Alternating Current (AC)
Download
Report
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