12: Electromagnetic
Induction
12.2 Alternating Current
Alternating Current
Demo: HEP demo or dynamo
Alternating Current and Voltage
Whenever a magnet rotates near a coil or wire, its
flux will move through the wire or coil inducing an
alternating EMF across the coil or wire as a result of
Faraday’s Law.
Rotating Coil in a Uniform Magnetic Field
A very simple AC generator can consist of a single
coil of copper wire being forced to rotate in a
uniform magnetic field as shown. At the each end
of the wire are connected circular ‘slip rings’.
Q1. Explain the
design and
purpose of the
‘slip rings’
Q2. Why is the
coil made from
copper wire?
Link: AC Generator Applet
This simplified diagram shows a coil ‘end-on’,
rotating anti-clockwise:
Q3. Explain using Faraday’s Law why the EMF will
vary from maximum to zero as angle θ (between the
normal to the coil and the field plane) goes from 90°
to zero (as shown in the diagrams).
Q4. Plot points on the graph of flux linkage against
time (for max positive flux linkage, max negative,
zero) and draw the line that goes through them.
Considering Faraday’s Law, similarly plot points on
the graph of EMF against time and draw the line.
As can be seen from the two graphs, if EMF (ε) is a
sinusoidal graph then flux linkage must give a
cosine graph.
In fact the equations for each are...
Nϕ = BAN cosθ
(or Nϕ = BAN cos ωt)
ε = BAN ω sin θ
(or ε = BAN ω sin ωt)
(You do not need to know these equations however
they should make sense to you).
Increasing the speed of rotation
If the coil is rotated at a greater angular speed, the EMF
generated will increase and the frequency of rotation will
also increase. Hence the graph will change in two ways.
Q5. The graph below shows the output of a coil rotating in a
fixed uniform magnetic field. On the same axes, sketch the
graph of
i.
a coil rotating with twice the frequency.
ii.
a coil rotating with half the frequency.
Root Mean Square Current
In mathematics the Root Mean Square (rms) is a
statistical method of determining the magnitude of a
quantity that is varying. It can be thought of as a
kind of ‘average’ value. In particular it is useful when
dealing with sinusoidal variations (that can be
positive or negative) such as induced EMF and
current from a rotating coil.
For discrete values of any quantity the following
formula can be applied:
Clearly the calculated value is the square root of the
mean of the squares of the discrete values.
Q6. Determine the rms value of current from the
following graph using eight successive discrete
values:
I (A)
For electrical output from a coil rotating at constant
speed in a uniform magnetic field, the following
formulae can be applied:
εrms = ε0
√2
Where...
ε0 = Maximum EMF (V)
I0 = Maximum current (A)
Irms = I0
√2
Power in AC circuits
When calculating the power dissipated in an AC
circuit, we use the rms values.
The rms value of an alternating current is identical to
the value of direct current that would dissipate power
at the same rate through a resistor.
Thus, for alternating current circuits...
Power = Irms x Vrms
Q7. Determine a formula for average power in an
alternating circuit in terms of ε0 and I0.
Q8. The rms voltage in Europe is about 230V.
Determine the peak voltage value.
What will be the rms current value through a 20W
fluorescent light bulb?
Transformers
If any two electrical circuits are near to each other, a
changing current in one can cause an induced EMF
in the other.
A transformer uses changing flux linkage produced
by one coil to induce an EMF in the second coil.
Primary
coil
Secondary
coil
The input current is a.c.
Plot a graph of current in the primary (Ip) against
time.
The flux in the core is proportional to Ip.
Plot a graph of flux in the core against time.
The EMF induced in the secondary is proportional
to the rate of change of flux linkage.
Plot a graph of Induced EMF in the secondary
against time.
Transformer Calculations
The flux passing through the primary and secondary
coils is identical in a 100% efficient transformer.
Q. Explain (using Faraday’s Law) why having more
turns in the secondary than the primary can lead to
the voltage being ‘stepped up’ (increased).
The ratio of the turns is equal to the ratio of the
voltages:
Vs
Ns
=
Vp
Np
Ideal Transformers
A 100% efficient transformer is known as an ‘ideal
transformer’. In this case all the power on the
primary side is transferred to the secondary side.
Thus...
IpVp = IsVs
(All values are
rms values)
Q. If Np < Ns, which of the following are true (rewrite the wrong statements):
a.
ϕp = ϕs
b.
flux linkage is equal in both coils
c.
Ip > I s
d.
Vp > Vs
Real Transformers
In reality the output power is less than the input
power. This could be due to:
- Resistance of wires (causing heat transfer)
- Eddy currents in core (causing heat transfer)
- Flux leakage (not linking into the secondary coil)
- Hysteresis (molecular friction) (causing heat
transfer)
Q. Suggest a way of decreasing each of the first
three losses.
(copper wires; laminated core; improved core design)
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