In any radio circuit, an important topic of interest is the anticipated or expected range of communication. There are several elements that affect the range of communication, namely

- transmitter output power
- transmitter transmission line loss
- transmitter antenna gain
- propagation path loss
- receiver antenna gain
- receiver transmission line loss
- receiver sensitivity
- desired signal margin.

Regarding signal power level, the comparison will be in decibels relative to one one-thousandths of a Watt or 0.001-Watt. This reference is called dBm (m for milliWatt).

Transmitter Power

For a VHF Marine Band ship station, the transmitter power output (Ptx) is limited to 25-Watts. Typically in order to achieve this power output the transmitter power supply must be maintained at the specified voltage and provide the specified current. The transmitter output must be connected to a properly matched antenna and transmission line. Any deficiency in supply power voltage or current or any mismatch in the antenna impedance and transmission line impedance will tend to reduce the power output. In order to not exceed the maximum power output, a manufacturer will typically aim for just under 25-Watts actual output. The effect of all these influences can reduce the actual power output to more like 20-Watts on average for a typical radio as installed on a typical boat. For better handling in the calculations, the power is converted to be in dBm, which is a ratio expressed is decibels to a power of 1-milliwatt. This produces Ptx = +43 dBm

Transmitter Transmission Line Loss

A general rule of thumb in selecting a transmission line is to aim for a loss in the transmission line (Ltx) of less than -1 dB at the frequency of operation. For 156.8-MHz and for a length of up to 20-feet, use of RG-58C/U transmission line will produce -1 dB or less of line loss. Thus Ltx= -1 dB

Transmitter Antenna Gain

Antenna gain is to be measured with reference to an isotropic radiator, indicated as dBi. Most VHF Marine Band antennas will exhibit about 3 dBi gain in their primary lobe of radiation. While antenna gain is difficult to measure, it can be done, and thus the transmitter antenna gain (Gtx) is a known value. A typical value will be Gtx= +3 dB

Propagation Path Loss

All radio signals lose intensity as they propagate, principally by beam spreading, and secondarily from the medium through which they are traveling. In some instances propagation loss can be increased or decreased by the a signal being reflected by terrain or other objects and recombining with the original signal in an additive or subtractive manner, depending on the phase relationship.

For propagation in a vacuum in completely empty or free space, theoretical analysis of propagation loss predicts that the intensity of the signal will decrease with the distance factor squared (second power exponent), also known as the inverse-square-law rule. In a real world in which the signal propagates through the atmosphere and there is terrain (land or sea) present, the effect of distance on signal strength is greater. Many approximations for propagation loss use a distance factor to the fourth exponent to better account of real-world propagation loss. Propagation loss is an estimated value, but it can be reasonably estimated and used in predicting range of communication. Path loss depends on three elements: frequency, distance, and propagation medium factor. Because the goal of this analysis is to find the maximum distance of communication, the frequency and the propagation medium factor must be known or be approximated as inputs to the calculation. A frequency of 156.8-MHz will be used. For the propagation medium factor, the fourth exponent of distance will be used as an approximation. The PathLoss which can be tolerated is yet to be deduced, as depends on the required it necessary receiver signal power, Prx.

Receiver Antenna Gain

As with transmitting antennas, the gain of a receiving antenna is the same as when transmitter. For this factor a gain of 3 dBi will again be used. Grx = +3 dB

Receiver Transmission Line Loss

Again, a value of -1 dB loss will be used for the receiver transmission line: Lrx = -1 dB

Receiver Sensitivity

All receivers have a limit to their useful sensitivity which is established by the internal electrical noise created by the receiver itself, by the bandwidth necessary to receive a modulated signal, and by the nature of the modulation. For a typical well-designed narrow-band FM receiver, the sensitivity is usually give as the needed signal power or voltage to produce a demodulated signal that is 10-times (10 dB) stronger than any noise or distortion in the recovered signal.

Receiver sensitivity (RxSens) is often stated in terms of microVolts at the 50-Ohm antenna input of the receiver. This power level can be converted to dBm. For advice on how to make the conversion, see a separate article at

Conversion of Receiver Sensitivity

From Microvolts to dBm

https://continuouswave.com/whaler/reference/dBm.html

A typical value for receiver sensitivity is 1-microvolt, and when converted to dBm gives an RxSens = -107 dBm.

Desired Margin

It is also useful to add some margin to the calculation to allow for variations in sensitivity caused by factors such as the presence of local radio-frequency noise interference or to allow for the possibility of signal fading due to variation in propagation loss. A margin for the signal to have a 100-times excess (20-dB) is useful to improve the reliability of the predicted communication range.

