Voltage Summation instead of Power summation - Am I going Mad?

[RFgeezer]

Let's assume I inject a radio signal into a receiver, this signal has a total power of 0 dBm (1mW)which is divided equally between two carrier frequencies F1 and F2 so that F1 and F2 are each -3dBm (1/2 mW). Now, let's assume that the receiver takes an FFT of the signal. I believe the voltages that it would report would be V_F1 and V_F2 = 0.158mV RMS assuming a 50 ohm system. Now, if we sum the POWER of F1 and F2 this gets us back to our 0dBm (1mW) signal....no problem with that (noise does of course increase by 3dB). However, if I were to be able to freqeuncy shift F1 to exactly the same freqeuncy as F2 and also make F1 and F2 have the same phase I would be able to coherently add them. Let's assume I do this. Is it then correct that I would have a 0.316mV rms sine wave as my resulting summed signal? The power of this in a 50 ohm system is then 2mW (+ 3dBm). Now that's twice as much power as the total signal power I injected into the recevier....That's a bit confusing? Again, I should also consider the noise which I believe in this case would increase by root 2 (1.5 dB). This all suggests that I could get a net SNR benefit of 4.5 dB, I like those apples but I can't help thinking that I've missed a fundamental issue somewhere?

Anyone care to comment on this one?

 

[VEBill]

There are several ways to slice this. Here's one example:

Two 50-ohm sources in parallel effectively make a 25-ohm source.

 

[Noway2]

However, if I were to be able to freqeuncy shift F1 to exactly the same freqeuncy as F2 and also make F1 and F2 have the same phase I would be able to coherently add them. Let's assume I do this.

In order to add these waves, would it not require an active component and would it not be that active component that is adding the power to the signal? Since you don't have the ability to cascade the sources to add the voltages at the source, I can't think of how you would add the signals any other way.

 

[RFgeezer]

HI VE1BLL and Norway 2.

The source is just one 50 ohm signal generator that has an output signal with two frequency components F1 and F2, each of the components is 1/2 mW.

The addition of power or votlage is done after the two component signal has been captured with the FFT. We are therefore actually adding up numbers. The simple model I have in my head is a 50 ohm load connected to the signal generator that is outputting the two carriers, each of which is 1/2 mW. We then take an FFT from voltage measured across the 50 ohm load. I'm assuming that FFT measurment instrument does not have any impact on the 50 ohm load.

 

[VEBill]

So now you have just one signal generator (one output recepticle) emitting "two" signals of the exact same frequency and phase. Hmmm...

To add the voltages, the two voltage sources need to be placed in series. If you connect a 12v battery in parallel with another 12v battery then you still have 12v. Same thing with your 2-in-1 signal generator. The voltages do not add in your example.

 

[LiteYear]

Interesting conundrum. Of course, you can't beat physics, even with clever tricks, so something has to give. Have the responses so far satisfied you? They're all correct but maybe it still hasn't clicked. In case it helps, here's a thought experiment that helped to clear it up in my head:

When the two sources are operating at different frequencies, their average power is 0.5mW each. But recall that the instantaneous power being delivered by each source is variable. It depends on where that particular source is in its cycle, and, where the other source is in its cycle. That's because at any point in time, each source sees the load, but also sees a voltage offset or current contribution due to the other source.

When the two sources differ in frequency, averaging the power over a long enough period will cancel the influence of the other source. This is because while the load always sinks power, the other source can alternate between sourcing or sinking power, averaging out to nothing.

When the two sources are identical in frequency and phase, this averaging does not sum to zero. At every instant in time, each source is sourcing power and spending no time sinking power from the other source. The sources, being constant voltage not constant power, will compensate to increase the average power in the load.

So the fact that two incoherent 0.158V sources add perfectly to provide 0.5mW is actually the exception! In general, the rule of superposition only works for voltage and current, not power.