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1- The radius of the outer conductor of a coaxial transmission line is 4 mm. a) Find the radius of the inner conductor so that the characteristic impedance of the line is Z o =50 and the capacitance per unit length is C=100pF/m. b) Which of the following materials must be used as dielectric filling: teflon (ε r =2.1), polyethylene (ε r =2.26), or polystyrene (ε r =2.56)? c) Find the inductance per unit length, L. μHm 2 0 0.25 L CZ = = 2- A transmission line is 80 cm long and operates at a frequency of 600 MHz. The line parameters are L=0.25 μH/m and C=100 pF/m. Find the characteristic impedance, the propagation constant and the input impedance for Z L =100 ..

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### Transcript of 1- The radius of the outer conductor of a coaxial transmission line is … 469/icerik/exam… ·... 1- The radius of the outer conductor of a coaxial transmission line is 4 mm. a) Find the radius of the inner conductor so that the characteristic impedance of the line is Zo=50 Ω and the capacitance per unit length is C=100pF/m.

b) Which of the following materials must be used as dielectric filling: teflon (εr=2.1), polyethylene (εr=2.26), or polystyrene (εr=2.56)?

c) Find the inductance per unit length, L.

μH m20 0.25L CZ= =

2- A transmission line is 80 cm long and operates at a frequency of 600 MHz. The line parameters are L=0.25 µH/m and C=100 pF/m. Find the characteristic impedance, the propagation constant and the input impedance for ZL=100 Ω.. 8- For a lossless transmission line with Z0 and L L LZ R jX= + .

(a) Derive andL LΓ Γ∡ in terms of 0, andL LR X Z .

2 2 2

0 0 02 2 2 2

0 0 0

2 210 0

2 2 2 2 20 0

2

( ) ( )

( ) 2, tan

( )

L L L L LL L L L

L L L L L L

L L LL L

L L L L

R Z jX R Z X X ZZ R jX j

R Z jX R Z X R Z X

R Z X X Z

R Z X R Z X

− + − += + → Γ = = +

+ + + + + +

− + → Γ = Γ = + + − + ∡

(b) Using the expression derived in (a) for the phase of the load reflection coefficient, deduce that for an inductive complex load it is 0≤θR ≤π , while for a capacitive load it is -π≤θR ≤0.

2- For load impedances that have a negative real part ( L L LZ R jX= − + ), show that the

magnitude of the load reflection coefficient is grater than 1. Assume Z0 is real.

2 2 2 20 0 00

2 22 20 0 00

( ) ( ) 21

( ) 2( )

L L L L LLL L L L

L L L LL L

R Z X R Z X Z RZ ZZ R jX

Z Z R Z X Z RZ R X

+ + + + +−= − + → Γ = → Γ = = ≥

+ + + −− +

4- Does the percentage of time-average incident power reflected by the load and power absorbed at the load change if the reactive (imaginary) part of the load impedance changed? 5- A lossless transmission line of characteristic impedance of 50-Ohm is terminated at an unknown resistive load. If 25% of the time-average incident power is dissipated at the load: (a) Calculate the magnitude of the reflection coefficient.

(b) What are the possible values of the resistive load?

(c) Which one of the two possible values in (b) results in a larger voltage magnitude at the load?

6- A lossless transmission line of characteristic impedance of 75 Ohm and length 1.5m is operated at a frequency of 100 MHz. The phase velocity on the line is 20 cm/ns. It is also given that the line is terminated at a resistive load of 50 Ohm. The voltage at the load is

3jRV e π= V.

(a) Calculate the time-average power dissipated at the load;

(b) Calculate the voltage and current (in phasor form) at the input of the line;

(c) For what lengths of the line will its input impedance be real?. 2- Aşağıda verilen devrede 50 50LZ j= + Ω olsun.

a) z l= − noktasındaki toplam empedansın tamamen reel sayı olması için AZ ne olmalıdır?

b) AZ endüktif midir, kapasitif midir?

c) Yukarıdaki AZ ’ yı kullanarak z l= − noktasındaki yansıma katsayısını bulun (?Γ = ). 2- For the lossless transmission line circuit shown below 50 50LZ j= + Ω .

/ 4λ

0z =z l= −

AZ0 50Z = Ω 0 50Z = Ω

a) What value of AZ is required to make the total impedance at z l= − purely real?

b) Is AZ inductive or capacitive?

c) Using the value of AZ found in part (a) calculate the reflection coefficient Γ at z l= − .

choose so that

b) inductive

c) with so

2(50) 50 1 1) , 50

1 50 50

150 , 50 . 0

1

50 50

in in A A

L A

A Total

j ja Z Y Z Z j

Z j Z

Z j Zj j

+ −= = → = = → = Ω

+

= Ω = = Ω Γ =+

1- A traveling sine wave at 100 MHz is propagating down a dispersionless transmission line with capacitance per unit length 100 pF mC′ = and a line dielectric is characterized by

2rε = and 1rµ = . a) What is the speed of this wave? b) What is the inductance per unit length? c) What is the characteristic impedance? d) What is the wavelength and phase constant? e) If the attenuation coefficient of the line is 0.1 dB mα = at 100 MHz, how far will the wave travel before loosing half of its initial power?

The distance must be :

8

7

2

0

-1

) 2.12 10 m s

) 1 2.22 10 H m

) 47.1

) 2.12 m, 2 2.96 m

) 3 dB (0.1 dB m) 30 m

p r r

rp r

a u c

b u L C c Lc C

c Z L C

d

e

ε µ

εε

λ β π λ

= = ×

′ ′ ′= = ⇒ = = ×′

′ ′= = Ω

= = =

= 2- A lossless coaxial transmission line terminated in a purely dissipative (i.e. resistive, non-reactive) load has a VSWR of 3.0. a) Use the attached Smith Chart (or algebra if you prefer) to determine the possible values of the line’s load impedance in Ohms, if its characteristic impedance is 50Ω .

b) If the line’s characteristic impedance is a constant, will its input impedance show any variation from point to point as you move along its length? (No explanation required here.) Yes c) Explain briefly (2 sentences referring to basic transmission-line ideas) why in general a transmission line’s input impedance can or cannot vary with position, given that its characteristic impedance is everywhere equal to a fixed constant.

d) The above coaxial line, part of an underwater link, now begins to leak. Its air insulator is replaced by fresh water, a material whose polarization P

comes chiefly from the rotation of polar molecules with large built-in dipole moments. As the line’s operating frequency increases, will the new insulator’s electrical permittivity ε tend in general to increase, decrease, or stay the same? State your reasoning in 1-2 short sentences. 4- A 50 Ω transmission line is terminated with a load ZL=60+j80 Ω. The propagation constant at 60 MHz is k = 0.2 π m-1. What is the shortest distance from the load to a point at which the impedance, Z(z) = R + jX is pure resistance (X=0). 5- The open and short-circuit impedances measured at the input of a transmission line of length 1.5 m, which is less than a quarter wavelength, are respectively: -j54.6 Ω and j103 Ω. a) Find the characteristic impedance, Z0.

b) Find the propagation constant, k.

c) How long should the short-circuited line be in order for it to appear as an open circuit at the input terminals? 1- Consider the following transmission line circuit. Calculate 1inZ of the line when

0, 0, and 2L L LZ Z Z Z= ∞ = = .

