33-1 (SJP, Phys 1120)AC Voltage and Current:Batteries produce a steady, fixed voltage, called DC, or direct current. (We shouldprobably call them DV, direct voltage, but never mind)The power company produces a time-varying voltage, AC, or alternating current.Here’s a sketch of voltage vs. time:Voltage (DC)Voltage (AC)timetimeV0 +V0 -V0Tperiod, TThe mathematical formula for AC voltage isV(t) = V0 sin (2 pi f t). (Your calculator MUST be in radian mode!)V0 is called “peak” or “maximum” voltage.In the USA the period T = 1/60 s, so frequency f = 1/T = 60 Hz.(In Europe, it’s closer to 50 Hz).A simple circuit diagram for a light bulb (which is basically a resistor) pluggedinto the wall might look like this: The squiggle on the left is the symbol for an AC voltagesource, rather than a battery.The current no longer has a definite direction: since thevoltage changes with time, so does I.In fact, we can figure out I(t) easily from Ohm’s law:I(t) = V(t)/R = V0Rsin(2π f t) = I0sin(2π f t).Here, we have found the maximum current I0 = V0/R.Clearly, current I alternates right along with V (hence the name AC)Ohm’s law continues to hold in AC circuits, and V0 = I0 R...The voltage V(t) is symmetric, it’s + as often as -, it averages to 0.We say V(ave)= 0 (or V =0) I V33-2 (SJP, Phys 1120)Similarly, I(t) is also oscillating about 0, I(ave) = 0.Recall that P = IV, so what is the average power, P ?(You might guess zero, but think about light bulbs: average power used by realbulbs surely can’t be zero, otherwise they’d be free)At any moment in time, P(t) = I(t)*V(t) = I0V0sin2 (2π f t).Let’s graph this, because the “sin squared” changes things a bit:Power (AC)time +I0V0Tperiod, T 0sin^2(anything) is always positive.sin^2(anything) runs from 0 up to 1 and back again.On average, it is 1/2.That means P = (1/2) I0 V0. (It is not zero.)Here’s an odd question: “what’s the average of the square of voltage?” (It maynot seem obvious why I’d care, but then remember P = V^2/R, so V^2 doesappear in formulas... We really will care about this.)Remember, V(ave)=0. So you might think the average of V^2 is zero - but no!Just like the power example above: V2(t) = V02sin2 (2π f t), and the average valueof sin^2 is 1/2, not 0: V2(average) = V02/ 2 .The “average of the square” is NOT the square of the average (which was zero)!Now we have another way to find average power: find the average of P=V^2/R,which (we just showed) is P(ave) = (1/2)V0^2/R.(It’s really the SAME result as the previous page, namelyP(ave) = (1/2) I0 V0, because remember I0 = V0/R.)33-3 (SJP, Phys 1120)People have even given a name to Sqrt[V^2(average)].They call this Vrms, the “root mean square” voltage.VRMS≡ V2( )AverageFrom its definition (just square both sides): Vrms^2 = V^2(ave).Now, plugging in my result above for V^2(ave) = (1/2) V0^2 givesVrms = V0 / Sqrt[2] .The rms voltage is not the average voltage (which is 0), but it’s kind of a“representative” voltage. After all, voltage runs from -V0 to +V0, it’s (almost)always less then V0, so V0/Sqrt[2] is kind of a more “typical” voltage...Similarly, Irms = I0/ Sqrt[2] gives the typical current.Now, remember, we had old (DC) formulas that saidP = IV = I^2 R = V^2/R.The new (AC, but averaged) formulas we just derived sayP(ave) = (1/2) I0*V0 = (1/2) I0^2 = (1/2) V0^2/R.That recurring factor of (1/2) is annoying, and perhaps confusing.It’s there because we’re writing average power in terms of maximum I and V.If instead we rewrote P(ave) in terms of rms values, we’d get a nicer result:P(ave) = Irms* Vrms. (Check that - convince yourself it’s right.)The AC average formula LOOKS like the old DC formula, exactly:no factors of 2 at all, if you just use rms values instead of “peak” values.P(ave) = Irms*Vrms = Irms^2 R = Vrms^2/R(Convince yourself that they’re all right)33-4 (SJP, Phys 1120)Example: US Wall sockets really have Vrms = 120 V. What is the peak voltage,V0?Answer: V0 = Sqrt[2]*Vrms = 170 V.The US wall voltage is NOT running from -120 V to +120 V.It’s really running from -170 V to +170 V.On average it is 0, but it has a “typical” value of 120 V.In particular, when computing POWER, you can just pretend it’s120 V DC, and just use the old familiar power formulas.That’s why people say “the wall is 120 V”; they really MEAN Vrms.In Europe, Vrms =240 V. This causes serious problems if you try to plug a USappliance into a European socket, or vice versa.E.g., consider a 100 W bulb purchased in the US. Plug it into the wall in Europe.The resistor, R, is the same of course, but V is different.Since P(ave) = Vrms^2/R, and Vrms is about 2 times bigger there,squaring gives 4 times more power. It becomes a 400 W bulb, but it’s notdesigned to dissipate all that heat - it’ll burn out immediately.(If you go the other way, and plug a European 100W bulb into the wall here, whatwill happen?)Example: Earlier, we computed R for a short piece of Cu wire, and got 0.01Ω.What’s the average power dissipated in this wire, if a current of Irms = 25 A runsthrough the wire into a big appliance?Answer: P(ave) = Irms^2*R = (25A)^2*(0.01Ω = 6.3 W.That’s quite a lot of power.(A toaster might dissipate 500 W or so) The wire willcertainly heat up. Even though R is small, it’s not 0. If you let appliancesconsume too much current, the wires in your house could easily start a fire. That’swhy you have fuses (typically rated at 15-20 A). A fuse is a little device that shutsoff all the current in a circuit if it ever exceeds the rated value.The circuit symbol for a 15 A fuse is 15 A33-5 (SJP, Phys 1120)Transformers:These are clever (and simple) devices to transform AC voltages.You need two coils:the “input wire” or “primary winding”,and the “output wire” or “secondary winding”.The wires are usually wrapped around iron. There are no moving mechanical partsneeded.IfV(in)issteady(I(in)issteady) thenB(induced)through thecoils issteady. The flux Φ through the secondary coil is therefore steady, and Faraday’slaw says V(out)=0: no output current, or voltage.Moral: transformers do nothing if the input is steady, or DC.But, if V(in) is AC, then the induced B keeps flipping direction, which means Φ ischanging, and Faraday says we will induce an EMF in the secondary coil. Thatmeans there is a V(out).Faraday says, specifically,V(in) = Np ΔΦ/Δt,V(out) = Ns ΔΦ/Δt.The flux Φ is
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