Understanding Resistor Connections: Series, Parallel, and Mixed Circuits

1️⃣ Introduction: Why Resistor Connections Matter

Resistors can be connected in different ways in circuits to control voltage, current, and power distribution. The three main types of resistor connections are:

✔ Series Connection – Resistors are connected end-to-end
✔ Parallel Connection – Resistors are connected across the same two points
✔ Mixed Connection – Combination of series and parallel

📌 Why It Matters: Understanding resistor connections is crucial for designing voltage dividers, current limiters, and power regulation circuits.

2️⃣ Series Resistor Connection

Definition: Resistors in a series circuit are connected end-to-end, meaning current flows through each resistor sequentially.

🔹 Key Properties of Series Connection

✔ Same Current Across All Resistors:

$$I_{\text{total}} = I_1 = I_2 = I_3 = ... = I_n$$

✔ Total Resistance is the Sum of Individual Resistances:

$$R_{\text{total}} = R_1 + R_2 + R_3 + ... + R_n$$

✔ Voltage Divides Proportionally Across Resistors:

$$U_{\text{total}} = U_1 + U_2 + U_3 + ... + U_n$$

✔ Voltage Ratio Matches Resistance Ratio:

$$\frac{U_1}{U_2} = \frac{R_1}{R_2}$$

🔹 Example Calculation

Three resistors (10Ω, 20Ω, and 30Ω) are connected in series with a 24V power supply.

✔ Total Resistance:

$$R_{\text{total}} = 10Ω + 20Ω + 30Ω = 60Ω$$

✔ Current Flowing Through Circuit:
Using Ohm’s Law ($I = \frac{U}{R}$):

$$I = \frac{24V}{60Ω} = 0.4A$$

✔ Voltage Drops Across Each Resistor:

$$U_1 = I \times R_1 = 0.4A \times 10Ω = 4V$$ $$U_2 = 0.4A \times 20Ω = 8V$$ $$U_3 = 0.4A \times 30Ω = 12V$$

✔ Check: $U_1 + U_2 + U_3 = 4V + 8V + 12V = 24V$ ✅

📌 Series Connection Application: Used in voltage divider circuits and current limiting applications.

3️⃣ Parallel Resistor Connection

Definition: In a parallel circuit, resistors are connected across the same two points, allowing current to split among them.

🔹 Key Properties of Parallel Connection

✔ Same Voltage Across All Resistors:

$$U_{\text{total}} = U_1 = U_2 = U_3 = ... = U_n$$

✔ Total Current is the Sum of Branch Currents:

$$I_{\text{total}} = I_1 + I_2 + I_3 + ... + I_n$$

✔ Total Resistance is Less than the Smallest Resistor:

$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}$$