Linear Algebra, 2020a

Topic
Analyzing Networks
The diagram below shows some of a car’s electrical network. The battery is on
the left, drawn as stacked line segments. The wires are lines, shown straight and
with sharp right angles for neatness. Each light is a circle enclosing a loop.
12V
Brake
Actuated
Switch
Off
Lo
Light
Switch
Dimmer
Hi
Dome
Light
Door
Actuated
Switch
L
R
L
R
L
R
L
R
L
R
Brake
Lights
Parking
Lights
Rear
Lights
Headlights
The designer of such a network needs to answer questions such as: how much
electricity flows when both the hi-beam headlights and the brake lights are on?
We will use linear systems to analyze simple electrical networks.
For the analysis we need two facts about electricity and two facts about
electrical networks.
The first fact is that a battery is like a pump, providing a force impelling
the electricity to flow, if there is a path. We say that the battery provides a
potential. For instance, when the driver steps on the brake then the switch
makes contact and so makes a circuit on the left side of the diagram, which
includes the brake lights. Once the circuit exists, the battery’s force creates a
flow through that circuit, called a current, lighting the lights.
The second electrical fact is that in some kinds of network components the
amount of flow is proportional to the force provided by the battery. That is, for
each such component there is a number, its resistance, such that the potential