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In this section we use the linear algebra we have developed to find the voltages and currents in a simple electrical circuit, such as the one shown in Figure 1. There are many reasons why this might be necessary; in this example we need to know the current flowing through the lamp to tell whether or not it will glow. Too little current will have no visible effect; too much current will cause the lamp to burn out. We will apply a few physical laws relating the voltages and currents in a circuit, turn these laws into systems of linear equations, and solve the equations for the voltages and currents.

**Current, Voltage, and Resistance.** We will use three physical
quantities in our analysis of electrical circuits: current, voltage, and resistance.
*Current* is the flow of electrical charge from one place to another. Electrons
flowing through a wire or through some other electronic device comprise a
current. *Voltage* is a difference in electric potential that makes electrons flow.
Voltage is sometimes called *electromotive force* because it is like a “force”
that moves electrons. *Resistance* is a property of the device through which
the electron current flows. The lower the resistance of a device, the more
easily current can flow through the device.

The analogy of water flowing through pipes can help you develop intuition about electrical circuits. In this analogy, electrical current corresponds to the flow rate of water. Voltage corresponds to the pressure that forces the water to flow, and resistance is the friction of flow. A small pipe would impede the flow of water more than would a large pipe, so the small pipe would correspond to a higher resistance. While this analogy can be helpful, keep in mind that electricity is not water. All analogies break down at some point.

We measure electrical current in *amperes*. The standard symbol for current is

A point in a circuit where several devices are connected together is called a *node*. The conservation law for current says that “what flows in must flow out of a node,” a principle known as *Kirchhoff*'*s current law*. Kirchhoff's current law states that *the sum of all currents leaving a node is zero*. In this law, a current entering the node is considered to be a negative current leaving the node.

Voltage is measured in *volts* and is usually written as

In an electrical circuit, one node is usually chosen as a *reference node* and is considered to have a voltage of zero. Then the voltage at every other node is measured with respect to the reference node. This saves us the trouble of always specifying pairs of nodes for voltage measurements and marking + and – signs for each voltage. Other names for the reference node are *common* and *ground*.

A *constant voltage source* is a device that always forces the voltage between its two terminals to be a constant value. In Figure 1 the circle at the left represents a constant voltage source of 5 volts, so that the voltage at the upper *always* exactly 5 volts higher than the voltage at the lower (-) end. A voltage source is something like a battery, but idealized. Real batteries do not maintain a constant output voltage under all conditions.

Resistance is measured in *ohms* and is denoted by *Ohm's law*, also called the *resistor law*, relates the voltage and current in a resistor. For the resistor shown in Figure 2, with reference directions assigned to

Note that current flows from + to - through the resistor.

Ohm's law and Kirchhoff's current law are the only principles we need to write equations that will allow us to find the voltages and currents in the resistive circuit of Figure 1. We begin by choosing a reference node and assigning variables to the voltages at every other node (with respect to the reference node). These choices are shown in Figure 3.

The constant voltage source forces

Next we write equations by applying Kirchhoff's current law to each
node in the circuit (except the reference node and

For current leaving through the 300 ohm resistor, the voltage is *leaving* the node
labeled

For the 100 ohm resistor, we can write

According to Kirchhoff's current law, the sum of these three leaving currents is zero:

Notice that when we wrote the equation for the node labeled

Note that this time it is

Equations 2, 6, and 7 give us a system of three equations in
the three unknown variables

(MATLAB) Use MATLAB to solve Equation 8. You should find

What is the determinant of the coefficient matrix

We can determine the current flowing through the lamp from

The visible effect will, of course, depend on the lamp. Let us assume that the specifications for our lamp indicate that 0.05 ampere or more is required before it will glow, and more than 0.075 ampere will cause it to burn out. In this case, our circuit would not make the lamp glow.

(MATLAB) Write and solve equations for the circuit in Figure 4. What are the voltages at the nodes labeled

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