2 Resistors in Parallel

Calculator and formula for calculating two resistors in a parallel circuit

2-Resistor Parallel Circuit

Calculation Modes

Calculate either the total resistance from two known resistors or an unknown parallel resistor R₂.

What should be calculated?
Result
Total resistance:
Resistor R₂:

2-Resistor Parallel Circuit

Circuit diagram: Parallel connection of two resistors R1 and R2

Circuit diagram: Parallel connection of two resistors

Two Calculation Modes
  • Calculate total resistance: From R₁ and R₂ → Rtotal
  • Calculate parallel resistor R₂: From R₁ and Rtotal → R₂
Product Formula
\[R_{total} = \frac{R_1 \cdot R_2}{R_1 + R_2}\]
Simplified formula for two resistors
Special Case
Equal resistors: Rtotal = R/2
For two 100Ω resistors → Rtotal = 50Ω

Formulas for Parallel Connection of Two Resistors

1. Calculate total resistance

Product formula for two resistors:

\[R_{total} = \frac{R_1 \cdot R_2}{R_1 + R_2}\]
When R₁ and R₂ are known
2. Calculate parallel resistor R₂

Rearranged product formula:

\[R_2 = \frac{R_1 \cdot R_{total}}{R_1 - R_{total}}\]
When R₁ and Rtotal are known
3. Derivation via conductances

The product formula can also be derived via conductances (G = 1/R):

\[G_{total} = G_1 + G_2 = \frac{1}{R_1} + \frac{1}{R_2}\]
\[R_{total} = \frac{1}{G_{total}} = \frac{R_1 \cdot R_2}{R_1 + R_2}\]

Practical Calculation Examples

Example 1: Calculate total resistance

Given: R₁ = 30Ω, R₂ = 20Ω

\[R_{total} = \frac{30 \times 20}{30 + 20} = \frac{600}{50} = 12Ω\]
Result: 12Ω

✓ Check: 12Ω < 20Ω (smallest resistor)

Example 2: Calculate parallel resistor R₂

Given: R₁ = 100Ω, Rtotal = 30Ω

\[R_2 = \frac{100 \times 30}{100 - 30} = \frac{3000}{70} = 42.86Ω\]
Result: about 43Ω

✓ Check: 30Ω < 43Ω < 100Ω

Example 3: Equal resistors (special case)

Special case: When both resistors are equal (R₁ = R₂ = R):

\[R_{total} = \frac{R \times R}{R + R} = \frac{R^2}{2R} = \frac{R}{2}\]
Example: Two 100Ω resistors → Rtotal = 50Ω

Memory rule: For two equal resistors in parallel, the resistance value is halved.

Current Distribution and Practical Applications

Current Distribution

In a parallel circuit, the total current divides inversely proportional to the resistance values. More current flows through the smaller resistor.

Current Distribution Formulas
\[I_1 = I_{total} \times \frac{R_2}{R_1 + R_2}\]
\[I_2 = I_{total} \times \frac{R_1}{R_1 + R_2}\]
Practical Applications
  • Reduce resistance values: Reduction of total resistance
  • Shunt resistors: Current measurement through parallel resistors
  • LED circuits: Series resistors for parallel LED strings
  • Bias circuits: Operating point adjustment in amplifiers
  • Voltage divider loading: Effects of load currents
Important Notes
  • Rtotal is always smaller than the smallest individual resistor
  • Total power increases (Ptotal = P₁ + P₂)
  • Different currents for different resistors
  • Consider load capacity of resistors
Memory Rules
  • For equal resistors: Rtotal = R/2
  • More current through smaller resistor
  • Same voltage across both resistors
  • Product formula only for two resistors


Circuits with resistors

Ohms Law  •  Total resistance of a resistor in parallel  •  Parallel- total resistance of 2 resistors  •  Series resistance for a voltmeter  •  Parallel resistance for an ampere meter  •  Voltage divider  •  Loaded voltage divider  •  Pi Attenuator  •  T Attenuator  •