lec17

# lec17 - Scott Hughes 14 April 2005 Massachusetts Institute...

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Unformatted text preview: Scott Hughes 14 April 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 17: AC Circuits; Impedance. 17.1 AC circuits: Intro As mentioned in Lecture 15, we use induction to generate the vast majority of electrical power in the world. Typically, we do this with some variation on the “vary the dot product” mechanism: B n ^ The loops of wire are rotated by some external source of work — a diesel engine, a waterfall, nuclear produced steam, etc. The magnetic field largely acts as a mechanism to transform that external work into electrical power. Doing so, we end up with a driving EMF that is sinusoidal: Φ B = BA cos ωt-→ E = ωBA c sin ωt . At least in part for this reason, the electricity that we get “from the wall” oscillates sinu- soidally. To insure some degree of uniformity, standard choices for the oscillation frequency have been agreed upon. To demonstrate that humans are illogical beings who will shoot themselves in the foot at any opportunity, two such standards are commonly used: ω = 2 π × 60Hz ω = 2 π × 50Hz . 60 Hz is used in North America, much of South America, South Korea, and western Japan; 50 Hz is used in Europe, most of Africa (except, for example, Liberia), the rest of South America, 156 and most of Asia (including eastern Japan). See http://kropla.com/electric2.htm for a summary. The main reason for going through this is to motivate how essential it is to understand how alternating current (AC) circuits work. Having spent as much time as we have studying direct current (DC) circuits, generalizing to AC is not too difficult — particularly if we are smart enough to recognize that our life is made a LOT easier by working with complex valued quantities. Our goal in this lecture will be to understand how to find the current which flows in the following circuit: C L R The symbol on the left is a source of AC EMF. This device supplies an EMF to the rest of the circuit which we will take to be E ( t ) = E cos ωt . In much of our analysis, we will find it very useful to take this be the real part of a complex EMF ˜ E ( t ): E ( t ) = Re h ˜ E ( t ) i where ˜ E ( t ) = E e iωt . (Note that E is a purely real number.) We will analyze this circuit by just generalizing the operating principles that we used for DC circuits. In particular, 1. The sum of the voltage drops in the loop must equal the driving EMF. This is true for both the real form of the voltage E ( t ) = V R ( t ) + V C ( t ) + V L ( t ) and the complex form: ˜ E ( t ) = ˜ V R ( t ) + ˜ V C ( t ) + ˜ V L ( t ) . 157 You may ask “What is the voltage drop across an inductor — doesn’t an inductor act as a source of EMF?” Indeed, you would be correct for asking this. To keep things uniform, what we do is move the EMF from the left-hand side to the right-hand side, introducing a minus sign. Hence, V L ( t ) =-E ind = LdI/dt . There’s nothing deep here — it’s just a trick we introduce in order to treat all circuit elements in a uniform way.— it’s just a trick we introduce in order to treat all circuit elements in a uniform way....
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lec17 - Scott Hughes 14 April 2005 Massachusetts Institute...

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