How to Calculate Mixed Resistor (Series-Parallel) Connections

✔ Current is Inversely Proportional to Resistance:

$$\frac{I_1}{I_2} = \frac{R_2}{R_1}$$

🔹 Example Calculation

Three resistors (10Ω, 20Ω, and 30Ω) are connected in parallel to a 12V power source.

✔ Total Resistance Calculation:

$$\frac{1}{R_{\text{total}}} = \frac{1}{10Ω} + \frac{1}{20Ω} + \frac{1}{30Ω}$$ $$\frac{1}{R_{\text{total}}} = 0.1 + 0.05 + 0.0333 = 0.1833$$ $$R_{\text{total}} = \frac{1}{0.1833} = 5.45Ω$$

✔ Total Current Flow:

$$I_{\text{total}} = \frac{12V}{5.45Ω} = 2.2A$$

✔ Current Through Each Resistor:

$$I_1 = \frac{12V}{10Ω} = 1.2A$$ $$I_2 = \frac{12V}{20Ω} = 0.6A$$ $$I_3 = \frac{12V}{30Ω} = 0.4A$$

✔ Check: $I_1 + I_2 + I_3 = 1.2A + 0.6A + 0.4A = 2.2A$ ✅

📌 Parallel Connection Application: Used in power distribution systems to ensure multiple devices receive the same voltage.

4️⃣ Mixed (Series-Parallel) Resistor Connection

Definition: A combination of series and parallel connections, used when circuit complexity increases.

🔹 Steps to Calculate Total Resistance in Mixed Circuits

1️⃣ Identify separate series and parallel sections
2️⃣ Calculate equivalent resistance for parallel groups
3️⃣ Combine with series resistances

🔹 Example Calculation

A circuit has three resistors:

✔ R1 = 10Ω (Series with Parallel Pair R2 = 20Ω and R3 = 30Ω)

Step 1: Calculate Parallel Resistance (R2 and R3)

$$\frac{1}{R_{\text{parallel}}} = \frac{1}{20Ω} + \frac{1}{30Ω}$$ $$R_{\text{parallel}} = \frac{1}{(0.05 + 0.0333)} = 12Ω$$

Step 2: Add Series Resistor (R1)

$$R_{\text{total}} = R1 + R_{\text{parallel}}$$ $$R_{\text{total}} = 10Ω + 12Ω = 22Ω$$

📌 Mixed Circuit Application: Found in filter networks, sensors, and complex voltage dividers.

5️⃣ Practical Applications of Resistor Combinations

✔ Voltage Divider (Series Resistors): Used in signal conditioning
✔ Current Sharing (Parallel Resistors): Used in power electronics
✔ Load Balancing (Mixed Resistors): Found in industrial automation

6️⃣ Conclusion: Choosing the Right Resistor Connection

✔ Use SERIES connections when you need to divide voltage
✔ Use PARALLEL connections to distribute current and reduce resistance
✔ Use MIXED connections for more advanced circuit control

🚀 Next Steps:
✔ Practice calculating equivalent resistance
✔ Experiment with resistor networks on a breadboard
✔ Use a multimeter to verify theoretical values

🔌 Mastering resistor connections is a fundamental step toward building complex electronic circuits!