Smith Chart Analysis: Calculating Reflection Coefficients, Impedances, and Stub Lengths, Assignments of Electrical and Electronics Engineering

Download Smith Chart Analysis: Calculating Reflection Coefficients, Impedances, and Stub Lengths and more Assignments Electrical and Electronics Engineering in PDF only on Docsity! ECE 450 - Fall 2007 DISCUSSION PROBLEMS ON TRANSMISSION LINES – PART III PROBLEM 1 A loss-less transmission line of characteristic impedance of 50 Ohm and length 0.55 wavelengths at the operating frequency is terminated at an unknown complex load ZR. The maximum of the voltage magnitude on the line is 1.4 V, while the minimum of the voltage magnitude is 0.5 V. The distance of the first voltage maximum from the load is 0.056 wavelengths. Using the Smith Chart, calculate the following: A) The magnitude, | |RΓ , and phase, θ , of the reflection coefficient at the load. [6 pts] o VSWR = 2.8. Identify the intersection of the circle 2.8 with the positive real axis ( 0 line) on the Smith Chart (Point A). The distance of this point from the center of the Chart equals | | . From R r θ = = Γ o max 40.32 the reflection coefficient scale, this length is found to be 0.48. The distance of the first V from the load is 0.056 or 0.056 (180/0.25)=40.32 . Hence, 0.48 oj R e λ ≈ ≈ × Γ ≅ B) The load impedance (in Ohms). [4 pts] ( ) 40.32 Draw the SWR circle on the Smith Chart and identify the point B with 0.48 . Identify the circle and arcs through this point: 1.6 1.2 50 1.6 1.2 80 60 Ohm. oj R R e r r j j Z j j χ χ Γ = + ≈ + ⇒ = + = + C) The values (in Ohms) of the line impedance at the position of a voltage minimum and at the position of a voltage maximum. [6 pts] max 0 m The value of the circle through the point of intersection of the SWR circle with the positive real axis gives us the value of the line impedance at the position of a voltage maximum: / 2.8 r Z Z Z= ⇒ ax min 140 Ohm. The value of the circle through the point of intersection of the SWR circle with the negative real axis gives us the value of the line impedance at the position of a voltage minimum: r Z = 0 min/ 0.36 18 Ohm.Z Z= ⇒ = D) The admittance (in Siemens) at the input of the line. [4 pts] 1 From the load move clockwise toward the generator by 0.55 . This amounts to a complete rotation (0.5 ) back to the load point, and an additional rotation, clockwise by 0.05 . Let D be the resulting poi λ λ λ 1 0 nt on the SWR circle. Its anti-diametric point is the one used to obtain the value of the normalized input admittance: /( ) 0.36 0.02 7.2 0.4 mS.in inY Z j Y j − ≅ − ⇒ ≅ − ProbQem 4 IMPEDANCE OR ADMITTANCE COORDINATES ee rarer Bebe 5° cenren PROBLEM 3 Consider the lossless transmission line network shown below. The operating frequency is 2000 MHz and the propagation velocity on the transmission line is 0.3 m/ns. (a) Using the Smith chart, determine the SWR on the section of line of length l. 9 0.3 15 2 10 o o v cm f λ = = = × 15 4015 40 0.3 0.8 50R jj z j++ Ω ⇒ = = + From Smith chart, VSWR=5.7 (b) Using the Smith chart find two values for the length l such that Z(l) is equal to Zo ± jX. [10 pts] l1=0.186λ-0.112λ=0.074λ=0.074×15=1.11 cm l2=0.313λ-0.112λ=0.201λ=0.201×15=3.015 cm Z(l1) = Zo + jX, l1 = 1.11 cm Z(l2) = Zo - jX, l2 = 3.015 cm (c) Determine the value of series inductance L and the proper length of the transmission line section (l1 or l2) that insures Zin = Zo [10 pts] Impedance is capacitive at l2 with z=1-j1.9 or Z=50+j95 Ω; hence, it can be compensated with inductor such that 9 95 7.55 2 2 10 L nH π = = × × IMPEDANCE OR ADMITTANCE COORDINATES RADIALLY SCALED PARAMETERS : : Bee Be Fe 8 8B AI oa eR PERS a Bee be 888 13g 2 a8 Sos 88". esgk CENTER PROBLEM 4 [25 points] NAME__SOLUTION In the above circuit all transmission lines are lossless with characteristic impedance 50 Ω. The load impedance is a pure resistance R that can be varied from 0 to ∞. The tuning stub (Line # 2) is shorted and can be varied in length from 0 to λ/2. (a) On a Smith chart, draw and label the locus of the normalized load impedance zL as R is varied from 0 to ∞. [5 pts] Locus is horizontal axis on the chart (b) Draw and label the locus of normalized load admittance yL as R is varied from 0 to ∞. [4 pts] Locus is also the horizontal axis on the chart (c) Draw and label the locus of y1' as R is varied 0 to ∞. [4 pts] Rotate axis 0.3λ toward the generator. (See chart) (d) There are two values of y1' for which the length of the tuning stub (Line # 2) can be adjusted to give no standing waves on Line # 3. They are: [4 pts] second locus of y1' is the unit circle y1' = _1+j0_____ or y1' = ___1-j2.7_________.