/ 4λ

LZ0 ,Z β

1inZ

21 0 1 1 1 00, , and, 2in L in in inZ Z Z Z Z Z Z= → = = ∞ =

2- Consider the following transmission line circuit. Determine 2inZ of the line when

0, 0, and 2L L LZ Z Z Z= ∞ = = .

/ 4λ

LZ02Z 0 ,Z β0 ,Z β

/ 4λ

2inZ

2 2 0 2 0, 2,in in inZ Z Z Z Z= ∞ = =

3- Kayıpsız bir iletim hattı için 0 50 , 0.4 , 40 30LZ l Z jλ= Ω = = + Ω ise girişteki

empedansı ve yükte harcanan gücün yüke giden güce oranını hesaplayın. 3- Find the input impedance and the percentage of power delivered to the load for a lossless transmission line with 0 50 , 0.4 , 40 30LZ l Z jλ= Ω = = + Ω .

00

0

220

0

tan (40 30) 50 tan(2 0.4)50 25.46 5.91

tan 50 (40 30)tan(2 0.4)

40 30 501 0.889 88.9%

40 30 50 3 3

Lin

L

j

LL L

L

Z jZ l j jZ Z j

Z jZ l j j

Z Z j j e PP P

Z Z j P

π

β π

β π

+

+

+ − + ×= = = + Ω

+ + − ×

− + −Γ = = = = → = − Γ = → =

+ + +

9- Karakteristik empedansı 0 50Z = Ω olan kayıpsız bir hatta, yükten 0.4λ uzaklıkta ölçülen

gerilim 4 2 VoltV j= + , akım ise 2 AI =− ’dir. Buna göre yük empedansını hesaplayın.

9- On a lossless transmission line with 0 50Z = Ω , the voltage at a distance 0.4λ away from

the load is 4 2 Voltj+ . The corresponding current is 2 AI = − . Determine LZ .

0 00

0 0

tan( ) 4 2 tan(0.8 )2 2.97 34.74

tan( ) 2 ( )tan(0.8 ) 1

L inin L

L in

Z jZ lV j jZ ZZ Z j Z j

I Z jZ l j Z Z

β π

β π

+ + −= = = =− − → = =− + Ω

+ − − 3- All transmission line segments in below figure are 4λ in length. Determine the input

impedance seen looking into transmission line #1, #2 and #3 .

2- A transmission line of 0 50Z = Ω is to be matched to a load of 40 10LZ j= + Ω through

a length L of another tr. line of 0Z ′ . Find the required L and 0Z ′ for matching at 50 MHz. 1- Twisted-pair copper wires are used in the U.S. to connect homes to the public telephone network. The plastic insulator separating the wires is a nonmagnetic material with relative permittivity of 2. In a wire at typical operating currents, the average electron drift velocity is on the order of 10-5 meters per second. a) How then is it possible to have a telephone conversation over twisted-pair phone lines? EM signals travel as waves b) What’s the ratio, approximately, between the velocity of a voice telecom signal over such a wire and the drift velocity of the electrons within it?

8, 3

5, 0 0 , ,

1 1 3 102.1 10

2 10

p EMWave

p drift r p drift r p drift

u c

u u uµ ε ε ε −

×= = = = ×

×

c) One of the more important properties of a twisted-pair copper telecom line is its characteristic impedance. What is the meaning of the characteristic impedance of such a line? (1 sentence). The characteristic impedance of transmission line is defined as the ratio of its forward traveling wave to its forward traveling current phasor. d) What’s the difference, from the point of view of basic definitions, between the characteristic impedance of a telephone line and its input impedance, a quantity that typically varies with position along the line? (1-2 sentences). Unlike characteristic impedance, the input impedance of a transmission line is defined as the total phasor voltage divided by the total phasor current including the forward and backward traveling waves. e) The impedance of a given line at a particular position z=z’ has a phase of 12o. What is the meaning of this statement, expressed in terms of basic measurable properties of signals traversing the line at z z ′= ? Measured at position z z ′= , the sinusoidal voltage leads the sinusoidal current by 12° f) If a twisted-pair phone line of characteristic impedance a is plugged into a telephone handset whose impedance is real but <a, will any current be reflected back into the line? Yes g) For the case described in (f), will the incident and reflected voltages at the telephone handset be in-phase or out-of-phase, or will they in general have some other phase relationship? Out of phase h) For the case described in (f), will the incident and reflected currents at the handset be in-phase or out-of-phase, or will they in general have some other phase-relationship? In phase 3- Once every nanosecond, but only once every nanosecond, the voltage on a lossless RG-58 coaxial transmission line is exactly zero everywhere along the line. a) What do you know about the line’s reflection coefficient? 1Γ =

b) What do you know about the position of the voltage maximum closest to the load? Nothing. (Except of course that its within 2λ of the load)

c) Given that 05 0oV + = ∠ V, what is the largest instantaneous real voltage value that ever appears anywhere along the line?

max max( ) , , 1 10j z j z

o o o oV z V e V e V V V V Vβ β+ − + + += + Γ = + Γ Γ = → = 7- Given 0 50Z = Ω , VSWR = 4, f = 500 MHz, the distance between two successive voltage

maximum on the air-filled line is 30 cm, and min 20 cml = . Find LΓ and LZ at the load.

7- Karakteristik empedansı 0 50Z = Ω olan bir kayıpsız iletim hattı LZ yükü ile

sonlandırılmıştır. Duran dalga oranı s = 4, frekans f = 500 MHz, iki voltaj maksimum arasındaki mesafe 30 cm ve min 20 cml = olduğunda yük empedansı LZ ’yi hesaplayın.

