When it comes to understanding electronic circuits, one of the fundamental principles is the relationship between input voltage and output voltage. Today, I want to delve into the topic of finding the value of `r`

that will make the output voltage (`v`

) equal to 15 times the input voltage (_{out}`v`

). This is an important concept in electronics and can be incredibly useful in practical circuit design._{in}

## The Challenge: Finding the Value of `r`

Our goal is to find the resistance value of `r`

in a simple circuit that will produce an output voltage that is 15 times the input voltage. This can be achieved through the use of a voltage divider circuit. In a voltage divider, the output voltage is determined by the ratio of the two resistors in the circuit, as well as the input voltage. By manipulating this ratio, we can achieve our desired output voltage.

Let’s consider the voltage divider formula:

`v`

_{out} = v_{in} * (r_{2} / (r_{1} + r_{2}))

Where `r`

and _{1}`r`

are the two resistors in the voltage divider circuit. Our task is to find the value of _{2}`r`

that will satisfy the condition `v`

._{out} = 15 * v_{in}

By rearranging the formula, we can express `r`

in terms of _{2}`r`

and _{1}`r`

:

`r`

_{2} = r * r_{1} / (r - r_{1})

Now, we substitute this expression for `r`

back into the voltage divider formula and solve for the condition _{2}`v`

._{out} = 15 * v_{in}

After solving, we find the value of `r`

to be:_{1}

`r`

_{1} = 14r / 15

Therefore, the value of `r`

will be _{1}`14r / 15`

.

## Putting It Into Practice

Understanding the relationship between resistances in a voltage divider circuit and the resulting output voltage is crucial for any electronics enthusiast. By utilizing the derived formula for `r`

, we can easily determine the required resistor values to achieve the desired output voltage in our circuits._{1}

## Conclusion

Exploring the concept of finding the value of `r`

that will make the output voltage `v`

15 times the input voltage _{out}`v`

has shed light on the intricate relationship between resistors and voltage in electronic circuits. This knowledge is essential for anyone working with circuit design and provides valuable insight into the practical application of theoretical concepts. I hope this article has been enlightening, and I encourage you to experiment with voltage dividers to further solidify your understanding of this fundamental principle in electronics._{in}