Understanding Gain in Cascaded Systems

In electronic systems, components are often connected in series, where the output of one stage becomes the input of the next. This is known as cascading. Examples include multi-stage amplifiers or signal processing chains. Each stage in a cascaded system contributes to the overall performance, including the gain or attenuation of the signal.

What is Gain?

Gain is a measure of how much a signal is amplified or attenuated by a circuit or system. It can be expressed as a ratio or in decibels (dB).

• Linear Gain: This is a simple ratio of the output signal power or voltage/current to the input signal power or voltage/current. For power, gain \(G = \frac{P_{out}}{P_{in}}\). For voltage, gain \(A_V = \frac{V_{out}}{V_{in}}\). For current, gain \(A_I = \frac{I_{out}}{I_{in}}\).
• dB Gain: The decibel is a logarithmic unit used to express ratios, often for power, voltage, or current gain. Using dB simplifies calculations involving very large or very small ratios.
  • Power Gain in dB: \(G_{dB} = 10 \log_{10} \left( \frac{P_{out}}{P_{in}} \right)\) or \(G_{dB} = 10 \log_{10} (G)\).
  • Voltage/Current Gain in dB: \(A_{V,dB} = 20 \log_{10} \left( \frac{V_{out}}{V_{in}} \right)\) or \(A_{I,dB} = 20 \log_{10} \left( \frac{I_{out}}{I_{in}} \right)\). The factor of 20 comes from the fact that power is proportional to voltage or current squared (\(P \propto V^2\) or \(P \propto I^2\)).

Calculating Total Gain in Cascaded Stages

Consider a system with two cascaded stages, Stage 1 and Stage 2. The output of Stage 1 is the input to Stage 2.

If the linear power gain of Stage 1 is \(G_1 = \frac{P_{out1}}{P_{in1}}\) and the linear power gain of Stage 2 is \(G_2 = \frac{P_{out2}}{P_{in2}}\), then the overall linear power gain \(G_{total}\) is the product of the individual gains:

\(G_{total} = \frac{P_{out2}}{P_{in1}}\)

Since the output of Stage 1 is the input to Stage 2 (\(P_{out1} = P_{in2}\)), we can write:

\(G_{total} = \frac{P_{out1}}{P_{in1}} \times \frac{P_{out2}}{P_{in2}} = G_1 \times G_2\)

For multiple cascaded stages (\(N\) stages), the total linear power gain is the product of the individual linear gains:

\(G_{total} = G_1 \times G_2 \times \dots \times G_N\)

Total dB Gain of Cascaded Stages

Now, let's find the total gain in dB. Using the definition of dB gain:

\(G_{total, dB} = 10 \log_{10} (G_{total})\)

Substitute the expression for total linear gain:

\(G_{total, dB} = 10 \log_{10} (G_1 \times G_2 \times \dots \times G_N)\)

Using the logarithm property \(\log(A \times B) = \log(A) + \log(B)\):

\(G_{total, dB} = 10 \left( \log_{10}(G_1) + \log_{10}(G_2) + \dots + \log_{10}(G_N) \right)\)

\(G_{total, dB} = 10 \log_{10}(G_1) + 10 \log_{10}(G_2) + \dots + 10 \log_{10}(G_N)\)

Each term \(10 \log_{10}(G_i)\) is the dB gain of the individual stage \(G_{i, dB}\). Therefore:

\(G_{total, dB} = G_{1, dB} + G_{2, dB} + \dots + G_{N, dB}\)

This shows that the total dB gain of cascaded systems is the sum of the dB gains of each individual stage.

Similarly, for voltage or current gains in dB:

\(A_{V, total, dB} = 20 \log_{10} (A_{V, total}) = 20 \log_{10} (A_{V,1} \times A_{V,2} \times \dots \times A_{V,N})\)

\(A_{V, total, dB} = 20 \left( \log_{10}(A_{V,1}) + \log_{10}(A_{V,2}) + \dots + \log_{10}(A_{V,N}) \right)\)

\(A_{V, total, dB} = 20 \log_{10}(A_{V,1}) + 20 \log_{10}(A_{V,2}) + \dots + 20 \log_{10}(A_{V,N})\)

\(A_{V, total, dB} = A_{V,1, dB} + A_{V,2, dB} + \dots + A_{V,N, dB}\)

Thus, regardless of whether it's power, voltage, or current gain (using the appropriate \(10 \log_{10}\) or \(20 \log_{10}\) formulas), the total dB gain is the sum of the individual stage dB gains.

Revision Table: Cascaded Gain Calculation

Additional Information: Why Use dB?

Using the dB scale for gain in cascaded systems offers several advantages:

• Simplification of Calculations: Multiplying linear gains becomes adding dB gains, which is simpler, especially with many stages.
• Handling Large/Small Numbers: Logarithmic scales compress wide ranges of values, making it easier to represent and work with very large amplifications or very small attenuations.
• Relating to Human Perception: The decibel scale relates somewhat to how humans perceive changes in signal strength (like loudness in audio).
• Standardization: Many specifications for electronic components (like amplifiers, filters) are given in dB.

This fundamental property of dB gain being additive is crucial in analyzing complex multi-stage electronic systems.