Question

Calculate the voltage gain in decibels of an amplifier where the input signal is \(0.15 \mathrm{~V}\) and the output signal is \(1.9 \mathrm{~V}\).

Short answer

Question: Calculate the voltage gain of an amplifier in decibels (dB) if the output signal is 1.9 V and the input signal is 0.15 V. Answer: The voltage gain of the amplifier in decibels is 22.06 dB.

Step-by-step solution

Calculate the voltage gain (ratio)

Find the ratio of the output voltage to the input voltage. This will give us the voltage gain of the amplifier. Gain (ratio) = \(V_{out} / V_{in}\) where \(V_{out}\) is the output signal (\(1.9 \mathrm{~V}\)) and \(V_{in}\) is the input signal (\(0 .15 \mathrm{~V}\)). Gain (ratio) = \(1.9 \mathrm{~V} / 0.15 \mathrm{~V}\) = \(12.67\)

Calculate the voltage gain in decibels (dB)

Now, we will change the gain ratio (unitless) to decibels (dB) using the standard formula. Voltage gain (dB) = \(20 * log_{10}\)(Gain ratio) where \(log_{10}\) is the base-10 logarithm and Gain ratio is 12.67. Voltage gain (dB) = \(20 * log_{10}(12.67)\) = \(20 * 1.103\) = \(22.06\,\text{dB}\) The voltage gain of the amplifier in decibels is \(22.06\,\text{dB}\).

Key concepts

Electrical Engineering

Electrical engineering is the branch of engineering that deals with the study and application of electricity, electronics, and electromagnetism. A fundamental concept within this field is the understanding of electronic circuits. Specifically, in the context of amplifiers, which are common in various electronic devices, the performance is often characterized by their gain.
Gain refers to the ability of an amplifier to increase the power level or amplitude of a signal from its input to output. In electrical terms, this could relate to voltage, current, or power gain. It's crucial for students and engineers to know how to calculate and express gain to ensure that the amplifiers are designed and functioning correctly within electronic circuits.

Decibels

The decibel (dB) is a logarithmic unit used to measure sound intensity, but it is also widely used in electronics, signals, and communication to express the magnitude of gain or loss within an amplifier or signal chain. Its logarithmic nature allows us to represent very large or small ratios of values in a manageable scale.

Because human perception of sound intensity is logarithmic, decibels are particularly effective in conveying changes in volume. However, in electrical engineering, a signal's power level, represented in decibels, allows engineers to quickly analyze and compare the effectiveness of amplifiers or attenuators. Understanding how to convert a ratio to decibels, as shown in your exercise, is essential for those hoping to excel in electrical engineering and related fields.

Logarithmic Functions

Logarithmic functions are inverses of exponential functions, and they play a significant role in various scientific and engineering disciplines. The logarithm to the base 10, typically denoted as \(log_{10}\), is particularly prevalent in calculations involving decibels.

Logarithmic scales are used to express large ranges of values in a condensed form—which is particularly useful when dealing with electronic signals that can vary by many orders of magnitude. For students, understanding logarithms is essential to grasp concepts such as pH in chemistry, Richter scale in seismology, and obviously, decibel calculations in electrical engineering.

Amplifier Gain

Amplifier gain is defined as the ratio of the amplifier's output signal to its input signal and is a dimensionless number. It can be expressed in terms of voltage, current, or power depending on which parameter of the signal is being amplified.

For instance, voltage gain is used when comparing the input and output voltages, as in your exercise. Calculating the gain allows designers to predict how the amplifier will behave in a circuit and ensures signals are amplified correctly without distortion or loss of quality. Understanding how to calculate amplifier gain, both as a ratio and in decibels, is crucial in designing reliable and efficient amplification circuits within various electronic devices. The process of improving this exercise would involve ensuring clarity on the distinction between the unitless gain ratio and the decibel representation which includes the logarithmic conversion.