LZ0 ,Z β

minl

max min max min

3 1.02min 0

10 10.6 m, , 0.6, , , 2 (2 1)

3 1 4 4

10 2 , 0.6 42.1 68.4 80

3 1

j j

L

sl l l l n

s

n l e Z Z j eπ

π λ λλ β θ β π

πθ π β θ

Γ

Γ Γ

−= = Γ = = < = + − = − +

+

+ Γ= → = − + = → Γ = → = = + = Ω

−Γ 1- Öz empedansı 50 ohm olan kayıpsız bir iletim hattında yükten itibaren ölçülen mutlak gerilimin grafigi aşağıda gösterilmektedir. Buna göre yük empedansını hesaplayın. 1- Consider the measured magnitude of the voltage shown in the figure for a loaded lossless air filled 50 ohm transmission line. Determine the load impedance.

min

max0

min

(2 1)2 3 cm 6cm, Min at load 0 , 0

2

1 2 310 1 2 2 2 15, , 50 10

2 1 3 3 3 1 1 2 3

LL

j LL L L

L

nz n

V ss e Z Z

V s

π

π φλ λ φ π

β

+ += → = ⇒ = = = ⇒ =−

−− + Γ= = = Γ = = Γ = = − → = = = Ω

+ −Γ − 1- 50 Ω öz empedansı olan kayıpsız bir iletim hattı LZ empedansı ile sonlandırılmıştır. Bu hatta duran dalga oranı s=3, ardışık iki minimum gerilim arasındaki mesafe 20 cm, ve yükten ilk minimum gerilime olan mesafe 5 cm’dir. Yük empedansını hesaplayınız. 1- The standing wave ratio of a 50 Ω lossless transmission line is 3s = . The distance between successive voltage minimum is 20 cm. First voltage minimum is located 5 cm from the load. Find ?LZ =

Voltage min. occurs when ( 2 )

0.5

0

2 s 10.4 m, 5 rad/m, 0.5

s 11 2 (2 1) 0,1,.....

2 2 0.5 rad 0.5 0.5

1 1 0.550 30 40

1 1 0.5

j l

j j

L

e l n n

l l e e j

jZ Z j

j

θ β

θ π

πλ β π

λθ β π

θ β π θ β π π

−= = = Γ = =

+=− → − =− + =

− = − → = − = − → Γ = Γ = = −

+ Γ −→ = = = − Ω

−Γ +

3- 50 Ω öz empedansı olan kayıpsız bir iletim hattı LZ empedansı ile sonlandırılmıştır. Bu hatta duran dalga oranı s=2, ardışık iki minimum gerilim arasındaki mesafe 25 cm, ve yükten ilk minimum gerilime olan mesafe 5 cm’dir. Yük empedansını hesaplayınız. 3- The standing wave ratio of a 50 Ω lossless transmission line is 2s = . The successive voltage minima are 25 cm apart and the first voltage minimum occurs at 5 cm from the load. Find ?LZ =

minmin

0.613 0

2 s 1 1 4 30.5 m, 4 rad/m, , 1 1

s 1 3 4 5

10.103 0.317 33 24.1

1j j

L

ll

e e j Z Z jθ π

π λ θλ β π θ π π

λ π λ

Γ

ΓΓ

− = = = Γ = = = + → = − = − +

+ ΓΓ = Γ = = − − → = = − Ω

−Γ

3- 50 Ω öz empedansı olan kayıpsız bir iletim hattı LZ empedansı ile sonlandırılmıştır. Bu hatta duran dalga oranı VSWR=3, ardışık iki minimum gerilim arasındaki mesafe 20 cm, ve yükten ilk minimum gerilime olan mesafe 15 cm’dir. Yük empedansını hesaplayınız. 3- The standing wave ratio of a 50 Ω lossless transmission line is VSWR=3. The successive voltage minima are 20 cm apart and the first voltage minimum occurs at 15 cm from the load. Find ?LZ =

minmin

2102

42 s 10.4 m, 5 rad/m, 0.5, 1 1

s 1 4 2

10.5 30 40 50 53.1

1

jj

L

ll

e e j Z Z jπθ

θπ λ πλ β π θ π

λ π λ

Γ

ΓΓ

− = = = Γ = = = + → = − = + + Γ

Γ = Γ = = → = = + Ω = ∠ °Ω−Γ 2- Öz empedansı 50 ohm olan bir iletim hattında yükten itibaren ölçülen gerilim duran dalgası aşağıda gösterilmektedir. Buna göre yük empedansını hesaplayın. 2- If the characteristic impedance of the transmission line is Z0 =50 Ω and the measured standing-wave pattern is the one shown in the figure, find the impedance of the load, ZL.

max

min

( 2 )max min

0.8 38.660

1.5 1 3 1 13

0.5 1 3 1 2

2occur when 1or 2 2 , 0,1,2

2

2 12 2 (0.2 ) 0.8 0.5 23.43 18.3 14.6

1

L

j z

j jLL L

L

V ss

V s

nV and I e z n n z

z e Z Z e j

θ β

π

θ πθ β π

β

πθ β λ π

λ

′−

°

− −= = = → Γ = = =

− +

+′ ′= − = − = ⋅⋅ ⋅→ =

+ Γ′= = = → Γ = → = = = + Ω

−Γ 7- The magnitude of the voltage measured along a 75 ohm transmission line is shown below. Find the load impedance. 7- Öz empedansı 75 ohm olan bir iletim hattında yükten itibaren ölçülen gerilimin mutlak degeri asagidaki gibi ölçulmüştür. Buna göre yük empedansını hesaplayın.

max

min

5.3max max

0

12.774 11.3436 (0.347) 0.9966 2.0m, 1.768 0.2774

2 7.226 1

First occurs at 0.8436 5.3 rad 0.2779 0.1541 0.23134

190 45

1

L

jLL L

LL

L

V ss

V s

V z e j

Z Z j

λλ

θλ θπ

−= − = → = = = = → Γ = =

+

=− =− → = → Γ = = −

+ Γ→ = = − Ω

−Γ

2- Öz empedansı 50 ohm olan kayıpsız bir iletim hattında yükten itibaren ölçülen gerilim duran dalgası aşağıda gösterilmektedir. Buna göre yük empedansını hesaplayınız. 2- Consider the measured magnitude of the voltage shown in the figure for a loaded, lossless, air-filled, 50 ohm transmission line. Determine the load impedance.

min max

0

50 VSWR 1 2VSWR 5 , 25 cm 1m 300MHz

10 VSWR 1 3 4

41 2 32 2 1

50 103 3 1 1 2 3

L

j j

L

cf

l l l

e e Z Zθ π

λλ

λ

π λθ β π θ π θ β π

λ

−= = → Γ = = = → = → = =

+

→ − =− → = → =− → = = =

−+ ΓΓ = Γ = = − → = = = Ω

−Γ + 2- Öz empedansı 50 ohm olan bir iletim hattında yük sıfır iken (ZL=0 ohm) yapılan ölçümde voltaj duran dalgasındaki iki minimum arasındaki mesafe 20 cm’dir. Kısa devre yük, bilinmeyen bir yük ile değiştirildi ğinde VSWR=3 olup voltaj minimumlar yüke doğru 5 cm yaklaşmıştır. Buna göre yük empedansını hesaplayın. 2- Measurement are taken on slotted line with Z0=50 ohm characteristic impedance and a ZL=0 ohm shorted load. The minima in the standing wave pattern are located 20 cm apart. When the shorted line is replaced with an unknown load, the resulting standing wave pattern has a VSWR=3 and the voltage minima have moved 5 cm closer to the load. What is the unknown load impedance?

