Resistors in Series and Parallel Examples (Circuits for Beginners #9)

Summary

This video explains how to calculate the equivalent resistance in various resistor configurations. Starting with simple parallel and series combinations, it highlights tricks to simplify calculations, such as multiplying in the numerator and adding in the denominator for two resistors in parallel. The video then explores more complex circuits with resistors in series and parallel, demonstrating how to redraw circuits for clarity and using Ohm's law to find voltage drops. The video concludes with an advanced example, showing how to solve tricky resistor networks, paving the way for more rigorous methods in future lessons.

Takeaways

• ⚡ Equivalent resistance allows multiple resistors in a circuit to be replaced with a single resistor that produces the same voltage-current relationship.
• 🔋 When two resistors are in series, their resistances simply add together (R_total = R1 + R2).
• 🔀 When two resistors are in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of each resistor (1/R_eq = 1/R1 + 1/R2).
• 🧮 A helpful shortcut for two parallel resistors is: multiply in the numerator and add in the denominator (R_eq = R1 * R2 / (R1 + R2)).
• 🧠 If two resistors have equal resistance and are connected in parallel, the equivalent resistance is half of either resistor’s value.
• 📉 Example: Two 6-ohm resistors in parallel yield an equivalent resistance of 3 ohms; two 20-ohm resistors in parallel give 10 ohms.
• 🔎 Redrawing a circuit can make it easier to identify which resistors are in series or parallel, as physical layout does not affect electrical connectivity.
• 💡 In one example, a 9-ohm resistor in parallel with a 6-ohm resistor simplifies to 3.6 ohms, which then adds in series with a 3-ohm resistor to yield 6.6 ohms total.
• ⚙️ Not all resistor networks can be simplified using only series and parallel combinations—some require more advanced techniques, like delta-wye transformations or circuit analysis methods.
• 🔧 The script demonstrates a conceptual trick for handling complex resistor arrangements by using equivalent voltage and current reasoning.
• 🧩 In the complex example, analyzing voltage drops across each resistor helps show that certain connections carry no current because their voltage differences are zero.
• 📘 The final example concludes with an equivalent resistance of 4 ohms using Ohm’s law (R_eq = V / I), though the instructor notes a more rigorous method will be shown later.

Q & A

What is the goal when calculating equivalent resistance in a circuit?

The goal is to find a single resistor that can replace a set of resistors in a circuit, such that the current and voltage through the circuit remain the same.

How do you combine resistors in series and parallel?

In series, the resistances add up directly. In parallel, the reciprocal of the total resistance is the sum of the reciprocals of each resistor's resistance.

What is the formula for combining two resistors in parallel?

The formula for the equivalent resistance of two resistors R1 and R2 in parallel is: 1/R_eq = 1/R1 + 1/R2. This can be simplified by multiplying the resistances and dividing by their sum, i.e., R_eq = (R1 * R2) / (R1 + R2).

Why is combining two resistors in parallel easier when they have the same resistance?

When two resistors in parallel have the same resistance, the equivalent resistance is simply half the value of either resistor. This is a shortcut because the sum of the reciprocals is straightforward.

How do you calculate the equivalent resistance of two resistors with different values in parallel, like 9 ohms and 3 ohms?

For two resistors in parallel, multiply their resistances (9 * 3 = 27) and divide by their sum (9 + 3 = 12). The equivalent resistance is 27/12, or 2.25 ohms.

Can you apply the shortcut for parallel resistors when there are more than two resistors?

No, the shortcut for multiplying and adding only works when there are two resistors. For three or more resistors in parallel, you must use the full reciprocal formula.

What challenges are faced when combining more than two resistors in complex configurations?

In more complex circuits where resistors are neither simply in series nor parallel, more advanced methods are required to calculate the equivalent resistance, as the simple series and parallel rules do not apply.

What trick is used to simplify the process of finding the equivalent resistance in certain complex circuits?

In some cases, you can apply Ohm’s law to find the voltage drops across resistors and use voltage balance between two parts of the circuit to reduce the complexity, allowing you to eliminate redundant sources or combine parallel paths.

Why is the current zero when two circuits are connected and the voltage drops are equal across them?

When two circuits are connected and the voltage drops across each part are equal, no current will flow through the connection because there is no potential difference to drive the current.

What does the final example with two 12-volt sources and two sets of resistors demonstrate?

The final example shows that two voltage sources supplying the same voltage to two parts of a circuit are effectively in parallel, and only one source is needed to drive the current, which can be calculated using Ohm's law.