Resistors in Parallel Formula

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Parallel Resistor Calculator

The parallel resistor calculator is a tool for determining the equivalent resistance of a circuit with up to five resistors in parallel. Read on or jump to the series resistor calculator.

Resistors in parallel formula

A parallel circuit is characterized by a common potential difference (voltage) across the ends of all resistors. The equivalent resistance for this kind of circuit is calculated according to the following formula:

1/R=1/R1​+1/R2​+…+1/Rn

where:

  • RRR is the equivalent parallel resistance
  • R1,R₂,…,RnR_1, R₂, …, R_nR1​,R₂,…,Rn​ are the resistances of individual resistors numbered 1,2,…,n1, 2, …, n1,2,…,n.

The units of all values are Ohms (symbol: Ω). 1 Ohm is defined as electrical resistance between two points that, when applied with a potential difference of 1 volt, produces a current of 1 ampere. Hence, 1 Ω = 1 V / 1 A or, in SI base units, Ω = kg·m²/(s³·A²).

How to calculate parallel resistance

The parallel resistor calculator has two different modes. The first mode allows you to calculate the total resistance equivalent to a group of individual resistors in parallel. In contrast, the second mode allows you to set the desired total resistance of the bunch and calculate the one missing resistor value, given the rest.

To keep it simple, we only show you a few rows to input numbers, but new fields will magically appear as you need them. You can input up to 10 resistors in total.

Let’s look at an example for the second, slightly more complicated, mode:

  1. Select Calculate missing resistor under Mode.
  2. Now input the total resistance you want your circuit/collection of resistors to have.
  3. Start by introducing the values of the resistors you already know (new fields will appear as needed).
  4. The calculator automatically gives you the required missing resistor after each input.

Knowing how the parallel resistors arrangement work makes it possible to apply the current divider rule in the circuit.

Other uses of the parallel resistor calculator

The principle is the same as when determining capacitance in series or induction in parallel – you can use it for these calculations too. Just remember that the units are not the same!

If you would like to find out the value of power dissipated in the resistor, try the Ohm’s law calculator or Resistor wattage calculator.

FAQ

How do you calculate two resistors in parallel?

Take their reciprocal values, add the two together and take the reciprocal again. For example, if one resistor is 2 Ω and the other is 4 Ω, then the calculation to find the equivalent resistance is 1 / (1/2 + 1/4) = 1 / (3/4) = 4/3 = 1.33.

Is the voltage the same in a parallel circuit?

Yes, the voltage across all the components is the same in a parallel circuit.

Why does resistance decrease in parallel?

This phenomenon happens because the current has many more paths that it could take. Imagine a shop opens up several new check-out tills. The overall resistance to people going through the check-out will decrease as the workload is shared in parallel.

How do you find an unknown resistor in a parallel circuit?

Rearrange the parallel resistor formula 1/R = 1/R₁ + 1/R₂ + … + 1/Rn in terms of Rn, given that you know the desired overall resistance. That gives you Rn = (1/R – 1/R₁ + 1/R₂ + …)-1.

For example, if you have R1 = 4, R2 = 2 and want R = 1, then R4 = 1 / (1 – 1/4 – 1/2) = 4 Ω.

Resistors in Parallel

Resistors are said to be connected together in parallel when both of their terminals are respectively connected to each terminal of the other resistor or resistors

Unlike the previous series resistor circuit, in a parallel resistor network the circuit current can take more than one path as there are multiple paths for the current. Then resistors in parallel circuits are classed as current dividers.

Since there are multiple paths for the supply current to flow through, the current may not be the same through all the branches in the parallel network. However, the voltage drop across all of the resistors in a parallel resistive network IS the same. Then, Resistors in Parallel have a Common Voltage across them and this is true for all parallel connected elements.

So we can define a parallel resistive circuit as one where the resistors are connected to the same two points (or nodes) and is identified by the fact that it has more than one current path connected to a common voltage source. Then in our parallel resistor example below the voltage across resistor R1 equals the voltage across resistor R2 which equals the voltage across R3 and which equals the supply voltage. Therefore, for a parallel resistor network this is given as:

In the following resistors in parallel circuit the resistors R1, R2 and R3 are all connected together in parallel between the two points A and B as shown.

Parallel Resistor Circuit

In the previous series resistor network we saw that the total resistance, RT of the circuit was equal to the sum of all the individual resistors added together. For resistors in parallel the equivalent circuit resistance RT is calculated differently.

