Eddy Currents - Michigan State University

Eddy Currents l Suppose I have a pendulum with a copper plate on the end that swings through a magnetic field l Do I expect anything to happen? u there’s a change in magnetic flux through the copper as it swings into the magnetic field, so an emf will be induced u and copper is a good conductor u eddy currents u which direction? u where would it be useful to use eddy current braking?

Search on eddy current braking on google l Ignore links to: u u Mary Baker Eddy Fish’s Eddy lIn addition to these generator brakes, the braking function of the vehicle is assured by the modular eddy-current brakes. The individual eddy-current braking magnets act on the guidance rails of the guideway and guarantee the braking of the vehicle.

Energy stored in inductor L R - e L I As I increases, I is flowing from to – side of L, so energy is being extracted from circuit (battery). Thus inductor is gaining energy that is being stored in the B field. Energy store in inductor 1 2 UL L I 2 Energy store in capacitor 1 U C C DV 2 2

“Discharging” an inductor R - e L I

“Discharging” an inductor R - e L I I R L

“Discharging” an inductor - e R R L I L I I will decay as UL dissipated as heat in R DI \ 0 Dt

“Discharging” an inductor R L - L I R I L I e R I will decay as UL dissipated as heat in R DI \ 0 Dt

“Discharging” an inductor L L - I I R L I e R R I will decay as UL dissipated as heat in R DI \ 0 Dt Lenz’s law says that inductor will have an emf across itself to oppose the reduction of I.

“Discharging” an inductor I e -t / t I e R L t R t

Zap! R - e L I

Zap! L I e - - e S R R L

Zap! L I e - - e S R R L 0 I

Zap! L I e - - e S R R L 0 I DI very large Dt

Zap! 0 I eL I e - L R L - e S R DI very large Dt DI eL L Dt Huge!

Zap! 0 I eL I e - L R L - e S R spark DI very large Dt DI eL L Dt Huge!

Back to our simple circuit l The battery is the first source of emf that we’ve come across l The circuit on the right has an electric current I e/R that is constant with time l And we remember that an electric current really consists of the electrons in the conductor moving with a speed vd (a few tenths of a mm/sec) opposite the direction of the electric field

A new type of circuit: an AC circuit If I turn the generator at a rate of 60 hz, then f is 60 hz e emax sin wt emax N B A w This is something new. An emf that changes with time. not only in magnitude but in direction.

An electric field that changes direction E l As the emf traces out its sinusoidal path, the electric field inside the conductor is constantly changing magnitude and direction l This means the electrons are also changing direction u 120 times a second for a 60 hz current l We thought vd was small, but now they’re really not getting anywhere because they’re constantly changing direction l Yet an AC current must still accomplish something useful u remember that the electrons didn’t have to travel all of the way to the end of the conductor to light a light-bulb

Current and voltage l Consider the current and voltage in the simple circuit shown to the right l The voltage produced by the generator (or electrical outlet) appears across the resistor u DvR Dv l The current in the resistor is the same as the current in the battery u iR Imax sin 2pft l Note that I’m using small letters for quantities that vary with time, and capital letters for quantities that are constant u I’m going to keep doing that

Current and voltage l Let’s consider something obvious u u u u whenever the voltage is zero, the current is zero whenever the voltage is at a maximum the current is at a maximum whenever the voltage changes direction, so does the current I say that the current and voltage are in phase with each other l Why consider the obvious? u because the above points won’t be true whenever we start to add other circuit elements (such as capacitors and inductors) into the mix

Current and voltage l Consider the AC current going through the resistor as shown l Does the resistor get hot? Yes. u the electrons are still colliding with the atoms in the resistor no matter which way they’re going. l Does the resistor care that the electrons can’t make up their minds which way to go? No. l Can AC currents do just as useful stuff as DC currents? Yes.

Current and power What’s the average current? iav 0 What’s the average power? P i2R Pav (i2)avR (i2)av 1/2I2max Fig. 21.3, p.654

rms current l (i2)av 1/2I2max l Note that an AC current of Imax is not the same as a DC current of Imax l Let me define a new quantity, the rms current u u u The rms current is the direct current that would dissipate the same amount of energy in a resistor as dissipated by the actual alternating current Irms sqrt[(i2)av] Imax/sqrt(2) P I2rmsR l I can also define other rms quantities u Vrms Vmax/sqrt(2) So when we say that there’s an outlet voltage of 120 V, that refers to the rms value Vpeak 170 V

Notation Ohm’s law still applies, to the relevant quantities Dv i R DVmax Imax R DVrms Irms R

And now for something completely different l An AC circuit with a capacitor Note that the current and the voltage across the capacitor are no longer in phase. How can we understand this behavior? Fig. 21.4, p.656

And now for something completely different l I can’t have any voltage across the capacitor until I accumulate some charge l And for that, the current has to flow from some time so the voltage across the capacitor always lags behind the current by 90o l When the current goes to 0, the voltage is at a maximum l When the current reverses direction, the capacitor starts to discharge l And eventually will charge up in the reverse direction

Capacitive reactance l Note that the presence of a capacitor in the circuit impedes the flow of current, just as the presence of a resistor would l We define the impedance for a capacitor as u XC 1/(2pfC) l Something new; a frequency dependent impedance l Capacitors like high frequencies; don’t like low frequencies l At high frequencies, less time for charge to accumulate on the capacitor; less opposition to current flow Version of Ohm’s law DVC,rms Irms XC

We’re on a roll l An AC circuit with an inductor Now the current and voltage are out of phase but in a different way. How can we understand this behavior? Fig. 21.6, p.657

Search on eddy current braking on google lIgnore links to: u Mary Baker Eddy u Fish's Eddy lIn addition to these generator brakes, the braking function of the vehicle is assured by the modular eddy-current brakes. The individual eddy-current braking magnets act on the guidance rails of the guideway and guarantee the braking of the vehicle.