Combining Resistors in Series & Parallel: Videos & Practice Problems

Understanding how to combine resistors is essential in circuit analysis. Resistors can be connected in series or parallel. In series, the equivalent resistance (R_eq) is the sum of individual resistances: R = R₁+R₂+R₃. In parallel, the equation is more complex: 1Rₑquivalence = 1R₁+1R₂. Shortcuts exist for two resistors in parallel, allowing for quicker calculations. Combining resistors effectively simplifies circuit analysis and enhances understanding of electrical resistance.

Combining Resistors in Series & Parallel

Combining resistors is an essential skill in circuit analysis, allowing you to simplify complex circuits into a single equivalent resistor. Resistors can be connected in two primary configurations: series and parallel. In a series connection, resistors are aligned end-to-end, creating a single path for current flow. Conversely, in a parallel connection, resistors are positioned on separate branches, allowing current to split between them.

To find the equivalent resistance in series, you simply add the individual resistances together. The formula for this is:

\[ R_{eq} = R_1 + R_2 + R_3 \]

For example, if you have three resistors with values of 1 ohm, 2 ohms, and 3 ohms, the equivalent resistance would be:

\[ R_{eq} = 1 + 2 + 3 = 6 \text{ ohms} \]

In contrast, the equivalent resistance for resistors in parallel is calculated using the following formula:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]

This equation indicates that the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances. For instance, if you have two resistors of 1 ohm and 4 ohms in parallel, the calculation would be:

\[ \frac{1}{R_{eq}} = \frac{1}{1} + \frac{1}{4} = 1 + 0.25 = 1.25 \]

To find \( R_{eq} \), you would then take the reciprocal:

\[ R_{eq} = \frac{1}{1.25} = 0.8 \text{ ohms} \]

When combining resistors, it's important to note that the equivalent resistance in a series connection is always greater than the individual resistances, while in a parallel connection, the equivalent resistance is always less than the smallest individual resistance. This behavior is due to the nature of current flow; in series, the current must pass through each resistor sequentially, while in parallel, the current can choose multiple paths, effectively reducing the overall resistance.

To illustrate this with an example, consider a circuit with four resistors. You would first identify which resistors can be combined based on their configuration. If two resistors are in parallel, you would calculate their equivalent resistance using the parallel formula, then combine that result with any series resistors that follow. This systematic approach allows you to simplify the circuit step by step until you arrive at a single equivalent resistor.

In summary, mastering the combination of resistors in series and parallel is crucial for analyzing electrical circuits effectively. Understanding the underlying principles and formulas will enable you to tackle complex circuit problems with confidence.

What is the equivalent resistance of the following combination of resistors?

• A. 0.69 Ω
• B. 1.45 Ω
• C. 5.8 Ω
• D. 6.2 Ω

Answer: (D)

Shortcut Equations for Resistors Parallel

When working with resistors in parallel, understanding the equivalent resistance is crucial for simplifying circuit calculations. The general formula for calculating the equivalent resistance (\(R_{\text{eq}}\)) of two resistors (\(R_1\) and \(R_2\)) in parallel is given by:

\[\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2}\]

However, a more efficient shortcut can be used specifically for two resistors. By rearranging the formula, the equivalent resistance can be calculated using:

\[R_{\text{eq}} = \frac{R_1 \cdot R_2}{R_1 + R_2}\]

This shortcut is particularly useful because it avoids the complexity of dealing with fractions. A helpful mnemonic to remember the formula is to visualize the multiplication in the numerator as being close together (indicating multiplication) and the addition in the denominator as being spaced apart (indicating addition), forming a triangle shape.

It is important to note that this shortcut only applies to two resistors. Attempting to extend this formula to three or more resistors is incorrect. For example, if you have three resistors, you cannot simply add another resistor to the existing formula. Always revert to the general equation for more than two resistors.

For instance, if you have a 4-ohm and a 6-ohm resistor in parallel, you can quickly find the equivalent resistance using the shortcut:

\[R_{\text{eq}} = \frac{4 \cdot 6}{4 + 6} = \frac{24}{10} = 2.4 \, \text{ohms}\]

Additionally, when dealing with multiple resistors of the same value, a different shortcut can be applied. For example, if you have three 6-ohm resistors in parallel, the equivalent resistance can be calculated as:

\[R_{\text{eq}} = \frac{6}{3} = 2 \, \text{ohms}\]

This method simplifies calculations significantly when resistors are identical. By combining these strategies, you can efficiently solve more complex resistor networks. For example, if you have three 9-ohm resistors and two 12-ohm resistors, you can first find the equivalent resistance of the 9-ohm resistors:

\[R_{\text{eq}} = \frac{9}{3} = 3 \, \text{ohms}\]

Then, for the two 12-ohm resistors:

\[R_{\text{eq}} = \frac{12}{2} = 6 \, \text{ohms}\]

Finally, you can combine these two results using the initial shortcut for two resistors:

\[R_{\text{final}} = \frac{3 \cdot 6}{3 + 6} = \frac{18}{9} = 2 \, \text{ohms}\]

By mastering these shortcuts, you can streamline your calculations and enhance your understanding of parallel resistor circuits.

