6 01 Fall Semester 2007 Lecture 7 Notes 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Fall Semester 2007 Lecture 7 Notes Constraint Systems and Circuits Circuits Electrical circuits are made up of components such as resistors capacitors inductors and transistors connected together by wires You can make arbitrarily amazing complicated devices by hooking 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 at the 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 such as 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 separately designed parts of a circuit tend to influence one another when they are connected together unless you design very carefully We ll explore a number of examples of when and how the abstractions can help us but also when they can leave out important detail and require different models Constraint Models So far we have looked at a number of different models of systems We have thought of software procedures as computing functions of a robot brain as performing a transduction from a stream of inputs to a stream of outputs and of linear systems as a special subclass of transductions that we can analyze for stability and other properties In each case we were able to construct or analyze the behavior of sub parts of the system as functions or transductions and then abstract away from their implementations use them to build more complex systems and use the understanding of the components 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 happen in a local piece of it We will be able to view the sub parts as putting constraints on the overall global behavior of the system once enough pieces are put together and their constraints are taken together 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 and the way in which they re connected together we have described a set of constraints on the positions of all the rods If for example we connect 4 rods of length 1 in a square then the positions of the other rods are not completely specified because the square can be squashed into a number of different rhombuses On the other hand if we connect only three rods into a triangle then the position of the third vertex will be completely specified 6 01 Fall Semester 2007 Lecture 7 Notes 2 We will use this way of thinking about and specifying the behavior of a system to understand simple electrical circuits as systems of constraints Voltage and current Voltage 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 relative concept we could pick any point in the circuit and call it ground and we would still get the same results Current is a flow of electrical charge through path in the circuit A positive current in a direction is generated by negative charges electrons moving in the opposite direction 1 We re not going to worry about the details of what particles are doing what until we get to semiconductors in another class We ll just have to be careful when we draw and describe circuits to label the directions of the currents we re talking about Static circuit model A circuit is made up of a set of components wired together in some structure Each component has a current flowing through it and a voltage difference across its two terminals points at which it is connected into the circuit Each type of component has some special characteristics that govern the relationship between its voltage and current In general circuits have dynamic behavior That is the voltages and currents in the system change over time Models for the dynamic behavior of circuits are usually in the form of difference or differential equations For now we will consider a simpler case one where we have assumed that the dynamic behavior has settled to an equilibrium state In this equilibrium setting we will consider the case where combinations components and the way they are connected provides a set of constraints on the equilibrium state of the circuit We ll work through this view by starting with the constraints that come from the structure and then examining constraints for two simple types of components Conservation laws One set of constraints in circuit problems stems from enforcing a conservation law on the circuit currents often referred to as Kirchoff s Current Law KCL This conservation law holds true no matter what kinds of components we use in our circuit We ll describe the conservation law using the circuit in figure 1A For now don t worry about what s in the components labeled A through D You can see that we ve labeled the current through each component with an arrow and named it ix We can choose these arrows to point in any direction we like as long as we treat them consistently we ll say this more precisely later For each component we can also talk about the voltage drop across the component which we ve labeled vx It is the potential difference between 1 At the semi conductor level it can also be viewed in an oversimplified way as as holes or positive charges moving in the direction of the current 6 01 Fall Semester 2007 Lecture 7 Notes 3 iA iA n1 vA n1 n2 RA n2 A iB vB B C vC i C iB RB VC iC D n4 vD n3 n4 iD RD n3 iD Figure 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 direction of the current arrow for the component flowing from to Kirchhoff s Current Law KCL Each place in a circuit where
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