How to generate j*cos(wt) in a single physical wire?

[vlsi_design2]

Hi, How can one generate j*coswt in the lab from coswt available in single physical wire? Please note that intention is to multiply both positive and negative frequency content with +j. I am not asking about hilbert transform which is -90 degree phase shift and rotates positive frequency content with -j and negative content with +j. Also, I am not asking about a anticlockwise phasor rotation of 90 degree or phasor multiplication by +j because phasor is e^jwt and not coswt. Is it necessary to have 2 physical wires and force orthogonality?

 

[BradtheRad]

In the Ghz realm there are Youtube videos of experimenters who confirm the speed of light by sending electricity through a wire 12 inches long. A very fast oscilloscope measures the delay between the two ends. The fraction of a second delay (1 nSec) is the time we expect.

Old-fashioned color tv's received chroma and luma signals a minuscule time apart, then synchronized them by sending one of them through a coil of wire hundreds of feet long, thus introducing the correct delay.

The above strategies work better in rf frequency ranges.

Your post seems to rule out (or does it?) the automatic 90 degree phase shift seen in a series capacitor when a sine wave is applied. The difference is voltage across the capacitor compared to current through it.

 

[KlausST]

Hi,

on a single physical wire ... you can not detect any phase shift at all. You would need a second signal as reference (and GND). So three wires in total.

So if this is a task given to you ... I guess "j" is just a variable for the amplitude. Can this be?

Klaus

 

[vlsi_design2]

In the Ghz realm there are Youtube videos of experimenters who confirm the speed of light by sending electricity through a wire 12 inches long. A very fast oscilloscope measures the delay between the two ends. The fraction of a second delay (1 nSec) is the time we expect.

Old-fashioned color tv's received chroma and luma signals a minuscule time apart, then synchronized them by sending one of them through a coil of wire hundreds of feet long, thus introducing the correct delay.

The above strategies work better in rf frequency ranges.

Your post seems to rule out (or does it?) the automatic 90 degree phase shift seen in a series capacitor when a sine wave is applied. The difference is voltage across the capacitor compared to current through it.

Dear BradtheRad,
Thanks. All these techniques generate phase shift/delay including the series cap. Lets say one wire has coswt and the other wire has sinwt. If we sum then it becomes coswt + sinwt which is another sinusoid and not coswt + j*sinwt. I guess keeping them seperate in two wires and maintaining orthogonal phase relationship signifies the j in coswt + j*sinwt.

--- Updated Oct 22, 2024 ---

Hi,

on a single physical wire ... you can not detect any phase shift at all. You would need a second signal as reference (and GND). So three wires in total.

So if this is a task given to you ... I guess "j" is just a variable for the amplitude. Can this be?

Klaus

Thanks KlausST. I think two wires can tell you the relative phase shift and maintain the orthogonality. This way we can have coswt in wire 1 and sinwt in another wire. What confuses me is how do we achieve coswt + j*sinwt? How does j show up? Multiplication by j rotates both positive and negative sidebands of sinwt and on addition we end up with e^jwt which has a single sided spectrum which can be understood mathematically. May be "j" signifies keeping cos and sin in two seperate wires and using them as a complex signal?

 

[BradtheRad]

One method is by multiplexing two signals. Here's a simplified schematic of my dual-trace scope adapter. Separating them is easy if you make them opposite polarities before combining them. I set my switches to alternate On-Off.

Another method is to apply amplitude-modulation to your signals using carriers of different frequencies. This way one wire can carry many channels. Filter them if you want to split them up at the receiving end.

 

[NigthMoth]

coswt + sinwt which is another sinusoid and not coswt + j*sinwt

Hello

I believe you are mixing two concepts:
- Pure math where Euler's formula links sinusoidal waveforms with complex numbers;
- Circuit analysis where imaginary number "j" is a useful math tool to calculate phase shifts;

As for cos(wt) + j*sin(wt) waveform:

When you looking at cos(wt) signal on your oscilloscope screen, actually you see projection of function e^(j*w*t) onto real number axis.
According to Euler's formula e^(j*w*t) = cos(w*t) + j*sin(w*t),
therefore cos(wt) is just real part of complex function e^(j*w*t).
So, in some sence, j*sin(w*t) is already here when you observe cos(w*t) waveform.