Adding power flux densities from multiple sources

[Wotwot]

The Power density (Pd in W/m^2) at a distance r (in meters) for a transmitting antenna with gain Gt is given by:

Pd = ( Pt * Gt) / 4 * Pi * r^2 where Pt (in Watts) is the power transmitted from the antenna.

All these variables are scalar values so if there were multiple antennas and we wanted to calculate the total power density due to all of them at a given point, you could calculate the individual power density due to each one and add them together.

Now, the power density in the far field is also given by:

Pd = Erms^2 / 377

where Erms is the RMS electric field (in V/m) at the location and 377 ohms is the intrinsic impedance of free space.

Having already calculated the total power density, we can substitute it into the above equation and find a value for the total RMS electric field due to all the sources at the point of interest.

This is where things seem to go wrong.

The electric field is a vector quantity and so when multiple fields meet, they have to be added using vector addition. You can't add two RMS electric field values together unless they are totally in phase. However, using the above methodology, we seemed to have arrived at a value for the total RMS electric field. It would seem the answer has to be wrong because we haven't taken the phase differences into account. But since these equations are scalar, where do you do that?

Does anyone have any suggestions?

 

[berkeman]

The Power density (Pd in W/m^2) at a distance r (in meters) for a transmitting antenna with gain Gt is given by:

Pd = ( Pt * Gt) / 4 * Pi * r^2 where Pt (in Watts) is the power transmitted from the antenna.

All these variables are scalar values so if there were multiple antennas and we wanted to calculate the total power density due to all of them at a given point, you could calculate the individual power density due to each one and add them together.

Now, the power density in the far field is also given by:

Pd = Erms^2 / 377

where Erms is the RMS electric field (in V/m) at the location and 377 ohms is the intrinsic impedance of free space.

Having already calculated the total power density, we can substitute it into the above equation and find a value for the total RMS electric field due to all the sources at the point of interest.

This is where things seem to go wrong.

The electric field is a vector quantity and so when multiple fields meet, they have to be added using vector addition. You can't add two RMS electric field values together unless they are totally in phase. However, using the above methodology, we seemed to have arrived at a value for the total RMS electric field. It would seem the answer has to be wrong because we haven't taken the phase differences into account. But since these equations are scalar, where do you do that?

Does anyone have any suggestions?

When the source signals are at the same frequency, then the relative phases of the receive signals at various points need to be taken into account. It's not so much a vector addition, it is a magnitude and phase addition problem.

If the sources are not at the same frequency, then adding the RMS powers would seem to work.

 

[Wotwot]

When the source signals are at the same frequency, then the relative phases of the receive signals at various points need to be taken into account. It's not so much a vector addition, it is a magnitude and phase addition problem.

If the sources are not at the same frequency, then adding the RMS powers would seem to work.

If we assume the frequencies of the multiple sources are the same, where in the above equations would you take phase information into account. OR is it implied that simple power density additions cannot be done for same frequency sources?

 

[berkeman]

If we assume the frequencies of the multiple sources are the same, where in the above equations would you take phase information into account. OR is it implied that simple power density additions cannot be done for same frequency sources?

You would calculate the each E(r) magnitude and phase, add the resulting E(t) components at the receiving point, and use your equation for the power.

You will end up with nulls and anti-nodes for a radiation pattern from an antenna array:

http://en.wikipedia.org/wiki/Phased_array

.