Question

Calculate the voltage gain in decibels of an amplifier where the input voltage is \(17 \mathrm{mV}\) and the output voltage is \(300 \mathrm{mV}\).

Short answer

Question: Calculate the voltage gain in decibels (dB) of an amplifier with an output voltage of 300 mV and an input voltage of 17 mV. Answer: The voltage gain in decibels of the amplifier is approximately 25.02 dB.

Step-by-step solution

Compute the voltage ratio

To compute the voltage ratio, divide the output voltage by the input voltage: $$ Voltage \: Ratio = \frac{Output \: Voltage}{Input \: Voltage} $$ Plug in the given values of output voltage (\(300 \mathrm{mV}\)) and input voltage (\(17 \mathrm{mV}\)): $$ Voltage \: Ratio = \frac{300 \mathrm{mV}}{17 \mathrm{mV}} $$ Now, calculate the voltage ratio: $$ Voltage \: Ratio ≈ 17.65 $$

Compute the voltage gain in decibels

To compute the voltage gain in decibels (dB), use the formula for voltage gain in decibels: $$ Voltage \: Gain \: (dB) = 20 \log_{10}\left(Voltage \: Ratio\right) $$ Plug in the calculated voltage ratio value: $$ Voltage \: Gain \: (dB) = 20 \log_{10}\left(17.65\right) $$ Now, calculate the voltage gain in decibels: $$ Voltage \: Gain \: (dB) ≈ 25.02 \mathrm{dB} $$ Therefore, the voltage gain in decibels of the amplifier is approximately \(25.02 \mathrm{dB}\).

Key concepts

Voltage Ratio

Understanding the concept of voltage ratio is crucial when analyzing electrical circuits, especially in the context of amplifiers. The voltage ratio is simply a comparison of two voltages: the output voltage of a circuit with respect to its input voltage. Mathematically, it's a division of the output voltage by the input voltage, represented as \( \frac{V_{out}}{V_{in}} \).

In the given exercise, the output voltage of the amplifier is 300 mV and the input voltage is 17 mV. Therefore, the voltage ratio is calculated to be approximately 17.65. This value is dimensionless and indicates how many times the input voltage has been amplified. It's the fundamental step before we can convert this ratio into a more comprehensible logarithmic scale, known as decibels.

Decibel Calculation

The decibel (dB) is a logarithmic unit used to express ratios, like the voltage ratio, and it is widely used for stating power and intensity gains in electronic circuits. The formula to calculate the voltage gain in decibels is \( 20 \cdot \log_{10}(Voltage \: Ratio) \). This formula incorporates the base-10 logarithm, reflecting the logarithmic nature of human hearing, and the factor of 20, which converts the power ratio to a voltage ratio.

A decibel is essentially a relative unit of measure; it compares the level of a physical quantity to a specified or implied reference level. For instance, if an amplifier increases the voltage 10 times, this is a gain of 20 dB, because \( 20 \cdot \log_{10}(10) = 20 \). The calculation of 25.02 dB in the exercise demonstrates the voltage gain from the input to the output, allowing us to quantify the amplification effect.

Logarithmic Expressions

Logarithmic expressions are indispensable in understanding decibel calculations. A logarithm answers the question: to what exponent do we need to raise a specific base number to obtain another number? Specifically, a base-10 logarithm, denoted as \(\log_{10}\), is used in the context of decibels.

The formula \(20 \cdot \log_{10}(Voltage \: Ratio)\) takes the logarithm of the voltage ratio and multiplies it by 20 to convert the logarithm into decibels. For the provided exercise, by taking the log of the voltage ratio (17.65), we can then multiply this logarithmic result by 20 to find the voltage gain in decibels. Understanding logarithms helps in interpreting the results; a small increase in logarithmic value corresponds to a large increase when converted back from the logarithmic scale. For example, a difference of 3 dB represents approximately a doubling or halving of power in acoustic measurements.

Amplifier Gain Analysis

Amplifier gain analysis involves comprehending how well an amplifier boosts a signal from its input to output. Gain can be expressed in terms of voltage, current, or power, with voltage gain being the ratio of output voltage to input voltage, as we've discussed previously.

In real-world applications, knowing the amplifier's gain is crucial for designing systems that require precise levels of signal amplification, such as audio equipment or radio transmitters. For instance, an amplifier with a high gain might introduce noise or distortion if not properly controlled. The value calculated in the exercise, approximately 25.02 dB, allows engineers to understand the capability of the amplifier and to match it with other system components for optimum performance. Moreover, comparing gains in decibels helps in quickly assessing the efficiency or effectiveness of different amplifiers.