Solving parallel circuits is a simple process once you know the basic formulas and principles. When you connect two or more resistors side by side, the current can "choose" which way to go (this is like when cars tend to change lanes and drive alongside each other when the road splits into two parallel lanes). After reading these steps, you will be able to find out the voltage or voltage, current, and resistance between two or more resistors connected in parallel.
Help sheet
- Total resistance R_{T} for resistors in parallel: ^{1}/_{RT = 1/R1 + 1/R2 + 1/R3 + …}
- The voltage between the different branches is always the same: V_{T} = V_{1} = V_{2} = V_{3} = …
- Current or total intensity: I_{T} = I_{1} + I_{2} + I_{3} + …
- Ohm's Law: V = IR
Steps
Part 1 of 3: Introduction to Parallel Circuits
Step 1. Identify the parallel circuits
A parallel circuit has two or more branches and they all lead from point A to point B. A simple stream of electrons splits to pass through several branches and then joins back into a single stream on the other side. In most problems that involve working with parallel circuits, you will be asked to identify the total voltage, resistance, or current across the circuit (from point A to point B).
Components "connected in parallel" are those that are located on a separate branch
Step 2. Understand how current and resistance work in parallel circuits
Imagine a highway with many lanes that has toll booths in each lane, which slow down traffic. If new lanes are built, the cars will have a new way to go, so traffic will always speed up even if you also add new toll booths in each of them. Similarly, adding a branch to a parallel circuit gives the current an alternate path to take. No matter how much resistance the new branch has, the total resistance of the circuit will decrease and the total current of the circuit will increase.
Step 3. Add the currents for each branch to find the total current
If you know the current of each of the branches, just add them together and you get the total current. This is the amount of current that flows through the circuit once all the branches are rejoined. In terms of a formula it would be: I_{T} = I_{1} + I_{2} + I_{3} + …
Step 4. Calculate the total resistance
To find the total resistance (RT) across the circuit, solve the equation ^{1}/_{RT = 1/R1 + 1/R2 + 1/R3 +… Where each R on the right side represents the resistance of one of the branches of the circuit.}
- For example, if the circuit has two resistors in parallel, each with 4Ω of resistance, then ^{1}/_{RT = 1/ 4Ω + 1/ 4Ω → 1/RT = 1/ 2Ω → RT = 2Ω. In other words, two branches of equal strength are exactly twice as easy to traverse as a single branch.}
- If one of the branches has no resistance (0Ω), all the current goes through the branch. Total resistance is 0.
Step 5. Remember what voltage describes
Voltage is the difference in electrical potential between two points. Since it is the result of comparing two points, and not of examining the path of motion, the voltage will remain the same regardless of which branch is observed: V_{T} = V_{1} = V_{2} = V_{3} = …
Step 6. Find the missing values with Ohm's law
Ohm's Law describes the relationship between voltage or voltage (V), current or intensity (I), and resistance (R): V = IR. If you know two of those values, use the formula to find the third.
- Make sure all the values refer to the same section of the circuit. You can use Ohm's Law to examine the entire circuit (V = I_{T}R_{T}) or just one branch (V = I_{1}R_{1}).
Part 2 of 3: Sample Circuit
Step 1. Create a table to keep track of your work
If you have a parallel circuit with several unknown values, a table will help you organize your information. Here is an example table to create a circuit with three parallel branches. Note that branches are often indicated by an R followed by a subscript number.
R_{1} | R_{2} | R_{3} | Total | Units | |
---|---|---|---|---|---|
V | volts | ||||
I | amps | ||||
R | ohms |
Step 2. Complete all the information given by the problem
For this example, a circuit powered by a 12 volt battery will be used. The circuit has three parallel branches with 2Ω, 4Ω, and 9Ω resistors. Add this information to the table:
R_{1} | R_{2} | R_{3} | Total | Units | |
---|---|---|---|---|---|
V |
Step 12. |
volts | |||
I | amps | ||||
R |
Step 2. |
Step 4. |
Step 9. |
ohms |
Step 3. Copy the stress value in each of the branches
Remember that the voltage of the entire circuit is equal to the voltage that passes through each of the branches of the parallel circuit.
