Networks Worksheet (Faithful to Original)

Worksheet 1.4 Networks Worksheet (Faithful to Original)

Basic laws for electrical circuits.

This two-page worksheet was generously donated to the sample article by Virgil Pierce at a CuratedCourses workshop in August 2018.

Theorem 1.4.1. Ohms Law.

The current through a resistor is proportional to the ratio of the Voltage to the Resistance

\begin{equation*} I = \frac{V}{R}\qquad\text{or for our purposes,}\qquad I R = V \end{equation*}

Theorem 1.4.2. Kirchoffs Current Law.

The sum of the currents in a network meeting at a point is zero.

\begin{equation*} \sum_{k=1}^n I_k = 0 \end{equation*}

Example 1.4.3. Kirchoff's Current Law.

For the circuit below \(I_1 + I_2 = I_3 \text{.}\)

Theorem 1.4.4. Kirchoffs Voltage Law.

The sum of the voltages around any closed circuit (or subcircuit) is zero.

\begin{equation*} \sum_{k=1}^n V_k = 0 \end{equation*}

Kirchoffs Current Law and Kirkoffs Voltage Law combined with Ohms Law gives for any circuit of resistors and sources a linear system that may (or may not) determine the currents.

1.

For the simple network pictured, calculuate the amperage in each part of the network by setting up a system of linear equations for the amperages.

2.

Compare it to a parallel circuit network. Calculate the amperage in each part of the network by setting up a system of linear equations for the amperages.

3.

Now for a more complicated network. Calculate the amperage in each part of the network by setting up a system of linear equations for the amperages.

4.

Now generalize these ideas to a context outside of electrical circuits. Consider the network of streets given in the diagram (with one-way directions as indicated).

A traffic engineer counts the hourly flow of cars into and out of this network at the entrances. They get (EB = East Bound; WB = West Bound):

Table 1.4.5. Estimated hourly traffic flow for the road network

	EB Winooski	WB Winooski	Shelburne St	Willow	Jay
into	50	400	0	10	50
out of	55	390	20	15	30

Use a variable for each segment inside of the network and set up a system of linear equations restricting the flow. Solve the system. Note that you should not get a unique solution as traffic should be able to flow through the network in various ways.