Finite difference method, coupled wave equations, chickens & eggs

[apchar]

I'm reading a book (Numerical Techniques in Electromagnetics by Sadiku) & just finished the section on finite difference methods. As what I thought would be an easy exercise, I tried to apply what I'd learned to the telegraphers equations that describe the voltage, V(x, t), and current, I(x, t), on a transmission line with some realistic (meaning ugly) boundary conditions:
Picture: A voltage source Vg, a resistor Rg, a long transmission line of length L with characteristic impedance Z0, and a load resistor RL.

The equations are 2 coupled (by the boundary conditions) wave equations:


$$\frac{\partial^2 V}{\partial t^2} = u^2 \frac{\partial^2 V}{\partial x^2}$$

$$\frac{\partial^2 I}{\partial t^2} = u^2 \frac{\partial^2 I}{\partial x^2}$$

Where u is the velocity of propagation and 0 < x < L, t > 0
The boundary & initial conditions are
$$V(0, 0) = V_g(0) \frac{Z_0}{Z_0 + R_g}$$

$$I(0, 0) = V_g(0) \frac{1}{Z_0 + R_g}$$

$$V(0, t) = V_g(t) - R_g I(0, t)$$

$$V(L, t) = R_L I(L, t)$$

V(x, 0) = I(x, 0) = 0 for x>0 & everything(t<0) = 0
RL, Rg, & Z0 are real positive constants. Vg(t) is a known function of time only.
It's those last 2 boundary conditions that are confounding me.

So I turn each into a difference equations using the centralized 2nd order approximation

$$\frac{V(i, j+1) - 2 V(i, j) + V(i, j-1)}{\Delta t^2} = u^2 \frac{V(i+1, j) - 2 V(i, j) + V(i-1, j)}{\Delta x^2}$$

same for I. I solve for V(i, j+1) & I(i, j+1). With j (time) in my outer loop & i (x) in my inner loop I start stepping across x for each time t. Everything is fine until I reach that last boundary condition. I have a chicken & egg problem. First impulse is to step V forward and calculate I or vice versa. But neither will give me the right answer. I'm going to have the same problem at x=0 after that first step.
The problem is easy if RL = Rg = 0. I can just crank the the equation for V since V(L, t) = 0 is a nice fixed condition.
Likewise it works fine for RL=infinity & Rg = 0. I can crank the equation for I.
It's also works for RL = Rg = Z0.
The fact that it works in these 3 situations leads me to believe it can work for others.

But, for RL & Rg > 0 but not = Z0, how do I handle those two boundary conditions that relate V & I at x=0 and x=L?

I know there are other (& probably better) ways to solve this, even analytically or just intuitively. But I need the practice with FD.

Thanks,
Apchar

 

[gtoguy97]

The solution to this problem lies in the fact that V and I are related by the boundary conditions. You need to use the boundary condition at x=0 (V(0, t) = Vg(t) - Rg I(0, t)) in order to calculate I(0, t). Then, you can use the boundary condition at x=L (V(L, t) = R_L I(L, t)) in order to calculate V(L, t). By alternating between solving for V and I, and using the boundary conditions to relate them, you can iteratively solve for V and I across the entire domain.