Here, the reciprocal ( 1/R ) value of the individual resistances are all added together instead of the resistances themselves with the inverse of the algebraic sum giving the equivalent resistance as shown.

Parallel Resistor Equation

Then the inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistances.

If the two resistances or impedances in parallel are equal and of the same value, then the total or equivalent resistance, RT is equal to half the value of one resistor. That is equal to R/2 and for three equal resistors in parallel, R/3, etc.

Note that the equivalent resistance is always less than the smallest resistor in the parallel network so the total resistance, RT will always decrease as additional parallel resistors are added.

Parallel resistance gives us a value known as Conductance, symbol G with the units of conductance being the Siemens, symbol S. Conductance is the reciprocal or the inverse of resistance, ( G = 1/R ). To convert conductance back into a resistance value we need to take the reciprocal of the conductance giving us then the total resistance, RT of the resistors in parallel.

We now know that resistors that are connected between the same two points are said to be in parallel. But a parallel resistive circuit can take many forms other than the obvious one given above and here are a few examples of how resistors can be connected together in parallel.

Various Parallel Resistor Networkss

The five resistive networks above may look different to each other, but they are all arranged as Resistors in Parallel and as such the same conditions and equations apply.

Resistors in Parallel Example No1

Find the total resistance, RT of the following resistors connected in a parallel network.

The total resistance RT across the two terminals A and B is calculated as:

This method of reciprocal calculation can be used for calculating any number of individual resistances connected together within a single parallel network.

If however, there are only two individual resistors in parallel then we can use a much simpler and quicker formula to find the total or equivalent resistance value, RT and help reduce the reciprocal maths a little.

This much quicker product-over-sum method of calculating two resistor in parallel, either having equal or unequal values is given as:

Resistors in Parallel Example No2

Consider the following circuit which has only two resistors in a parallel combination.

Using our formula above for two resistors connected together in parallel we can calculate the total circuit resistance, RT as:

One important point to remember about resistors in parallel, is that the total circuit resistance ( RT ) of any two resistors connected together in parallel will always be LESS than the value of the smallest resistor in that combination.

In our example above, the value of the combination was calculated as: RT = 15kΩ, where as the value of the smallest resistor is 22kΩ, much higher. In other words, the equivalent resistance of a parallel network will always be less than the smallest individual resistor in the combination.

Also, in the case of R1 being equal to the value of R2, that is R1 = R2, the total resistance of the network will be exactly half the value of one of the resistors, R/2.

Likewise, if three or more resistors each with the same value are connected in parallel, then the equivalent resistance will be equal to R/n where R is the value of the resistor and n is the number of individual resistances in the combination.

For example, six 100Ω resistors are connected together in a parallel combination. The equivalent resistance will therefore be:  RT = R/n = 100/6 = 16.7Ω. But note that this ONLY works for equivalent resistors. That is resistors all having the same value.

Currents in a Parallel Resistor Circuit

The total current, IT entering a parallel resistive circuit is the sum of all the individual currents flowing in all the parallel branches. But the amount of current flowing through each parallel branch may not necessarily be the same, as the resistive value of each branch determines the amount of current flowing within that branch.

For example, although the parallel combination has the same voltage across it, the resistances could be different therefore the current flowing through each resistor would definitely be different as determined by Ohms Law.

Consider the two resistors in parallel above. The current that flows through each of the resistors ( IR1 and IR2 ) connected together in parallel is not necessarily the same value as it depends upon the resistive value of the resistor. However, we do know that the current that enters the circuit at point A must also exit the circuit at point B.

Kirchhoff’s Current Laws states that: “the total current leaving a circuit is equal to that entering the circuit – no current is lost“. Thus, the total current flowing in the circuit is given as:

IT = IR1 + IR2

By using Ohm’s Law, we can calculate the current flowing through each parallel resistor shown in Example No2 above as being:

The current flowing in resistor R1 is given as:

IR1 = VS ÷ R1 = 12V ÷ 22kΩ = 0.545mA or 545μA

The current flowing in resistor R2 is given as:

IR2 = VS ÷ R2 = 12V ÷ 47kΩ = 0.255mA or 255μA

thus giving us a total current IT flowing around the circuit as:

IT = 0.545mA + 0.255mA = 0.8mA or 800μA

and this can also be verified directly using Ohm’s Law as:

IT = VS ÷ RT = 12 ÷ 15kΩ = 0.8mA or 800μA (the same)

The equation given for calculating the total current flowing in a parallel resistor circuit which is the sum of all the individual currents added together is given as:

Then parallel resistor networks can also be thought of as “current dividers” because the supply current splits or divides between the various parallel branches. So a parallel resistor circuit having N resistive networks will have N-different current paths while maintaining a common voltage across itself. Parallel resistors can also be interchanged with each other without changing the total resistance or the total circuit current.