2 1 3 120 cm 0.4m 5 rad/m, 3 0.5

2 1 3 1L

ss

s

λ πλ β π

λ

− −= → = → = = = → Γ = = =

− +

The first min for a short is at z=0. If the minima move 5 cm closer to the load, then the first minimum with the unknown load is at zmin= -15 cm.

occurs when

( 2 )min

0.5

0

1 2 (2 1) 0,1,.....

2 2 2.5 0.5 0.5 0.5

1 1 0.550 30 40 50 53.1

1 1 0.5

L

j l

L

j j

L L L L

LL

L

V e l n n

l l e e j

jZ Z j

j

θ β

θ π

θ β π

θ β π θ β π π π

− =− → − =− + =

− = − → = − = = → Γ = Γ = =

+ Γ +→ = = = + Ω = ∠ ° Ω

−Γ −

4- A long transmission line cable is cut at some point. We know that the cable is distortionless, but it exhibits some small attenuation coefficient α = 0.02 km-1. Find the location of this cut, knowing the time-domain-reflectometer (TDR) method showed that at the generator end the ratio of the incident step wave to the reflected one 1 1| | | | 2.72V V+ − = .

1 121 1 1

50

ln(| | | |) ln2.12| | | | 2.72 25 km

2 2l

V VV V e lα

α

+ −

+ − −= = ⇒ = = =

1- A 50 m long transmission line is shorted on one end. The voltage standing wave ratio at the other end is s = 9.5 and frequency 100 MHzf = .

for short

Np/m

2

-3

1( ) ln , 1

2 ( )

s 1 9.5 1 1 | 1 |( ) 0.810 ln 2.107x10

s 1 9.5 1 2(50) 0.810

Ll

L Ll el l

l

α α

α

− Γ Γ = Γ → = Γ = − Γ

− − −Γ = = = → = =

+ + 4- If a transmitter can deliver 30 W to a 50 Ω load, how much power is delivered to a load

impedance of 80 40LZ j= + Ω with a 50 Ω coaxial cable? 4- Bir verici 50 ohm’luk bir yüke bağlandığında yükte harcanan güç 30 W ise, bu verici 50 ohm coax kablo ile 80 40LZ j= + Ω yüküne bağlandığında yükte harcanan güç nedir?

2 20.367 36 (1 ) 30 [1 (0.367) ] 25.96L L LP P W+Γ = ∠ ° → = − Γ = − =

5- A 75 , 25WΩ transmitter is connected to a load impedance 40 20LZ j= + Ω through a

transmission line with 0 75 , 0.3Z l λ= Ω = . Find the power delivered to the load.

2 20.345 140.4 (1 ) 25[1 (0.345) ] 22L L inc LP P WΓ = ° → = − Γ = − =∡

1- A 50 , 10WΩ transmitter is connected to a load impedance of 75 25LZ j= + Ω through

a transmission line with 0 50Z = Ω . Find the power delivered to the load.

2

21) 0.231 0.154 0.2774 33.69 (1 ) 10 [1 (0.277) ] 9.23L L L

j P P W+Γ = + = ∠ ° → = − Γ = − = 4- A lossless tr. line is terminated in a non-ideal short circuit such that the VSWR = 200. a) What is the power dissipated in this circuit as a percentage of the incident power? Compare this result to that of an identical transmission line terminated in an ideal short circuit. b) If there is now an additional 0.1 dB distributed attenuation between the short circuit termination and the input reference plane, what will be the new VSWR seen at the input? 4- Aşağıdaki toplam kaybı 0.1 dB olan bir iletim hattında yükte ölçülen duran dalga oranı VSWR = 200 olup girişteki duran dalga oranını hesaplayın.

(i.e. %98 of incident power is reflected)

%2 of power must be dissipated in the load (compared to %0 for an ideal short circuit)

2

2

1 200 1) 0.99 0.98

1 200 1

) 0.99, , 0.1 1

L L L L

l

L in L

sa P P

s

b e α

− +

− −Γ = = = → = Γ =

+ +

Γ = Γ = Γ − =0.1 102 2

0.1 10

0 log 10

110 0.967464848, 60.47

1

l l

in

in L in

in

e e

s

α α −− −

→ =

+ ΓΓ = Γ = = =

+ Γ

5- Aşağıdaki toplam kaybı 3 dB olan bir iletim hattında yükte harcanan güç W1LP = .

a) Girişteki yansıma katsayısını bulun ?inΓ = b) Girişteki net gücü bulun ?inP =

3 10 3 102 2 2

2 22

22

2

0.99, 3 10 log 10 , 0.99 10 0.496

(1 ) 50.25, 100.26, (1 ) 75.6

(1 )75.6

(1 )

l l l

L in L

l

L L L L in L in in in

l

L in

in

L

e e e

P P P P P e P P W

e PP W

α α α

α

α

− −− − −

+ + + + +

Γ = − = → = Γ = Γ = × =

= − Γ → = = = = − Γ =

− Γ= =

− Γ 6- Aşağıdaki iletim hattı için yükte harcanan gücü hesaplayın. 6- What is the power delivered to the load for the lossless transmission line below?

015 (rms), 75 , 60 40 , 0.7S S LV V Z Z Z j l λ= = = Ω = − Ω = .

2 2

2 2 20

0 02

2

1) 0.303 94 (1 ) (1 ) (1 ) 0.681 W4

2) 55.4 29.5 Re 0.681 W

g

L L inc L L L

G

in L in in in in

G in

V VP P

Z Z

VZ P P I R Z

Z Z

+

Γ = − ° → = − Γ = − Γ = − Γ =

= ∠ ° → = = = =+

1- Consider a coaxial cable with characteristic impedance Z0 = 75.0 Ω and solid plastic insulator with εr = 4.0 and α=0.01904 N/m as shown in the figure. Consider f = 250 MHz. a) Find the wavelength, the load and the input reflection coefficient and the input impedance. b) Find the phasor voltage and current and average power at the input of the cable line. c) Find the average power delivered to the load and the average power dissipated by the line.

