Lec16 - Scott Hughes Massachusetts Institute of Technology...

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Scott Hughes 8 April 2004 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2004 Lecture 16: RL Circuits. Undriven RLC circuits; phasor representation. 16.1 RL circuits Now that we know all about inductance, it’s time to start thinking about how to use it in circuits. A simple circuit that illustrates the major concepts of inductive circuits is this: V L R S2 S1 Suppose that at t = 0, we close the switch S 1, leaving S 2 open. How does the current evolve in the circuit after this? Let’s first think about this physically. Lenz’s law tells us that the magnetic flux through the inductor does not “want” to change. Since this flux is initially zero, the inductor will impede any current that tries to flow it — the current will initially try to remain at zero. As time passes, current will gradually leak through, and the magnetic flux will build up. Eventually, it should saturate at I = V/R — at this point the most current possible is flowing through the circuit. To show this explicitly, we turn to Kirchhoff: V - IR - L dI dt = 0 . V is the EMF supplied by the battery, - IR is the voltage drop at the resistor, - LdI/dt is the voltage drop at the inductor. Rearrange: dI dt = V - IR L = R L V R - I .
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Now divide; we end up with something that is of the form du/u : dI I - V/R = - R dt L so it is easy to integrate up, using the boundary conditions I = 0 at t = 0, I = I ( t ) at some general time t : ln " I ( t ) - V/R - V/R # = - Rt L or I ( t ) = V R 1 - e - Rt/L · . As we guessed on physical grounds, the current builds up, with a “growth timescale” L/R . The timescale L/R plays a role in LR circuits similiar to that played by the timescale RC in circuits with capacitors. Suppose we let this circuit sit in this state for a very long time, so that the current saturates at its equilibrium value, I = V/R . What happens if we now simultaneously open S 1 and close S 2? Physically, our expectation is that Lenz’s law will try to hold the current flow constant — the inductor says “I don’t want the flux through me to change!” However, we know that the resistor dissipates energy, so the current must bleed away — we should see some kind of exponential decay. We test this with Lenz’s law: going around the circuit, we have - L dI dt - IR = 0 leading to the easily integrated equation dI I = - R dt L leading to the exponential solution I ( t ) = I ( t = 0) e - Rt/L . (Note that we have redefined the meaning of t = 0 — this is now the time at which we open S 1 and close S 2.) When thinking about any circuit with inductance, the key thing you should keep in mind is that inductors cause currents to have a kind of “inertia”. If there is no current flowing, the inductor will force the current to build up gradually; if there is current flowing, the inductor will force it to change gradually. If current is already flowing and there is a sudden break in the circuit, the inductor will do whatever it takes to keep the current flowing. A consequence of this is that when we open switches or break circuits very suddenly, we may see a large spark! That spark is because of Lenz’s law — inductors in the circuit refuse to let the flux
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