Problem 30 - Applied Physics

Question

Solve each formula for the quantity given. $$ \frac{V_{P}}{V_{S}}=\frac{N_{P}}{N_{S}} \text { for } N_{S} $$

Step-by-Step Solution

Answer

\( N_{S} = \frac{V_{S} \cdot N_{P}}{V_{P}} \)

Step 1: Identify the Equation

The equation given is \( \frac{V_{P}}{V_{S}} = \frac{N_{P}}{N_{S}} \). We need to solve for \( N_{S} \).

Step 2: Cross-Multiply the Fractions

To eliminate the fractions, cross-multiply: \( V_{P} \cdot N_{S} = V_{S} \cdot N_{P} \). This step rearranges the equation to facilitate solving for \( N_{S} \).

Step 3: Isolate \( N_{S} \)

Solve for \( N_{S} \) by dividing both sides of the equation by \( V_{P} \): \( N_{S} = \frac{V_{S} \cdot N_{P}}{V_{P}} \). This isolates the variable we are solving for.

Key Concepts

Voltage Ratio

When dealing with transformers, understanding the voltage ratio is essential. The voltage ratio compares the voltages on the primary side (\( V_{P} \)) and the secondary side (\( V_{S} \)) of a transformer. This ratio is vital because it shows how voltages are transformed between the two sides.
Knowing the voltage ratio helps predict how the voltage changes from input to output.

• Primary voltage (\( V_{P} \)) is the voltage applied to the input winding.
• Secondary voltage (\( V_{S} \)) is the voltage detected in the output winding.

If the voltage ratio is high, it means there is a significant difference between the two voltages, often resulting in a higher step-up or step-down transformation. Understanding this ratio is key for designing efficient electrical systems.

Turns Ratio

The turns ratio is another critical concept in transformer equations. It involves the number of turns in the primary and secondary windings of a transformer, denoted as \( N_{P} \) and \( N_{S} \) respectively. In calculations, the turns ratio is given by \( \frac{N_{P}}{N_{S}} \).
This ratio directly affects the voltage ratio of a transformer since they are proportional.

• A higher turns ratio indicates a greater difference in the voltage transformation between primary and secondary sides.
• It also determines how much a transformer can "step up" or "step down" the voltage.

Understanding the turns ratio is crucial for predicting how a transformer will interact with an electrical circuit, ensuring the transformer effectively meets the system's needs.

Cross-Multiplication

Cross-multiplication is an algebraic method used to solve equations involving two fractions set equal to one another. This technique eliminates the fractions by multiplying across the equals sign, which simplifies the equation.
Using cross-multiplication, the equation \( \frac{V_{P}}{V_{S}} = \frac{N_{P}}{N_{S}} \) becomes \( V_{P} \cdot N_{S} = V_{S} \cdot N_{P} \).This rearrangement is essential because:

• It transforms the equation into a more straightforward form to manipulate.
• It lays the foundation for isolating a specific variable.

Cross-multiplying is a fundamental algebra skill that simplifies complex equations, making it easier to find solutions effectively.

Isolating a Variable

Isolating a variable means rearranging an equation so that one variable stands alone on one side of the equality. This process is crucial when solving equations to find the specific value of a variable you're interested in.
In our example, we need to solve for \( N_{S} \). After cross-multiplying, we get \( V_{P} \cdot N_{S} = V_{S} \cdot N_{P} \).To isolate \( N_{S} \), divide both sides of the equation by \( V_{P} \):\[ N_{S} = \frac{V_{S} \cdot N_{P}}{V_{P}}.\]

• This step ensures that \( N_{S} \) is alone on one side, making it straightforward to calculate.
• Isolating a variable is often the final step when solving for a specific unknown in algebra.

Mastering this technique will make solving complex equations much more manageable and less daunting.