# Reflection coefficient of a transmission line with discontinuity

I have a question, let's say that we want to calculate the Reflection Coefficient that we see by watching from left to right of the red-line this circuit:

Let's say that:

• Za= Equivalent Impedence of the circuit to the right of the red-line
• Zb= Equivalent Impedence of the circuit to the left of the red-line

My question is which of these is the formula of Γ:

1. Γ=(Za-Zb)/(Za+Zb)
2. Γ=(Za-Z1)/(Za+Z1) (I think that this is the answes but I'm not sure)
3. Γ=(Zb-Z2)/(Zb+Z2)

What is the correct formula for calculate Γ?Why?

None of your formulas are correct. For the reflection coefficient at the red line only Z1 and Z2 are relevant.

So it's simply Γ=(Z2-Z1)/(Z2+Z1) for a wave hitting Z2 from the left.

You have to remember one of the fundamental differences of transmission lines compared to every other circuit element: Time delay. If a wave is at the red line, it's no longer at Zg and not yet at Z.

Also for the reflection coefficient by definition only the impedance directly before and directly after an impedance discontinuity are relevant. This means when a wave hits the load at Z there is another reflection coefficient (calculated only with Z2 and Z).

So what happens to a wave during a full transit time (from the source to the load and back to the source) depends on many different reflection coefficients.

• @feynman Can I ask why the Reflection Coefficient dosn't depend on Za, I mean the wave that went through Z2 then interact with the load and we have another reflected wave, why this reflected wave dosn't interact with Z1? Commented Sep 13, 2023 at 19:46
• @feynman Also, the impedance that we see at the right of Z1 is ZA, in theory if substitute the whole circuit at the right of Z1 we will obtain the same circuit, but if we do this substitusion and we calculate Γ as Γ=(ZA-Z1)/(ZA+Z1) we obtain a different Γ, why there is no equivalence between the 2 Γ? electronics.stackexchange.com/users/322841/feynman Commented Sep 13, 2023 at 19:53
• @MartinoPistis I edited my answer that should explain this hopefully. Commented Sep 14, 2023 at 6:06

Here is what happens the "very first" time the wave starts from the generator ...

FROM 1 TO 2 ...

Write that the Voltages of incident wave + reflected wave = voltage of transmitted wave : $$\Wi+Wr = W12= 1 + (Z2-Z1)/(Z2+Z1)\$$
$$\=(Z2+Z1+Z2-Z1)/(Z2+Z1)\$$ $$\=2*Z2/(Z2+Z1)\$$.

And when the wave come from the other side ...

FROM 2 to 1 ...

$$\Wi+Wr = W21= 2*Z1/(Z2+Z1)\$$.

• Grazie mille Antonio! So we have the reflection of the wave, but matematically what happen? Commented Sep 13, 2023 at 20:26
• Just write that the voltage is "conservative". So, in the first picture, write that the Voltages of incident wave + reflected wave = voltage of transmitted wave : Wi+Wr = W12= 1 + (Z2-Z1)/(Z2+Z1)=(Z2+Z1+Z2-Z1)/(Z2+Z1)=2*Z2/(Z2+Z1). Commented Sep 14, 2023 at 5:59