Final answer:
To find the directivity of an antenna in dB, efficiency is first calculated using the ratio of power radiated to power input, then the given gain in dB is converted to its linear value, and finally the directivity in linear terms is found and converted back to dB, resulting in a directivity of approximately 11.76 dB.
Step-by-step explanation:
The question asks to find the directivity of an antenna, given that the power radiated (Pradiated) is 10W, the power input (Pin) is 15W, and the gain of the antenna is 10 dB. To solve this problem, we first need to understand the relationship between directivity, gain, and efficiency. Gain, which is given in dB, can be calculated using the formula:
G = 10 log10(Pradiated/Pin)
However, we are given the gain in dB, so we'll work backwards to find the efficiency (η). The gain in linear terms (not dB) is the efficiency (η) multiplied by the directivity (D). Since the efficiency can be calculated as the ratio of power radiated to power input, we have:
η = Pradiated/Pin = 10W/15W = 0.6667 (or 66.67%)
Next, we convert the given gain in dB to its linear value:
Glinear = 10(Gain in dB/10) = 10(10/10) = 10
Now we have:
Glinear = ηD
Plugging in the values, we get:
D = Glinear/η = 10/0.6667 ≈ 15
To express the directivity in dB, we use:
DdB = 10 log10(D) = 10 log10(15) ≈ 11.76 dB
The directivity of the antenna in dB is approximately 11.76 dB.