What is the equivalent resistance of the following combination of resistors?

• A. 0.4 Ω
• B. 2.5 Ω
• C. 7.0 Ω
• D. 10 Ω

Answer: (C)

Weird Arrangement (Re-Drawing Resistors)

Understanding complex resistor networks can be simplified by rearranging the circuit to make it more familiar. When faced with a network of resistors, the first step is to visualize the arrangement clearly. For instance, if resistors are positioned at unusual angles, you can mentally or physically adjust their positions to create a more straightforward layout.

Consider a scenario with resistors of values 4 ohms and 5 ohms, along with a 3-ohm resistor. By moving the 4-ohm resistor to a vertical position and adjusting the layout, you can create a clearer representation of the circuit. This adjustment allows you to identify branches where resistors are connected in parallel or series more easily.

In this example, once the resistors are rearranged, you can see that the 4-ohm and 3-ohm resistors are in parallel. The formula for calculating the equivalent resistance \( R_{eq} \) of two resistors in parallel is given by:

\[ R_{eq} = \frac{R_1 \cdot R_2}{R_1 + R_2} \]

Applying this to the 4-ohm and 3-ohm resistors, you find:

\[ R_{eq} = \frac{4 \cdot 3}{4 + 3} = \frac{12}{7} \approx 1.71 \, \text{ohms} \]

After combining these resistors, the circuit can be redrawn with the equivalent resistance replacing the original resistors. This simplification continues as you identify other resistors in the circuit. For example, if you have a 5-ohm resistor in series with the newly calculated 1.71-ohm resistor, you can again apply the series resistance rule, which states that resistors in series simply add together:

\[ R_{total} = R_1 + R_2 \]

Thus, the total resistance becomes:

\[ R_{total} = 2 + 1.71 = 3.71 \, \text{ohms} \]

It’s essential to remember that when combining resistors in parallel, the total resistance will always be less than the smallest individual resistor. This principle serves as a useful check for your calculations. By carefully rearranging and simplifying the circuit, you can effectively determine the equivalent resistance, making complex networks manageable.

If every resistor below has resistance R, what is the equivalent resistance of the combination, in terms of R?

• A. 0.5R
• B. R
• C. 4R
• D. 5R

Answer: (A)

Here’s what students ask on this topic:

How do you calculate the equivalent resistance of resistors in series?

To calculate the equivalent resistance of resistors in series, you simply add their resistances together. The formula is:

$R_{\mathrm{eq}}=R_1+R_2+R_3+...+R_n$

For example, if you have three resistors with resistances of 2Ω, 3Ω, and 5Ω, the equivalent resistance would be:

$R_{\mathrm{eq}}=2+3+5=10\Omega$

What is the formula for calculating the equivalent resistance of resistors in parallel?

The formula for calculating the equivalent resistance of resistors in parallel is:

$\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+...+\frac{1}{R_n}$

For example, if you have three resistors with resistances of 2Ω, 3Ω, and 6Ω, the equivalent resistance would be:

$\frac{1}{R_{\mathrm{eq}}}=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$

After finding a common denominator and summing the fractions, you would get:

$\frac{1}{R_{\mathrm{eq}}}=\frac{6}{6}=1$

So, $R_{\mathrm{eq}}=1\Omega$.

What is the shortcut for calculating the equivalent resistance of two resistors in parallel?

The shortcut for calculating the equivalent resistance of two resistors in parallel is:

$R_{\mathrm{eq}}=\frac{R_1\times R_2}{R_1+R_2}$

For example, if you have two resistors with resistances of 4Ω and 6Ω, the equivalent resistance would be:

$R_{\mathrm{eq}}=\frac{4\times 6}{4+6}=\frac{24}{10}=2.4\Omega$

How do you combine resistors of the same resistance in parallel?

When combining resistors of the same resistance in parallel, you can use a simple shortcut. If you have $n$ resistors, each with resistance $R$, the equivalent resistance is:

$R_{\mathrm{eq}}=\frac{R}{n}$

For example, if you have three resistors, each with a resistance of 6Ω, the equivalent resistance would be:

$R_{\mathrm{eq}}=\frac{6}{3}=2\Omega$

Why is the equivalent resistance of resistors in parallel always lower than the smallest individual resistance?

The equivalent resistance of resistors in parallel is always lower than the smallest individual resistance because the current has multiple paths to flow through. This effectively reduces the overall resistance. When resistors are in parallel, the total current is divided among the parallel branches, making it easier for the current to flow. Mathematically, this is represented by the formula:

$\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}$

Since the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances, the equivalent resistance is always less than the smallest individual resistance.