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.
L = CZ 02 = 0.25 μH m
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 Z L = RL + jX L .
(a) Derive ΓL and ∡ΓL in terms of RL , X L and Z 0 .
RL − Z 0 + jX L
RL2 − Z 02 + X L2
2X LZ 0
Z L = RL + jX L → ΓL =
=
+j
2
2
RL + Z 0 + jX L
(RL + Z 0 ) + X L
(RL + Z 0 )2 + X L2
→ ΓL =


(RL − Z 0 )2 + X L2
2X LZ 0
, ∡ΓL = tan−1  2
2
2
2
2

(RL + Z 0 ) + X L
 RL − Z 0 + X L 
(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 ( Z L = − RL + jX L ), show that the
magnitude of the load reflection coefficient is grater than 1. Assume Z0 is real.
( RL + Z 0 )2 + X L2
( RL + Z 0 )2 + X L2 + 2Z 0RL
ZL − Z 0
Z L = − RL + jX L → ΓL =
→ Γ=
=
≥1
ZL + Z 0
( RL + Z 0 )2 + X L2 − 2Z 0RL
(Z 0 − RL )2 + X L2
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
VR = e jπ 3 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 Z L = 50 + j 50 Ω olsun.
a) z = −l noktasındaki toplam empedansın tamamen reel sayı olması için Z A ne olmalıdır?
b) Z A endüktif midir, kapasitif midir?
c) Yukarıdaki Z A ’ yı kullanarak z = −l noktasındaki yansıma katsayısını bulun ( Γ = ? ).
2- For the lossless transmission line circuit shown below Z L = 50 + j 50 Ω .
λ/4
Z 0 = 50 Ω
ZA
Z 0 = 50 Ω
z=0
z = −l
a) What value of Z A is required to make the total impedance at z = −l purely real?
b) Is Z A inductive or capacitive?
c) Using the value of Z A found in part (a) calculate the reflection coefficient Γ at z = −l .
(50)2
50
1+ j
1
−j
=
→ Yin =
, choose Z A so that
=
→ Z A = j 50 Ω
ZL
1+ j
50
ZA
50
b) inductive
a ) Z in =
c) with Z A = j 50 Ω, ZTotal =
1
= 50 Ω. so Γ = 0
1+ j
j
−
50
50
1- A traveling sine wave at 100 MHz is propagating down a dispersionless transmission line
with capacitance per unit length C ′ = 100 pF m and a line dielectric is characterized by
ε r = 2 and µr = 1 .
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 α = 0.1 dB m at 100 MHz, how far will the wave
travel before loosing half of its initial power?
a)up = c
εr µr = 2.12 × 108 m s
b) u p = 1
L ′C ′ = c
εr
⇒ L′ =
εr
= 2.22 × 10−7 H m
c C′
2
c) Z 0 = L ′ C ′ = 47.1 Ω
d ) λ = 2.12 m, β = 2π λ = 2.96 m -1
e ) The distance must be : 3 dB (0.1 dB m) = 30 m
2- A lossless coaxial transmission line terminated in a purely dissipative (i.e. resistive, nonreactive) 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 Z in1 of the line when
Z L = ∞, Z L = 0, and Z L = Z 0 2 .
λ/4
Z0 , β
ZL
Z in1
Z in 1 = Z 02 Z L → Z in 1 = 0, Z in 1 = ∞, and, Z in 1 = 2Z 0
2- Consider the following transmission line circuit. Determine Z in 2 of the line when
Z L = ∞, Z L = 0, and Z L = Z 0 2 .
λ /4
λ/4
Z0 , β
2Z 0
Z0 , β
ZL
Z in 2
Z in 2 = ∞, Z in 2 = Z 0 2, Z in 2 = Z 0
3- Kayıpsız bir iletim hattı için Z 0 = 50 Ω, l = 0.4λ, Z L = 40 + 30 j 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 Z 0 = 50 Ω, l = 0.4λ, Z L = 40 + 30 j Ω .
Z in = Z 0
Z L + jZ 0 tan βl
(40 − j 30) + j 50 tan(2π × 0.4)
= 50
= 25.46 + j 5.91 Ω
Z 0 + jZ L tan βl
50 + j (40 − j 30) tan(2π × 0.4)
j
π
Z − Z0
40 + j 30 − 50
j
e 2
P
2
ΓL = L
=
= =
→ + = 1 − ΓL = 0.889 → P = 88.9%P +
ZL + Z 0
40 + j 30 + 50 3
3
P
9- Karakteristik empedansı Z 0 = 50 Ω olan kayıpsız bir hatta, yükten 0.4λ uzaklıkta ölçülen
gerilim V = 4 + j 2 Volt , akım ise I = −2 A ’dir. Buna göre yük empedansını hesaplayın.
