DC Transmission Line II (5-?)
Wow, the previous post sure puts my puny efforts in perspective.
Dec. 3 - Since no one has yet pointed out my lapses I will fix them here in italics.
I am going to try a more generic approach to balancing the forces in the transmission line.
Equation 40 on page 138 of the “Discharges book” gives an equivalency between the L and C in a given transmission line:
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/ELoEDWaIaOT-1914_138_40.png)
Where C is the capacitance in farads per cm and L1 is inductance external to the conductors in henrys per cm. k is the specific capacitance or permittivity and µ the permeability of the medium. v is the velocity of light in cm per second. You can calculate L from C and C from L:
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/LC_by_v_squared.png)
Starting from the dielectric and magnetic energy of the transmission line circuit. Length is the length of the line in cm.
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/energy_dielectric_magnetic2.png)
The force is the change in energy with respect to changes in s. Note that s being measured in a direction parallel to the direction of the force. Set the sum of the forces to zero.
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/force_balance2.png)
The Lengths cancel out. Next substitute L for C and apply chain rule. The derivatives of L will cancel out. Then substitute C for L.
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/force_substitution.png)
Taking the square root of both sides:
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/z0_2.png)
which has just been shown to be the same ratio of e to i which causes the magnetic and dielectric forces to balance.
Wow, the previous post sure puts my puny efforts in perspective.
Dec. 3 - Since no one has yet pointed out my lapses I will fix them here in italics.
I am going to try a more generic approach to balancing the forces in the transmission line.
Equation 40 on page 138 of the “Discharges book” gives an equivalency between the L and C in a given transmission line:
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/ELoEDWaIaOT-1914_138_40.png)
Where C is the capacitance in farads per cm and L1 is inductance external to the conductors in henrys per cm. k is the specific capacitance or permittivity and µ the permeability of the medium. v is the velocity of light in cm per second. You can calculate L from C and C from L:
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/LC_by_v_squared.png)
Starting from the dielectric and magnetic energy of the transmission line circuit. Length is the length of the line in cm.
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/energy_dielectric_magnetic2.png)
The force is the change in energy with respect to changes in s. Note that s being measured in a direction parallel to the direction of the force. Set the sum of the forces to zero.
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/force_balance2.png)
The Lengths cancel out. Next substitute L for C and apply chain rule. The derivatives of L will cancel out. Then substitute C for L.
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/force_substitution.png)
Taking the square root of both sides:
![](http://i1123.photobucket.com/albums/l552/z377/E_F/EF_PWHWEPD_616/z0_2.png)
which has just been shown to be the same ratio of e to i which causes the magnetic and dielectric forces to balance.
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