# EM Waves & T-Lines

#### idontknoweverythingyet

Joined Aug 7, 2013
6
I asked this on physics.SE, but I figured I would ask here as well seeing as though engineers tend to have a different, but useful and interesting perspective on electromagnetic theory.

http://physics.stackexchange.com/questions/75533/power-delivered-by-an-electromagnetic-wave-versus-power-delivered-by-a-transmiss

To add on to the question(s) posed above, one thing that I did not ask on physics.SE originally that I feel may be fruitful to explore here is what is the physical significance of the complex conjugation operator in determining power? e.g.

$$P_{\mathrm{avg}}=VI^*$$​

#### LDC3

Joined Apr 27, 2013
924
My brain hurts! Sorry, I couldn't resist.

#### Papabravo

Joined Feb 24, 2006
12,472
The physical significance is that the complex conjugate of a function has the same magnitude but the phase is changed by 180°. The average value of a periodic waveform, such as the sine or the cosine, over any integral number of periods is zero. For real functions we use RMS to calculate the effective non zero power delivered by an AC waveform. To compute magnitude squared of a complex function we use VV* to compute |V|^2. I'm thinking something similar is going on in your question, but I don't quite see the reasoning behind the assertion nor do I see the factor of 1/2 being explained by the use of peak values.

Some other things I don't quite follow are that an ideal transmission line is lossless and consumes no power so all of it is nominally delivered to the load. Free space is not like a transmission line however and signal propagating through free space are indeed attenuated quite significantly.

#### t_n_k

Joined Mar 6, 2009
5,447
The physical significance is that the complex conjugate of a function has the same magnitude but the phase is changed by 180°.
This needs qualification - the conjugate is the complex number having the same real part as the original number but having an imaginary part of the opposite sign to that of the original number.

Hence the conjugate of (a+jb) is (a-jb).

The meaning of $$\text{VI^*}$$ is elaborated in the attachment.

Which would lead me to the conclusion ...

$$\text{P_{avg}=Re(VI^*)}$$

Where Re() indicates the real part of the complex result.

#### Attachments

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#### WBahn

Joined Mar 31, 2012
24,848
... what is the physical significance of the complex conjugation operator in determining power? e.g.

$$P_{\mathrm{avg}}=VI^*$$​

Particularly the material above Eqn 10.

#### Papabravo

Joined Feb 24, 2006
12,472
This needs qualification...
Hence the conjugate of (a+jb) is (a-jb).
My mistake. A phase change by 180 degrees in NOT the same thing as changing the sign of the imaginary part? I was thinking of the imaginary unit as a rotation operator.

From the article referenced by WBahn:
This can be easily remedied by noting that the conjugate of a complex quantity has the same magnitude but the additive inverse (i.e., negative) of the phase.
just above equation (10). That is what I should have said.

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