Resistors in parallel
Calculator and formulas for calculating resistors in parallel
In a parallel resistor circuit, all resistors are connected parallel to each other. This means that the same voltage is applied across all resistors, but the total current divides according to the individual resistance values.
Key characteristics of parallel circuits:
- Same voltage across all resistors
- Total current is the sum of individual currents
- Total resistance is always smaller than the smallest individual resistor
Calculate total resistance
Input guidelines:
- Exponents are not allowed (e.g., use 1000 instead of 1E3)
- Enter all values in the same unit (e.g., all in Ω or all in kΩ)
- The result will be displayed in the same unit
- Use decimal comma (,) or decimal point (.) according to your system settings
Input format: Enter the values of the individual resistors separated by semicolons.
Example: 33; 12.1; 22
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Formulas for parallel resistor circuits
Basic formula
To calculate the total resistance of several parallel-connected resistors, their conductance values (reciprocals of resistance) are added. The conductance G is the reciprocal of the resistance R:
\[ G = \frac{1}{R} \]
The general formula for n parallel-connected resistors is:
\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n} \]
Solved for Rtotal:
\[ R_{total} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}} \]
Special case: Two resistors
For two parallel resistors, there is a simplified formula:
\[ R_{total} = \frac{R_1 \cdot R_2}{R_1 + R_2} \]
This formula is often called the "product formula" and is particularly useful for two equal resistors.
Special case: Equal resistors
When all n resistors have the same value R, the formula simplifies to:
\[ R_{total} = \frac{R}{n} \]
Example: Four parallel 100Ω resistors result in a total resistance of 25Ω.
General formula for n resistors
For n resistors R₁, R₂, ..., Rₙ, the general formula applies:
\[\frac{1}{R_{total}} = \sum_{i=1}^n \frac{1}{R_i}\]
Practical calculation examples
Example 1: Three different resistors
Given: Three resistors in parallel: 4Ω, 6Ω and 12Ω
Calculation:
\[\frac{1}{R_{total}} = \frac{1}{4} + \frac{1}{6} + \frac{1}{12}\]
\[\frac{1}{R_{total}} = 0.25 + 0.1667 + 0.0833 = 0.5\]
Result:
\[R_{total} = \frac{1}{0.5} = 2Ω\]
✓ Check: 2Ω is smaller than the smallest individual resistor (4Ω) - this is correct!
Example 2: Two equal resistors
Given: Two 100Ω resistors connected in parallel
Using the product formula:
\[R_{total} = \frac{100 \times 100}{100 + 100} = \frac{10000}{200} = 50Ω\]
Using the simplified formula:
\[R_{total} = \frac{100}{2} = 50Ω\]
Example 3: Practical application
Task: A 470Ω resistor should be reduced to about 150Ω. What parallel resistor is needed?
Given: R₁ = 470Ω, Rtotal = 150Ω, find: R₂
Rearranging the product formula:
\[R_2 = \frac{R_1 \times R_{total}}{R_1 - R_{total}} = \frac{470 \times 150}{470 - 150} = \frac{70500}{320} = 220Ω\]
Answer: A 220Ω resistor in parallel with 470Ω gives approximately 150Ω.
Current distribution in parallel circuits
In a parallel circuit, the total current divides inversely proportional to the resistance values:
\[ I_{total} = I_1 + I_2 + I_3 + ... + I_n \]
The current through each individual resistor is calculated according to Ohm's law:
\[ I_i = \frac{U}{R_i} \]
Rule: In parallel circuits, more current flows through smaller resistors than through larger ones.
Common applications
- Electrical lighting: Lamps in household installations
- Electronics: Voltage dividers and bias circuits
- Measurement technology: Shunt resistors for current measurement
- Power matching: Reducing total resistance
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