The word 'resistance' has many different meanings - air resistance is the force that slows down objects moving through the air, your body has a resistance to many illnesses thanks to your immune system, and a group of people may form a resistance to a political regime. In the case of electric circuits, resistance is the opposition to current flow. The resistance of a circuit can be increased or decreased by adding circuit components called resistors. They can either be added in a series connection or a parallel connection. In this article, we will explore both of these types of connections and their consequences in electric circuits.
Adding Resistors in Series and Parallel
The circuits we will be considering in this article will only contain batteries, wires and resistors. Batteries are the power source that drives the current around a circuit and the resistors provide resistance to this current.
Electrical resistance is a measure of an electrical component's opposition to current flow. It is measured in Ohms, \( \mathrm\Omega \).
The circuit symbol for a resistor is shown in figure 1. This is actually called a fixed resistor and there are various other types of resistors. The total resistance of a circuit depends on how the different resistors are connected together.
Fig. 1 - The circuit symbol for a resistor.
Resistors can be combined in series, which is when they are added one after the other - they are on the same branch of a circuit. In this setup, we say that they only share one node.
A node is a region on a circuit between two circuit elements.
Fig. 2 - Resistors in series are connected on the same branch.
Resistors can also be added in parallel, which is when they are added across from each other - they are on different branches of a circuit. In this case, the resistors share nodes at both ends.
Fig, 3 - Resistors in parallel are on different branches but share the same nodes.
Voltage across Resistors in Series and Parallel
We can learn more about resistors in series and parallel by considering the voltage drops across them when a current flows.
Voltage is the energy transferred per unit of charge that is passing.
The total voltage drop across resistors connected in series is equal to the sum of the voltage drops across each resistor. For example, consider a circuit with a battery and two resistors of the same resistance \( R \) connected in series. If the voltage supplied by the battery is \( V \) then the voltage drop across each resistor will be \( \frac V2 \).
Fig. 4 - The total voltage drop across resistors in series is equal to the sum of the voltage drops across each resistor. The voltage drops can be measured by using voltmeters.
On the other hand, the same voltage will be supplied to each resistor for a circuit with a battery and two resistors connected in parallel. If the battery's voltage is \( V \), then the voltage drop across each resistor will also be \( V \).
Fig. 5 - All of the resistors in a parallel connection take all of the voltage from the battery.
These two cases generalize easily to any number of resistors. For resistors in series, the supplied voltage is given by
$$V=V_1+V_2+...+V_N,$$
where the subscripts indicate the resistor. For resistors in parallel:
$$V=V_1=V_2=...=V_N.$$
Current Through Resistors in Series and Parallel
The amount of current flowing through resistors is different when they are connected in series and when they are connected in parallel.
A flow of charge carriers in an electrical circuit is called a current. It is measured in units of Amps, \( \mathrm A \).
For resistors connected in series, the same current flows through all of them, as there are no junctions between the resistors where the current can split.
$$I=I_1=I_2=...=I_N.$$
For resistors connected in parallel, the current splits between them. For \( N \) resistors in parallel, the total current through them is given by
$$I=I_1+I_2+...+I_N.$$
Resistors in Series and Parallel Formulae
To find the formulae for the resistance of resistors in series and parallel, we need to use Ohm's law, which states that, for an Ohmic conductor, the relationship between its voltage, current and resistance is
$$V=IR.$$
Fixed resistors are Ohmic conductors and obey Ohm's law. For resistors in series, the supplied voltage is given in terms of the voltage drops across the resistors by
$$V=V_1+V_2+...+V_N.$$
The total resistance of the resistors will be given by the rearranged version of Ohm's law:
$$R_T=\frac VI.$$
The current is the same through each resistor for a series combination, so
$$R_T=\frac{V_1}{I}+\frac{V_2}{I}+...\frac{V_N}{I}.$$
Each term is simply the resistance of each resistor, so the total resistance for a series combination is equal to the sum of the resistances of the resistors!
