- #1
biker.josh07
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Homework Statement
Find the general solution for the current I(z,t) associated with the voltage V(z,t).Do this by substituting [1] into [2] and [3], integrate with respect to time, and then take the derivative with respect to z.
Homework Equations
V(z,t)= f+(t-z/vp) + f-(t+z/vp) [1] where f is some arbitrary function.
∂V/∂z= -L(∂I/∂t) [2]
∂I/∂z= -C(∂V/∂t) [3]
The Attempt at a Solution
Ok,so I tried starting with [3] and integrating with respect to time: ∫(∂I/∂z)∂t= ∫(-C(∂V/∂t))∂t
which gives (I think) V=-1/C ∫(∂I/∂z)∂t
now differentiating this with respect to z: ∂V/∂z= ∂/∂z[-1/C ∫(∂I/∂z)∂t]
and substituting the RHS of this into [2]:∂/∂z[-1/C ∫(∂I/∂z)∂t]=-L(∂I/∂t)
now I'm stuck and not sure where to go from here. The solution is the well known equation
I(z,t)= (1/Z0)[f+(t-z/vp) + f-(t+z/vp)] where Z0=√(L/C)
I would appreciate knowing the steps to get there.
Thanks very much