Question

Calculate the voltage gain in decibels of an amplifier where the input signal is \(0.5 \mathrm{~V}\) and the output signal is \(2.2 \mathrm{~V}\).

Short answer

Question: Calculate the voltage gain in decibels of an amplifier with an input signal of 0.5 V and an output signal of 2.2 V. Answer: The voltage gain in decibels of the amplifier is approximately 12.86 dB.

Step-by-step solution

Identify the given values

In this exercise, we are given the input voltage (\(0.5 \mathrm{~V}\)) and output voltage (\(2.2 \mathrm{~V}\)). We will use these values to calculate the voltage gain.

Calculate the ratio of output to input voltage

We need to find the ratio of the output voltage to the input voltage. This can be done by dividing the output voltage (\(V_{out}\)) by the input voltage (\(V_{in}\)): \(\frac{V_{out}}{V_{in}} = \frac{2.2 \mathrm{~V}}{0.5 \mathrm{~V}}\).

Solve for the output-to-input voltage ratio

Divide the output voltage by the input voltage to get the ratio: \(\frac{V_{out}}{V_{in}} = \frac{2.2 \mathrm{~V}}{0.5 \mathrm{~V}} = 4.4\)

Calculate the voltage gain in decibels

Now we can calculate the voltage gain in decibels using the formula \(G_v = 20 \log_{10} \left( \frac{V_{out}}{V_{in}} \right)\). Substitute the calculated ratio (4.4) into the formula: \(G_v = 20 \log_{10}(4.4)\)

Solve for the voltage gain in decibels

Calculate the voltage gain in decibels using a calculator or logarithm table: \(G_v = 20 \log_{10}(4.4) \approx 12.86\)

Write the final answer

The voltage gain in decibels of the amplifier is approximately \(12.86 \mathrm{dB}\).

Key concepts

Decibels Calculation

The decibel (dB) is a unit used to express the magnitude of a physical quantity, such as the gain of an amplifier, relative to a specified or implied reference level. It is logarithmic by nature, meaning it simplifies large ratios through the use of logarithms. When calculating voltage gain in decibels, we use the formula:\[ G_v = 20 \log_{10}\left(\frac{V_{out}}{V_{in}}\right) \]where \( V_{out} \) is the output voltage and \( V_{in} \) is the input voltage. This formula helps to convert the linear voltage ratio into a logarithmic scale.
Converting a voltage gain into decibels allows for easier comparison and analysis, especially when dealing with amplifiers and audio equipment, as decibels are more manageable numbers compared to large ratios.

Amplifier Circuit

An amplifier circuit is an essential part of many electronic devices. It increases the power of a signal, meaning it increases its voltage, current, or both. Amplifiers are critical in audio devices, broadcasting, and many types of communications equipment.

• **Input and Output:** In our example, the input signal is 0.5 V, and the output is 2.2 V.
• **Purpose:** The main goal of this circuit is to amplify or increase the strength of an input signal.
• **Types of Amplifiers:** These can be classified by their function, such as voltage amplifiers, current amplifiers, or power amplifiers.

Knowing how to calculate the voltage gain in decibels helps in designing and analyzing these circuits for desired performance.

Electronics Education

Understanding fundamental concepts like voltage gain and decibels is crucial in electronics education. These concepts form the basis for more advanced topics in the field.
Voltage gain quantifies how much an input voltage is amplified by an amplifier circuit. Educating students about this involves teaching them how to compute ratios, use logarithms, and understand decibels.

• **Practical Skills:** Students must learn to use calculators or software tools to evaluate logarithmic equations.
• **Conceptual Understanding:** Grasping why and how we use decibels to measure gain is key for future learning.

Educational exercises like this one reinforce these principles through practical application, enhancing problem-solving skills.

Voltage Ratio

The voltage ratio is a basic but important concept in many electronics calculations. It is defined as the ratio between the output voltage and the input voltage of a system.\[ \text{Voltage Ratio} = \frac{V_{out}}{V_{in}} \]
This ratio is dimensionless and provides a straightforward way to measure how much an amplifier can increase the input signal.

• **Calculation:** In our example, the voltage ratio is calculated as \( \frac{2.2 \text{V}}{0.5 \text{V}} = 4.4 \).
• **Importance:** Understanding voltage ratios is essential before moving on to more complex concepts like gain in decibels.

By mastering voltage ratios, students can better understand and predict the performance of electronic circuits.