$$R_T=R_1+R_2+...R_N.$$ The same process can be repeated to find the total resistance of a parallel combination. For resistors in parallel, the current is shared between them and the total current is equal to
$$I=I_1+I_2+...+I_N.$$
The total current will be equal to the voltage supplied divided by the total resistance:
$$I=\frac {V}{R_T}.$$
The voltage drop across each resistor is equal to \( V \) so the current through each resistor can be expressed in a similar way. For example, for the first resistor:
$$I_1=\frac{V}{R_1},$$
so the following expression can be written (both terms are equal the total current):
$$\frac {V}{R_T}=\frac{V}{R_1}+\frac{V}{R_2}+...+\frac{V}{R_N}.$$
The \( V \) on either side cancels out and leaves the equation for the resistance of a combination of parallel resistors.
$$\frac 1{R_T}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_N}.$$
Difference Between Resistors in Series and Parallel
There are several key differences between resistors in parallel and series connections:
- The total voltage drop across a series connection is equal to the sum of the individual voltage drops. There is the same voltage drop across each resistor in a parallel connection.
- The current is the same through each resistor in a series connection. The current is shared between the resistors in a parallel connection.
- Adding more resistors in series increases the total resistance because the current has to flow through each resistor. Adding more resistors in parallel decreases the total resistance because there are more paths for the current to pass through.
Rules for Resistors in Series and Parallel
The rules for resistors in series and parallel connections are summarized in the table below and should be remembered.
| Series | Parallel |
| The total resistance is equal to \( R_T=R_1+R_2+...R_N \). | The total resistance is equal to \( \frac 1{R_T}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_N} \). |
| The current is the same through the resistors. | The current is shared between the resistors. |
| The voltage drop is the sum of the voltage drops of the resistors. | The voltage drop across each resistor is the same. |
| Adding more resistors increases the resistance. | Adding more resistors decreases the resistance. |
Equivalent Resistors in Series and Parallel
Consider two resistors in parallel that each have a resistance of \( 2R \), as shown in the circuit below.
Fig. 6 - Two \( 2R \) resistors placed in parallel have the same resistance as one resistor of resistance \( R \).
The total resistance of the combination can be calculated using the parallel resistors formula with \( N=2 \):
$$\frac 1{R_T}=\frac{1}{R_1}+\frac{1}{R_2}.$$
Both of the resistances are \( R \) so this becomes
$$\frac 1{R_T}=\frac{1}{2R}+\frac{1}{2R}=\frac 2R.$$
Rearranging this expression gives the total resistance as \( R_T=R \). This shows that two resistors with a resistance \( 2R \) connected in parallel have the same total resistance as a single resistor of resistance \( R \).
Examples of Resistors in Series and Parallel
The formulae for resistors in series and parallel connections can be useful in practice problems. In the following practice problems, the resistance of the battery will be assumed to be negligible.
A battery supplies a voltage of \( 6\,\mathrm V \) to a circuit with a \( 5\,\mathrm\Omega \) resistor and a resistor of unknown resistance. The circuit is shown below. If the current flowing through the circuit is \( 1\,\mathrm A \), what is the resistance of the second resistor?
Fig. 7 - The current is measured using an ammeter in a circuit with a resistor of unknown resistance.
We can find the total resistance of the circuit by using Ohm's law,
$$V=IR,$$
which can be rearranged to
$$R=\frac VI.$$
The voltage supplied by the battery is \( 6\,\mathrm V \) and the current in the circuit is \( 3\,\mathrm A \), so the total resistance is
$$R=\frac{6\,\mathrm V}{1\,\mathrm A}=6\,\mathrm\Omega.$$
We have learned that the total resistance of resistors in series is the sum of their resistances, so the unknown resistance will be equal to the total minus the other resistance:
$$6\,\mathrm\Omega-5\,\mathrm\Omega=1\,\mathrm\Omega.$$
The two resistors in the circuit are reconnected in parallel to each other. What is the current through each resistor? What is the total resistance in the circuit?
Fig. 8 - The current through each resistor changes when they are connected in parallel.
