Resistor networks are used to control current, divide voltage, protect components, and set operating points in circuits. Engineers often replace a group of resistors with one equivalent resistor to make a circuit easier to analyze. Series and parallel connections are the two basic patterns that appear inside larger networks.
Understanding them is essential for using Ohm's law accurately in real circuits.
Key Facts
- Ohm's law: V = IR
- Series equivalent resistance: R_eq = R1 + R2 + R3 + ...
- Parallel equivalent resistance: 1/R_eq = 1/R1 + 1/R2 + 1/R3 + ...
- For two resistors in parallel: R_eq = (R1 R2)/(R1 + R2)
- In series, the same current flows through every resistor and voltages add: V_total = V1 + V2 + ...
- In parallel, the same voltage is across every branch and currents add: I_total = I1 + I2 + ...
Vocabulary
- Equivalent resistance
- The single resistance value that would draw the same total current from the same voltage source as the entire resistor network.
- Series circuit
- A circuit connection in which components are connected end to end so there is only one path for current.
- Parallel circuit
- A circuit connection in which components are connected across the same two nodes so each branch has the same voltage.
- Node
- A point or connected region in a circuit where two or more component terminals meet at the same electric potential.
- Voltage divider
- A series resistor arrangement that splits a supply voltage into smaller voltage drops across the resistors.
Common Mistakes to Avoid
- Adding parallel resistors directly is wrong because parallel resistance is found using a reciprocal sum, not simple addition.
- Assuming current is the same in every parallel branch is wrong because branch current depends on each branch resistance through I = V/R.
- Assuming voltage is the same across every series resistor is wrong because series voltage drops divide in proportion to resistance.
- Reducing the wrong pair of resistors first is wrong because only resistors that are truly in series or truly in parallel can be combined at that step.
Practice Questions
- 1 Three resistors of 4 ohms, 6 ohms, and 10 ohms are connected in series to a 12 V battery. Find the equivalent resistance and the total current.
- 2 A 12 ohm resistor and a 6 ohm resistor are connected in parallel across a 9 V battery. Find the equivalent resistance, the total current, and the current through each branch.
- 3 A network has one resistor in series with a parallel pair. Explain why the series resistor has the total circuit current, while the two parallel resistors share that current.