2 2

0

12

2 22 2

2

1150 75 1a) 0.6m, 0.211 115.13

150 75 3 1

b) 42.81V 371.87 mA Re 7.96W

c) (1 ) (1 ) 41

l j l in

R in R in

inr

gin

in g in in in in

g in g in

l lin

L in R R

in

ce e Z Z

f

VZV V I P V I

Z Z Z Z

PP P e e

α β

α α

λε

− −

+ − −

+ Γ−= = Γ = = → Γ =Γ = → = = Ω

+ −Γ

= = → = = → = = + +

= − Γ = − Γ =− Γ

.689W 3.271Wline in L

P P P→ = − = 7- For a transmission line circuit 020 (rms), 100 , 500 MHz, 4 mS SV V Z Z f l= = = Ω = =

Calculate the input power and power delivered to the load when the power is attenuated by

dB m and

dB m and

dB m and

a) 0.0 150

b) 0.5 150

c) 0.5 100

L

L

L

Z

Z

Z

= Ω

= Ω

= Ω

.

final fin

initial init

Np/m

W

2 2

2 2 1m

10

22 8 2

a) 0.2, (1 ) 1 (1 (0.2) ) 9.6 W

0.5b) 0.5 10 log 10 log( ) 0.0576

20 log

0.2 0.13 (1 ) 1 (1 (0.13) ) 0.984

L in L L

l

l

in L in in in

L

P P P

P Pe e

P P e

e e P P

P

α α

α α

α

+

− − ×

− − +

Γ = = = − Γ = − =

= ⇒− = = ⇒ = =

Γ = Γ = = ⇒ = − Γ = × − =

W

W dBm dBm

22 2(0.23) 2(1 ) 1 (1 (0.2) ) 0.606

c) 0, 1 30 , 28

l

in L

L in L

P e e

P P

α+ − −

+ +

= − Γ = − =

Γ = = = =

3- A 50 Ω transmission line is matched to a 10 W source, and feeds a load 100LZ = Ω . If

the line is 2.3 λ long and has an attenuation constant dB λ0.5α = , find the power

delivered by the source, lost in the line and delivered to the load.

dB Np 2(0.1324)

2

20 2

0

2

20 2 2(0.23) 2

0

0.333, 0.5 dB/ 1.15 0.1324 0.333 0.256

(1 ) 10(1 (0.256) ) 9.34 W2

(1 ) 10 (1 (0.333) ) 6.85 W 2.49W2

L in

in in

l

L L lost in L

l l e

VP

Z

VP e e P P P

Z

α

α λ α α −

+

+

− −

Γ = = → = → = → Γ = =

= − Γ = − =

= − Γ = − = → = − = 5- A resistive load reflects 5% of the incident power when it is connected to an ideal line of characteristic impedance Z0. What is the input return loss when the same load is connected to a lossy transmission line of the same characteristic impedance with length 30 cml = and a distributed attenuation of 0.2 dB/cm? (Hint: 1 Np = 0.0183 dB)

2 2

2(0.023 Np cm) (30 cm)

0.2 dB cm 0.023 Np cm, (0) 0.05 (0) 0.224 ( ) (0)

( ) 0.224 0.056 20 log | | 25 dB

ll e

l e RL

γα −

= = Γ = → Γ = → Γ = Γ

Γ = = → =− Γ =

4- Aşağıdaki iletim hattı için f = 200 MHz ve 82 10 m spu = × ise ( 2) ?V z l= − =

4- Consider the lossless transmission line circuit shown below. It is assumed that f = 200 MHz, 82 10 m spu = × . Find the input impedane of the transmission line, the input voltage,

the forward travelling the voltage at load and the phasor voltage at 2z l=− .

00 0

0

2 2

0 0

21m 2 46.1 11.5 4.07 8.37 V

0.227 31.4 , ( ) ( ) ( ) ( )

5 90 V ( 2) ( )( )

p g in

in in

g in

j z j z j l j lLL L L in

L

j l j linLj l j l

L

u V ZZ j V

f Z Z

Z ZV z V e e V l V e e V

Z Z

VV V z l V e e

e e

β β β β

β β

β β

πλ β π

λ

+ − + −

−+ +

= = → = = → = + Ω→ = = ∠ °+

−Γ = = − ° = + Γ → − = + Γ =

+

→ = = ∠ ° → =− = + Γ+ Γ

5.68 54.8 V= ∠ ° 2- For the following transmission line circuit determine the voltage across the terminals where the two quarter-wave lines are joined together and the average power delivered to each antenna.

in AB

ave Each antenna gets half the power

21

1 2

(50)22.94 6.88 11.45 3.44

100 30 2

10 10(11.47 3.44)0.073 0.035A, 1 0.96 0.15V

100 50 11.47 3.44 100 50 11.47 3.44

1Re 0.0377 W

2AB in

ZZ Z j Z j

j

jI j V j

j j j j

P V I P∗

= = = + → = = + Ω−

+= = − = = −

+ + + + + +

= = → W0.01882ave

ant

P= =

1- İçi hava ile dolu aşağıdaki kayıpsız bir iletim hattının parametreleri aşağıda verilmektedir. Yükteki yansıma katsayısını, girişteki empedans, gerilim ve akımı ve yükte harcanan gücü hesaplayın. 072 40 , 300 , 50 , 10 0 , 100MHz, 1.75mL S SZ j Z Z V f l= + Ω = Ω = Ω = ∠ ° = = .

1- An air filled lossless transmission line circuit below has the following parameters. Determine the reflection coefficient at load, the impedance, voltage and current at input, and the power delivered to load.

072 40 , 300 , 50 , 10 0 , 100MHz, 1.75mL S SZ j Z Z V f l= + Ω = Ω = Ω = ∠ ° = = .

0

0

00

0

0.595 0.171, 3m, 2 1.17 3.67 rad

tan 72 40 300 tan(3.67)300 111.88 218.79

tan 300 (72 40)tan(3.67)

8.88 1.5 V, 22.4 29.9mA

LL

L

Lin

L

g in g

in in

G in G in

Z Zj c f l l

Z Z

Z jZ l j jZ Z j

Z jZ l j j

V Z VV j I j

Z Z Z Z

λ β π λ π

β

β

−Γ = = − + = = = = =

+

+ + += = = + Ω

+ + +

= = + = = − →+ +

[ ]12 Re 0.077 Win L in inP P V I ∗= = = 3- Consider the transmission line. Find the voltage, current and the forward traveling voltage at input. 0 50 , 100 , 4, 75 25 , 25 , 1 0 .L in S SZ Z l Z j Z Vλ= Ω = Ω = = + Ω = Ω = °∡

20 0 0 0

2 2

0 0 0 0

0

0.76 0.05 V 9.7 14 mA, 1 3, 2

( ) ( ) (1 )

( ) (1 ) (1 )4 (1 )

g in

in in in in L

in S

j z j z j z j z j z j z

L L

j jj inin L L in

L

V ZV j I V Z L

Z Z

V z V e V e V e e V e e

VV V z V e e jV V V V e

jV

β β β β β β

π ππ

β π

λ

+ − − + − + −

+ − + + + +

+

= = + → = = ∠− Γ = =+

= + = + Γ = + Γ

= = − = + Γ = −Γ → = → =−Γ

5- Consider the transmission line with 0 50 , 75 , 0.15 , 100 0 V.S L SZ Z Z l Vλ= = Ω = Ω = = °∡

a) Compute the input impedance, input current, voltage and power. b) Compute load current, voltage and power. How does input power compare to load power?

