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Definition of parallel and series circuit
If we want to connect two circuit components together in a circuit, then we have two ways of doing it, in series and in parallel.
A series circuit consists of components that are connected in series, i.e. they are connected one after the other in a sort of "train" of components.
A parallel circuit consists of components that are connected in parallel. For this, we split the circuit in two and we put the components side by side on multiple different branches, after which we merge the branches again.
So what is exactly the difference between these two types of circuits, and how do we see it in the circuit diagrams?
Difference between series and parallel circuits
In the image below, we can see very clearly what the difference is between electrical components that are connected in series or in parallel.
The difference between a series and a parallel circuit is in what configuration the components are connected to each other.
Series and parallel circuits formulas and rules
The three basic quantities corresponding to circuits are voltage \(V\), current \(I\), and resistance \(R\). In general, the voltage can be seen as the 'force' pushing the charged particles through the circuit, the current can be seen as how many charged particles can pass through the circuit, and the resistance can be seen as a narrowing of the road or a small door: the larger the resistance, the smaller the door the charged particles have to pass through.
With these comparisons, we can make sense of Ohm's law if we state it as follows:
\[I=\dfrac{V}{R}\]
If we increase the push (increase the voltage) or make the road wider (decrease the resistance), more particles will be able to pass (the current will increase).
Series circuit rules
For a series circuit, we are in the situation of the figure below, in which two (or more) resistors with resistances \(R_1\) and \(R_2\) are connected in series over a voltage \(V\).
The rules of series circuits are:
- The total resistance over all of the resistors is the sum of the resistances of the individual resistors. Thus, the total resistance \(R_{tot}\) in the circuit above is \(R_{tot}=R_1+R_2\).
- The current \(I\) is the same through all of the resistors. With this knowledge, we can now calculate this current using Ohm's law, \(\text{voltage}=\text{current}\times \text{resistance}\), to be \(I=\dfrac{V_{tot}}{R_{tot}}=\dfrac{V}{R_1+R_2}\).
- As a result of the previous point, we can calculate the voltage over the individual resistors (\(V_1\) and \(V_2\)) to be \(V_1=IR_1\) and \(V_2=IR_2\). Thus, the voltage over the resistors is proportional to their resistances, and the sum of the individual voltages is equal to the total voltage.
Remember these rules by making sense of them! Here's a way to look at the rules and formulas for series circuits.
- If you are a charged particle, you have to go through both barriers. The total barrier you experience is the sum of the two individual barriers because you are slowed down twice on your way.
- If the current is not the same everywhere, then there will be a build-up or a loss of charge somewhere, which is impossible in an ideal circuit. One batch of charged particles cannot overtake another, because there is only one branch. Adding more resistors will make the total barrier larger, so the current will be smaller.
- This rule follows Ohm's law.
Suppose the two resistors are actually two lamps. The power over a component in an electrical circuit can be calculated by \(P=VI\), so the power over lamp 1 is:
\[\begin{aligned} P_1&=V_1 I \\ P_1&=IR_1 I \\ P_1&=I^2R_1\end{aligned}\]
and we can do a similar calculation for lamp 2. We see that the lamp with a larger resistance uses more power from the series circuit.
Parallel circuit rules
For a parallel circuit, we are in the situation of the figure below, in which two (or more) resistors with resistances \(R_1\) and \(R_2\) are connected in parallel over a voltage \(V\).
A parallel circuit, Wikimedia Commons CC BY-SA 4.0.
The rules of parallel circuits are:
- The voltage over all branches is the same, namely the full voltage \(V\), and the current through the individual resistors can be calculated using Ohm's law.
- The total current is the sum of the currents over the individual resistors. Therefore, the total resistance is smaller than the resistance of the individual resistors.
Here's a way to make sense of these rules and formulas for parallel circuits.
- Both resistors are directly connected to the electrical cell with both ends, so the voltage over them must be the voltage of the cell.
- The charged particles now have multiple ways of getting to the other end of the cell: there are two doors side-to-side, and particles can choose for which one they want to queue up. Thus, more charged particles can be let through the doors.
