Express the power dissipated in a 100 Ω resistor in dB relative to 1 mW, when the voltage across the resistor is 1.0 Vrms
10 dB
Power Dissipation in Resistor: Understanding dBm Calculation
The question asks us to express the power dissipated in a 100 Ω resistor in dB relative to 1 mW, given that the voltage across the resistor is 1.0 Vrms. This involves calculating the power first in Watts, then converting it to milliWatts, and finally expressing it in decibels relative to 1 mW (dBm).
Calculating Power in Watts
To find the power dissipated in a resistor, we can use the formula that relates voltage and resistance. The formula for power (P) when voltage (V) and resistance (R) are known is:
$$ P = \frac{V_{rms}^2}{R} $$
- Given voltage ($V_{rms}$) = 1.0 Vrms
- Given resistance (R) = 100 Ω
Now, let's substitute the given values into the formula:
$$ P = \frac{(1.0 \text{ V})^2}{100 \text{ Ω}} $$
$$ P = \frac{1.0 \text{ V}^2}{100 \text{ Ω}} $$
$$ P = 0.01 \text{ W} $$
So, the power dissipated in the 100 Ω resistor is 0.01 Watts.
Converting Power to Milliwatts (mW)
The question requires us to express the power in dB relative to 1 mW. This means we first need to convert the calculated power from Watts to milliWatts. We know that 1 Watt is equal to 1000 milliWatts.
$$ P_{mW} = P_{W} \times 1000 \frac{\text{mW}}{\text{W}} $$
Substitute the power in Watts we calculated:
$$ P_{mW} = 0.01 \text{ W} \times 1000 \frac{\text{mW}}{\text{W}} $$
$$ P_{mW} = 10 \text{ mW} $$
Thus, the power dissipated is 10 milliWatts.
Expressing Power in dB relative to 1 mW (dBm)
The formula to express power in decibels relative to a reference power of 1 mW (which is denoted as dBm) is:
$$ P_{dBm} = 10 \log_{10}\left(\frac{P_{mW}}{1 \text{ mW}}\right) $$
Here, $P_{mW}$ is the power in milliWatts, which we calculated as 10 mW.
Substitute the value into the dBm formula:
$$ P_{dBm} = 10 \log_{10}\left(\frac{10 \text{ mW}}{1 \text{ mW}}\right) $$
$$ P_{dBm} = 10 \log_{10}(10) $$
Since $\log_{10}(10) = 1$:
$$ P_{dBm} = 10 \times 1 $$
$$ P_{dBm} = 10 \text{ dBm} $$
Therefore, the power dissipated in the 100 Ω resistor, when the voltage across it is 1.0 Vrms, is 10 dB relative to 1 mW.
| Step | Description | Result |
|---|---|---|
| 1 | Calculate power in Watts ($P = V_{rms}^2 / R$) | 0.01 W |
| 2 | Convert power to milliWatts ($P_{mW} = P_{W} \times 1000$) | 10 mW |
| 3 | Express power in dBm ($P_{dBm} = 10 \log_{10}(P_{mW}/1 \text{ mW})$) | 10 dBm |
Important Questions from Power
An elevator weighing 500 kg is to be lifted up at a constant velocity of 0.25 m/s. What would be the minimum horsepower of the motor to be used? G = 9.8 m/s2 and there is no frictional loss.
A man of mass 80 kg climbs up 4 m high stairs in 10 sec. Find the power spent by the man. (Take g = 10 m/sec2)
How many units of electric power will be consumed by 4 motors of 0.5 HP each in 8 hours?
The power of a water pump is 2 kW. The amount of water (in litres) it can raise in one minute to a height of 10 m will be :
(g = 10 m/s 2 )
Which of the following is not equivalent of Watts?
