Resistor Combinations: Series and Parallel
The Building Blocks: What is a Resistor?
Before we combine them, let's understand what a resistor is. Imagine water flowing through a pipe. If the pipe is narrow, it resists the flow of water. A resistor acts similarly in an electrical circuit; it is a component that resists or opposes the flow of electric current. This resistance is measured in a unit called Ohms, symbolized by the Greek letter Omega: $ \Omega $. The higher the resistance, the harder it is for current to flow. Resistors are used to control current levels, divide voltages, and protect sensitive components.
Resistors in Series: A Single File Path
When resistors are connected end-to-end, like links in a chain, they are said to be in series. There is only one path for the electric current to take. If any resistor in a series connection breaks or is disconnected, the entire circuit is broken, and current stops flowing everywhere—just like old-fashioned Christmas tree lights.
• The same current flows through each resistor.
• The total voltage across the combination is the sum of the voltages across each individual resistor.
• The total or equivalent resistance is simply the sum of all the individual resistances.
The formula for calculating the equivalent resistance ($ R_{eq} $) of resistors in series is straightforward:
$ R_{eq} = R_1 + R_2 + R_3 + \cdots + R_n $
Where $ R_1, R_2, R_3, \dots, R_n $ are the resistances of the individual resistors.
Resistors in Parallel: Multiple Highways for Current
When resistors are connected side-by-side, providing multiple separate paths for current to flow, they are in a parallel combination. Think of a major highway splitting into several smaller roads. If one road is blocked, traffic can still use the others. Similarly, if one resistor in a parallel connection fails, current continues to flow through the other paths.
• The voltage across each resistor is the same.
• The total current is the sum of the currents through each parallel branch.
• The total or equivalent resistance is less than the smallest individual resistance in the combination.
The formula for the equivalent resistance of resistors in parallel is a bit more complex:
$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_n} $
For two resistors in parallel, a handy shortcut formula can be used:
$ R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} $
Comparing Series and Parallel Connections
The table below provides a clear, side-by-side comparison of these two fundamental connection methods.
| Feature | Series Circuit | Parallel Circuit |
|---|---|---|
| Current (I) | Same through all components: $ I_{total} = I_1 = I_2 = I_3 $ | Divides among branches: $ I_{total} = I_1 + I_2 + I_3 $ |
| Voltage (V) | Divides across components: $ V_{total} = V_1 + V_2 + V_3 $ | Same across all branches: $ V_{total} = V_1 = V_2 = V_3 $ |
| Equivalent Resistance ($ R_{eq} $) | $ R_{eq} = R_1 + R_2 + R_3 $ (Increases) | $ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} $ (Decreases) |
| If One Component Fails | The entire circuit is broken. All components turn off. | The other branches continue to work. Only that branch turns off. |
Step-by-Step Circuit Analysis
Most real-world circuits are not purely series or parallel; they are combinations of both. To analyze these, we break them down step-by-step, simplifying sections until we find a single equivalent resistance.
Example 1: Simple Series Circuit
Imagine a circuit with a $ 9V $ battery connected to three resistors: $ R_1 = 10 \Omega $, $ R_2 = 20 \Omega $, and $ R_3 = 30 \Omega $ in series. What is the total resistance and current flowing from the battery?
Step 1: Find Equivalent Resistance.
Since they are in series: $ R_{eq} = R_1 + R_2 + R_3 = 10\Omega + 20\Omega + 30\Omega = 60\Omega $.
Step 2: Find Total Current.
Using Ohm's Law ($ V = I \times R $), we can find the total current: $ I_{total} = \frac{V}{R_{eq}} = \frac{9V}{60\Omega} = 0.15A $ (or $ 150mA $).
Example 2: Simple Parallel Circuit
Now, let's connect the same three resistors in parallel to the same $ 9V $ battery. What is the new total resistance and total current?
Step 1: Find Equivalent Resistance.
Using the parallel formula: $ \frac{1}{R_{eq}} = \frac{1}{10} + \frac{1}{20} + \frac{1}{30} = \frac{6}{60} + \frac{3}{60} + \frac{2}{60} = \frac{11}{60} $.
Therefore, $ R_{eq} = \frac{60}{11} \approx 5.45\Omega $. Notice how the total resistance ($ 5.45\Omega $) is less than the smallest resistor ($ 10\Omega $).
Step 2: Find Total Current.
$ I_{total} = \frac{V}{R_{eq}} = \frac{9V}{5.45\Omega} \approx 1.65A $. By providing multiple paths, the circuit allows much more current to flow than the series configuration.
Example 3: Mixed Combination Circuit
Consider a circuit where $ R_1 = 4\Omega $ is in series with a parallel combination of $ R_2 = 6\Omega $ and $ R_3 = 12\Omega $, all connected to a $ 12V $ battery. Let's find $ R_{eq} $.
Step 1: Simplify the parallel section ($ R_2 $ and $ R_3 $).
$ R_{parallel} = \frac{R_2 \times R_3}{R_2 + R_3} = \frac{6 \times 12}{6 + 12} = \frac{72}{18} = 4\Omega $.
Step 2: Simplify the series section.
Now we have $ R_1 $ ($ 4\Omega $) in series with $ R_{parallel} $ ($ 4\Omega $).
$ R_{eq} = R_1 + R_{parallel} = 4\Omega + 4\Omega = 8\Omega $.
The total current from the battery would be $ I_{total} = \frac{12V}{8\Omega} = 1.5A $.
Practical Applications in Everyday Electronics
The principles of series and parallel resistor combinations are everywhere in the devices we use daily.
1. Home Lighting: The lights in your house are wired in parallel. This is why you can turn on the kitchen light without turning on the living room light, and why one burned-out bulb doesn't plunge your entire home into darkness. If they were in series, you would need all lights on at once, and one failure would break the whole circuit.
2. Voltage Dividers: A series combination of two resistors is commonly used to create a voltage divider. This circuit takes a higher input voltage and outputs a lower, specific voltage. For example, if you have a $ 9V $ battery but need $ 3V $ to power a small sensor, you can use two resistors in series. The voltage drop across the smaller resistor will be your desired $ 3V $. The formula for the output voltage ($ V_{out} $) is: $ V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2} $.
3. USB Hub and Power Strips: When you plug multiple devices into a USB hub or a power strip, you are connecting them in parallel to the power source. Each device gets the same voltage (e.g., $ 5V $ for USB), and the total current drawn is the sum of the currents required by each device.
Common Mistakes and Important Questions
Q: Why is the total resistance in a parallel circuit always less than the smallest resistor?
Q: What is a common calculation error when working with parallel resistors?
Q: Can a circuit have both series and parallel parts?
Mastering series and parallel resistor combinations is a cornerstone of electronics. By understanding that series connections add resistance and share voltage, while parallel connections reduce resistance and share current, you gain the ability to analyze and design a vast array of simple and complex circuits. Remember to use the step-by-step simplification process for combination circuits, and always double-check your calculations, especially the reciprocal for parallel resistances. With this knowledge, you are well-equipped to explore more advanced topics in electricity and circuit design.
Footnote
1 Ohm ($ \Omega $): The SI2 unit of electrical resistance. One ohm is defined as the resistance between two points of a conductor when a constant potential difference of one volt applied to these points produces a current of one ampere.
2 SI: International System of Units (from the French "Système International d'Unités"), the modern form of the metric system and the world's most widely used system of measurement.
3 Ohm's Law: A fundamental law of electronics stating that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. The formula is $ V = I \times R $.