telegrapher.jl

# Obtained from https://en.wikipedia.org/wiki/Telegrapher%27s_equations#Lossy_transmission_line
"
    telegrapher(L,C,R=0,G=0,n=100)

returns an array of differential equations that can be
used to solve for the travelling voltages and currents as a function of time.
All quantities are expected to be per-length quantities (in S.I. units).

The problem is solved by discretizing the transmission line into n sections. If
the solution is too coarse then try changing n.
"
function telegrapher(L, C, R=0, G=0 ;n=100, vs=t->1, is=t->0)
    if L == 0 || C == 0
        error("Both the inductance and capacitance need to be non-zero.")
    end
    return function(t,u,du)
        u[1] = vs(t)
        u[n+1] = is(t)
        dx = 1/(n-1)
        for i = 2:n
            # voltage equation
            du[i] = (-1/C)*((u[n+i-1]-u[n+i])/dx + G*u[i])
            # current equation
            du[n+i] = (-1/L)*((u[i-1]-u[i])/dx + R*u[n+i])
        end
    end
end

"
    tline_solve(f,L,C,R=0,G=0,t=(0,1),n=100)

Solves for the time-domain response of a transmission line to an incident
voltage waveform f(t).

### Arguments
u0: initial voltage applied to the line\n
t: solution interval\n
n: number of tline discretizations\n
"
function tline_solve(u0,vs,is,L,C,R=0,G=0;t=(0.,1.),n=100,dt=1e-3)
    prob = DifferentialEquations.ODEProblem(telegrapher(L,C,R,G,n=n,vs=vs),u0,t)
    solver = DifferentialEquations.Euler()
    sol = DifferentialEquations.solve(
                                      prob,
                                      solver,
                                      reltol=1e-8,
                                      abstol=1e-8,
                                      dt=dt
                                     )
    return sol
end