Transformations on the Complex Γ Plane

Transcription

1 2/12/27 Transformations on the Complex 1/7 Transformations on the Complex Γ Plane The usefulness of the complex Γ plane is apparent when we consider aga the termated, lossless transmission le: z = z = Z, β Γ Z, β Γ Recall that the reflection coefficient function for any location z along the transmission le can be expressed as (sce z = ): ( z ) Γ =Γ e j 2βz e j ( θ 2βz ) Γ + And thus, as we would expect: -j 2β Γ ( z = ) = Γ and Γ ( z = ) = Γ e = Γ Recall this result says that addg a transmission le of length to a load results a phase shift θ Γ by 2β radians, while the magnitude Γ remas unchanged.

2 2/12/27 Transformations on the Complex 2/7 Q: Magnitude Γ and phase θ Γ --aren t those the values used when plottg on the complex Γ plane? θ A: Precisely! In fact, plottg the transformation of Γ to Γ along a transmission le length has an terestg graphical terpretation. et s parametrically plot from z = z (i.e., z = ) to z = z (i.e., z = ): ( z ) = Γ Γ Γ= 1 ( z ) =Γ θ = θ 2β Sce addg a length of transmission le to a load Γ modifies the phase θ Γ but not the magnitude, we trace a circular arc as we parametrically plot! This arc has a radius Γ and an arc angle 2β radians. Γ

3 2/12/27 Transformations on the Complex 3/7 With this knowledge, we can easily solve many terestg transmission le problems graphically usg the complex Γ plane! For example, say we wish to determe Γ for a transmission le length = λ 8 and termated with a short circuit. z = z = Γ Z β Z β 1,, = λ 8 j The reflection coefficient of a short circuit is Γ = 1 = 1e π, and therefore we beg at that pot on the complex Γ plane. We then move along a circular arc 2β = 2( π 4) = π 2 radians (i.e., rotate clockwise 9 ). 1 e + j π 2 1 e

4 2/12/27 Transformations on the Complex 4/7 When we stop, we fd we are at the pot for Γ ; this case 2 1e j π o (i.e., magnitude is one, phase is 9 ). Now, let s repeat this same problem, only with a new transmission le length of = λ 4. Now we rotate clockwise 2 β = π radians (18 ). 1 e + j 1 e For this case, the put reflection coefficient is the reflection coefficient of an open circuit! j Γ = 1e = 1 : Our short-circuit load has been transformed to an open circuit with a quarter-wavelength transmission le! But, you knew this would happen right?

5 2/12/27 Transformations on the Complex 5/7 z = z = Z β 1 Z β 1, (open), (short) = λ 4 Recall that a quarter-wave transmission le was one of the special cases we considered earlier. Recall we found that the put impedance was proportional to the verse of the load impedance. Thus, a quarter-wave transmission le transforms a short to an open. Conversely, a quarter-wave transmission can also transform an open to a short: 1 e + j 1 e

6 2/12/27 Transformations on the Complex 6/7 Fally, let s aga consider the problem where Γ = 1 (i.e., short), only this time with a transmission le length = λ 2 ( a half wavelength!). We rotate clockwise 2β = 2 π radians (36 ). Hey look! We came clear around to where we started! 1 e 1 e Thus, we fd that Γ = Γ if = λ 2--but you knew this too! Recall that the half-wavelength transmission le is likewise a special case, where we found that Z = Z. This result, of course, likewise means that Γ =Γ.

7 2/12/27 Transformations on the Complex 7/7 Now, let s consider the opposite problem. Say we know that the put impedance at the begng of a transmission le with length = λ 8 is:.5e j 6 Q: What is the reflection coefficient of the load? A: In this case, we beg at Γ and rotate COUNTER- COCKWISE along a circular arc (radius.5) 2β = π 2 radians (i.e., 6 ). Essentially, we are removg the phase shift associated with the transmission le! θ = θ + 2β θ.5 5. e j 6 5. e j 15 Γ= 1 The reflection coefficient of the load is therefore:.5e j 15