#### Transcript Alternating Current (AC)

Warm-up—1/15/14 What happens when you rotate a coil of conductive material through a uniform magnetic field (not just move it linearly)? Assess. State. 12.2.1 – 12.2.9 Assess. State. 12.3.1 – 12.3.5 Due Friday, 1/16/15 AC Generator As the ring rotates within the magnetic field, what happens? AC generators Video link (it’s old…and if you watch the whole thing, it’s long, but it’s good)—we’re just going to watch the first few minutes today http://www.youtube.com/watch?v=LisefA_YuVg&safe=active In a nutshell… DC generators—use a split-ring commutator to ensure that the direction of the induced emf (and resulting current) is always in the same direction upon output from the generator AC generators—use a set of slip rings to provide constant contact with the brushes, resulting in an induced emf and current that are alternating in magnitude and in direction Peak voltage Flux linkage: Φ = 𝑁𝐵𝐴𝑐𝑜𝑠(𝜃) How is the angle in that equation related to the rotation of the coil? 𝜃 = 𝜔𝑡 (at some time, t, the angle of the coil in the magnetic field is q, which depends on how quickly the coil is rotating) So…a little use of Faraday’s Law, and a little calculus later… Peak Voltage 𝜀 = 𝜔𝑁𝐵𝐴 ∙ 𝑠𝑖𝑛(𝜔𝑡) Peak Voltage: the maximum induced emf that is generated by an AC generator (i.e. coil rotating in a magnetic field) Peak Voltage Peak Current We are going to safely assume Ohm’s Law works, so the peak current (maximum current induced) through a resistor in an AC circuit is: 𝜀 𝜀0 sin(𝜔𝑡) 𝐼= = 𝑅 𝑅 𝐼 = 𝐼0 sin(𝜔𝑡) Power in an AC Circuit Just like AC voltage and current, not constant with time: 𝑃 = 𝜀𝐼 𝑃 = 𝜀𝑜 𝐼𝑜 sin 2 𝜔𝑡 Peak Power is the product of peak voltage and peak current Power is always a positive value, and will be equal to zero Watts every half rotation of the coil. Average power is ½ the peak power: rms Voltage Root Mean Square (rms) Voltage and Current: The best way we have of measuring an average voltage or current in AC circuits Step 1: Square the Current(or voltage) 𝐼 2 = 𝐼0 2 sin2 𝜔𝑡 2 𝐼 0 𝐼2 = 1 − cos 2𝜔𝑡 2 Step 2: average this (now always positive) quantity In 1 cycle, the cosine term averages to zero! 2 𝐼 0 𝐼2 = 2 Step 3: Take that average’s square root 𝐼𝑟𝑚𝑠 = Same thing for voltage: 𝜀𝑟𝑚𝑠 = 𝐼0 2 𝜀0 2 Average Power: 𝑷 = 𝜺𝑟𝑚𝑠 𝑰𝑟𝑚𝑠 Transformers A tool used to take advantage of the fact that an alternating current generates an alternating magnetic flux in a coil. An iron core connects two separate coils Primary coil the coil that is the “input” to the transformer Incoming alternating current generates an ever-changing flux Secondary coil the coil that delivers the “output” Because of the iron core, the flux from the primary coil induces an emf in the secondary coil and, therefore, a current Transformers--quantified The induced emf in the secondary coil, as well as the amount of magnetic flux rate of change is dependent on Faraday’s Law. The primary coil generates a magnetic flux changing at ∆Φ a rate shown by 𝑉𝑝 = 𝑁𝑝 Δ𝑡 The secondary coil generates an induced emf: ∆Φ 𝑉𝑠 = 𝑁𝑠 Δ𝑡 Transformers--continued ∆Φ Δ𝑡 is a constant, which leaves us the following ratio: 𝑉𝑝 𝑁𝑝 = 𝑉𝑠 𝑁𝑠 Knowing that an ideal transformer will have no power loss between the coils, so 𝐼𝑝 𝑉𝑝 = 𝐼𝑠 𝑉𝑠 , this can also be written as: 𝐼𝑠 𝑉𝑝 𝑁𝑝 = = 𝐼𝑝 𝑉𝑠 𝑁𝑠 Example: Step-down and Step-up Transformers Step-Down: A transformer designed to have a high input voltage and a low output voltage There will be fewer loops in the secondary coil Step-up: A transformer designed to have a low input voltage and a higher output voltage More loops in secondary coil