a. The total decibel gain of a three-stage system is \(120 \mathrm{~dB}\). Determine the decibel gain of each stage if the second stage has twice the decibel gain of the first and the third has \(2.7\) times the decibel gain of the first. b. Determine the voltage gain of each stage.

Understanding decibel gain is crucial for anyone delving into the field of electronics, especially when dealing with amplifiers. Decibel (dB) is a logarithmic unit used to express ratios—commonly power or intensity ratios in the context of electronic circuits. It is a convenient way of quantifying gain since it compresses the range of values into a manageable scale.

For amplifiers, which are devices that increase the power of a signal, gain is a measure of how much an input signal's power has been increased, and it's often expressed in decibels. Calculating decibel gain involves using the formula: \( G_{db} = 20 \cdot \log_{10}(G_v) \), where \( G_{db} \) is the decibel gain and \( G_v \) is the voltage gain.

When cascading multiple amplifier stages, as in multi-stage amplifiers, the total gain is the product of the gains of each stage, but when using decibels, we can simply add their gains together. This property simplifies calculations, particularly when ascertaining the gain of complex systems. It's crucial to remember the importance of keeping a consistent reference when comparing decibel gains across different stages or systems.

Converting between voltage gain and decibel gain is a fundamental skill in electronics that allows for easier comparison and combination of gains. Voltage gain (\( G_v \)) is a linear measure and represents the ratio of the output voltage to the input voltage of an amplifier. On the other hand, decibel gain is a logarithmic representation that can express large changes in gain more compactly.

The conversion from decibel gain to voltage gain is done using the expression: \( G_v = 10^{(G_{db}/20)} \) as seen in the exercise. Conversely, to convert from voltage gain to decibel gain, the reverse calculation \( G_{db} = 20 \cdot \log_{10}(G_v) \) is used.

In practice, this conversion allows technicians and engineers to work with more intuitive numbers, as handling very large or small ratios can be cumbersome and prone to error. It's especially useful when dealing with amplifier designs or when adjusting equipment to suit certain specifications.

Analyzing a three-stage amplifier involves an understanding of how each stage's gain contributes to the overall system gain. An amplifier stage is a single amplification unit that can be combined with others to form a more complex amplifying system. The analysis of a multi-stage amplifier typically centers around gaining an overall picture of its performance based on the gains of the individual stages.

In the scenario where the decibel gain of subsequent stages is a multiple of the first, we set up proportional relationships, as seen in the exercise. By knowing the total desired gain and the relationship between stages, we can determine the specific gain of each stage to meet our design criteria. This step-by-step approach, starting from total system gain and working down to individual stage gains, is fundamental for designing and understanding complicated electronic systems.

The exercise illustrates that even with varying gains across the stages, the total system gain could be strategically distributed to achieve a balanced output or to meet specific criteria, such as noise performance or bandwidth limitations, that each stage might have. By mastering these calculations, one can deftly handle the intricacies involved in amplifier design and signal processing.