Tru Physics

Chapter 12: Resistors in Series and Parallel

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12.1 Introduction to Resistors

In electrical circuits, resistors are often connected in various configurations to control current flow and voltage distribution. Two common resistor arrangements are series and parallel. In this chapter, we will discuss the properties of resistors in series and parallel configurations, as well as how to analyze and calculate the equivalent resistance for each arrangement.

12.2 Resistors in Series

Three resistors are connected in series with one another.
Three resistors are connected in series with one another. The equivalent resistance (measured in ohms) is just R1 + R2 + R3.

Resistors are in series when they are connected end-to-end, and the current flows through each resistor sequentially. In a series configuration, the current remains constant throughout the resistors, while the voltage drop across each resistor depends on its resistance. The equivalent resistance for resistors in series (R_eq) can be calculated using the following formula:

R_{\text{eq}} = R_1 + R_2 + R_3 + ... = \displaystyle\sum_{i=0}^{n}R_i

where the sum is over all of the individual resistor values (R_i). Essentially, the equivalent resistance is just the sum of the n number of individual resistances.

12.3 Resistors in Parallel

Parallel combinations allow the current several different "options" or "branches" through which it can travel. In the diagram above, the current has three different possible paths from A to B making this a parallel combination of resistors.
Parallel combinations allow the current several different “options” or “branches” through which it can travel. In the diagram above, the current has three different possible paths from A to B making this a parallel combination of resistors.

Resistors are in parallel when their terminals are connected together, creating multiple paths for the current to flow. In a parallel configuration, the voltage across each resistor remains constant, while the current through each resistor depends on its resistance. The equivalent resistance for resistors in parallel (R_eq) can be calculated using the following formula:

R_eq = \dfrac{1}{\dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} + ...} = \dfrac{1}{\displaystyle\sum_{i=0}^{n}R_i}

This formula tends to cause more confusion. However, it’s not really as complicated as it might seem, especially if you have an inverse button (\box^{-1}) on your calculator. Then, you can simply input the calculation as:

R_{\text{eq}} = ((R_1)^{-1} + (R_2)^{-1} + (R_3)^{-1})^{-1} = \left( \displaystyle\sum_{i=0}^{n} (R_i)^{-1}} \right)^{-1}

As you can see, by using the inverse function on your calculator, you can eliminate many careless errors when making this type of calculation.

12.4 Analyzing Series and Parallel Circuits

To analyze a circuit containing resistors in series and parallel configurations, follow these steps:

  1. Identify the series and parallel resistor combinations within the circuit.
  2. Calculate the equivalent resistance for each combination using the appropriate formulas.
  3. Replace the original resistor combinations with their equivalent resistances, simplifying the circuit.
  4. Analyze the simplified circuit using Ohm’s law, Kirchhoff’s laws, and other relevant principles.
  5. Work backward through the circuit, replacing the equivalent resistances with the original resistor combinations to find the current, voltage, and power values for each resistor.

12.5 Power in Series and Parallel Circuits

The power consumed by a resistor can be calculated using the following formula:

P = IV = I^2R = \dfrac{V^2}{R}

where P is the power in watts (W), I is the current in amperes (A), V is the voltage in volts (V), and R is the resistance in ohms (\Omega). In series and parallel circuits, the power dissipation in each resistor can be found by substituting the appropriate values for current and voltage.

Chapter Summary

In this chapter, we explored the properties of resistors in series and parallel configurations, including their effect on current flow and voltage distribution within a circuit. We discussed the formulas for calculating equivalent resistance in both configurations and how to analyze series and parallel circuits. Understanding resistors in series and parallel is essential for designing and analyzing electrical circuits that control current flow and voltage distribution.

Continue to Chapter 13: Kirchhoff’s Rules

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