Finding the equivalent resistance of a system involves simplifying a network of resistors into a single equivalent resistor that would have the same effect on the circuit. The method you use depends on how the resistors are connected—whether in series, parallel, or a combination of both. Here’s a detailed guide:
### 1. **Series Resistors**
When resistors are connected end-to-end in a series, the same current flows through each resistor, and their resistances add up directly. The formula for equivalent resistance (\( R_{eq} \)) in series is:
\[ R_{eq} = R_1 + R_2 + R_3 + \cdots + R_n \]
**Example:** For resistors \( R_1 = 4 \, \Omega \), \( R_2 = 3 \, \Omega \), and \( R_3 = 2 \, \Omega \) connected in series, the equivalent resistance is:
\[ R_{eq} = 4 \, \Omega + 3 \, \Omega + 2 \, \Omega = 9 \, \Omega \]
### 2. **Parallel Resistors**
When resistors are connected across the same two points, they are in parallel. The voltage across each resistor is the same, but the current divides among them. The formula for equivalent resistance in parallel is:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_n} \]
To find \( R_{eq} \), take the reciprocal of the sum of the reciprocals of the individual resistances.
**Example:** For resistors \( R_1 = 4 \, \Omega \), \( R_2 = 3 \, \Omega \), and \( R_3 = 6 \, \Omega \) connected in parallel, the equivalent resistance is:
\[ \frac{1}{R_{eq}} = \frac{1}{4} + \frac{1}{3} + \frac{1}{6} \]
First, find a common denominator and add:
\[ \frac{1}{R_{eq}} = \frac{3}{12} + \frac{4}{12} + \frac{2}{12} = \frac{9}{12} = \frac{3}{4} \]
So:
\[ R_{eq} = \frac{4}{3} \, \Omega \approx 1.33 \, \Omega \]
### 3. **Combination of Series and Parallel**
For circuits with a combination of series and parallel resistors, follow these steps:
1. **Identify Simple Series and Parallel Groups**: Break down the circuit into simpler parts. Find the equivalent resistance for these parts step-by-step.
2. **Simplify Step-by-Step**: Start by simplifying the most straightforward series or parallel combinations first. Replace these combinations with their equivalent resistances, and then solve the remaining parts of the circuit.
3. **Repeat as Necessary**: Continue simplifying until you are left with a single equivalent resistor.
**Example:** Consider a circuit with resistors \( R_1 \) and \( R_2 \) in series, and this series combination in parallel with resistor \( R_3 \).
- First, find the equivalent resistance of \( R_1 \) and \( R_2 \) in series:
\[ R_{s} = R_1 + R_2 \]
- Then find the equivalent resistance of \( R_{s} \) in parallel with \( R_3 \):
\[ \frac{1}{R_{eq}} = \frac{1}{R_s} + \frac{1}{R_3} \]
\[ R_{eq} = \frac{1}{\left(\frac{1}{R_s} + \frac{1}{R_3}\right)} \]
### 4. **Special Cases and Practical Considerations**
- **Delta (Δ) and Wye (Y) Transformations**: In complex circuits, you may encounter Δ (delta) or Y (wye) configurations. You can use transformation formulas to convert between these configurations to simplify analysis.
- **Measurement and Verification**: For real-world circuits, you can measure the resistance directly using a multimeter after simplifying the circuit.
By carefully applying these methods, you can determine the equivalent resistance of virtually any resistor network.