[ ]

00

0

12

2 2

2 0.15 tana) 0.942 rad, 0.2, 41.25 16.35

tan

1.08 10.1 A 47.86 11.46 V Re 24 W

b) ( ) (1 ) (1 ) 50

LL in

L

g

in in in in in in in

G in

j l j l j L j L

L L in L L L

Z jZ ll Z Z j

Z jZ l

VI V I Z P V I

Z Z

V l V e e V V e e Vβ β β β

π λ ββ

λ β

+ − + − +

× += = Γ = = = − Ω

+

= = ∠ ° → = = ∠− ° → = =+

= + Γ → = + Γ → =

[ ]

54 54

2 54 12

0

V, 60 V

( ) (1 ) 0.8 A Re 24 W (lossless)

j j

L

j l j l jLL L L L L L in

e V e

VI l e e I e P V I P P

Z

β β

− ° − °

+− − ° ∗

=

= −Γ → = → = = → = 1- Aşağıda verilen iletim hattı devresi için giriş empedansını, girişteki gerilim ve akımı, ve yükteki ortalama gücü hesaplayın. 025 , 50 , 75 , 10 0 ,L S SZ Z Z V l λ= Ω = Ω = Ω = ∠ ° =

1- A transmission line circuit below has the following parameters:

025 , 50 , 75 , 10 0 ,L S SZ Z Z V l λ= Ω = Ω = Ω = ∠ ° =

[ ]1225 , 2.5 V, 0.1 A, Re 0.125Wg in in

in in in in L in in

in S in

V Z VZ V I P P V I

Z Z Z

∗= Ω = = = = = = =+

1- Aşağıdaki şekilde 1.2 cm uzunluğunda öz empedansı Zo olan ve ZL empedansı ile sonlandırılan bir iletim hattı görülmektedir. Dalga boyu 5 cm olup giriş empedansı ise Zin = 50 + j20 ohm’dur. Hattın uzunluğu 3.7 cm olduğunda giriş empedansı ne olur? 1- A 1.2 cm long lossless transmission line has characteristic impedance Zo and is terminated by a load ZL. The wavelength is 5 cm and the input impedance is Zin = 50 + j20 ohm. What is the input impedance if the length of the transmission line is increased to 3.7 cm.

Zin does not change because the length of the line increases by half wavelength. 2- What is the value of Z02 to impedance match the antenna to transmission line Z01? 2- Z02 empedans değeri ne olmalıdır ki dipole antenin empedansı verici ile uyumlu olsun?

202

0275 150300

ZZ= → = Ω 1- Şekildeki devre için 0 50Z = Ω olsun. a) 1VSV = ve 0SZ Z= olduğunda yükteki gerilimi

hesaplayın. b) Yükte harcanan gücün maximum olması için ZS ’in değeri ne olmalıdır? 1- For the following transmission line circuit assume that 0 50Z = Ω . a) Determine the voltage across the load when 1VSV = and 0SZ Z= . b) Determine the source impedance ZS required to maximize’ the power delivered to the load. Does this imply that there are no reflections on the line? Explain.

a)

V

b)

22 0

0

220

4 0 020

20

0 020

( ) (1 ), , , 0.15 0.76

( ) (1 ) (1 )

0.5 (1 ) 0.38 0.425 0.572 132 V1

j l j l S inL in in L

S in L

j jSin L L

S L

SL L

S L L

s in

V Z ZV l V e e V Z j

Z Z Z

V ZV V l V e e jV

Z Z Z

Z jVV j V V j

Z Z Z

Z Z

β β

π πλ

+ −

+ − +

+ +

= + Γ = = Γ = − −+

= = = = + Γ = −Γ+

−= = − → = +Γ = − − = − °

+ −Γ

= =

There are reflections but they add up properly to achieve

maximum power transfer

20 14.7 58.8 .LZ Z j∗ = − Ω

2- Find the load reflection coefficient, the input impedance and power delivered to the load. (peak) GHz cm m s8

02V , 30 , 50 , 50 , 100 , 1 , 3 10g g pV f Z Z R l u= = = Ω = Ω = Ω = = ×

0

0

2

00

0

1(0) , 0.01, 0.01 2 tan2 0

3

tan 2 4 16100 V 0.0089W

tan 3 2 1800

p

inL inin L in L

L in g in

VR Zl l

R Z f

VZ jZ l ZZ Z Z V P

Z jZ l Z Z R

λ β π π

β

β

−Γ = = = = = → = ⇒ =

+

+= = = Ω→ = = → = = =

+ + 5- Aşağıda 3 dB/λ kaybı olan bir iletim hattında yükteki giden voltaj değeri 10 V (peak)’tir. a) Yüke giden, yükten yansıyan ve yükte harcanan gücü hesaplayın. b) Giden ve yansıyan voltajın mutlak değerleri ile yansıma katsayısını girişte hesaplayın. c) Girişteki net gücü ve iletim hattında harcanan gücü hesaplayın. 5- Consider a lossy transmission line (a loss of 3 dB per λ) with ZL=30-j50 ohm with a length of 1λ of characteristic impedance Zo=50 ohm. The incident voltage at load is 10 Volt (peak). a) Determine the incident, reflected power and power delivered to the load. b) Find the magnitude of incident and reflected voltage, and reflection coefficient at the input. c) Find the input power and the power lost in the line.

2

2 2

0

0 0

2

0 0

a) 0.57 80 , 1W, 0.325W (1 ) 0.675W2

5.7b) 2 10 2 V, V, 0.285 80

2 2 2

c) 2 2W (1 ) 1.83755W P 1.16245W

L

L L L L L L L L

L LL in

L in in line in L

VP P P P P

Z

VV V V

P P P P P P

+

+ − + +

−+ + −

+ + +

Γ = − ° = = = Γ = → = − Γ =

Γ= = = = Γ = = − °

= = → = − Γ = → = − =

7- Aşağıdaki iletim hattı için 0 50 , 6 , 0.5 dB/ , 30 50S LZ Z l Z jλ α λ= = Ω = = = − Ω . Yükte harcanan güç 0.675 WLP = ise hattın girişindeki net gücü hesaplayın. 7- For the given transmission line 0 50 , 6 , 0.5 dB/ , 30 50S LZ Z l Z jλ α λ= = Ω = = = − Ω . The power delivered to load is 0.675 W,LP = Calculate the input power.

dB/λ Np

W

W W

22 2(0.35)

22

0.57 80 , 0.5 3 dB 0.35

0.57 0.285 (1 ) 0.675W 1

2 (1 ) 1.84

L

l

in L L L L L

l

in L in in in

l l

e e P P P

P P e P P

α

α

α α α

− − + +

+ + +

Γ = − ° = → = → =

Γ = Γ = = → = − Γ = → =

= = → = − Γ = 7- Aşağıdaki iletim hattı için 0 50 , 6 , 0.5 dB/ , 30 50S LZ Z l Z jλ α λ= = Ω = = = − Ω .

Voltaj kaynağının gücü 2 W ise yükte harcanan gücü ve hatta harcanan gücü bulun. 7- For the given transmission line 0 50 , 6 , 0.5 dB/ , 30 50S LZ Z l Z jλ α λ= = Ω = = = − Ω .

