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.
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 (Vā) is connected to the positive terminal (+) of the second voltage source (Vā).
- Result: The sources work together, "pushing" in the same direction. Their individual voltages add up to create a larger total voltage.
- Formula: V_total = Vā + Vā
- 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.
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 togetherāeither 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 = |Vā - Vā|
- Polarity: The polarity of the total voltage is determined by the larger of the two voltage sources.
- In Diagram (b): The formula (Vā - Vā)
implies that Vā is greater than Vā. Therefore, the resulting polarity matches Vā, with the top terminal being positive (+).
- In Diagram (c): The formula (-Vā + Vā)
, which is the same as (Vā - Vā)
, implies that Vā is greater than Vā. In this scenario, the effective "push" from Vā 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.
Identifying an Incorrect Calculation
It's crucial to apply the correct formula based on the connection type.
Diagram (d):
- Connection: The sources Vā and Vā are connected positive-to-positive, which is a series-opposing configuration.
- The Error: The formula provided is (Vā + Vā)
. 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 |Vā - Vā|.
Summary Table
| Connection Type | Terminal Connection | Total Voltage (V_total) |
| :--- | :--- | :--- |
| Series-Aiding | Negative (-) to Positive (+) | Vā + Vā (Voltages add) |
| Series-Opposing | Positive (+) to Positive (+) or Negative (-) to Negative (-) | \|Vā - Vā\| (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.
