electricity and magnetism 29 alternating currents and power transmission chapter 29 alternating...
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Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
Chapter 29
Alternating Currents and Power Transmission
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
29.1 Alternating current
• In a direct current (d.c.) circuit, electric current flows in one
direction only.
Alternating currents and alternating voltages
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• In an alternating current (a.c.) circuit, current reverses its
direction periodically.• In general, a.c. are produced by alternating e.m.f.s (voltages)
from a.c. sources.
Square waveform Sinusoidal waveform
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• A sinusoidal alternating voltage can be expressed as
where V0 is its peak value and is its angular frequency.
• The angular frequency is
related to the frequency f
and the period T of the
voltage by
V = V0 sin t
V0
Tf
π2π2
π2T
−V0
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
Resistive circuit
• In a purely resistive circuit,
V and I are in phase.
Example 29.1 Checkpoint (p.418) O
V = V0 sin t I = I0 sin t
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• However, what we really concern is the average power of an
a.c. supplied to a resistive load, which is given by
where the symbol denotes the time-average of the
enclosed quantity.
Root-mean-square (r.m.s.) values
Experiment 29.1
RIRIP 22
• Applying P = VI, for sinusoidal
voltage and current, the
instantaneous power supplied to
the load is
P = V0I0 sin2t
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• For a sinusoidal voltage, we have
• The root-mean-square (r.m.s.)
value of a current I is defined as
• Hence, for sinusoidal voltages, we have .
• Since , we have
2rms II
20
2
2
1II
tII sin20
2
20
rms
II
RIP 2rms
RIP 2
R.m.s. values of alternating currents of square waveform
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• In other words,
• The average power can be expressed in the following forms:
• For sinusoidal alternating currents and voltages, we have
The root-mean-square (r.m.s.) value of an a.c. is the
steady d.c. which delivers the same average power
as the a.c. to a resistive load.
RIP 2rms
R
VP rms rmsrmsIVP
20
rms
II
20
rms
VV
Example 29.2Checkpoint (p.423) O
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
29.2 Transformer
• A transformer is a device that can change the value of an
alternating voltage.
A transformer for notebook computers
Substations contain a lot of transformers.
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• A change in current through a coil induces a voltage in
another nearby coil due to the change in magnetic field
through the latter coil.• This effect is called mutual induction.
Structure and working principle of a simple transformer
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• The primary coil of a transformer is connected to an a.c.
source and the secondary coil gives the output voltage.• If the magnetic flux through each turn of the coils in a
transformer is the same, there is perfect flux linkage (i.e. no
flux leakage) between the coils.
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• The voltage ratio and turns ratio of a transformer, with perfect
flux linkage and negligible coil resistance, are related by
Voltages and currents in transformers
p
s
p
s
N
N
V
V
Experiment 29.2
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• Apply Faraday’s law, the alternating magnetic flux through
the coils induce e.m.f.s in the coils.
• With the iron core, the magnetic flux through each turn of the
two coils is the same.
tN
ppp
tN
sss
tt
sp
sp p
s
p
s
N
N
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• For a step-up transformer, Ns > Np, and so Vs > Vp.
• For a step-down transformer, Ns < Np, and so Vs < Vp.
• The efficiency of a transformer can be expressed as
% 100% 100powerinput
poweroutput efficiency
pp
ss VI
VI
Transformer
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• An ideal transformer has 100% efficiency. For such a
transformer, we have
• Since , we have
• Hence, for an ideal transformer, whenever the voltage is
stepped up (or stepped down), the current through the
secondary coil decreases (or increases) by the same factor.
p
s
s
p
V
V
I
I
p
s
p
s
N
N
V
V
p
s
s
p
N
N
I
I
Example 29.3Checkpoint (p.431) O
Example 29.4
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29 Alternating Currents and Power Transmission
• Transformers have very high efficiency, but are never ideal.
Energy loss in transformers and practical transformer designs
improvements made in practical transformers
reasons for energy loss
use high-qualitymagnetic materials tomake the core
continuous magnetization and demagnetization of the iron core
use thick copper wires to make the coils
resistance in the coils
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
improvements made in practical transformers
reasons for energy loss
use a laminated coreeddy currents in the iron core
Checkpoint (p.433) O
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
29.3 Transmission and distribution of electricity
• One of the problems in
transmitting electricity is the
power loss in the transmission
lines due to the heating effect of
current.
Overhead transmission lines
Power loss in transmission lines
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• If a transmission line carries a current I and has a resistance
R, the power dissipated as heat is given by
• The lower the current, the lower
the power dissipated.• Using transformers, a.c. voltage
can be stepped up easily and
efficiently, and the current can be
stepped down accordingly.• Hence, alternating current is
used to transmit mains electricity
in order to reduce the power
loss.
P = I2R
Experiment 29.3
Overhead transmission lines
Example 29.5
Electricity and MagnetismElectricity and Magnetism
29 Alternating Currents and Power Transmission
• A grid system is a transmission and distribution network of
mains electricity.
Grid system in Hong Kong
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29 Alternating Currents and Power Transmission
• The voltage of the electricity generated at a power station is
stepped up before transmission and stepped down
successively at substations in populated areas.
Checkpoint (p.441) O
Schematic diagram of the grid system in Hong Kong
Birds on transmission lines