Resistors in Parallel

What you’ll learn

Combine branch conductances and use common voltage.

គណនារេស៊ីស្តង់ខ្នែង និងប្រើតង់ស្យុងដូចគ្នាតាមសាខា។

Meaning and use

Parallel branches connect to the same pair of nodes, so each has the same potential difference in the ideal-wire model. Branch currents add at a junction. Equivalent resistance uses the reciprocal sum. For positive finite branch resistances, the equivalent is smaller than the smallest branch resistance.

រេស៊ីស្តង់ខ្នែងតភ្ជាប់ទៅចំណុចពីរដូចគ្នា ដូច្នេះមានតង់ស្យុងដូចគ្នា។ ចរន្តតាមសាខាបូកគ្នា។ រេស៊ីស្តង់សមមូលប្រើផលបូកចំនួនច្រាស។

Formulas

Parallel resistance

\[\frac1{R_p}=\frac1{R_1}+\frac1{R_2}+\cdots\]

Two-branch shortcut

\[R_p=\frac{R_1R_2}{R_1+R_2}\]

Symbols and units

Symbol	Meaning	Unit or restriction
Rp, R₁, R₂	Equivalent and branch resistances	Ω
V	Common branch voltage	V
Itotal	Sum of branch currents	A

Conditions and limits

• The shortcut is for exactly two branches.
• Use positive finite resistances for these numerical examples and ideal connecting wires.
• កុំច្រឡំផលបូកចំនួនច្រាសជាមួយ R សមមូលផ្ទាល់។

Worked examples

Two branch resistances

Combine 6 Ω and 3 Ω in parallel.

1. Use \(R_p=(6\times3)/(6+3)\).
2. Calculate \(18/9=2\) Ω.

2 Ω

សមមូលតូចជាងទាំង 3 Ω និង 6 Ω។

Branch currents

The 6 Ω and 3 Ω branches have 6 V across them.

1. First current \(6/6=1\) A.
2. Second current \(6/3=2\) A.
3. Total \(1+2=3\) A.

1 A and 2 A; total 3 A

តង់ស្យុងស្មើគ្នា តែចរន្តអាស្រ័យតាម R។

Common mistakes

Instead of	Use	Why
Parallel resistance is R₁ + R₂.	Add reciprocals, then take the reciprocal.	Direct addition is the series rule.
Parallel branches always carry equal current.	They share voltage; currents depend on resistance.	Equal current requires equal branch resistance here.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Combine 4 Ω and 4 Ω in parallel.
2. Find current in a 6 Ω branch with 12 V across it.
3. For positive finite branches, should Rp exceed both branch resistances?

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 2 Ω

   4 × 4/(4 + 4) = 2.

2. 2 A

   12/6 = 2.

3. No.

   It must be smaller than the smallest branch resistance.

Quick recap

• Check that branches share both nodes.
• Use common voltage, not common current.
• Sanity-check equivalent resistance.