Lecture 19: Electric Power Distribution

Electric Power Distribution

Introductory Question

• Electric power reaches our city via high-voltage transmission lines.

• What fraction of the electric charges traveling on those transmission lines reaches your home?

  • About 1%

  • About 0.01%

  • Exactly 0%

Observations about Electric Power Distribution

• Household electricity is alternating current (AC).

• Household voltages are typically

  • 120V

  • 240V

• Power is distributed at much higher voltages.

• Power transformers are common around us.

• Power substations exist but are harder to find.

Questions about Electric Power Distribution

1. Why isn’t power transmitted at low voltages?

2. Why isn’t power delivered at high voltages?

3. What is “alternating current” and why use it?

4. How does a transformer transfer power?

Question 1: Why Isn't Power Transmitted at Low Voltages?

Electric Power and a Wire

• The resistance in a wire converts a current’s electrical power into thermal power, which is wasted energy.

  • Power lost can be represented as:
    $ext{Power wasted} = ext{current} imes ext{voltage drop in wire}$

  • Using Ohm's Law, the voltage drop in the wire can be expressed as:
    $ext{Voltage drop} = ext{resistance} imes ext{current}$

  • Thus,
    $ext{Power wasted} = ext{resistance} imes ext{current}^2$

• Doubling the current in the wire results in quadrupling the wasted power, emphasizing the inefficiency of sending high currents through long wires.

Concept Question

• Two long wires will carry electrical power most efficiently from a generator to a community if the voltage difference between the wires is:

  • Large and the current they carry is large.

  • Large and the current they carry is small.

  • Small and the current they carry is large.

  • Small and the current they carry is small.

Large Currents are Wasteful

• The goal of a power distribution system is to transmit substantial electric power to a city with minimal losses.

  • Power transmitted equation:
    $ext{Power transmitted} = ext{current} imes ext{voltage drop at city}$

  • Power wasted equation:
    $ext{Power wasted} = ext{resistance} imes ext{current}^2$

• A good balance of energy efficiency versus energy transmission can be achieved by:

  • Using a small current in the wires.

  • Maximizing the voltage drop at the city.

  • Utilizing low-resistance wires.

Power Lines

• The distances that power lines must cover often range from tens to hundreds of miles.

• The resistance of any wire increases with length, leading to significant energy losses in power lines.

• To minimize resistance:

  • Use thicker wires which have less resistance.

  • Use copper, recognized as a very good conductor.

  • Power lines are often constructed using thick cables made of numerous wound strands of copper wiring.

Question 2: Why Isn't Power Delivered at High Voltages?

High Voltages are Dangerous

• When large voltage drops are available:

  • Strong electric fields are present.

  • Charges experience enormous forces.

  • Currents tend to flow through unexpected paths.

• High-voltage electrical power in a home introduces risks:

  • Spark hazard.

  • Fire hazard.

  • Shock hazard.

The Voltage Hierarchy

• To manage these risks, electric power distribution employs a hierarchy:

  • High voltage for transmission across long distances.

  • Medium voltage for distribution within cities.

  • Low voltage for usage in neighborhoods and homes.

• Transformers are crucial as they transfer voltage and current between circuits.

Question 3: What is “Alternating Current” and Why Use It?

Direct Current (DC)

• In Direct Current (DC):

  • The voltage remains steady and unchanging over time.

  • Resulting currents are also steady and unchanging.

  • This type of electricity is commonly provided by batteries.

  • Difficulties arise in transforming DC voltages for efficient transmission.

Alternating Current (AC)

• Alternating Current (AC) entails:

  • The voltage of the power delivery wires alternates regularly, cycling between +V and -V in a harmonic manner.

  • Based on this regular variation, the currents also typically alternate periodically.

Characteristics of AC

• In the U.S., the alternating voltage completes 60 cycles per second (60 Hz) and reverses the flow of current every 1/120 second.

• AC power is:

  • Naturally produced by electric generators in power stations.

  • Easier to transform to higher or lower voltages, making it highly efficient for power distribution.

AC and Transformers

• AC has negligible effects on basic electric devices (e.g., lightbulbs, space heaters, toasters) as their operation relies on simple Ohmic resistance, which does not discriminate current flow direction.

• However, AC can present challenges for electronic devices (e.g., computers, televisions, sound systems), which may require a steady current to function.

  • This may lead to rapid flickering of devices (often too fast for the human eye) and results in electronic noise and other peculiar electrical effects.

AC and Transformers

• AC facilitates the simple use of transformers, which can transfer power between circuits:

  • From a low-voltage circuit to a high-voltage circuit.

  • From a high-voltage circuit to a low-voltage circuit.

Question 4: How Does a Transformer Transfer Power?

Electromagnetism (Version 2)

• Magnetic fields are produced by:

  • Magnetic poles (though free poles don’t seem to exist),

  • Moving electric charges,

  • Changing electric fields.

• Electric fields originate from:

  • Electric charges,

  • Moving magnetic poles,

  • Changing magnetic fields.

Electromagnetic Induction

• Moving poles or changing magnetic fields generate electric fields that drive currents through conductors; this phenomenon is known as magnetic induction of current.

• Changing magnetic fields can induce currents in conducting loops (circuits).

  • Example: Moving a magnet inside a coil induces current flow, as indicated by movement in an ammeter needle.

