Voltage sources in series

Question

Understanding Voltage Sources in Series: Aiding vs. Opposing Connections

Answer 1

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Understanding Voltage Sources in Series: Aiding vs. Opposing Connections

When working with electrical circuits, it's common to connect multiple voltage sources, like batteries, in series to achieve a desired total voltage. The way you connect them determines whether their voltages add up or subtract. The image provided illustrates the two primary configurations: series-aiding and series-opposing.

Let's break down each case shown in the diagrams.

What Does "In Series" Mean?

Connecting components in series means they are connected end-to-end, creating a single path for the current to flow through. For voltage sources, this involves connecting the terminal of one source to a terminal of another.

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Case 1: Series-Aiding Connection (Voltages Add)

This is the most common way to connect batteries to get a higher voltage.

Diagram (a):

• Connection: The negative terminal (-) of the first voltage source (V₁) is connected to the positive terminal (+) of the second voltage source (V₂).
• Result: The sources work together, "pushing" in the same direction. Their individual voltages add up to create a larger total voltage.
• Formula: V_total = V₁ + V₂
• Polarity: The overall polarity is determined by the unconnected terminals. In this case, the top terminal is positive (+) and the bottom terminal is negative (-).

For example, connecting two 1.5V AA batteries in a series-aiding manner results in a total voltage of 1.5V + 1.5V = 3V.

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Case 2: Series-Opposing Connection (Voltages Subtract)

This configuration is less common in everyday applications but is an important concept in circuit analysis.

Diagrams (b) and (c):

• Connection: Like terminals are connected together—either positive-to-positive as shown here, or negative-to-negative.
• Result: The sources work against each other. The net voltage is the difference between the individual voltages.
• Formula: V_total = |V₁ - V₂|
• Polarity: The polarity of the total voltage is determined by the larger of the two voltage sources.
  • In Diagram (b): The formula (V₁ - V₂)
    implies that V₁ is greater than V₂. Therefore, the resulting polarity matches V₁, with the top terminal being positive (+).
  • In Diagram (c): The formula (-V₁ + V₂)
    , which is the same as (V₂ - V₁)
    , implies that V₂ is greater than V₁. In this scenario, the effective "push" from V₂ is stronger, but the diagram keeps the output polarity reference the same as in (b). A more intuitive representation would show the bottom terminal as positive. However, both (b) and (c) correctly represent the calculation for a series-opposing setup.

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Identifying an Incorrect Calculation

It's crucial to apply the correct formula based on the connection type.

Diagram (d):

• Connection: The sources V₁ and V₂ are connected positive-to-positive, which is a series-opposing configuration.
• The Error: The formula provided is (V₂ + V₁)
  . This is incorrect. Because the sources are opposing, their voltages should be subtracted, not added. The correct formula for the magnitude of the voltage would be |V₂ - V₁|.

Summary Table

| Connection Type | Terminal Connection | Total Voltage (V_total) |
| :--- | :--- | :--- |
| Series-Aiding | Negative (-) to Positive (+) | V₁ + V₂ (Voltages add) |
| Series-Opposing | Positive (+) to Positive (+) or Negative (-) to Negative (-) | \|V₁ - V₂\| (Voltages subtract) |

By understanding these fundamental rules, you can correctly analyze and design circuits with multiple voltage sources in series. Always check the polarity of the connection before calculating the equivalent voltage.