Transmission Line Equations
Transmission Line Equations:
In a long electrical cable or a telephone wire both the current and voltage depend upon position along the wire as well as the time.
It is possible to show, using basic laws of electrical circuit history, that the electrical current i(x,t) satiesfies the PDE
Where the constants R, L, C and G are for unit length of cable, respectively the resistance, inductance, capacitance and leakage conductance. The voltage v (x, t) also satiesfies the above equations. Special cases of equation arise in particular situations. For a submarine cable G is negligible and frequencies are low so inductive effects can also be neglected. So it becomes
Which is called the submarine equations or telegraph equations. For high frequencies alternating current, again with negligible likage can be approximated by:
Which is called High Frequency Line Equations.
Other Important PDE:
1. Poisson's Equations:
Where f(x,y) is agiven functions. This equation arises in electrostatics, electricity theory.
2. Helmholtz's equations:
Which arises in the wave theory.
3. Schrodinger's equations:
Which arises in the quantum mechanics.
4. Transverse Vibration in a Homogeneous rod:
Where u (x, t) is the displacement at time t of the cross- section through x.