Problem 112 - Elementary and Intermediate Algebra

Question

Find the dB gain of an amplifier if the output voltage is 2.8 volts when the input voltage is 0.05 volt. Round to the nearest dB.

Step-by-Step Solution

Answer

The dB gain of the amplifier is approximately 35 dB.

Step 1: Understanding the Decibel (dB) Formula

To find the gain in decibels (dB) of an amplifier based on input and output voltages, we use the formula: \\[\text{Gain (dB)} = 20 \cdot \log_{10}(\frac{V_{\text{out}}}{V_{\text{in}}})\] \where \(V_{\text{out}}\) is the output voltage and \(V_{\text{in}}\) is the input voltage.

Step 2: Plugging in the Values

Given that \(V_{\text{out}} = 2.8\) volts and \(V_{\text{in}} = 0.05\) volt, plug these values into the formula: \\[\text{Gain (dB)} = 20 \cdot \log_{10}\left(\frac{2.8}{0.05}\right)\]

Key Concepts

Decibel Gain Calculation

When working with amplifiers, one of the most common measurements you'll encounter is the decibel (dB) gain. This measurement allows us to understand how much an amplifier boosts a signal. The concept of decibel gain is particularly useful because it provides a logarithmic scale to represent the ratio of two values, making it easier to comprehend large differences in voltage levels.

To calculate the decibel gain, we use the formula: \[\text{Gain (dB)} = 20 \cdot \log_{10}\left(\frac{V_{\text{out}}}{V_{\text{in}}}\right)\] where:

• \(V_{\text{out}}\) is the output voltage.
• \(V_{\text{in}}\) is the input voltage.

This formula allows you to compute the gain by comparing the output voltage to the input voltage in a logarithmic scale, which is ideal for audio and electronic applications due to its sensitivity to changes in power levels. Understanding this formula is crucial for accurately assessing how an amplifier impacts an incoming signal.

Amplifier Voltage Gain

The concept of amplifier voltage gain refers to the ability of an amplifier to increase the amplitude of an input voltage signal. This gain is a measure of how much an amplifier increases the power of a signal. In electronic terms, it's the ratio of output voltage to input voltage expressed as a factor or percentage.

While working with amplifiers, the voltage gain can be calculated using:

• \( \text{Voltage Gain (A}_v\text{)} = \frac{V_{\text{out}}}{V_{\text{in}}} \)

This formula shows that the voltage gain is simply the output voltage divided by the input voltage, giving a straightforward measure of the amplifier's performance before converting it to decibels.An amplifier with a higher voltage gain can make a weak signal significantly stronger, which is essential in applications where signal strength is vital, such as audio equipment, broadcasting, and communication devices.

Input and Output Voltage

Understanding input and output voltage is fundamental when dealing with any amplifier or electronic circuit. The input voltage \(V_{\text{in}}\) is the initial electrical signal introduced to the circuit. Meanwhile, the output voltage \(V_{\text{out}}\) is the signal after it has been amplified.

Consider the role each plays:

• Input Voltage: The raw or original signal that needs amplification.
• Output Voltage: The enhanced or amplified signal that leaves the amplifier.

In our original exercise, the input voltage was given as 0.05 volts, and the output voltage was measured at 2.8 volts. These values demonstrate how effective an amplifier can be. Analyzing these voltages helps in understanding the gain and efficiency of an amplifier, ensuring that the device meets the requirements for its intended application, whether that be in audio systems, signal processing, or other electronic innovations.