# Simple circuits - series and parallel connection of resistors

**Level 2**requires school mathematics. Suitable for pupils.

## Table of contents

- Construction of a simple circuit Here you will learn the necessary ingredients for the simplest circuit.
- Series connection of resistors Here you will learn what characterizes a series circuit and how to determine the total resistance of such a circuit.
- Parallel connection of resistors Here you will learn what characterizes a parallel circuit and how to determine the total resistance of such a circuit.

If you want to understand how a complicated circuit of a smartphone, a computer or any other electronic device works, you first need to understand the structure and functioning of simple circuits. Here you will learn the very first and simplest basics. In particular, you will learn about parallel and series connections of resistors, which are used, for example, to build *ammeters* (current measuring device) and *voltmeters* (voltage measuring device).

Probably the most important ingredient for analyzing circuits and building them yourself is Ohm's law:

**Ohm's law as a formula**

Ohm's law states how current and voltage are related. You can use it to find out what current flows through a wire when you apply a certain voltage to the conductor. Or the other way around. Ohm's law formula contains:

\(U\) is the

**voltage**between any two points. In terms of a circuit, it is the voltage between two points in a circuit. Measured in volts (\(\mathrm{V}\)).\(R\) is the constant

**electrical resistance**. For example, it can be the resistance of a component in a circuit. Measured in Ohms (\(\Omega \)).\(I\) is the

**electric current**flowing through a circuit. This is measured in Amps (\(\mathrm{A}\)).

## Construction of a simple circuit

For example, a simple circuit may look like this:

For a circuit to work at all, you need a **voltage source** - this is used to generate a voltage \(U\) called **source voltage**. A voltage source has a positive and a negative terminal and could be, for example, a battery or a generator. As you know, a voltage *always* refers to two points! Therefore, the next question you should ask yourself is:

*Between WHICH points is the source voltage applied?*

In our circuit (Illustration 2), the source voltage \(U\) is applied *between the two ends* of the **resistor** that has the resistance \(R\). This resistor can represent any component, such as a lamp or a hair dryer, because they have a certain electrical resistance.

What does it mean that 'there is a voltage between the ends of a resistor'? It means that at one end of the resistor is the positive (+) pole and at the other end is the negative (-) pole. So there is a *charge difference* between the two ends of the resistor. However, nature wants to equalize this charge difference. Consequently, the charges flow from one pole to the other, creating an electric current \(I\). This current then can power the resistor (e.g. a lamp).

A voltage source constantly tries to maintain the charge difference so that the current does not drop. Of course, a battery will eventually run down, so the source voltage will decrease over time. An AC socket, on the other hand, will provide you with a voltage continuously as long as you pay your electricity bill. If the voltage at the resistor is not maintained, it will naturally decrease to zero over time because the charge difference decreases over time.

What are these dark lines in illustration 2, which start from the negative and positive pole and go to the resistor? You can think of these lines as **wires** (conductors). One wire connects the positive terminal to one end of the resistor and the other wire connects the negative terminal of the voltage source to the other end of the resistor.

Note that in this case these wires are thought to be *ideally conducting*. That is: they have *no or very low resistance*! Of course, for high quality circuits, it may be important to consider the wire resistance as well, even if it is quite small (less than 1 Ohm). To do this, you just have to add another resistor to your circuit, which then represents the wire resistance.

Of course, we can also connect several components and thus several resistors to the circuit. For example, let's take three resistors with \(R_1\), \(R_2\) and \(R_3\). There are two ways we can connect these resistors: Either in *series* or *parallel*.

## Series connection of resistors

Let's connect them first in series. By 'in series' we mean that we connect them one after the other. This could look like this:

This is a so-called **series connection of resistors**. This is characterized by the fact that the positive pole of the voltage source is connected to only one end of *a single* resistor, in our case to one end of \(R_1\). Subsequent resistors \(R_2\) and \(R_3\) are then connected in series like in a chain. The negative pole of the voltage source is also connected to only one end of *a single* resistor, namely to the end of the chain, to \(R_3\).

There are also other characteristics by which you can identify whether you have a series circuit in front of you. One characteristic is the current \(I\) that flows through the resistors. The current \(I\) flows from the positive pole to the negative pole. Since all the resistors are connected in series, the same current \(I\) flows through *all* three resistors. There are no branches or similar that the charges of the current could take to get from positive to the negative pole.

What about the voltage across the resistors? According to Ohm's law \( U = R \, I\) we know that there is a voltage \(U\) between the two ends of the resistor \(R\) when a current \(I\) flows through it. We also know that the current \(I\) through all three resistors is the SAME. Since the resistors can in general have different restistances, we expect by Ohm's Law that there will be a different voltage across each resistor:

Voltage \( U_1 = R_1 \, I \) is across the first resistor.

Voltage \( U_2 = R_2 \, I \) is across the second resistor.

Voltage \( U_3 = R_3 \, I \) is across the third resistor.