Required Minimum Signal Power at Receiver

Next, the necessary minimum signal power at the receiver site (Prx) to produce a useful demodulated signal is calculated. The receiver antenna gain improves sensitivity (allows weaker signals at the antenna) ; line loss decreases sensitivity (requires more signal), and the desired margin also effectively decreases receiver sensitivity (requires more signal) so we algebraically sum as follows:

Prx = RxSens + Margin

Now we find the maximum Path Loss (Lp) that can be tolerated, that is, when the desired receiver minimum power (Prx) will occur based on the transmitter power, antenna gains, transmission line losses, and the path losses .

Maximum tolerable propagation loss occur when

Prx = (Ptx + Ltx + Gtx) + LpathMax + (Grx + Lrx). (Here Ltx and Lrx are expressed as negative dB)

Collecting terms

Prx = (Ptx + Ltx + Gtx + Grx + Lrx) + LpathMax

Solving for LpathMax by subtracting (Ptx + Ltx + Gtx + Grx + Lrx) from both sides

LpathMax = Pr -(Ptx + Ltx + Gtx + Grx + Grx)

We have now found the maximum path loss that can be tolerated. Once this value has been estimated, the distance of communication can be calculated.

Calculating Path Distance As Function of Path Loss

The formula for path loss can be derived from radio theory for propagation in free space for the unit values to be MHz and Miles to be:

Path Loss = -36.6 - 20log(f) - 20log(d) (f in MHz, d in miles)

To see how this equation using deciBel ratios was derived, read my article "Marine VHF Radio Communication" and see the section under the heading "Converting Path Loss to deciBel Equation” at

https://continuouswave.com/whaler/reference/VHF.html

The influence of distance on loss is provided by the factor 20log(d), which is a way to express the familiar inverse-square-law of decrease in intensity with distance to a decibel form. To account for the greater loss over actual terrain and in the atmosphere, a formula with greater influence of distance is used, increasing the coefficient of the log(d)factor to 40 or higher:

Path Loss = -36.6 - 20log(f) - 40log(d) (f in MHz, d in miles)

Solving the above for distance (d) gives

Path Loss + 36.6 + 20log (f) = 40log(d)

log(d) = (Path Loss + 36.6 + 20log(f) ) / 40

d = 10^ [ (Path Loss + 36.6 + 20log(f) ) / 40]

To put some real numbers into these calculations, I will demonstrate a typical boat-to-boat VHF Marine Band situation. For the example, I will use the following inputs to deduce the allowed path loss Lpath:

f = 156.8 Mhz

Ptx = +43 dBm

Ltx = -1 dB

Gtx = 3 dB

Grx = 3 dB

Lrx = -1

RxSen = -107 dBm

Margin = 20

Prx = -87 dBm

Distance Loss coefficient = 40

Prx = Ptx + Gtx + Grx + Ltx + Lrx + Lpath

Path Loss = Prx - Ptx - Gtx - Gtx - Ltx - Lrx

Path Loss = -87 - (43) - (3) - (3) - (-1) - (-1)

Path Loss = -134 dB

Now that the maximum tolerable path loss has been calculated, we can use the formula for predicting path loss and solve it for distance. Again, path loss in decibel notation as a function of frequency and miles is

Path Loss = -36.6 - 20log(f) - 40log(d) (f in MHz, d in miles)

Plugging our value for maximum path loss of of -134 dB and frequency 156.8 and solving for d gives

-134 = -36.6 - 20log(156.8) - 40log(d)

40log(d) =134 -36.6 - 43.91

40log(d)=53.49

d=10^(53.49/40)

d = 21.7-miles

By creating a spreadsheet, the various input values can be changed to suit. Typical changes would be antenna gains, transmission line losses, and the coefficient for the distance factor. The latter change will have the most impact. For example, with all other variable as shown above, if the distance factor is changed to the free space model of 20log(d), the distance calculation becomes enormously longer: 473-miles.

-134 = -36.6 - 20log(156.8) - 20log(d)

20log(d) =134 -36.6 - 43.91

20log(d)=53.49

d=10^(53.49/20)

d = 473-miles

If we further reduce reducing the signal margin required to zero, the range estimate becomes a rather fantastic 4,728-miles.

Getting back to real-world conditions, we restore the margin to 20 dB and increase the distance factor to 46log(d), which yields a predicted range of only 14.6-miles. This is probably the most reasonable estimate and most likely to actually be achieved in the real world of boat-to-boat communication.