3 References [] M. Khoo. Physologcal Control Systems: Analyss, Smulaton, Estmaton. IEEE Press, New York, [2] K. Ogata. System Dynamcs. Prentce-Hall, New Jersey, 978.