Transformations on the Complex Γ Plane
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1 2/4/2010 Transformations on the Complex G plane present.doc 1/8 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 = 0 Z, β Γ 0 Z, β 0 Γ Recall that the reflection coefficient function for any location z along the transmission le can be expressed as (sce z = 0 ): ( ) ( 2 ) Γ z =Γ e = Γ e θ β j2 β z j Γ + z And thus, as we would expect: ( 0) and ( ) -j 2β Γ z = = Γ Γ z = = Γ e = Γ
2 2/4/2010 Transformations on the Complex G plane present.doc 2/8 Transformg Γ to 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. 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 = 0 ) to z = z (i.e., z = ): 1 ( z 0) =Γ Γ ( z ) = Γ θ = θ 2β
3 2/4/2010 Transformations on the Complex G plane present.doc 3/8 Graphically Transformg Γ to Γ 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 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 = 0 Γ Z β Z β 1 0, 0, = λ 8
4 2/4/2010 Transformations on the Complex G plane present.doc 4/8 Example: Graphically Transformg Γ to Γ The reflection coefficient of a short circuit is 1 = 1 j e π 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 90 ). ( ), and therefore we beg at + j π 2 Γ=1 When we stop, we fd we are at the pot for Γ ; this case o is one, phase is 90 ). 2 Γ 1 j = e π (i.e., magnitude
5 2/4/2010 Transformations on the Complex G plane present.doc 5/8 Example: Now with l = λ/4 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 (180 ). 0 1 e + j Γ=1 For this case, the put reflection coefficient is of an open circuit! j 0 Γ = 1e = 1 : the reflection coefficient Our short-circuit load has been transformed to an open circuit with a quarterwavelength transmission le!
6 2/4/2010 Transformations on the Complex G plane present.doc 6/8 You re not surprised are you? z = z = 0 Z β 1 Z β 1 0, (open) 0, (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. 0 1 e + j 1 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:
7 2/4/2010 Transformations on the Complex G plane present.doc 7/8 Example: Now with l = λ/2 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 (360 ). Hey look! We came clear around to where we started! Thus, we fd that Γ = Γ if = λ 2--but you knew this too! Recall that the halfwavelength transmission le is likewise a special case, where we found that Z = Z. This result, of course, likewise means that Γ = Γ. 1
8 2/4/2010 Transformations on the Complex G plane present.doc 8/8 Example: Now transform Γ to Γ 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: 0.5e j 60 Q: What is the reflection coefficient of the load? θ A: In this case, we beg at Γ and rotate COUNTER-COCKWISE along a circular arc (radius 0.5) 2β = π 2 θ = θ + 2β 0.5 radians (i.e., 60 ). Essentially, we are removg the phase shift associated with the transmission le! 05. e j e j 60 The reflection coefficient of the load is therefore: e 0.5 j 150 1
Transformations on the Complex Γ Plane
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 =
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