9- On a lossless transmission line with Z 0 = 50 Ω , the voltage at a distance 0.4λ away from
the load is 4 + j 2 Volt . The corresponding current is I = −2 A . Determine Z L .
Z L + jZ 0 tan(βl ) 4 + j 2
V
jZ 0 tan(0.8π) − Z in
= Z in = Z 0
=
=− 2 − j → Z L =
=− 2.97 + j 34.74 Ω
I
Z 0 + jZ L tan(βl )
−2
j (Z in Z 0 ) tan(0.8π) − 1
3- All transmission line segments in below figure are λ 4 in length. Determine the input
impedance seen looking into transmission line #1, #2 and #3 .
2- A transmission line of Z 0 = 50 Ω is to be matched to a load of Z L = 40 + j 10 Ω through
a length L of another tr. line of Z 0′ . Find the required L and Z 0′ 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?
u p,EM Wave
1
1
c
3 × 108
=
=
=
= 2.1 × 103
u p,drift
µ0 ε0 εr u p,drift
εr u p,drift
2 × 10−5
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 inphase 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?
Γ =1
b) What do you know about the position of the voltage maximum closest to the load?
Nothing. (Except of course that its within λ 2 of the load)
c) Given that Vo+ = 5∠00 V, what is the largest instantaneous real voltage value that ever
appears anywhere along the line?
V (z ) = Vo+e − j βz + ΓVo+e j βz ,Vmax = Vo+ + Γ Vo+, Γ = 1 → Vmax = 10V
7- Given Z 0 = 50 Ω , VSWR = 4, f = 500 MHz, the distance between two successive voltage
maximum on the air-filled line is 30 cm, and lmin = 20 cm . Find Γ L and Z L at the load.
7- Karakteristik empedansı
Z 0 = 50 Ω olan bir kayıpsız iletim hattı Z L yükü ile
sonlandırılmıştır. Duran dalga oranı s = 4, frekans f = 500 MHz, iki voltaj maksimum
arasındaki mesafe 30 cm ve lmin = 20 cm olduğunda yük empedansı Z L ’yi hesaplayın.
λ 2
lmin
Z0 , β
ZL
10π
s −1
λ
λ
, Γ =
= 0.6, lmax < , l min = l max + , θΓ − 2βl min = −(2n + 1)π
3
s +1
4
4
π
1+ Γ
jπ 3
→ ZL = Z 0
= 42.1 + j 68.4 = 80 e j 1.02 Ω
n = 0 → θΓ = −π + 2βl min , θΓ = → Γ = 0.6e
3
1− Γ
λ = 0.6 m, β =
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.
(2n + 1)π + φL
, n = 0 ⇒ φL = −π
2β
1−2 3
2
1 + ΓL
= − → ZL = Z 0
= 50
= 10 Ω
3
1 − ΓL
1−2 3
λ 2 = 3 cm → λ = 6 cm, Min at load ⇒ z min = 0 =
s=
Vmax
10
s −1 2
2
=
= 5, ΓL =
= , ΓL = e − j π
Vmin
s +1 3
2
3
1- 50 Ω öz empedansı olan kayıpsız bir iletim hattı Z L 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 s = 3 . The distance
between successive voltage minimum is 20 cm. First voltage minimum is located 5 cm from
the load. Find Z L = ?
2π
s −1
= 5π rad/m, Γ =
= 0.5
λ
s +1
Voltage min. occurs when e j (θ−2 βl ) = −1 → θ − 2βl = −(2n + 1)π
λ = 0.4 m, β =
jθ
θ − 2βl = −π → θ = 2βl − π = −0.5π rad → Γ = Γ e = 0.5e
→ ZL = Z0
n = 0,1,.....
− j 0.5 π
= −j 0.5
1+Γ
1 − j 0.5
= 50
= 30 − j 40 Ω
1− Γ
1 + j 0.5
3- 50 Ω öz empedansı olan kayıpsız bir iletim hattı Z L 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 s = 2 . The successive
voltage minima are 25 cm apart and the first voltage minimum occurs at 5 cm from the load.
Find Z L = ?
 4l

2π
s −1 1
λ
θ 
3
= 4π rad/m, Γ =
= , l min = 1 + Γ  → θΓ = π  min − 1 = − π
λ = 0.5 m, β =




λ
s +1 3
4
π
λ
5
1+Γ
Γ = Γ e j θΓ = 13 e − j 0.6 π = −0.103 − j 0.317 → Z L = Z 0
= 33 − j 24.1 Ω
1−Γ
3- 50 Ω öz empedansı olan kayıpsız bir iletim hattı Z L 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 Z L = ?
