Calculating reflections

Question

I just solved a problem in my homework. I had to calculate the reflection waves and I was given the following data:

Resistance
R₀ = 120 Ω line impedance
R_i = 90 Ω resistance at input
R_b = 1 kΩ termination resistance

Line
l = 0.5m length of the line
δ = 6 ns/m

Source voltage
U₀ = 0.4 V logical 0 - low voltage
U₁ = 4.8 V logical 1 - high voltage

I was also given the following graph:

Showing I have to calculate reflections for when signal goes from 0>1 / low to high voltage basically when we have a pulse.

First I calculated Tau, T = l * δ = 3 ns, and then I calculated the reflection coefficients at R₀ and R_b using formula ρ_x = \$ \frac{Rx - R0}{Rx = R0} \$

With that I was able to calculate for the part before the pulse:

u_i(0^-) = u_b(T^-) = U₀ \$ \cdot \$ \$ \frac{Rb}{Rb + Ri} \$

Followed by the rise:

u_i(0⁺) = u_i(0^-) + ΔU \$ \cdot \$ \$ \frac{R0}{R0 + Ri} \$

And then I was simply able to calculate reflection voltages at certain time for instance first traveling voltage from the change:

u₀(1) = ΔU \$ \cdot \$ \$ \frac{R0}{R0 + Ri} \$

And then reflection voltage at T on the end of the line:

u_b(T⁺) = u_b(T^-) + u₀(1) + u₀(2)

After that I can just calculate the reflections until 5 Tau to see if the signal stabilizes anyhow.

• My question is how do my calculations and formulas change when the graph provided would be inverse showing a drop in the pulse from 4.8V to 0.4V ?

Answer 1

First of all, you wait for 5 tau for the transient to extinguish, provided that tau is the time constant of an exponential signal. When transmission lines are involved, the decay at each reflection is dependent on the reflection coefficient at the load and the generator. In general, you do not wait 5 line delays! (Consider the limiting case of a negligible impedance source driving a line terminated in an open or short circuit, where you have theoretically infinite ringing).

Then, as Ignacio explains, you can use superposition. Keep in mind that the solution for a constant signal is what you would obtain from classic circuit theory: the input and output ports collapse into lumped nodes. You can get the solution as a voltage divider. If you are able to compute vo(t) as the response to a unit step u(t), the response to A·u(t) is A·vo(t). And the response to a sum of inputs is the sum of their individual responses...

Answer 2

You can use linearity of the wave equation. A pulse going from 4.8V to .4V is like the original pulse inverted, plus a constant 5.2V. So take your previous solution, invert it and add 5.2V .