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1- A lossless transmission line of characteristic impedance 50 ohm is to be matched to a load impedance of 100+j50 ohm using short circuited stub tuner. Find the required lengths using Smith Chart. Attach your Smith Chart!

3- Design a low pass lumped element L-network to match the load 75 100LZ j= − Ω to a 50 ohm

transmission line. Determine the actual component values. Attach your Smith Chart.

150 (75 100) 1.5 2 0.24 0.32L Lz j j y j= − Ω = − ⇒ = + . The SWR circle will intersect the rotated 1 jX+

circle into two points 1 0.24 0.43y j= + and 2 0.24 0.43y j= − . 1st solution: 0.43 0.32 0.11jb j j j= − = , we have to add a (+) suseptance 0(2 ) 6.37 pFC b fZπ⇒ = = .

Converting 1y to 1 1 1.77z j= − , then we have to add (+) reactance 0 ( 1.77) 1.77jx j j j= − − = to move

point 1z to 1 0.0j+ . 0 (2 ) 282nHL xZ fπ→ = = . Low pass.

2nd solution: 0.43 0.32 0.75jb j j j=− − =− , we have to add a (-) suseptance

0 ( 2 ) 212nHL Z b fπ→ = = . Converting 2y to impedance 2 1 1.77z j= + , then we have to add (-)

reactance 0 1.77 1.77jx j j j= − = − to move point 1z to 1 0.0j+ .

01 ( 2 ) 34.97 pFC x fZπ→ = = . High Pass.

3- Design a high pass lumped element L-network to match a 50 ohm transmission line to a load impedance of 70+j100 ohm at 900 MHz. The component next to load must be an inductor. Determine the actual component values. Attach your Smith Chart. 3- 900 MHz’te 50 ohm bir iletim hattını 70+j100 ohm yüke uyumlandırmak için bir L-devresi tasarlayın ve devre elemanlarının tipini belirtin. Yüke en yakın eleman bobin olmalı. Smith Chart’i kağıdınıza ekleyin.

0

0

1.4 2.0 inside the unity circle

0.42 ( 0.33) 0.09 inductor, (2 ) 0.09 98.24 nH

1.9 capacitor,1 ( 2 ) 1.9 1.86pF

Lz j

B Z fL L

X Z fC C

π

π

= +

= − − − = − ⇒ = ⇒ =

= − ⇒ = ⇒ =

3- 150 ohm’luk empedansı olan bir verici 1 GHz’te 75 ohm olan bir antene öz empedansı Zo=150 ohm ve dalga hızı 3x108 m/s olan bir koaksiyel kablo ile bağlanmıştır. Aynı öz empensa sahip ucu açık parallel bağlı bir iletim hattı kullanarak empedans uyumlandırma yapın. Smith’i kağıdınıza ekleyin.

1- (10 points) The Smith chart shown below depicts the design of a L matching network. Draw the matching network providing all values and indicate whether the elements are capacitors or inductors. 1- Aşağıda verilen “Smith Chart” L-devresiyle yapılan bir uyumlandirmayı gösterir. Bu çalışmaya uygun olan uyumlandırma devresini bütün değerleriyle gösterin.

3λ/8

λ/4

λ/8

0 105.0

1.0

2.0

0.2

0.5

1.0

10

2.0

3.0

5.0

0.5

0.2

-10

λ/8

λ/4

3λ/8

0

-2.0

-3.0

-5.0

-0.5

-0.2

tow

ard

toward

gene

rato

r

load

LZ

Solution: 0.2 0.3Lz j= + Ω . 2b = is a capacitor, 0.1x = is an inductor .

LZ0Z jb

jx

1- Bir LC devresi kullanarak 50 MHz’te 90-jl00 ohm’luk bir yükü 50 ohm’luk bir iletim hattına uyumlandırma devresi tasarlayın. Smith Chart’i ekleyin. 1- Design an LC network that will match 90 - j l00 ohms to 50 ohms at 50 MHz. Sketch your network, give all values in pF and nH. Attach Your Smith Chart.

