6.01, Fall Semester, 2007—Lecture 7 Notes 1MASSACHVSETTS INSTITVTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.01—Introduction to EECS IFall Semester, 2007Lecture 7 NotesConstraint Systems and CircuitsCircuitsElectrical circuits are made up of components, such as resistors, capacitors, inductors, and tran-sistors, connected together by wires. You can make arbitrarily amazing, complicated devices byhooking these things up in different ways, but in order to help with analysis and design of circuits,we need a systematic way of understanding how they work.As usual, we can’t comprehend the whole thing at once: it’s too hard to analyze the system atthe level of individual components, so, again, we’re going to build a model in terms of primitives,means of combination, and means of abstraction. The primitives will be the basic components, suchas resistors and op-amps; the means of combination is wiring the primitives together into circuits.We’ll find that abstraction in circuits is a bit harder than in software or linear systems: separatelydesigned parts of a circuit tend to influence one another when they are connected together, unlessyou design very carefully. We’ll explore a number of examples of when and how the abstractionscan help us, but also when they can leave out important detail and require different models.Constraint ModelsSo far, we have looked at a number of different models of systems. We have thought of softwareprocedures as computing functions, of a robot “brain” as performing a transduction from a streamof inputs to a stream of outputs, and of linear systems as a special subclass of transductions thatwe can analyze for stability and other properties. In each case, we were able to construct or analyzethe behavior of sub-parts of the system, as functions or transductions, and then abstract away fromtheir implementations, use them to build more complex systems, and use the understanding of thecomponents to understand the larger system.Now we’re going to consider a different class of systems that has a kind of modularity, but where,typically, you have to have a description of the entire system in order to say what is going to happenin a local piece of it. We will be able to view the sub-parts as putting “constraints” on the overallglobal behavior of the system; once enough pieces are put together and their constraints are takentogether, the behavior of the entire system will be specified.One intuitive example is a set of rigid rods connected together with pins, all resting flat on a table.If we specify the x, y coordinates of the end points of one rod, and the lengths of the other rods, andthe way in which they’re connected together, we have described a set of constraints on the positionsof all the rods. If, for example, we connect 4 rods of length 1 in a square, then the positions ofthe other rods are not completely specified, because the square can be squashed into a number ofdifferent rhombuses. On the other hand, if we connect only three rods into a triangle, then theposition of the third vertex will be completely specified.6.01, Fall Semester, 2007—Lecture 7 Notes 2We will use this way of thinking about and specifying the behavior of a system to understandsimple electrical circuits as systems of constraints.Voltage and currentVoltage is a difference in electrical potential between two different points in a circuit. We will,generally speaking, pick some point in a circuit and say that it is “ground” or has voltage 0.Now, every other point has a voltage defined with respect to ground. Because voltage is a relativeconcept, we could pick any point in the circuit and call it ground, and we would still get the sameresults.Current is a flow of electrical charge through path in the circuit. A positive current in a directionis generated by negative charges (electrons) moving in the opposite direction.1We’re not going toworry about the details of what particles are doing what (until we get to semiconductors, in anotherclass). We’ll just have to be careful when we draw and describe circuits to label the directions ofthe currents we’re talking about.Static circuit modelA circuit is made up of a set of components, wired together in some structure. Each component hasa current flowing through it, and a voltage difference across its two terminals (points at which it isconnected into the circuit). Each type of component has some special characteristics that governthe relationship between its voltage and current.In general, circuits have dynamic behavior. That is, the voltages and currents in the system changeover time. Models for the dynamic behavior of circuits are usually in the form of difference ordifferential equations. For now, we will consider a simpler case, one where we have assumed thatthe dynamic behavior has settled to an equilibrium state.In this equilibrium setting, we will consider the case where combinations components, and the waythey are connected, provides a set of constraints on the equilibrium state of the circuit. We’llwork through this view by starting with the constraints that come from the structure, and thenexamining constraints for two simple types of components.Conservation lawsOne set of constraints in circuit problems stems from enforcing a conservation law on the circuitcurrents, often referred to as Kirchoff’s Current Law (KCL). This conservation law holds true, nomatter what kinds of components we use in our circuit. We’ll describe the conservation law usingthe circuit in figure 1A. For now, don’t worry about what’s in the components labeled A throughD. You can see that we’ve labeled the current through each component with an arrow, and namedit ix. We can choose these arrows to point in any direction we like, as long as we treat themconsistently (we’ll say this more precisely later). For each component, we can also talk about thevoltage drop across the component, which we’ve labeled vx. It is the potential difference between1At the semi-conductor level, it can also be viewed in an oversimplified way as as “holes” or positive chargesmoving in the direction of the current.6.01, Fall Semester, 2007—Lecture 7 Notes 3n1n4 n3n2ADB CiAiDiCiB+ -vA+-vD+-vC+-vBn1n4 n3n2iAiDiBRA+-VCRBRDiCFigure 1: A. Circuit with four components. B. Circuit with three resistors and a voltage source.the terminal labeled ’+’ and the terminal labeled ’-’, which should agree with the
View Full Document
Unlocking...