Resistors in Parallel Example No3

Calculate the individual branch currents and total current drawn from the power supply for the following set of resistors connected together in a parallel combination.

As the supply voltage is common to all the resistors in a parallel circuit, we can use Ohms Law to calculate the individual branch current as follows.

Then the total circuit current, IT flowing into the parallel resistor combination will be:

This total circuit current value of 5 amperes can also be found and verified by finding the equivalent circuit resistance, RT of the parallel branch and dividing it into the supply voltage, VS as follows.

Equivalent circuit resistance:

Then the current flowing in the circuit will be:

Resistors in Parallel Summary

So to summarise. When two or more resistors are connected so that both of their terminals are respectively connected to each terminal of the other resistor or resistors, they are said to be connected together in parallel. The voltage across each resistor within a parallel combination is exactly the same but the currents flowing through them are not the same as this is determined by their resistance value and Ohms Law. Then parallel circuits are current dividers.

The equivalent or total resistance, RT of a parallel combination is found through reciprocal addition and the total resistance value will always be less than the smallest individual resistor in the combination. Parallel resistor networks can be interchanged within the same combination without changing the total resistance or total circuit current. Resistors connected together in a parallel circuit will continue to operate even though one resistor may be open-circuited.

Thus far we have seen resistor networks connected in either a series or a parallel combination. In the next tutorial about Resistors, we will look at connecting resistors together in both a series and parallel combination at the same time producing a mixed or combinational resistor circuit.

Parallel resistors

Resistors are in parallel if their terminals are connected to the same two nodes. The equivalent overall resistance is smaller than the smallest parallel resistor. 

Components are in parallel if they share two nodes, like this:

In this article we will work with resistors in parallel, to reveal the properties of the parallel connection. Later articles will cover capacitors and inductors in series and parallel.

Resistors in parallel

Resistors are in parallel when their two terminals connect to the same nodes.

Resistors in parallel share the same voltage on their terminals.

The resistors in the following image are not in parallel. There are extra components (orange boxes) breaking up the common nodes between resistors.

Properties of resistors in parallel

Figuring out parallel resistors is a little trickier than series resistors. Here is a circuit with resistors in parallel. (This circuit has a current source. We don’t get to use those very often, so this should be fun.)

With just this little bit of knowledge, and Ohm’s Law, we can write these expressions:

This is enough to get going. Rearrange the three Ohm’s Law expressions to solve for current in terms of voltage and resistance:

appearing in a double-reciprocal in place of of a single resistor.

We conclude:

For resistors in parallel, the overall resistance is the reciprocal of the sum of reciprocals of the individual resistors.

(This sounds complicated, but we will derive something simpler before we are done.)

Equivalent parallel resistor

The previous equation suggests we can define a new resistor, equivalent to the parallel resistors. The new resistor is equivalent in the sense that, for a given i, the same voltage v appears.

The equivalent parallel resistor is the reciprocal of the sum of reciprocals. We can write this equation another way by rearranging the giant reciprocal,

If you have multiple resistors in parallel, the general form of the equivalent parallel resistance is,

Current distributes between resistors in parallel

We worked out the voltage vvvv across the parallel connection, so what’s left to figure out is the currents through the individual resistors.

Do this by applying Ohm’s Law to the individual resistors.

Special case – two resistors in parallel

Two resistors in parallel have an equivalent resistance of:

It’s possible do a bit of manipulation to eliminate the reciprocals and come up with another expression with just one fraction. Rather than just telling you the answer, it is a rite of passage to work through the algebra the first time. The answer is tucked away so you can try this on your own before peeking.

This is the equation for two resistors in parallel.

The product over the sum. This is worth memorizing.

Special special case – two equal resistors in parallel

Two identical resistors in parallel have an equivalent resistance half the value of either resistor. The current splits equally between the two.

Summary

Resistors in parallel share the same voltage.

The general form for three or more resistors in parallel is,

Current distributes amongst parallel resistors, with the largest current flowing through the smallest resistor.

How to Calculate the Equivalent Resistance of Resistors Connected in Parallel 

Calculating the equivalent resistance (Req) of resistors in parallel (Figure 1) by hand can be tiresome. 