- A current is just how many charged particles pass a checkpoint per second. This quantity is the same everywhere, so the sum of the individual currents must be the total current. It is as if two streams of runners join together on one large street.
Suppose the two resistors are actually two lamps. The power over lamp 1 is then \(P_1=VI_1=V\dfrac{V}{R_1}=\dfrac{V^2}{R_1}\), and we can do a similar calculation for lamp 2. We see that the lamp with a smaller resistance gets more power from a parallel circuit.
We can calculate the total resistance of a parallel circuit. We calculate:
\[R_{tot}=\dfrac{V_{tot}}{I_{tot}}=\dfrac{V}{I_1+I_2}=\dfrac{V}{\frac{V}{R_1}+\frac{V}{R_2}}=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\]
In words, we can say that the total resistance of a parallel circuit is equal to the inverse of the sum of the inverses of the individual resistances. We see that, indeed, the total resistance is smaller than the resistance of the individual resistors. This results in a larger total current than if there were only one branch. This means that creating parallel branches in a circuit will reduce the resistance and increase the current through the circuit. This makes sense within our analogy with the doors.
Series circuit and parallel circuit example and calculations
Let's look at a hard example that combines series and parallel circuits. See the figure below for the setup. In most practical situations, you can determine the voltage V1 yourself and choose the resistors you use, and your job is to find the other quantities. This is what we will do.
Notice how the voltages over resistances are measured by an instrument that is connected in parallel to the resistances. This is because the voltage over a parallel circuit is the same over all branches, so the voltage the voltmeter measures is the same as the voltage over the resistance it is connected to in parallel!
Suppose we connected the voltmeter in series with the resistance that we want to measure the voltage over. Then the voltage will be separated between the voltmeter and the resistance and the voltmeter would only be measuring the voltage over itself, which will be close to the total voltage supplied by the battery because voltmeters are made to have extremely high resistances.
Notice also how the current through the circuit is measured by an instrument that is connected in series to the resistances that we want to measure the current through. This is because the current through a series circuit is the same everywhere, so the current the ammeter measures is the same as the current through the resistances it is connected to in series!
Suppose we connected the ammeter in parallel with the resistance that we want to measure the current through. Then the current will be divided between the ammeter and the resistance and the ammeter would only be measuring the current through its own branch and not through the branch of the interesting resistance at all! This current will be very high because ammeters are made to have extremely low resistances.
The question gives the values of \(V_1\), \(R_1\), \(R_2\), and \(R_3\).
We see that we have a parallel circuit, but one of the branches of the parallel circuit contains two resistors that are connected in series. Let's give the total resistance of resistors \(R_2\) and \(R_3\) the name \(R_{below}\), and the total voltage over \(R_{below}\) the name \(V_{below}\).
From the rules of parallel circuits, we know that \(V_2=V_1\) and \(V_{below}=V_1 \). We also know that the sum of the individual currents is the total current, so \(A_2+A_3=A_1\).
According to the rules of series connections, we know that \(R_{below}=R_2+R_3\) and that \(V_{below}=V_3+V_4\). So far so good.
We can now use Ohm's law to conclude that
\[A_2=\dfrac{V_2}{R_1}=\dfrac{V_1}{R_1}\]
And
\[A_3=\dfrac{V_{below}}{R_{below}}=\dfrac{V_1}{R_2+R_3}\]
The total current is then
\[A_1=A_2+A_3=\dfrac{V_1}{R_1}+\dfrac{V_1}{R_2+R_3}\]
We use Ohm's law again to discover what the voltages \(V_3\) and \(V_4\) are:
\[V_3=A_3R_2=\dfrac{V_1R_2}{R_2+R_3}\]
And
\[V_4=A_3R_3=\dfrac{V_1R_3}{R_2+R_3}\]
We have now successfully expressed the unknown quantities in terms of the known quantities, so we are done! In the process, we used the rules from both series and parallel circuits, because this circuit is a combination of the two.
