Re: Poynting Vector
Hi, Dr_Mas:
Yes. The formula I postered is for the same as yours:
P = Real [ V_rms * Conj( I_rms ) ] for TEM transmission lines only. However, I do think the Imag part may mean something. Basically, I can see that, when Imag = 0, the power is propagationg in one direction. It seems to me that the Real and Imag parts do indicate how good the system is matched.
Again, the true definition of P is:
P = Real{ Integral_On_Cross_Section[ E_rms X Conj( H_rms ) ] }
for a general case. When the field is a TEM field, the E in the cross-section is conservative and the double integral on the cross-section can be separated and you can write it as:
P = Real{ Integral_On_One_Croodinate( E_rms ) X Integral_On_Another_Coordinate[ Conj( H_rms ) ] }
V_rms = Integral_On_One_Croodinate( E_rms )
I_rms = Integral_On_Another_Coordinate[ Conj( H_rms ) ] with a different direction
Please understand that, from E and H to V and I is only good for TEM wave.
Hi, Dowejones: I hope we could find everything from textbooks. However, I have not found this derivation from textbook, and I have not found detailed explanation why and when the P = V * Conj( I ) can or can't be used. Same thing is true on complex Zc. All the textbooks assume Zc or the E/H in waveguide is a real number. You just can't find any text book to discuss it. That is the reason we documented it in the IE3D User's Manual. In fact, when E/H is no longer a pure real number, you will find that the whole waveguide theory is an approximation. A typical example is that walls of a rectangular waveguide (RWG) is no longer PEC. You will find that the field inside it can't be separated into incident wave (Inc) and reflected wave (Ref). When you try to find the power consumed in the material inside of a section of RWG, you will need to find it as:
P1 = Integration_on_cross_section_on_left + Integration_on_cross_section_on_right + Integration_on_all_side_walls
instead of:
P2 = Integration_on_cross_section_on_left + Integration_on_cross_section_on_right
which is documented in all Electromagnetic textbook.
The P2 formula leads:
P = |Inc*Inc| - |Ref*Ref|
and this formula is valid only conditionally. More precisely, Inc and Ref in a waveguide are valid only conditionally. There are many things involved when you allow losses on the side walls or RWG or there is loss in the transverse direction of a waveguide. Unfortunately, it is the real case in reality. Maybe I should write a book on it when I have time. Thanks!