In parallel circuits, each branch receives all of the voltage from the battery. Ohm's law rearranged for current is
$$I=\frac VR.$$
The voltage across both resistors is \( 6\,\mathrm V \), so the current through the \( 5\,\mathrm\Omega \) resistor is
$$I_5=\frac{6\,\mathrm V}{5\,\mathrm\Omega}=1.2\,\mathrm A$$
and the current through the \( 1\,\mathrm\Omega \) is
$$I_1=\frac{6\,\mathrm V}{1\,\mathrm\Omega}=6\,\mathrm A.$$
The total resistance of the circuit can be found from the parallel resistors formula:
$$\frac 1{R_T}=\frac 15+\frac 11=0.2\,{\Omega}^{-1}+1\,{\Omega}^{-1}=1.2\,{\Omega}^{-1}.$$
This leads to
$$R_T=\frac {1}{1.2\,{\Omega}^{-1}}=0.83\,\mathrm\Omega.$$
This value can also be obtained by dividing the battery voltage by the total current. The total current is equal to the sum of the currents in the branches
$$I=1.2\,\mathrm A+6\,\mathrm A=7.2\,\mathrm A$$
and hence
$$R_T=\frac VI=\frac{6\,\mathrm V}{7.2\,\mathrm A}=0.83\,\mathrm\Omega.$$
Combining Resistors in Series and Parallel
Throughout this article, it may have seemed that we have been assuming that there is one resistor on each branch in parallel combinations. However, even if there is more than one, the series connection formula can be used to find the overall resistance of multiple resistors on a branch so that they can be treated as one resistor when using the parallel resistor formula.
Calculate the total resistance of the circuit in figure 9.
Fig. 9 - Resistors can be joined together in parallel and in series with other resistors.
In the first branch of the circuit, a \( 4\,\mathrm\Omega \) resistor and a \( 6\,\mathrm\Omega \) resistor are connected in parallel. We can use the parallel resistors formula to find their total resistance:
$$\frac 1{R_T}=\frac 1{R_1}+\frac 1{R_2}.$$
Let's call the total resistance of these resistors \( R_P \).
$$\frac1{R_P}=\frac 1{4\,\mathrm\Omega}+\frac 1{6\,\mathrm\Omega}=\frac5{12\,\mathrm\Omega}$$
and
$$R_P=\frac{12}{5}\,\mathrm\Omega=2.4\,\mathrm\Omega.$$
The resistance of the first branch, \( R_{B1} \), is equal to this added to the resistance of the other resistance on the branch, which is \( 2\,\mathrm\Omega \), so
$$R_{B1}=2.4\,\mathrm\Omega+2\,\mathrm\Omega=4.4\,\mathrm\Omega.$$
The total resistance of the second branch is \( 5\,\mathrm\Omega \). We can find the total resistance of the circuit, \( R_C \), with the parallel resistors formula:
$$\frac 1{R_C}=\frac 1{4.4\,\mathrm\Omega}+\frac 1{5\,\mathrm\Omega}=0.43\,\mathrm\Omega,$$
which leads to
$$R_C=\frac{1}{0.43}\,\mathrm\Omega=2.3\,\mathrm\Omega.$$
Resistors in Series and Parallel - Key takeaways
- Electrical resistance is a measure of an electrical component's opposition to current flow. It is measured in Ohms, \( \mathrm\Omega \).
- Resistors in series are added to a circuit one after the other and they only share one node.
- Resistors in parallel are added to a circuit across from each other and they share both nodes.
- A node is a region on a circuit between two circuit elements.
- Resistors in series have the same current passing through them.
- Resistors in parallel each have the same voltage drop across them.
- Adding more resistors in series increases the total resistance.
- Adding more resistors in parallel decreases the total resistance.
References
- Fig. 1 - "Resistor circuit diagram", StudySmarter Originals
- Fig. 2 - "Resistors in series", StudySmarter Originals
- Fig. 3 - "Resistors in parallel", StudySmarter Originals
- Fig. 4 - "Series resistors circuit", StudySmarter Originals
- Fig. 5 - "Parallel resistors circuit", StudySmarter Originals
- Fig. 6 - " Parallel resistors circuit example", StudySmarter Originals
- Fig. 7 - "Series resistors circuit example", StudySmarter Originals
- Fig. 8 - "Parallel resistors circuit with ammeters", StudySmarter Originals
- Fig. 9 - "Circuit with resistors in parallel and series", StudySmarter Originals
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