Lenz’s Law

• Lenz’s law states: "When a changing magnetic field induces a current in a conductor, the magnetic field from that current opposes the change that induced it."

Examples of Electromagnetic Induction

• When a credit card's magnetic strip is swiped through a reader, the movement alters the magnetic field, inducing current in the reader's coil.

  • The machine decodes the information based on these current pulses.

• In a microphone, sound waves induce movement in a diaphragm which, through a magnet's motion near a coil, generates an AC current that encodes sound information.

  • The frequency of the induced current matches the sound wave's frequency, with amplitude correlating to the sound wave's amplitude.

• In an electric generator, a coil of wire rotates in a magnetic field, inducing voltage rise in the coil.

  • Torque is required to keep coils turning, with magnetic induction being the mechanism for conversion of mechanical energy to electrical energy, primarily yielding AC voltage.

• In an electric motor, a magnetic field causes a wire current to rotate.

  • Voltage drives current through the coil, enabling it to act as an electromagnet that experiences torque from the magnetic field, thus performing work.

  • The rotor continues to flip the coil’s dipole, sustaining torque.

Transformer Mechanism

• A transformer can produce an alternating current in one circuit that induces an alternating current in a secondary circuit through induction without transferring charges between the circuits.

  • Electrical power is transferred by induction, not conduction.

Transformer Operation

• An input AC voltage causes AC current to flow through the primary circuit, generating a magnetic field.

  • Since the current is alternating, so is the magnetic field, which is changing in nature.

• An iron core enhances and channels the magnetic field.

• A coil within the second circuit, influenced by the changing magnetic field from the iron core, generates an electric field within it, inducing voltage and prompting current in the secondary coil.

  • This induced AC voltage and current will differ from the input, having been transformed.

Current and Voltage Relationship in Transformers

• Transformers must adhere to conservation of energy principles, such that: $ext{Power in} = ext{Power out}$

  • Expressed as:
    $V<em>p I</em>p = V<em>s I</em>s$

• Since power is a product of voltage and current, one transformer can exchange voltage for current or vice versa.

  • When one increases, the other decreases accordingly.

• It is vital to understand that there always exists some dissipative losses in real transformers due to resistive heating, implying the power output is lesser than the input.

  • For practical problem-solving, these losses can often be disregarded.

Example Problem 11.2-1

• A transformer receiving 1 Ampere at a voltage of 120 Volts. If the secondary “steps-up” the voltage to 240 Volts, the question arises: How much current flows in the secondary circuit?

Step-Up Transformer

• A step-up transformer:

  • Contains more turns in its secondary coil than in the primary.

  • Each coil’s turns contribute to the induced voltage.

  • More turns yield a higher overall voltage, leading to a smaller current at that higher voltage in the secondary circuit.

Step-Down Transformer

• A step-down transformer features:

  • Fewer turns in its secondary coil.

  • Experiences proportionately less overall voltage increase.

  • Allows larger current at lower voltage to flow in the secondary circuit.

Power Distribution System

• A step-up transformer raises the voltage for efficient long-distance transmission, whereas a step-down transformer reduces voltage to ensure safe delivery to communities and homes.

Current and Voltage Ratio in Transformers

• The factor by which a transformer modifies voltage between the primary and secondary coils is equivalent to the turns ratio of those coils:
  $rac{V<em>p}{V</em>s} = rac{N<em>p}{N</em>s}$

• This relationship can also translate to primary versus secondary current ratios:
  $rac{I<em>s}{I</em>p} = rac{N<em>p}{N</em>s}$

Example Problem 11.2-2

• A transformer acquires 1 Ampere at a voltage of 120 Volts; when the secondary steps up to 240 Volts with the primary coil having 50 turns, how many turns does the secondary coil possess?

Introductory Question (Revisited)

• Electric power reaches cities via high voltage transmission lines. However, what fraction of the electric charges traveling on those lines actually reaches the home?

  • Answer: Exactly 0%

  • Note: Electric power from the power station travels through multiple transformers to reach the home without the actual charges moving through those transformers.

Magnetic Energy

• A transformer receives electrical power at its primary and transfers that power to the secondary without direct contact between the two.

  • The magnetic field through the iron core is the connecting feature conveying the energy.

• Conclusion: The magnetic field itself stores and carries energy!

Magnetic Energy Density

• The magnetic energy stored by a magnetic field in a volume of space is given by: $U = rac{B^2}{2 imes ext{μ}_0}$

  • Units: J/m³

Example Problem 11.2-3

• How much energy is stored in the Earth’s magnetic field, which has a strength of $B$Tesla inside a room measuring 5×10^{-4} cubic meters (dimensions: 15 feet by 15 feet by 8 feet high)?

Example Problem 11.2-4

• It requires 333 kiloJoules of energy to melt 1 kilogram of ice (latent heat of ice). A kilogram of ice occupies a volume of 1 liter. What strength must a magnetic field be to contain that much energy within a liter?

  • This would generate a field 56,000 times greater than the Earth’s magnetic field.

  • Note: Such strong fields can only be achieved with large superconducting electromagnets, and simply having the field alone would not guarantee melting ice unless there is a mechanism converting that magnetic energy into heat.

Summary about Electric Power Distribution

• Electric power is transmitted at high voltages.

• Electric power is delivered at low voltages.

• Transformers facilitate the transfer of power between circuits.

• Transformers require AC power to function efficiently.

• The overall power distribution system is based on alternating current (AC) principles.