If the equations are not displayed correctly, please use the desktop view
You will know whether resistors in parallel or series simply from its terminal connection. The connection of parallel resistors is their terminals are connected together respectively from resistor to resistor.
If series resistors only have one path for electric current, parallel resistors have multiple paths for electric current since they have at least one node. This fulfills Kirchhoff’s laws where the sum of electric currents entering a node is equal to the electric currents leaving that node.
Parallel Circuit Definition
This parallel resistor is different from the series resistor where there is only one straight path passing the resistors. In the parallel resistor there will be many paths from head to head and tail to tail.
From this term, the parallel resistors are used as a current divider, while the series resistor is a voltage divider.
Since the current is divided to multiple paths or branches, the current for each branch may be different from each other. But, the voltage drop for each resistor will be the same for each other.
This concludes that
Resistors in parallel may have different current through each of them but always have the same voltage drop across each of them.
If we are asked to define parallel circuit, the best answer is:
Parallel circuit definition is a circuit where the elements are connected together to the same nodes and produce more than one current path connected to the same voltage source.
If you are still confused about how we make a parallel resistive circuit, we can find it below.
Parallel Circuit Diagram
The example below is the simple parallel resistor in a circuit. We use three resistors R1, R2, and R3. All of the three resistors are connected together between A and B.
From the explanation above, the voltage drop across the resistors in the parallel connection will be the same for each other and equal to the voltage source.
Different from series resistive circuits where the equivalent resistance is the sum of all the resistors, the parallel resistor is calculated differently. We are using the reciprocal of resistance (1/R) for each resistor added together.
Parallel Resistors Formula
Just like we mentioned above,
The equivalent resistance of parallel resistors is the sum of reciprocals of each resistor.
If the resistors connected together in parallel have the same value then it will be very easy.
Just as we mentioned above, we can look at the example below:
If two resistors connected in parallel have the same resistance then the equivalent resistance, Req is half the resistance of one resistor. Hence the equivalent resistance of two resistors connected in parallel if they have the same resistance is R/2. If there are three resistors in parallel then the equivalent resistance is R/3, and so on.
Aside from series resistors, this type of circuit is the most common type of electric circuit. For analyzing a parallel resistor circuit we can use the same method for the series resistor, using Kirchhoff’s laws and Ohm’s law.
Consider the circuit below,
where two resistors are connected in parallel and therefore have the same voltage across them.
From Ohm’s law
Applying KCL at node a gives the total current i as
Substituting equations for voltage into current, we have
where Req is the equivalent resistance of the resistors in parallel :
The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum.
Above is the simplest parallel resistor equation we can use every time we need it.
Must be noted that the Req equation above only works for two resistors in parallel.
We can expand the equivalent parallel resistors resistance to the general case of a circuit with N resistors in parallel. The equivalent resistance is
Take a note that Req is always smaller than the resistance of the smallest resistor in the parallel connection. If R1 =R2 = … = RN = R, then
For example, if 4 resistors with 100 Ω are connected in parallel, their equivalent resistance is 25 Ω.
Keep in mind that,
The equivalent resistance of parallel resistors is always less than the smallest resistor connected to that network. Hence, the equivalent resistance, Req, will decrease every time we have additional parallel resistors.
How to Find Current in a Parallel Circuit
Because the current in a parallel circuit is based on its resistance, now we will learn about how to find current in a parallel circuit.
The currents I1, I2, …, In entering the parallel path of resistors depend on the resistance of that branch. The total current, IT is the sum of the currents in parallel branches. If the resistance between branches is equal then the currents will be also equally divided.
If R1 = R2 then the I1 = I2 = 0.5IT. It means the total current IT is divided equally for two branches. If the R1 has different resistance from R2 then we need to calculate the I1 and I2 differently. Even if the voltage across the branches are equal, the current may differ according to Ohm’s law.
For parallel circuit example let us see the circuit below and try to find all the current in parallel,
Since the R1 and R2 have different values, then the currents I1 and I2 are guaranteed to have different values. Remember one of the Kirchhoff laws?
Kirchhoff’s current laws states:
The total current leaving a node is equal to the current entering that same node.
The total current in the circuit can be expressed as:
After that, we will use Ohm’s law to calculate the current entering each branch through the resistors. The I1 is the current entering R1 while the I2 is the current entering R2. The voltage source Vs has 12V and we get:
And we get the total current,
To clarify this, we use Ohm’s law to calculate IT from Vs and Req.
The equivalent resistance Req is
Then the total current IT is
Hence this clarifies our calculations.
We conclude that
Current Divider Equation
After learning about parallel resistors, we will learn about the current divider rule.
If you have learnt about voltage divider, the current divider is somewhat similar and somewhat not similar.
A voltage divider equation can solve voltage drop calculation regardless of the number of resistors. But for the current divider, it is quite different.
The current divider formula can be used for two resistors. We will find the reason why the current divider rule for 3 resistors in parallel needs us to solve with a different approach.
Will give us current divider equation
which shows the total current i is shared by the resistors in inverse proportion to their resistances.
This is known as the principal of current division, and the circuit above is known as current divider.
Take a note that the larger current flows through the smaller resistance.
Suppose one of the resistors in circuit is zero, say R2 = 0; so R2 is a short circuit, as can be seen in circuit below.
From the current divider equation, R2 = 0 implies that i1 = 0, i2 = i. This means that the entire current i bypasses R1 and flows through the short circuit R2 = 0, the path with least resistance.
When a circuit is short-circuited as can be seen above, take note that :
The equivalent resistance Req = 0
The entire current flows through the short circuit.
For another extreme example where R2 = ∞ , that is, R2 is an open circuit as can be seen in the circuit below.
The current still flows through a path with least resistance, R1.
The current divider equation becomes
In general, if a current divider has N conductors in parallel with source current i, the Nth conductor will have current
It is very convenient to combine resistors in series and parallel into single equivalent resistance Req.
Such equivalent resistance must have the same values of current and voltage as the original network at the terminal.
After reach this point, we can conclude that,
Current divider rule for 3 resistors in parallel needs us to find the equivalent resistance, parallel voltage drop, and finally the corresponding currents.
Parallel Resistors Example
Let us review the example below for a better understanding. We will try more than 3 resistors in parallel for an example.
Find Req for the circuit below.
6 Ω and 3 Ω in parallel
1 Ω and 5 Ω in series
2 Ω and 2 Ω in series
6 Ω and 4 Ω in parallel
Three resistors in series
The equivalent resistance for parallel resistors is