Circuit Models

Energy

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

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.

$P = IV$

Ideal elements

Idealized circuit representation

• Lines have no resistance and uniform voltage

• Real elements (heaters, filaments, batteries) are represented as ideal

  models

Example

• Simple light

• Sketch physical representation

• Sketch wiring diagram

• Sketch schematic

• Apply mathematical models

Ideal Voltage Source

• An ideal voltage source can deliver any amount of current while

  holding its voltage constant.

• 

Ideal Resistor

An ideal resistor obeys Ohm's Law, $V=IR$

Ohm's Law

$V = IR$ The current in a resistor is proportional to the voltage across it. The constant of proportionality is the resistance.

Ideal Current Source

• An ideal current source delivers a fixed amount of current no matter

  the voltage at its terminals

• 

Kirchhoff's Current Law

• The sum of currents flowing into a node must equal the sum of the

  currents leaving the node at any instant.

• Water flow or automobile traffic provide some intuition.

Kirchhoff's Voltage Law

• The sum of the voltages around any loop of a circuit at any instant is

  zero.

• Hiking a trail provides some intuition.

Equivalent resistances

• We can use Ohm's Law and Kirchoff's Laws to determine an equivalent

  resistance to model a network of resistors.

Resistors in series

• Resistors in series have the same current flowing through each one

  

Resistors in parallel

• Resistors in parallel have the same voltage across their terminals

• 

Relationship to energy

• One coulomb of charge raised to a potential of one volt gains one

  Joule of energy

Relationship to power

• 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.

Voltage and current directions

Delivered power

• Power delivered by an ideal voltage source

Consumed power

• Power consumed by an ideal resistor

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

Tariff

• The utility charges proportional to the amount of kWh consumed

• Some consumers are also charged according to the maximum power

  observed

Wire resistance

• Resistivity - property of the material - intensive

• Resistance - property of the wire - extensive

Wire resistance

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

• 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

Units

• To get proper units of resistance in ohms

• Resistivity is expressed in Ohm/meter

• Length in meters

• Area in square meters