įor the steady-state conditions, when the waves travel in the same direction with the same phase velocity, it can be considered as one travelling wave. ![]() As this is an infinite process, the total reflection coefficient is the sum of all the partially reflected waves, which gives the result Γ = Γ 1 – T 1 * T 2 * Γ 3 * ∑ n = 0 ∞ ( – Γ 2 Γ 3 ) n. The propagation and reflection coefficients are the following: Γ 1 = Z 1 – Z 0 Z 1 + Z 0, Γ 2 = Z 0 – Z 1 Z 1 + Z 0 = – Γ 1, Γ 3 = R – Z 1 R + Z 1, T 1 = 2 Z 1 Z 0 + Z 1, T 2 = 2 Z 0 Z 0 + Z 1. Figure 2 depicts the reflected and propagated waves. The reflected part of the wave goes the length of the quarter-wave transformer λ 4 again and partially reflects on the Z 0 with reflection coefficient Γ 3, partially propagates Z 0 with the coefficient T 3, and so on. The reflected part of the wave has the reflection coefficient Γ 2 , the propagated wave has the coefficient T 2. The propagated wave passes the length of the quarter-wave transformer reflecting on the load and passing the load partially. When it reaches the quarter-wave transformer, part of the wave reflects with the reflection coefficient Γ 1, another part of the wave propagate with the coefficient T 1. There is two transmission lines made with blue dashed circles There is a wave that falls from the side of the Z 0. ![]() Let’s consider the case of multiple reflection of the waves. The real quarter-wave impedance transformer experiences a variety of reflected and propagated waves. T h e Z i n = Z 0, for the lossless transmission line. And the characteristic impedance of the arbitrary transmission line is Z 1 = Z 0 R. For the β = 2 π λ, l = λ 4, s o Z i n → Z 1 2 R. The input impedance here is Z i n = Z 1 R + j Z 1 tan β l Z 1 + j R tan β l. ![]() The matching transmission lines are assumed to be lossless. The characteristic impedance of the quarter-wave transformer is Z 1, the length is λ 4. Let’s consider the random transmission line with the characteristic impedance Z 0, and the load with resistance R. The quarter-wave impedance transformer is a device that matches the transmission line and the impedance and is shown in Figure 1.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |