The voltage input to an amplifier is \(30 \mathrm{mV}\). (a) Calculate the output voltage if the amplifier has a gain of \(16 \mathrm{~dB}\). (b) Calculate the output voltage if the amplifier has a gain of \(32 \mathrm{~dB}\).

Understanding how to convert decibels (dB) to a linear scale is crucial in electrical engineering, especially when dealing with amplifier gains. Decibels are a logarithmic unit used to express ratios, like gain, in a compact and convenient form. To convert gain from decibels to a linear scale, you can use the formula:

Gain(dB) = 20 \(\log_{10}\)(Gain)

Solving this for the linear gain gives us:

Gain = \(10^{\frac{Gain(dB)}{20}}\)

This formula is essential because it allows engineers to work with real-world quantities like voltage or power, starting from a dB value. In the context of the given exercise, knowing this conversion is the first step before you can calculate the amplifier's output voltage based on its gain in decibels.

The voltage gain ratio of an amplifier is a measure of how much the amplifier increases the input signal's voltage. It's defined as the output voltage \(V_{out}\) divided by the input voltage \(V_{in}\):

Gain = \(\frac{V_{out}}{V_{in}}\)

This simple formula underpins many calculations in an amplifier's design and analysis. A higher gain means a stronger amplification of the input signal. In practical applications, the gain can be controlled or set as per the requirements of the system within which the amplifier operates. The exercise provided leads to the calculation of output voltage by multiplying the gain ratio by the input voltage, a fundamental application of the voltage gain concept.

Electrical engineering mathematics involves a mix of algebraic manipulation, calculus, and logarithmic functions. These mathematical tools help in designing, analyzing, and understanding electrical circuits. The exercise demonstrates the application of logarithms in converting decibel values to linear scale gain and then using algebra to calculate the output voltages of an amplifier. Mastery of these mathematical principles is essential for any aspiring electrical engineer, as they form the bedrock for circuit design, signal processing, and system analysis. Incorporating concepts like these in practical problems not only enforces the theory but also prepares students for real-world engineering challenges.

It is advisable for students to practice these operations to become proficient. Hands-on exercises, such as the amplifier gain calculation, solidify their understanding and ability to apply mathematics in the context of electrical engineering.