When a charge travels from the positive terminal to the negative terminal of the voltage source, it passes through the voltage \(U\). If you look at the circuit in illustration 4, you will see that \(U\) is applied between one end of \(R_1\) and one end of \(R_3\). So \(U\) is the *total voltage* applied to the entire chain of three resistors. So for the charge to pass through the voltage \(U\), it must pass through the voltage \(U_1\), then the voltage \(U_2\), and then the voltage \(U_3\). So we can write the total voltage as follows:

**Total voltage in a series connection**

What is the **total resistance** \(R\) (also called **equivalent resistance**) of a series circuit? We can find this out if we insert Ohm's law into the equation 2

for the total voltage and also for the individual voltages \(U_1\), \(U_2\) and \(U_3\):

**Rewritten voltage sum with Ohm's law**

R \, I &~=~ R_1 \, I ~+~ R_2 \, I ~+~ R_3 \, I \end{align} $$

Now we can simply cancel out the current \(I\) and get:

**Total resistance of a series circuit**

In principle, we can simplify the series circuit by combining the three resistances \(R_1\), \(R_2\) and \(R_3\) into a total resistance \(R\) and drawing only a single resistor element in the circuit. In this way, we can significantly simplify circuits and make them easier to understand:

## Parallel connection of resistors

Another way to connect resistors is to connect them in *parallel*. Here you connect the positive pole of the voltage source *to the three ends of all three* resistors and the negative pole to the other three ends of the resistors.

Thus, in a parallel connection, a charge passes through the same voltage \(U\) when it travels through resistor \(R_1\), \(R_2\) or through \(R_3\). By connecting resistors 'in parallel' like this, we ensure that there is *the same* voltage \(U\) across all three resistors. And this is already the first important difference to a series connection.

When resistors are connected in parallel, the same voltage drops across each resistor.

What about the current through the resistors? If you look at the circuit (illustration 8), you will see that the total current \(I\) splits at the branches. You can also understand this with the help of Ohm's law. The same voltage \(U\) is applied to all three resistors. However, the resistors have generally different resistances. Therefore, the currents must be generally different, because they depend on the chosen resistance values.

The current \( I_1 = \frac{U}{R_1} \) flows through the first resistor.

The current \( I_2 = \frac{U}{R_2} \) flows through the second resistor.

The current \( I_3 = \frac{U}{R_3} \) flows through the third resistor.

When different resistors are connected in parallel, different currents flow through the resistors.

The only thing we need to figure out is the total resistance (equivalent resistance) \(R\) in a parallel circuit. Again, Ohm's law \( U = R \, I\) helps us here. The total current \(I\) is the sum of the individual currents that flow through the branches into the resistors:

**Total current of a parallel circuit**

Now we have to express the currents with the resistances. The total current \( I \) can be expressed by the total voltage and total resistance using Ohm's law: \( \frac{U}{R} \). As you know, the same voltage \(U\) is applied to the resistors. Therefore, we rewrite the individual currents as follows:

**Total current of parallel circuit rewritten with Ohms law**

If you now divide both sides of the equation by \(U\), you get the relationship between the total resistance \(R\) and the individual resistances of a parallel circuit:

**Total resistance for a parallel circuit**

This is another difference between a parallel and series circuit. While in a series circuit the total resistance is simply the sum of the individual resistances, in a parallel circuit the total resistance is a bit more complex.

The advantage of a parallel circuit over a series circuit is that if one component (e.g. a lamp) breaks down, other components connected in parallel still work. This is exploited, for example, in electrical wiring in residential buildings.

If your kitchen stove explodes, the computer in your room won't go out. With a series connection, which is often used for fairy lights, only one non-functioning lamp is enough and the entire fairy light chain stops working...

Three resistances \(100 \, \Omega \), \(50 \, \Omega \) and \(200 \, \Omega \) are connected in parallel and a total voltage of \(12 \, \mathrm{V} \) is applied.

What is the

**total resistance**\( R \) of the circuit?What is the

**total current**\( I \)?Which

**currents**\(I\), \(I_1\), \(I_2\) and \(I_3\) flow through the resistors?

The reciprocal of the total resistance \(R\) of a parallel circuit is given by the equation 10

:

**Example of the reciprocal of the total resistance of a parallel circuit**

&~=~ \frac{1}{100 \, \Omega} ~+~ \frac{1}{50 \, \Omega} ~+~ \frac{1}{200 \, \Omega}\\\\

&~=~ 0.035 \, \frac{1}{\Omega} \end{align} $$

This is the reciprocal of the total resistance \(R\). To obtain the total resistance, we must form the reciprocal of it:

**Example for the reciprocal value of the total resistance**

&~=~ 28.6 \, \Omega \end{align} $$

With this example you can see that the total resistance of a parallel circuit is *much smaller* than the total resistance of a series circuit with the same individual resistances!

To get the total current \(I\), we have to divide the total voltage \(U = 12 \, \text{V} \) by the total resistance \(R\) according to Ohm's law:

**Example of the total current of a parallel circuit**

&~=~ \frac{12 \, \text{V} }{ 28.6 \, \Omega} \\\\

&~=~ 0.42 \, \text{A} \end{align} $$

As you know, the same source voltage \(U = 12 \, \mathrm{V} \) is applied to all three resistors. With the given resistance, Ohm's law tells you what current flows through this single resistor. The following current flows through the first resistor:

**Example of the first current through a parallel resistor**

&~=~ \frac{12 \, \text{V}}{100 \, \Omega} \\\\

&~=~ 0.12 \, \text{A} \end{align} $$

Through the second resistor:

**Example of the second current through a parallel resistor**

&~=~ \frac{12 \, \text{V}}{50 \, \Omega} \\\\

&~=~ 0.24 \, \text{A} \end{align} $$

Through the third resistor:

**Example of the third current through a parallel resistor**

&~=~ \frac{12 \, \text{V}}{200 \, \Omega} \\\\

&~=~ 0.06 \, \text{A} \end{align} $$

Of course, the total current \(I = 0.42 \, \mathrm{A} \) must be obtained when you add the three individual currents together.