Figure 1. Circuit schematic for resistors connected in parallel

This tool was designed to help you quickly calculate the equivalent resistance of up to 6 resistors connected in parallel. To use it, just specify how many parallel resistors there are and the resistance value for each one. If you have more than 6 resistors, simply use the calculator to determine the equivalent resistance of the first 6 resistors, and then plug that value in for R1 and add values for R7, R8, …, R11 into the calculator in the R2, R3, …, R6 input fields.

Calculating Resistance in Parallel Using Ohm’s Law

The voltage (V) across all of the resistors in a parallel circuit is identical. This can be seen by observing that the parallel resistors share the same nodes. The current through each individual resistor, Rx, can be calculated using Ohm’s law:

The total current through the parallel resistors is the sum of the individual currents:

The current through each individual resistor does not change when you add resistors in parallel because adding resistors in parallel does not affect the voltage across the resistors’ terminals. What changes in the total current delivered by the power supply, not the current through one particular resistor.    

Parallel Resistance Formula

From the total current equation, we can then derive an equation for the equivalent parallel resistance:

This is often expressed as:

When you have only two resistors in parallel, the equivalent resistance can be easily calculated using this equation:

Applications for Parallel Resistors

Resistors in parallel always result in an equivalent resistance that is lower than the resistance of each individual resistor. Resistors in series, on the other hand, are equivalent to one resistor whose resistance is the sum of each individual resistor. If you think about it, the lower equivalent resistance for parallel resistors makes sense. If you apply a voltage across a resistor, a certain amount of current flows. If you add another resistor in parallel with the first one, you have essentially opened up a new channel through which more current can flow. No matter how high the resistance of the second resistor is, the total current flowing from the power supply will be at least slightly higher than the current through the single resistor. And if the total current is higher, the overall equivalent resistance must be lower.

Multiple parallel resistors are often used to create a smaller effective resistance when you don’t have the desired resistor value readily available. This is handy when you need a specific resistance value and don’t have an appropriate part readily available. For example, you can easily calculate the equivalent resistance when you have two identical resistors in parallel: it is half of the individual resistance. If you need about 500 Ω to get the desired brightness out of an LED circuit, you can use two 1 kΩ resistors in parallel.

Using parallel resistors can also allow you to minimize the power consumption of any individual resistor in the circuit. Imagine our LED circuit example connected to a 16 V supply with 1 V dropped across the LED. A single 500 Ω would consume 450 mW (152 / 500). But, using two 1 kΩ resistors in parallel would limit the individual resistor power consumption to 225 mW, thereby allowing the use of standard 0.25 W resistors.

Solved Examples

Question 1: Calculate the resultant resistance of a parallel circuit containing three resistors; R1 = 2Ω, R2 = 4Ω and R3 = 6Ω?

Solution:Given value of resistors are,
R1 = 2Ω,
R2 = 4Ω,
R3 = 6Ω
The formula for resistors in parallel is,

1/Rp =(1/2) +(1/4)+(1/6)

1/Rp =11/12

Rp=12/11
Rp = 1.0909Ω

Question 2: Calculate total resistance of the given parallel connection; R1 = 4Ω and R2 = 5Ω?

Solution: Given the value of resistors are,
R1 = 4Ω,
R2 = 5Ω
Formula for resistors in parallel is,
1/Rp = (1/4)+(1/5)

1/Rp = 9/20

Rp = 20/9
So, Rp = 2.222Ω

Resistors In Series Formula

Resistors can be arranged in series form too such that the current flows through the resistors in series. Following is the table of formulas for parameters like the current, voltage and total resistance:

The flow of current through the resistorsI=I1=I2=I3…..=In
The voltage drop across the resistorsVtotal=V1+V2+V3…..+Vn
Total resistance equationRtotal=R1+R2+R3…..+Rn

Solved Examples

Q.1: What is the equivalent resistance of a 1000 kilo-ohm and a 250 kilo-ohm resistor connected in parallel?

Solution: The resistances are both expressed in kilo-Ohms. Thus, there is no need to change the units. We can find the equivalent resistance in kilo-ohm using the formula:

Therefore, the equivalent resistance of the 1000kilo−Ωand250.0kilo−Ωresistorsinparallelis200.0kilo−Ω.

Q.2: What is the equivalent resistance of a 10 ohm, 20 ohm and 30-ohm resistors connected in parallel?

Solution:

Given:

R1 = 10 ohm

R2 = 20 ohm

R3 = 30 ohm

We can find the equivalent resistance in kilo-ohm using the formula:

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