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confusion about path loss calculation for zigbee
Hello,
I've been trying to wrap my head about this problem for days and can't seem to figure it out. Zigbee technology allows you to transmit at 250 kbps. I'm comparing the required power to transmit at that rate to the required minimum power based on the path loss equations to reach the minimum receiver sensitivity of the 802.15.4 physical layer for Zigbee of roughly -94 dBm. Simple enough right? To analyze the required power to transmit at 250 kbps for Zigbee, I've just re-arranged the Shannon capacity formula of Rate = w * log_2 (1 + (Pt * G)/(w*N + I)). Here, w is the bandwidth, Pt is the transmit power, Pt * G is the received signal power, N is the white Gaussian noise and I is the sum of interference powers. For Zigbee, these values are as follows: w = 5 MHz N (in dB) = 10*log10(k) + 10*log10(T) where k is boltzmann's constant of 1.23E-23 and T is the temperature in Kelvin of 300 degrees. In Watts, N is approximately 3.6900e-021. I've assumed no interference at this stage, so I = 0. Simple enough so far right? So to calculate the required transmit power to reach a rate of 250 kbps, I've just re-arranged this equation to yield: Pt = [2^(250 kbps/w) - 1]*(w * N + I)/G and plugging in the above values gives me a reasonable transmit power of 1.6 uW. The reason I haven't talked about G yet is that I'm addressing it here. G is the channel quality (path loss) and is defined as follows: G(freq,D) = GrdBi+ GtdBi- 20*log10(4*pi*freq*d0/c) - 10*pathLossExponent*log10(D/d0) (in dB) I'm using unity gains so GrdBi = GtdBi = 0 dBi. Other parameters a freq = 2.4 GHz d0 = 1m as the reference distance c = 3.0 x 10^8 pathLossExponent = 2, and D = 500 metres Hence, G (in Watts) * Pt is the received signal power. G(2.4E9, 500) is the path loss transmitting at 2.4 GHz a distance of 500m away. G (2.4E9, 500) = -94.0254 dB Now the first thing that I mentioned was the receiver sensitivity for Zigbee of -94 dBm, Analyzing G (in dB). The required transmit power considering this path loss is easily calculated as: -94 dBm = P_required + G(2.4E9, 500) P_required is then easily calculated as -94 dBm - G(2.4E9, 500) = 0.0254 dBm. This translates to a transmit power of 1 mW. This required power is LARGER than the maximum permissible power to transmit at 250 kbps, which is the maximum data rate for Zigbee. So how does this make sense? However, 1mW happens to be the maximum transmit power of Zigbee of roughly 0 dBm, so there may be a range problem here. Testing other distances, say 300m, we get a maximum transmit power Pt = 5.9181e-007 = 0.59 uW to transmit at 250 kbps, but a P_required = -4.41 dBm = 3.6211e-004, which is our maximum transmit power Pt. So again, we run into the same problem at 300m. It just doesn't make sense to me. Can anybody shed some light on this? I hope my math is clear. Thanks in advance. |
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