12.2 Alternating current - Cobequid Educational Centre

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Transcript 12.2 Alternating current - Cobequid Educational Centre

12.2 Alternating current
2 hours
Alternating Current
• The induced emf in a coil rotated within a
uniform magnetic field is sinusoidal if the
rotation is at constant speed.
• The most important practical application of
the Laws of Electromagnetic Induction was the
development of the electric generator or
dynamo.
The AC Generator
Link to Java Applet
• http://www.walter-fendt.de/ph14e/generator_e.htm
Changing the Frequency of the
Generator
• The emf of a rotating coil can be calculated at
a given time. If a coil of N turns has an area A,
and its normal makes an angle θ with the
magnetic field B, then the flux-linkage F is
given by: F  NAB cos 
• Using calculus and differentiating cosθ, the
relationship for e.m.f. becomes:
• Remember from your knowledge of rotational motion that
Δ θ ÷ Δt = the angular velocity, w, in rad s-1 = 2 π f
• Also
θ = ωt = 2πft
• so that
ε = ω ×N×A×B ×sin(ωt) or ε = 2πf ×N×A×B × sin(2 π f t)
• When the plane of the coil is parallel to the magnetic field, sin ωt will have its
maximum value as ωt = 90°, so sin ωt = 1. This maximum value for the emf ε0 is
called the peak voltage, and is given by:
• Therefore:
ε0 = ω NBA
ε = ε0 sin(ω t)
• The frequency of rotation in North America is 60 Hz but the main frequency used
by many other countries is 50 Hz. Note that if the speed of the coil is doubled
then the frequency and the magnitude of the emf will both increase.
Root Mean Square (rms) Quantities
• An alternating current varies sinusoidally and can
be represented by the equation
I = I0sin(ωt)
• where I0 is the maximum current called the peak
current as shown in Figure 1223 for a 50 Hz mains
supply.
• In commercial practice, alternating currents are
expressed in terms of their root-mean-square
(r.m.s.) value.
• The r.m.s. value of the alternating current that
produces the power is equal to the d.c. value of the
direct current.
• For the maximum value in a.c., the power dissipated is
given by
P = V0 sin(ωt) × I0 sin(ωt) = V0I0 sin2(ωt)
• This means that the power supplied to the resistor in
time by an alternating current is equal to the average
value of I 2R multiplied by time.
P = I 2ave × R = I02 R sin2 ωt
• Because the current is squared the, the value for the
power dissipated is always positive.
• The value of sin2ωt will therefore vary between 0 and 1. Therefore
its average value = (0 + 1) / 2 = 1/2
• Therefore the average power that dissipates in the resistor equals:
or
• So the current dissipated in a resistor in an a.c. circuit that varies
between I0 and -I0 would be equal to a current I0 /√2 dissipated in a
d.c circuit. This d.c current is known as r.m.s. equivalent current to
the alternating current.
RMS Current and Voltage
• It can be shown that:
• Provided a circuit with
alternating current only
contains resistance
components, it can be
treated like a direct
current circuit.
The Transformer
• A useful device that makes use of
electromagnetic induction is the ac
transformer as it can be used for increasing or
decreasing ac voltages and currents.
The Transformer
• It consists of two coils of wire known as the primary and
secondary coils. Each coil has a laminated (thin sheets
fastened together) soft iron core to reduce eddy currents
(currents that reduce the efficiency of transformers). The coils
are then enclosed with top and bottom soft iron bars that
increase the strength of the magnetic field.
• A typical circuit for a simple transformer together with the
recommended circuit symbol.
Transformers
• When an ac voltage is applied to the primary coil, an ac
voltage of the same frequency is induced in the
secondary coil. This frequency in North America is 60
Hz.
• When a current flows in the primary coil, a magnetic
field is produced around the coil. It grows quickly and
cuts the secondary coil to induce a current and thus to
induce a magnetic field also. When the current falls in
the primary coil due to the alternating current, the
magnetic field collapses in the primary coil and cuts
the secondary coil producing an induced current in the
opposite direction.
Transformers
• The size of the voltage input/output depends
on the number of turns on each coil. It is
found that
• Where N = the number of turns on a
designated coil and I is the current in each
coil.
• It can be seen that if Ns is greater than Np then
the transformer is a step-up transformer. If the
reverse occurs and Ns is less than Np it will be a
step-down transformer.
• If a transformer was 100% efficient, the power
produced in the secondary coil should equal to
the power input of the primary coil. In practice
the efficiency is closer to 98% because of eddy
currents.
• This means that if the voltage is stepped-up by a
certain ratio, the current in the secondary coil is
stepped-down by the same ratio.
Transformer Java Applet
• http://micro.magnet.fsu.edu/electromag/java/transformer/
Tsokos, Section 5.8