R_{1} | R_{2} | R_{3} | Total | Units | |
---|---|---|---|---|---|
V |
Step 12. |
Step 12. |
Step 12. |
Step 12. |
volts |
I | amps | ||||
R | 2 | 4 | 9 | ohms |
Step 4. Use Ohm's Law to find the current in each branch
Each column of the graph shows voltage or voltage, current, and resistance. This means that as long as you have the other two values in the same column, you can calculate the remaining value. In case you need a reminder, Ohm's Law states that V = IR. The missing value in this example is current, so you can rearrange the formula as follows: I = V / R.
R_{1} | R_{2} | R_{3} | Total | Units | |
---|---|---|---|---|---|
V | 12 | 12 | 12 | 12 | volts |
I | 12/2 = 6 | 12/4 = 3 | 12/9 = ~1, 33 | amps | |
R | 2 | 4 | 9 | ohms |
Step 5. Solve the sum to get the total current
The total current is easy to find, because it is equal to the sum of the currents of each branch.
R_{1} | R_{2} | R_{3} | Total | Units | |
---|---|---|---|---|---|
V | 12 | 12 | 12 | 12 | volts |
I | 6 | 3 | 1, 33 | 6 + 3 + 1, 33 = 10, 33 | amps |
R | 2 | 4 | 9 | ohms |
Step 6. Solve the operations to obtain the total resistance
You can find it in two different ways. The first is to use the resistance row and calculate it using the formula ^{1}/_{RT = 1/R1 + 1/R2 + 1/R3However, it is usually easier to solve it using Ohm's Law and the total values of V and I. To calculate the resistance, you must rearrange the terms of Ohm's Law as follows: R = V / I}
R_{1} | R_{2} | R_{3} | Total | Units | |
---|---|---|---|---|---|
V | 12 | 12 | 12 | 12 | volts |
I | 6 | 3 | 1, 33 | 10, 33 | amps |
R | 2 | 4 | 9 | 12 / 10, 33 = ~1, 17 | ohms |
Part 3 of 3: Additional Calculations
Step 1. Calculate the power
In any circuit, the power P = IV. If you have already calculated the power together with each of the branches, the total power is equal to the sum of all the power values of the branches (P_{1} + P_{2} + P_{3} + …).
Step 2. Find the total resistance for a two-limb circuit
If there are exactly two resistors in parallel, you can simplify that equation by replacing it with the "product over sum" equation:
- R_{T} = R_{1}R_{2} / (R_{1} + R_{2})
Step 3. Find the total resistance when all resistors are identical
If all the resistors in the parallel circuit have the same resistance value, the equation becomes much simpler: R_{T} = R_{1} / N, where N is the number of resistors.
For example, two identical resistors in parallel provide ½ of the total resistance of a single resistor. Eight identical resistors provide ⅛ of the total resistance
Step 4. Calculate the current of the branches without voltage
This equation, called Kirchhoff's Current Divider Law, allows you to solve for the currents in each individual branch even if you don't know the circuit voltage. You must know the resistance of each branch and the total current of the circuit:
- Two resistors in parallel: I_{1} = I_{T}R_{2} / (R_{1} + R_{2})
- More than two resistors in parallel: To solve for I1, find the combined resistance of all resistors besides R_{1}. Remember to use the formula for resistors in parallel. Now use the equation above, but replacing R_{2} with your answer.
Advice
- In a parallel circuit, the same voltage is always applied across all resistors.
- You may have been taught that Ohm's Law states that E = IR or V = AR. They are just different notations, but they mean the same thing.
- Total resistance is also known as "equivalent resistance".
- If you don't have a calculator, it can be difficult to find the total resistance from R_{1}, R_{2}, etc. Instead, use Ohm's Law to find the current flowing through each branch.
- Whenever you must solve parallel and series circuits, solve the parallels first. Then you will have the series circuits, which are much easier to solve.