Audio Amplifier Output Power Calculation

This problem involves calculating the output power of an audio amplifier system when the input power and the gain of a subsequent amplifier stage are known. The gain is given in decibels (dB), which requires a specific formula to convert into a power ratio.

Understanding the Given Information

• Input Power ($P_{in}$) = 100 mW
• Gain of the cascaded amplifier = 10 dB

Calculating Power Gain from Decibels (dB)

The gain of an amplifier is often expressed in decibels (dB). The formula to convert power gain into decibels is:

$$ \text{Gain (dB)} = 10 \times \log_{10} \left( \frac{P_{out}}{P_{in}} \right) $$

Where:

• $P_{out}$ is the output power.
• $P_{in}$ is the input power.
• $\log_{10}$ is the logarithm base 10.

Determining the Power Ratio

We are given the gain in dB and need to find the ratio $\frac{P_{out}}{P_{in}}$. We can rearrange the formula:

1. Start with the formula: $10 \text{ dB} = 10 \times \log_{10} \left( \frac{P_{out}}{P_{in}} \right)$
2. Divide both sides by 10: $ \frac{10 \text{ dB}}{10} = \log_{10} \left( \frac{P_{out}}{P_{in}} \right) $
3. This simplifies to: $ 1 = \log_{10} \left( \frac{P_{out}}{P_{in}} \right) $
4. To find the power ratio, we take the antilogarithm (10 raised to the power of both sides): $ 10^1 = \frac{P_{out}}{P_{in}} $
5. Therefore, the power ratio is: $ \frac{P_{out}}{P_{in}} = 10 $

This means the power amplifier increases the power by a factor of 10.

Calculating the Output Power

Now we can calculate the output power ($P_{out}$) using the input power ($P_{in}$) and the power ratio:

$$ P_{out} = \text{Power Ratio} \times P_{in} $$

Substitute the known values:

$$ P_{out} = 10 \times 100 \text{ mW} $$

$$ P_{out} = 1000 \text{ mW} $$

Converting to Watts (W)

Since $1000 \text{ mW} = 1 \text{ W}$, the output power is:

$$ P_{out} = 1 \text{ W} $$

Final Output Power

The final output power of the cascaded amplifier system will be 1 W.