3- Design an L-section lumped element matching network to match a 50 ohm transmission line to 25 ohm load at 2 GHz. Give your matching circuit and label the component values. 3- 50 ohm bir iletim hattını 25 ohm yüke uyumlandırmak için bir L devresi tasarlayın ve devre elemanlarını belirtin. Since 0LR Z< we use a shunt susseptance followed by series reactance

0( ) 25(50 25) 25L L LX R Z R X= ± − − = ± − = ± and 0

0

( ) 1

50L LZ R R

BZ

−= ± = ±

+ sign 9

2525 1.99 nH

2 2 10j L j Lω

π= + → = ≅

× ×,

9

150 1.59 pF

50 2 2 10j C j Cω

π= → = ≅

× × ×

- sign 9

1 125 3.18 pF

25 2 2 10j C

j Cω π= − → = ≅

× × ×,

9

1 5050 3.98 nH

2 2 10j L

j Lω π= − → = ≅

× ×

smith chart solution 1st option: from load add +j0.5 (series inductor), add +j1.0 (parallel capacitor) 2nd option: from load add -j0.5 (series capacitor), add -j1.0 (parallel inductor)

3- Design an L-section lumped element matching network to match a 50 ohm transmission line to load impedance of 100 60j+ Ω at 2 GHz. Give the actual component values.

3- 2 GHz’te 50 ohm bir iletim hattını 100 60LZ j= + Ω yüke uyumlandırmak için bir L-devresi tasarlayın ve devre elemanlarını belirtin. Yüke en yakın eleman kondansatör olmalı.

b) What is the reflection coefficient at 1.8 GHz and 2.2 GHz?

3- Design an L-section lumped element matching network to match a 50 ohm transmission line to load impedance of 20 20LZ j= − Ω at 2 GHz. Give the actual component values.

3- 2 GHz’te 50 ohm bir iletim hattını 20 20LZ j= − Ω yüke uyumlandırmak için bir L-devresi tasarlayın

ve devre elemanlarını belirtin. Yüke en yakın eleman kondansatör olmalı.

b) What is the reflection coefficient at 1.8 GHz and 2.2 GHz?

1- Design an LC matching network at 10 GHz to match 12.5 Ω load to a 50 transmission line. Plot |S11|2 vs.

frequency in dB (5-20 GHz) and determine the –10 dB bandwidth. Attach your Smith Chart. First element next to load must be an inductor. 1- 10 GHz’te 12.5 Ω yükü 50 Ω iletim hattına uyumlandırmak için toplu eleman bir LC devresi tasarlayın. Smith Chart’ı ekleyin. Yüke bağlı ilk eleman bobin olsun.

0.44jx j= → 0.44 50 22X = × = , 2X fLπ= → 0.35nHL = 1.7jb j= → 1.7 0.02 0.034B = × = , 2B fCπ= → 0.541pFC =

3- Design an L-section lumped element matching network to match a 50 ohm transmission line to a load impedance of 100 50j+ Ω . The component next to load must be an inductor. Attach your Smith Chart.

3- 50 ohm bir iletim hattını 100 50LZ j= + Ω yüke uyumlandırmak için bir L-devresi tasarlayın ve devre

elemanlarının tipini belirtin. Yüke en yakın eleman bobin olmalı. Smith Chart’i kağıdınıza ekleyin.

6- The λ/4-matching network is called a "transformer" because it also acts as a voltage transformer (and not only an impedance transformer). For the λ/4-matching section shown below,

a) Calculate V2/V1 and determine the transformer ratio n. (Hint: Think of input and output powers and you will get it in 3 lines). b) Calculate the voltages at the first impedance step and the second impedance step (again, THINK, and do not calculate a lot).

1- Design an LC matching network at 10 GHz to match 12.5 Ω load to a 50 transmission line. Plot |S11|2 vs.

frequency in dB (5-20 GHz) and determine the –10 dB bandwidth. Attach your Smith Chart. First element next to load must be an inductor.

1- 10 GHz’te 12.5 Ω yükü 50 Ω iletim hattına uyumlandırmak için toplu eleman bir LC devresi tasarlayın. Smith Chart’ı ekleyin. Yüke bağlı ilk eleman bobin olsun.

