Question

How do you find the equivalent resistance between two points?

Answer 1

Finding the equivalent resistance between two points in a circuit depends on the arrangement of the resistors. Here’s a detailed guide to help you through the process:

### 1. **Series Resistors**

When resistors are connected end-to-end, they are in series. The total or equivalent resistance (\(R_{eq}\)) is the sum of the individual resistances.

**Formula:**
\[ R_{eq} = R_1 + R_2 + R_3 + \ldots + R_n \]

**Example:**
If you have three resistors with values 4Ω, 5Ω, and 6Ω in series, the equivalent resistance is:
\[ R_{eq} = 4Ω + 5Ω + 6Ω = 15Ω \]

### 2. **Parallel Resistors**

When resistors are connected such that both ends of each resistor are connected to the same two points, they are in parallel. The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.

**Formula:**
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n} \]

**Example:**
For resistors with values 4Ω, 5Ω, and 6Ω in parallel:
\[ \frac{1}{R_{eq}} = \frac{1}{4Ω} + \frac{1}{5Ω} + \frac{1}{6Ω} \]
\[ \frac{1}{R_{eq}} = 0.25 + 0.20 + 0.167 = 0.637 \]
\[ R_{eq} = \frac{1}{0.637} \approx 1.57Ω \]

### 3. **Series-Parallel Combinations**

In more complex circuits where resistors are neither purely in series nor purely in parallel, you’ll need to simplify the circuit step-by-step. Here’s how to approach it:

1. **Identify and Simplify:** Identify series and parallel groups of resistors and simplify them into equivalent resistances one group at a time.

2. **Combine Simplified Resistances:** Replace each simplified group with its equivalent resistance and then simplify the remaining circuit.

3. **Repeat Until Simplified:** Continue simplifying until you reduce the circuit to a single equivalent resistance between the two points of interest.

**Example:**
Imagine a circuit where:
- Resistor \( R_1 \) (5Ω) is in series with a parallel combination of \( R_2 \) (10Ω) and \( R_3 \) (15Ω).

First, find the equivalent resistance of \( R_2 \) and \( R_3 \) in parallel:
\[ \frac{1}{R_{parallel}} = \frac{1}{10Ω} + \frac{1}{15Ω} \]
\[ \frac{1}{R_{parallel}} = 0.1 + 0.067 = 0.167 \]
\[ R_{parallel} = \frac{1}{0.167} \approx 6Ω \]

Next, add \( R_{parallel} \) to \( R_1 \) in series:
\[ R_{eq} = R_1 + R_{parallel} \]
\[ R_{eq} = 5Ω + 6Ω = 11Ω \]

### 4. **Using Kirchhoff’s Laws**

In more complex circuits, Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) may be used to set up equations that can help find the equivalent resistance. These laws are particularly useful for circuits that are neither purely series nor purely parallel.

### Summary

1. **Series:** Add resistances directly.
2. **Parallel:** Use the reciprocal sum formula.
3. **Combination:** Simplify in steps, handling series and parallel sections separately.
4. **Complex:** Use Kirchhoff’s laws if necessary.

By following these methods, you should be able to find the equivalent resistance between any two points in a circuit.

Answer 2

Finding the equivalent resistance between two points in an electrical circuit is crucial for analyzing how the circuit will behave. This process involves combining resistors in series and parallel configurations. Here's a detailed breakdown of how to find the equivalent resistance:

### 1. **Understand the Circuit Configuration**

   - **Series Configuration:** Resistors are in series if the same current flows through each resistor. The total or equivalent resistance (\(R_{eq}\)) is simply the sum of the individual resistances.
   - **Parallel Configuration:** Resistors are in parallel if they are connected to the same two nodes, meaning the voltage across each resistor is the same. The total resistance can be found using the reciprocal formula.

### 2. **Calculate Equivalent Resistance for Series Resistors**

   When resistors are connected in series, their resistances add up directly.

   **Formula:**
   \[
   R_{eq} = R_1 + R_2 + R_3 + \cdots + R_n
   \]
   
   **Example:**
   If you have three resistors in series with resistances \(R_1 = 4 \, \Omega\), \(R_2 = 6 \, \Omega\), and \(R_3 = 3 \, \Omega\), the equivalent resistance is:
   \[
   R_{eq} = 4 \, \Omega + 6 \, \Omega + 3 \, \Omega = 13 \, \Omega
   \]

### 3. **Calculate Equivalent Resistance for Parallel Resistors**

   For resistors connected in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.

   **Formula:**
   \[
   \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_n}
   \]
   or
   \[
   R_{eq} = \frac{1}{\left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_n}\right)}
   \]
   
   **Example:**
   For two resistors in parallel with resistances \(R_1 = 4 \, \Omega\) and \(R_2 = 6 \, \Omega\), the equivalent resistance is:
   \[
   \frac{1}{R_{eq}} = \frac{1}{4 \, \Omega} + \frac{1}{6 \, \Omega} = 0.25 + 0.1667 = 0.4167
   \]
   Thus:
   \[
   R_{eq} = \frac{1}{0.4167} \approx 2.4 \, \Omega
   \]

### 4. **Combine Series and Parallel Resistors**

   In real circuits, resistors are often combined in both series and parallel. Here’s how to handle such cases:

   - **Simplify Step-by-Step:** Start by simplifying the most straightforward sections of the circuit. Combine resistors in series or parallel, then replace them with their equivalent resistance and simplify the remaining circuit.
   - **Redraw the Circuit:** After each simplification, redraw the circuit to better visualize the next steps.

   **Example:**
   Suppose you have a circuit with resistors \(R_1 = 4 \, \Omega\) and \(R_2 = 6 \, \Omega\) in parallel, and this combination is in series with \(R_3 = 3 \, \Omega\). First, find the equivalent resistance of \(R_1\) and \(R_2\):
   \[
   \frac{1}{R_{eq1}} = \frac{1}{4} + \frac{1}{6} = 0.4167 \Rightarrow R_{eq1} \approx 2.4 \, \Omega
   \]
   Then, add \(R_{3}\) in series:
   \[
   R_{eq} = R_{eq1} + R_{3} = 2.4 \, \Omega + 3 \, \Omega = 5.4 \, \Omega
   \]

### 5. **Check Your Work**

   After calculating the equivalent resistance, ensure it makes sense in the context of the circuit. For example, the equivalent resistance of a parallel combination should always be less than the smallest individual resistor.

By breaking down the circuit into simpler parts and applying these formulas, you can find the equivalent resistance between any two points in the circuit.