Identifying series and parallel circuits
It is pretty easy to identify a series circuit because series circuits only have one wire that the current can go through: there are no extra branches in series circuits. On the other hand, a parallel circuit is a circuit in which all the components are directly connected to both sides of the voltage source. The difficulties come when we need to identify components within a combined circuit. In short, series-connected components are back-to-back and parallel-connected components are side-to-side.
We can view a continuous bit of wire as a node: different nodes are separated by components within a circuit. Two components are connected in parallel if (and only if) they share two nodes, i.e. they are connected to the same two nodes.
Two components are connected in series if (and only if) they share exactly one node that does not branch out between the two components. It's a good exercise to try to identify the nodes in all the figures in this article. You can do this by giving them names or colours. Then see if you come to the right conclusions about how all the components are connected based on this method! See the example below of implementing this method.
The resistors both share the blue and the red node, so they are connected in parallel, adapted from by Paulgwilliamson, Wikimedia Commons CC BY-SA 4.0.
Uses of series and parallel circuits
In general, choosing between series and parallel circuits is simple. We connect a switch in series with a lamp such that cutting off the current at the switch by flicking it will also cut off the current through the lamp. We also connect a resistor in series with diodes such that the current through the diodes is not too high, preventing the overheating of the diodes.
On the other hand, we connect headlights in cars in parallel, such that if one of the branches of the circuit fails, the other branch still carries a current. This way, you will still have one working headlight if the other one fails: a parallel circuit adds a safety factor. We also connect household appliances in parallel, so they are all under the same voltage. By having switches connected in series with the individual appliances, we can manipulate the current through the individual appliances. If all appliances were connected in series, we would have to choose between everything on and everything off!
Series and Parallel Circuits - Key takeaways
- A series connection has components one after the other.
- A parallel connection has components parallel to each other.
- Rules for connections in series:
- The total resistance over all of the resistors is the sum of the resistances of the individual resistors.
- The current is the same through all of the resistors.
- The voltage over the resistors is proportional to their resistances, and the sum of the individual voltages is the total voltage.
- Rules for connections in parallel:
- The total resistance over all of the resistors is smaller than the resistance of the individual resistors and it is given by \(R_{total}=\dfrac{1}{\frac{1}{R_1}+\frac{1}{R_2}+\ldots + \frac{1}{R_n}}\).
- The total current is the sum of the currents over the individual resistors.
- The voltage over all resistors is the same, namely the total voltage.
- To identify series and parallel connections, you can use the method of nodes.
- Whenever a situation calls for a safety factor, you generally see parallel connections, just like in household appliances. Series connections are used for switches and prevention of components overheating.
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Frequently Asked Questions about Series and Parallel Circuits
What is a series circuit and a parallel circuit?
In a series circuit, the components are connected in series, so one behind the other. In a parallel circuit, the components are connected in parallel, so side by side on different branches of the circuit.
What is the difference between a series and a parallel circuit?
In a series circuit, the components are connected one behind the other, while in a parallel circuit, the components are connected side by side on different branches of the circuit. For series circuits, the current of every component is the same but the voltage over the components can differ according to their resistance values. For parallel circuits, the voltage over every branch of the circuit is the same but the current might differ between branches according to their resistance values.
What is an example of a series circuit and a parallel circuit?
An example of a series circuit is a circuit containing two lamps with one behind the other. An example of a parallel circuit is a circuit that branches into two branches and then merges again, but that has a lamp on each of the two branches.
What is the formula for series circuit and parallel circuit?
In series circuits, we have one total resistance R and a total voltage V. The current I through the circuit is then I=V/R everywhere, and the voltages over a component with a resistance Rc can be found from the formula Vc=IRc.
In parallel circuits, we have the same voltage V over every branch. The current Ib through a branch with resistance Rb is then Ib=V/Rb.
How do you determine if something is a series circuit or a parallel circuit?
To determine if a circuit is in series of in parallel, you need to look at how the components are connected. If the components are one behind the other, it is a series circuit, but if the components are all side by side on different branches of the circuit, then it is a parallel circuit.
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