0.44jx j= → 0.44 50 22X = × = , 2X fLπ= → 0.35nHL =

1.7jb j= → 1.7 0.02 0.034B = × = , 2B fCπ= → 0.541pFC =

2- Transfer the design of (1) into t-lines with ZH = 75 Ω and ZL = 33 Ω, and calculate the frequency response from 5-20 GHz. Compare with (1) and comment.

Because the model for ZH and ZL are not exactly one L and one C the frequency which we have S11 min is shifted from 10 GHz and the BW is decreased in comparison with (1)

3- Design a 3-stage LC matching network at 10 GHz for a 12.5 Ω load. Plot |S11|2 vs. frequency in dB (5-20

GHz) and determine the –10 dB bandwidth.

4- Transfer the design of (3) into t-lines with ZH = 75 Ω and ZL = 33 Ω, and calculate the frequency response from 5-20 GHz. Compare with (3) and comment.

In comparison with (3), we can see that the point with minimum S11 is shifted from 10 GHz and the BW is smaller.

8- Consider the quarter-wave matching transformer circuit shown below. Find V + and V − on the quarter-wave line section, in terms of iV .

0Z0 LZ R

/ 4λiV

V +

V −

LR

0z =z l= −

( ) , 0j z j zV z V e V e l zβ β+ − + += + Γ − < ≤ and 0

0

L L

L L

R Z R

R Z R

−Γ =

+.

At z l= − , ( ) [ ]i j l j lV l V V e eβ β+ −− = = + Γ .

[ ]

i

j l j l

VV

e eβ β+

−=+ Γ

and V V− += Γ . Assuming iV with a phase reference at z l= − .

9- A cellular phone at 1.9 GHz has 40 ohm antenna impedance. The feed line from the transmitter to this antenna has0 100Z = Ω . Design a quarter-wave line to match the antenna to the transmitter.

4λ

40LZ = Ω01 100Z = Ω 02 ?Z =

a) What length (in meters) of an air filled line can be used to match the antenna to the transmitter?

0.157 mc fλ = = , 4 0.039 mL λ= =

b) What should be the characteristic impedance of this matching line?

201 02 L inZ Z Z Z= = , 02 100 40 63.24Z = × = Ω

5- Design a matching network using a quarter-wave transformer to match a dipole antenna with input impedance 86 38aZ j= + Ω at the design frequency f =91.1 MHz to a twin-lead transmission line with

characteristic impedance 0 300Z = Ω . Using Matlab, determine the SWR=1.5 bandwidth of your design

assuming the input impedance of the antenna does not change.

6- A dipole antenna consisting of two highly conductive circular rods of combined length L=1.5 m and rod diameter 2a = L/400 is matched to a twin-lead transmission line (Z0=300Ω). Design a quarter-wave transformer matching network at the design frequency f=100 MHz. Using Matlab, determine the SWR=1.5 bandwidth of the system for both transformer solutions. Show your results in a plot of SWR vs. frequency. How can the bandwidth of the system be increased? BONUS: Design a matching network with increased bandwidth.

1- A cable-TV company has a 75Ω coaxial cable plant but wishes to use cable set-top boxes having (150 150)j− Ω input impedance. Using a Smith Chart, design a single-stub matching network using a shorted stub as close as possible to the load. Attach your Smith Chart. a) What is the location d of the stub in wavelength units?

(0.18 0.04) 0.14d λ λ= − = b) What is minimum length l of the stub in wavelength units?

(0.339 0.25) 0.089l λ λ= − =

3- In the circuit below the source impedance is composed of R1 and L1, and the load impedance is the combination of R2 and L2. At ω = 2 x1010 rad/s the load and the source impedances are matched by two segments of lossless T-Lines, and their characteristic impedance is 50 Ω.

a) With the aid of Smith chart find the shortest lengths (in λ) of the two segments of TLines, and specify the nodes A, B and C on the Smith Chart.

b) From the Smith chart find the reflection coefficient and the VSWR (looking to the right) of the points A and C in the circuit, respectively.

c) Suppose that the source impedance (R1 and L1) keeps the same, the lengths of thetwo segments of T-Lines can change, the load (R2 and L2) is replaced by a load impedance ZL; specify the forbidden region of ZL on the Smith chart.