Energy

We will focus on the elements of electric theory that are most closely related to the expenditure of energy in electrical systems.

Charge

  • charge is the net sum of electrons and protons
  • if there are more electrons than protons, an item has a negative charge
  • Charge is measured in coulombs
  • A coulomb is approximately 6.2 \cdot 10^{18} electrons of charge.
  • We also measure charge in ampere-hours when we are talking about batteries.

Current

  • current is the movement of electrons in a conductor
  • This has units of coulombs per second or amperes.
  • Water flow is a good mental model.

Voltage

  • when charge moves to a higher or lower voltage it gains or loses energy.
  • Voltage is the joules per coulomb of energy gained or lost
  • Height is a good mental model
  • One coulomb of charge raised to a potential of one volt gains one Joule of energy

Energy in a circuit element

  • Any circuit element with a given voltage and current is consuming power equal to the product of the current and the voltage
  • If the voltage is in volts, and the current is in amperes, the resulting power is in units of watts.

P = IV

To find the energy from this power we multiply power by time.

E = P \cdot t

This energy is expressed in watt-hours or kilowatt-hours.

  • A watt times an hour is a watt-hour.
  • A kilowatt times an hour is a kilowatt-hour. (This is the unit of electricity on your bill.)
  • A watt times a second is a joule.

We can also convert this to joules.

Power Units

  • Voltage is energy per charge
  • Current is charge per time
  • Voltage times current has units of energy per time or power
  • \frac{energy}{charge} \cdot \frac{charge}{time}
  • P = VI

Power

P = VI The power dissipated by a device is equal to the voltage across it multiplied by the current flowing through it.

Resistance

Ohms Law

For many materials, the relationship between current and voltage is linear.

V = IR

For a given voltage, less current flows as the resistance rises.

Wire resistance

We can calculate the resistance of a conductive object if we know its dimensions and properties.

R = resistivity \cdot \frac{length}{area} R = \rho \frac{l}{A}

  • The resistance of a wire is proportional to
    • the resistivity of the material
    • the length of the wire
  • It is inversely proportional to
    • the cross-sectional area

Wire resistance

Electrical energy converted to heat in a wire is usually considered wasted energy.

This energy loss is seen as a reduced voltage across the load or device.

  • Resistivity - property of the material - intensive
  • Resistance - property of the wire - extensive

Units

  • To get proper units of resistance in ohms
  • Resistivity is expressed in Ohm/meter
  • Length in meters
  • Area in square meters

Energy Dissipation in a Resistor

As current flows through a resistor, the potential energy of the charges is converted to heat.

The power converted to heat in the resistor is the voltage across the resistor multiplied by the current through the resistor.

P = IV

The voltage and current through the resistor is related by Ohm’s Law V=IR. So, I can substitute the voltage in the equation and get

P = I (IR) P = I^2R

The power is proportional to the square of the current. This squared relationship is important for many circuit designs. When the current is in amps and the resistance is in ohms, the power is in watts.

We can do the same thing but use Ohm’s Law to substitute for the current and we get

P = \frac{V^2}{R}

Note that the power is not linear with the current or voltage but has a squared relationship.

Energy

  • The energy consumed is equal to the power multiplied by the time.
  • The energy unit we use is kWh (kilowatt-hour)
  • A 1 kW device consuming power for 1 hour uses 1 kWh of electricity

Paying for Electricity (Tariffs)

  • The utility charges proportional to the amount of kWh consumed
    • This is called a volumetric charge ($/kWh).
  • Some consumers are also charged according to the maximum average power observed over an interval of time (often 15 or 30 minutes)
    • This is called a peak charge or demand charged.

Similarities to Thermal Conduction

The electrical resistance of a block of material is given by

\textrm{resistance} = \frac{\textrm{resistivity of material}\cdot\textrm{length of material}}{\textrm{cross sectional area of material}}

R = \frac{\rho \cdot l}{A}

The UA of a block of material has the formula

\textrm{UA value} = \frac{\textrm{thermal conductivity of material} \cdot \textrm{cross sectional area}}{\textrm{thickness of material}}

UA = \frac{k \cdot A}{t}

These are basically reciprocals of each other.