Find the power lost in the line and at the load if the line is connected to a 2 W source.

22 2(0.35)

2

0.57 80 , 0.5 dB/ 3 dB 0.35 Np

0.57 0.285, 2 (1 ) 1.84

1W (1 ) 0.675W 2 1 1W

L

l

in L in in in in

L L L L line in L

l l

e e P W P P W

P P P P P P

α

α λ α α

− − + +

+ + + +

Γ = − ° = → = → =

Γ = Γ = = = → = − Γ =

= → = − Γ = → = − = − =

1- Asağıdaki iletim hattında ZL değeri ne olmalıdır ki ZMID = j150 ohm olsun. Bu durumda Zin değeri nedir? 1- For the transmission line circuit below find the load impedance ZL that is required to make ZMID = j150 ohm at 1.5 GHz. And compute Zin.

2

1.5GHz 0.2m, 2.5 cm tan 18 4

30 (150)30 150 20 , 5 cm @1.5GHz, 150

30 4 150L

mid L in

L

cf l l

f

Z jZ j Z j Z j

jZ j

λ πλ β β

λ

= → = = = → = ⇒ =

+= = Ω⇒ = Ω = = = − Ω

+ 1- Aşağıdaki kayıpsız iletim hattı devresi maksimum güç transferi şartlarında çalışmaktadır. Buna göre yük empedansını, yükte harcanan ortalama gücü ve yükteki gerilim hesaplayın. 1- The circuit below is operating at the condition of maximum power transfer. The transmission line is lossless of characteristic impedance of 50 Ohm. Calculate the load impedance, the time-average power dissipated at load, and the voltage at the load.

max power since

2

2 3 4 3

2 20 0 0

0

50 50 0.25W8Re

( 1.5 ) 1 (1 )(1 )

(1 ) (5 5)V

G

in G R in L in

G

j jin

in R R

R

G inR R in

G in

VZ Z j Z Z P P

Z

VV V l V e e V V

V ZV V V j

Z Z

π λ π λ

λ λλ

−+ + +

+

⇒ = = + Ω→ = → = = =

= = = + Γ = − + Γ → =− + Γ

→ = + Γ = − = − =− ++

2- Aşağıdaki kayıpsız iletim hattı devresi maksimum güç transferi şartlarında çalışmaktadır. Buna göre yük empedansını, yükte harcanan ortalama gücü ve yükteki gerilim hesaplayın. 2- The lossless transmission line circuit below is operating at the condition of maximum power transfer. Calculate the load impedance, the time-average power dissipated at load, and the voltage at the load.

max power

220

2 22

4 40 0 0

0

0

50 50 25 25 0.25W8Re

( 4) 1 (1 )(1 )

(1 )(1 ) 5V

(1 )

G

in G R L in

in G

j jin

in R R

R

R LR R in in

R

VZZ Z j Z j P P

Z Z

jVV V z V e e jV V

ZV V jV jV j

Z

π λ πλ

λ λλ

+ + +

+

⇒ = = + Ω → = = − Ω→ = = =

= = − = + Γ = −Γ → = − − Γ

+ Γ→ = + Γ = − = − = −

−Γ 3- Consider the lossless transmission configuration shown in the figure. The 100 W time harmonic source has a 50 ohm impedance and a frequency equal to 300 MHz. a) Determine the equivalent impedance of the load. b) Determine the input impedance seen by the source c) Determine how much power is dissipated in the load.

0

0 0 0

0

tan 2, 0 tan , cot ,

tan 8 4

50 250, 50 25 25 1 0

50 2

sc ocLin L in L in

L

eqsc oc

in in eq in L L L

eq

Z jZ lZ Z Z Z jZ l Z Z jZ l l

Z jZ l

Z jZ j Z j Z j Z j P

Z j

β π λ πβ β β

β λ

+= = → = = ∞ → =− = =

+

− −= = → = → = → Γ = = → Γ = → =

+ +

2- For the lossless transmission line circuit below find V1, V2, and V0. if E1=10 Volt (peak) V0=50 ohm and l=λ/2. 2- Aşağıdaki kayıpsız iletim hattı için, V1, V2, ve V0 potansiyelini bulun. E1=10 Volt (peak) Z0=50 ohm ve l=λ/4.

0

0 0

0 0 1 0 10 0 1

0 0

2 1 1 12

232 2 2

3 90 2 0 123 3

20

3 3 2 3

( ) (1 ) ( 0)3 3 3

L L in

j l j l j l j l j l

L

Zj l

Z Z

Z Z E Z EZ Z Z Z V

Z Z

E E EV l V e e V e V e V V l e

VV V V E e

β β β β β

β

+ − + + − −

= + = → Γ = → = → = =+

= + Γ = = → = → = = =

= = → =+ 1- Aşağıdaki ikinci iletim hattında s=2 olup iki minimum gerilim arasındaki mesafe 50 cm ve yükten ilk max gerilime olan mesafe 20 cm’dir. 100 ohm direncin üzerindeki gerilimi bulun. İki iletim hattında da faz hızı aynidir. f= 200 MHz. 1- On line 2 the VSWR=2. The distance between successive minima on line 2 is 50 cm, and the distance from the load to the first maximum is 20 cm. Find the phasor voltage across the 100 ohm resistor. Phase velocity is the same on both lines. f= 200 MHz.

line m

2 max max

0.80 2 1

2 20150 70 1

50 70 6 1 2

1 1 22 50 1 , 2 , 2 2 0.8

1 31 1

26.93 11.87 , tan tan(1.4 ) 1003 1

(70)1 100 100 50 , , 3.75 3.75

50

L

j LL L in

L

L L in

L

ss z z

s

e Z Z j l Z

ZZ l Z

Z

λ

π

πλ θ β π

λ

β π

λ−+

−→ = → = = → Γ = = = = =

−+ Γ

Γ = → = = + Ω = → = Ω−Γ

→ = = Ω Γ = = − = = → = =

V

1 12 298 35 1 198 70 6 6 6

1 256 6

98

( ) (1 ) 10 (1 ) (1 ) 5

( 0) (1 ) 5(1 )

j l j l j l j l

L in

L

V l V e e V V e e jV V j

V l V j j

β β β β+ − + − + ++

+

= Ω

= + Γ → = = = − = − + → =

= = + Γ = − =

3- A 3.2ml= long transmission line has 0 75 0.94Z j= + Ω and 10.224 17.614mjγ −= + . It is driven by a generator, operating at 105.2MHzf = , with 16 0 VSV = ∠ ° and 75SZ = Ω . Using a network analyzer, the input impedance is measured to be 88.6 20.3inZ j= + Ω .

a) Find the input reflection coefficient and the phasor input voltage, b) Find the forward traveling voltage wave at the input, c) Find the load reflection coefficient and load impedance, d) Find the load voltage.

a)

b)

c)

d)

0

0

2 20

0.14341 47.5 , 8.82 5.8 V

(1 ) 8.01 0.3 V(1 )

