33 1 SJP Phys 1120 AC Voltage and Current Batteries produce a steady fixed voltage called DC or direct current We should probably 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 AC Voltage DC V0 V0 time T time V0 period T The mathematical formula for AC voltage is V 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 plugged into the wall might look like this The squiggle on the left is the symbol for an AC voltage source rather than a battery V I The current no longer has a definite direction since the voltage changes with time so does I In fact we can figure out I t easily from Ohm s law I t V t R V0 sin 2 f t I 0 sin 2 f t R 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 33 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 real bulbs surely can t be zero otherwise they d be free At any moment in time P t I t V t I 0 V0 sin2 2 f t Let s graph this because the sin squared changes things a bit Power AC I0V0 0 period T T time sin 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 may not seem obvious why I d care but then remember P V 2 R so V 2 does appear 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 V 2 t V0 2 sin 2 2 f t and the average value of sin 2 is 1 2 not 0 V 2 average V0 2 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 namely P 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 V Average 2 From 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 gives Vrms 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 said P IV I 2 R V 2 R The new AC but averaged formulas we just derived say P 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 s 120 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 US appliance 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 not designed 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 what will happen Example Earlier we computed R for a short piece of Cu wire and got 0 01 What s the average power dissipated in this wire if a current of Irms 25 A runs through the wire into a big appliance Answer P ave Irms 2 R 25A 2 0 01 6 3 W That s quite a lot of power A toaster might dissipate 500 W or so The wire will certainly heat up Even though R is small it s not 0 If you let appliances consume too much current the wires in your house could easily start a fire That s why you have fuses typically rated at 15 20 A A fuse is a little device that shuts off all the current in a circuit if it ever exceeds the rated value The circuit symbol for a 15 A fuse is 15 A 33 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 …
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