Problem 101 - Precalculus

Question

Decibels are used to quantify losses associated with atmospheric interference in a communication system. The ratio of the power (watts) received to the power transmitted (watts) is often compared. Often, watts are transmitted, but losses due to the atmosphere typically correspond to milliwatts being received: $$\mathrm{dB}=10 \log \left(\frac{\text { Power received }}{\text { Power transmitted }}\right)$$ If 1 W of power is transmitted and \(3 \mathrm{mW}\) is received, calculate the power loss in dB.

Step-by-Step Solution

Answer

The power loss is approximately -25.229 dB.

Step 1: Understand the units

Identify the given values. We have \(1 \text{ W}\) for the transmitted power and \(3 \text{ mW}\) for the received power. Convert the received power from milliwatts to watts.

Step 2: Conversion of units

Since 1 milliwatt (mW) is equal to \(0.001\) watts, convert the received power: \(3 \text{ mW} = 3 \times 0.001 \text{ W} = 0.003 \text{ W}\).

Step 3: Substitute into the formula

Substitute the transmitted and received power into the decibel formula: \[ \text{dB} = 10 \log\left(\frac{0.003 \text{ W}}{1 \text{ W}}\right) \].

Step 4: Calculate the ratio

Calculate the ratio of \(\frac{0.003}{1}\), which simplifies to 0.003.

Step 5: Compute the logarithm

Calculate the logarithm base 10 of 0.003. Use a calculator to find \( \log_{10}(0.003) \approx -2.5229 \).

Step 6: Calculate decibels

Substitute the logarithm value back into the decibels formula: \[ \text{dB} = 10 \times (-2.5229) \approx -25.229 \].

Step 7: Interpret the result

The power loss in the communication system is approximately \(-25.229\) dB, which indicates a loss.

Key Concepts

Power Ratio

When discussing communication systems, the power ratio is crucial. It's a measure of how efficiently power is transmitted from one point and received at another. In simple terms, the power ratio compares the power output at the receiving end to the power input at the transmitting end. This ratio is usually expressed as a fraction, where the numerator is the power received and the denominator is the power transmitted.
In our exercise, the power transmitted is 1 watt, and the power received is 3 milliwatts. Before using these values in calculations, we need to convert everything to the same unit to ensure accuracy. This is why understanding both ratios and conversions becomes vital in the context of decibels and power calculations.

Unit Conversion

Unit conversion is the process of converting a measure from one unit to another, which is essential when dealing with problems involving different measurement systems. In the context of power systems, converting between milliwatts (mW) and watts (W) is common because signals can experience significant losses over distances.
For example, 1 watt equals 1000 milliwatts. So, to convert milliwatts to watts, you multiply by 0.001. In our exercise, 3 milliwatts of received power is converted to watts as follows:

• 3 mW times 0.001 equals 0.003 W.

Understanding how to convert these units properly is essential for accurate calculations in further steps, especially when dealing with logarithmic functions.

Logarithmic Calculation

Logarithmic calculations are fundamental in computing decibels, a logarithmic unit that measures the power level of a signal in comparison to a reference level. The logarithmic scale is used because it can manage very large or very small numbers more conveniently than a linear scale.
To compute decibels (dB), we use the formula:

• \( \text{dB} = 10 \log\left(\frac{\text{Power received}}{\text{Power transmitted}}\right) \).

To solve the exercise, once you've converted your power measurements to the same unit (watts), you proceed by calculating the ratio, which is 0.003 in this instance. You then compute the logarithm base 10 of this ratio, influencing the results significantly. Understanding this logarithmic approach helps in gauging how much power loss occurs over communication systems.

Communication Systems

Communication systems are intricate setups involving equipment and technology that transmit information from one place to another. Signals in these systems can sometimes face interference from external factors like the atmosphere, causing power loss. The concept of decibels is particularly useful here to easily understand and measure how much power is lost.
Decibels provide a standardized way to express power loss in a communication system, making it easier to identify inefficiencies. In our example, a significant loss of approximately -25.229 dB reveals how certain conditions can impact power reception. Through decibels, engineers and technicians can develop strategies to minimize these losses, thus ensuring more effective and reliable communication systems.