10.6 26.4 , 166.36 143.22

1

(1 ) 5.71 2.155 6

in G inin in

in in G

inin in in in

in

l l Lin L L in L

L

l

L in L

Z Z V ZV

Z Z Z Z

VV V V

e e Z Z j

V V e j

γ γ

γ

+ +

+ −

−Γ = = ∠ ° = = ∠ °

+ +

= + Γ ⇒ = = ∠ °+ Γ

+ ΓΓ = Γ ⇒ Γ = Γ = ∠ ° = = + Ω

−Γ

= + Γ = + = .10 20.7 V∠ ° 2- Aşağıdaki kayıpsız iletim hattında faz hızı m s82 10pV = × ise dalga boyunu, yükteki

yansıma katsayısını, giriş empedansını, girişteki gerilimi ve yükte harcanan gücü bulun. Volt7

0 150 , 6m, 150 50 , ( ) 5 cos(8 10 )S L SZ Z l Z j V t tπ= = Ω = = − Ω = ×

2- A 6ml = long lossless TEM transmission line having 0 150Z = Ω is driven by a source with Volt7( ) 5 cos(8 10 )gV t tπ= × and 0gZ Z= . If the line has a relative permittivity of 2.25 (corresponding to a phase velocity of m s82 10pV = × ) and is terminated in a load

150 50LZ j= − Ω . a) Find the wavelength on the line. b) Find the reflection coefficient at the load. c) Find the input impedance. d) Find the input voltage. e) Find the power available from the source and power delivered to the load.

a)

b)

c)

d)

7

0

0

00

0

28 10 40MHz, 0.4 rad m 5m

0.027 0.162 0.164 80.5

tan 150 50 (150)(3.078)150 116 27 119 13.3

tan 150 (150 50)(3.078)

5(11

p

LL

L

Lin

L

g in

in

in g

fV

Z Zj

Z Z

Z jZ l j jZ Z j

Z jZ l j j

V ZV

Z Z

ω πω π β π λ

β

β

β

= × → = = = → = =

−Γ = = − = ∠− °

+

+ − += = = + = ∠ °Ω

+ + −

= =+

e) is when i.e.when

f)

22

0

0

2

2

9 13 )2.23 8.5 V

266 27

1, 2 20.8mW

2 8

120.3mW check (1 ) 20.3mW

2

ginav in g in g av in

in

inL in L in L L av

in

j

VVP Z Z Z V V P R

Z Z

VP P P R P P

Z

∠ °= ∠ °

+

= = = → = = =

= → = = = − Γ = 2- 0.725λ uzunluğunda kayıpsız bir iletim hattında V( ) 100 cos( )SV t tω= , 0 50Z = Ω , 10 10SZ j= + Ω 40 30LZ j= + Ω ise yükteki yansıma katsayısını, girişteki empedansı ve

gerilimi, ve yükte harcanan gücü hesaplayın. 2- A 0.725λ long lossless transmission line having 0 50Z = Ω , 10 10SZ j= + Ω

40 30LZ j= + Ω is driven by a source with V( ) 100 cos( )SV t tω= . Compute , , , ,L in in in LZ I V PΓ .

a)

b)

c)

d)

0

0

00

0

2

190

3

tan(1.45 )49.10 35.03 60.31 35.5

tan(1.45 )

1.56 22.95 A, 91.72 20.41V 93.97 12.55 V

1 1Re 59.6W

2 2

LL

L

Lin

L

S S inin in

in S in S

inL in in

in

Z Z

Z Z

Z jZZ Z j

Z jZ

V V ZI V j

Z Z Z Z

VP P VI R

Z

π

π

−Γ = = ∠ °

+

+= = − = ∠− °Ω

+

= = ∠ ° = = − = ∠− °+ +

= = = =

5- 0.725λ uzunluğunda kayıpsız bir iletim hattında V( ) 100 cos( )SV t tω= , 0 50Z = Ω ,

10 10SZ j= + Ω 30 40LZ j= + Ω ise yükteki yansıma katsayısını, girişteki empedansı, akım ve gerilimi, ve yükte harcanan gücü hesaplayın. 5- Consider the transmission line circuit below with ( ) 100cos( )SV t tω= , 10 10SZ j= + Ω ,

0 50Z = Ω 30 40LZ j= + Ω . Compute , , , ,L in in in LZ I V PΓ .

a)

b)

c) A

d) V

e) 12

0.5 90

1.45 64.36 51.74

( ) 1.55 39.11

( 0.725 , ) 101.165 cos( 12.63 )

Re 48.26

L

in

in s s in

in

L

l Z

I V Z Z

V t t

P VI W

β π

λ ω

Γ = °

= → = ∠− Ω

= + = ∠

− = − °

= = 2- Aşağıdaki iletim hattı devresinde giden, yansıyan ve 50 ohm’luk hatta iletilen ortalama güçleri hesaplayın. 2- For the circuit shown below, calculate the average incident power, the average reflected power, and the average power transmitted into the infinite 50 ohm line. You may assume that the infinitely long line is slightly lossy.

mW

0

0

250 60 1 150 60 11 11

2 22

01 01

50 , , , (1 ) , 0.25

(1 ) 11 1V 28.01

(1 ) 2(1 ) 6 2 8 1

G

G

Z Zj l G inZ Zin L in L in in in G

G in

in GG in G G Gin in

in g in G in G in

in

V ZZ e V V

Z Z

V VV Z VV P

Z Z Z Z

P P

β −− +−+ +

+

+ +

= Ω Γ = = − Γ =Γ = − = + Γ = Γ = =+

− Γ −Γ = = = → = = = + Γ + −Γ Γ −Γ Γ

= mW mW2 20.232 , (1 ) 27.78in in inP P+ +Γ = = − Γ =

5- Aşağıdaki kayıpsız iletim hattında girişteki ve yükteki gerilimi hesaplayın. 5- Determine the input voltage and load voltage for the following lossless transmission line.

V, m s80 75 , 30km, 100 200 , ( ) 15 cos(8000 ) 2.5 10S L S pZ Z l Z j V t t uπ= = Ω = = + Ω = = ×

4

00

0

8000 1.005 10 rad m 3.0159, 0.628 0.425 0.758 34 ,

tan43.6 136 , ( ) ( ), 0( )

tan

10.9 4.7 V 11.87 23.3 V ( )

L

p

j z j zLin L L

L

j L j LS inin L L L

in g

l ju

Z jZ lZ Z j V z V e e z

Z jZ l

V ZV j V e e V

Z Z

β β

β β

ωω π β β

β

β

+ −

+ − +

= → = = × → = Γ = + = ∠ °

+= = + Ω = + Γ =

+

= = + ° = ∠ ° = + Γ → =+

7.28 1.26V

11.32 5.15 V 12.4 155.6 VL

j

V j

− −

= − − ° = ∠− °