# Series DC Circuits

Now we will look at some simple DC circuits.  Before we begin, we will define some symbols that will be used on schematic diagrams. A schematic diagram is a diagram that shows all the devices in a circuit as well as all interconnections between those devices.

First, we need a symbol for a source of DC current. There are three that we will use at various points in the lecture. They are:

Battery:

The positive terminal of the battery is indicated by the longer line.

Ideal Voltage Source:

An ideal voltage source can provide a specified value of voltage at any required current.

Ideal Current Source:

An  ideal current source can provide a specified current at any voltage.

Next, we need a symbol for a resistor:

Wires interconnecting circuit components are shown as lines.

Here is a simple circuit consisting of a battery and a resistor:

Note that we define current as flowing from positive to negative. This may seem a bit odd, but unfortunately, Ben Franklin thought that electricity was made up of positive charge carriers rather than electrons, and as a result, he defined electrical current as flowing from the positive to the negative terminal of a source. By the time that the end of the nineteenth century rolled around and engineers discovered that electricity in wires was actually made of negative charge carriers that really flowed in the opposite direction to what Franklin thought, the direction of current flow was too well established and engineers have kept their convention. The flow of current from positive to negative is called "conventional current flow" to distinguish it from the actual electron flow, which is in the other direction. In my opinion, electrical engineers should just admit that Franklin was wrong, redefine current flow in schematics to match reality and move on. Oddly enough, physicists did just that shortly after the electron was discovered, about 100 years ago, which can make things very confusing for physicists who cross over into electrical engineering (like me!).

We will use conventional current flow because everyone else does and we will ignore the fact that it is wrong.

Something else that is noteworthy about this circuit is that the same current, I, flows through all the components in the circuit. A circuit that has this property is called a "series circuit". No matter how may resistors, batteries and other components are in a series circuit, the same current flows through all of them. The picture below shows a more complex series circuit:

The current flowing through all the resistors and the battery is I, so this is a series circuit. As we shall see shortly, if we know the values of all the resistors, and the voltage of the battery, we can compute the current by using Ohm's Law.

Let us examine a simpler circuit, one with only two resistors:

Intuitively, it should seem that the total resistance offered by resistors R1 and R2 in series should just be the sum of their resistances, that is:

RTOTAL = R1 + R2

In fact, this is exactly what happens in a series circuit. The total resistance in a series circuit is just the sum of the individual resistances, regardless of how many there are. Once we have the total resistance, we can find the current from Ohm's Law:

I = V/RTOTAL

Before we go any farther, let's work out some examples to become more familiar with series circuits.

Example 1: Three resistors are connected in series with a battery as shown below. What is the current flowing in the circuit?

Solution:

The total resistance in the circuit is the sum of the three individual resistors:

RTOTAL = R1 + R2 + R3 = 3 + 7 + 2 = 12 ohms

Now we use Ohm's Law to find the current:

I = V/R = 9V/12ohms = 0.75 A = 750 mA.

A current of 750 mA flows through the battery and all three resistors.

Example 2:

Find the battery voltage in the circuit below, if the current is 2 A.

Solution:

If we look at our Ohm's Law pie, we know that:

V = IR.

All we need to do is find the total resistance of the circuit and multiply it by the current in order to get the battery voltage. The total resistance is:

RTOTAL = R1 + R2 + R3 + R4 = 12+47+39+82 = 180 ohms.

Now we can use Ohm's Law:

V = IR = (2A)(180ohms) = 360 V.

Before we leave series circuits, we need to explore the concept of voltage drop. Consider the circuit below, which contains two resistors in series.

It should be obvious that the voltage between points A and C is equal to the battery voltage, V.  The voltage between points A and B, is the "voltage drop" across R1, VR1. By similar reasoning, the voltage between points B and C is called the voltage drop across R2, VR2. How do we find these two voltage drops?

In order to determine VR1and VR2 we only need to find the value of the series current I. Once we have I, the voltage drops across he resistors are given by:

VR1=IR1

VR2=IR2

Now things are going to get very interesting. We know that I is given by:

I = V/RTOTAL = V/(R1+R2).

Thus the voltage across each resistor is:

VR1=IR1 = {VR1/(R1+R2)}

VR1=IR2 = {VR2/(R1+R2)}

Watch what happens if we add them up:

VR1+VR2 = {VR1/(R1+R2)}+{VR2/(R1+R2)} = (V)(R1+R2)/(R1+R2) = V

As we noted earlier, the sum of the voltage drops across the individual resistors is equal to the source voltage.

We define the voltage drop across a resistor as positive, because resistors consume power. Since a battery, current source or voltage source supplies power to a circuit, its voltage drop is negative. We can use this fact to write the following equation:

VR1+VR2 + (-V) = V - V = 0.

In plain English, the sum of the voltage drops around a series circuit (also called a loop) is zero. This is known as as Kirchoff's Voltage Law( KVL), and was named in honor of the German physicist Gustav Kirchoff, who discovered it.

KVL can help us simplify complex circuits, as we shall see later on.

That is pretty much all there is to know about series circuits. We can compute the series current from the supply voltage and the total resistance. We can compute the voltage dropped across each resistor from its resistance and the series current, using Ohm's Law.

One last thing I want to show you before we move on is the concept of an equivalent circuit. Here is a question:  are the two circuits below equivalent?

The answer is yes, because they both  have the same current for a given value of battery voltage. The simpler circuit on the right is called the equivalent circuit. It behaves the same way as the original circuit on the left, but is simpler to understand, because it has only one component other than the voltage source. As we proceed to more complex circuits, we will frequently simplify them by replacing parts or all of the complex circuit with a simpler equivalent circuit.

Next we move onto parallel DC circuits.

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