 4l
 π
θ 
2π
s −1
λ
λ = 0.4 m, β =
= 5π rad/m, Γ =
= 0.5, l min = 1 + Γ  → θΓ = π  min − 1 =
 2
 λ
λ
s +1
4 
π 
1+ Γ
jπ 2
jθ
Γ = Γ e Γ = 12 e
= 0.5 j → Z L = Z 0
= 30 + j 40 Ω = 50∠53.1° Ω
1− 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.
s=
V max
1.5
s −1 3 −1 1
=
= 3 → ΓL =
=
=
V min
0.5
s −1 3 + 1 2
′
Vmax and I min occur when e j (θ−2 βz ) = 1or θ − 2β z ′ = −2n π, n = 0,1, 2 ⋅ ⋅ ⋅→ z ′ =
θ = 2β z ′ = 2
θ + 2n π
2β
2π
1 + ΓL
(0.2λ) = 0.8π → ΓL = 0.5e j 0.8 π → Z L = Z 0
= 23.43 e j 38.66° = 18.3 + j 14.6 Ω
λ
1 − ΓL
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.
V
λ
12.774
s −1
= 1.3436 − (0.347) = 0.9966 → λ = 2.0 m, s = max =
= 1.768 → ΓL =
= 0.2774
2
V min
7.226
s +1
θ
FirstVmax occurs at z max =− L λ =− 0.8436 → θL = 5.3 rad → ΓL = 0.2779e j 5.3 = 0.1541 − j 0.2313
4π
1 + ΓL
→ ZL = Z0
= 90 − j 45 Ω
1 − ΓL
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.
50
VSWR − 1 2 λ
c
= 5 → ΓL =
= ,
= 25 cm → λ = 1m → f = = 300 MHz
10
VSWR + 1 3 4
λ
2π λ
Vmin → θ − 2βl = −π → l = 0 → θ = −π rad, Vmax → θ = 2βl = 2 ( ) = π rad
λ 4
1
−
2
3
2
2
1+ Γ
Γ = Γ e jθ = e j π = − → Z L = Z 0
= 50
= 10 Ω
3
3
1− Γ
1+2 3
VSWR =
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 değiştirildiğinde 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?
λ
2π
s −1 3 −1
= 20 cm → λ = 0.4 m → β =
= 5π rad/m, s = 3 → ΓL =
=
= 0.5
2
λ
s −1 3 +1
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.
Vmin occurs when e j (θ−2 βl ) = −1 → θL − 2βl = −(2n + 1)π
n = 0,1,.....
θL − 2βl = −π → θL = 2βl − π = 2.5π = 0.5π rad → ΓL = ΓL e j θL = 0.5e j 0.5 π = j 0.5
→ ZL = Z0
1 + ΓL
1 + j 0.5
= 50
= 30 + j 40 Ω = 50∠53.1° Ω
1 − ΓL
1 − j 0.5
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 | V1+ | | V1− | = 2.72 .
+
1
−
1
| V | | V | = 2.72 = e
−2 αl
⇒l =
ln(| V1+ | | V1− |)
2α
=
ln 2.12
= 25 km
2 501
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 f = 100 MHz .
1  Γ 
Γ(l ) = ΓL e −2αl → α = ln  L  , ΓL = −1 for short
2l  Γ(l ) 
Γ(l ) =
s − 1 9.5 − 1
1
| −1 |
=
= 0.810 → α =
ln
= 2.107x10-3 Np/m
s + 1 9.5 + 1
2(50) 0.810
4- If a transmitter can deliver 30 W to a 50 Ω load, how much power is delivered to a load
impedance of Z L = 80 + j 40 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 Z L = 80 + j 40 Ω yüküne bağlandığında yükte harcanan güç nedir?
2
ΓL = 0.367∠36° → PL = P + (1 − ΓL ) = 30 [1 − (0.367)2 ] = 25.96 W
5- A 75 Ω, 25 W transmitter is connected to a load impedance Z L = 40 + j 20 Ω through a
transmission line with Z 0 = 75 Ω, l = 0.3 λ . Find the power delivered to the load.
2
ΓL = 0.345∡140.4° → PL = Pinc (1 − ΓL ) = 25[1 − (0.345)2 ] = 22 W
1- A 50 Ω, 10 W transmitter is connected to a load impedance of Z L = 75 + j 25 Ω through
a transmission line with Z 0 = 50 Ω . Find the power delivered to the load.
2
1) ΓL = 0.231 + j 0.154 = 0.2774∠33.69° → PL = P + (1 − ΓL ) = 10 [1 − (0.277)2 ] = 9.23 W
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.
s − 1 200 − 1
2
=
= 0.99 → PL− PL+ = ΓL = 0.98 (i.e. %98 of incident power is reflected)
s + 1 200 + 1
∴ %2 of power must be dissipated in the load (compared to %0 for an ideal short circuit)
a ) ΓL =
−0.1 10
b ) ΓL = 0.99, Γin = ΓL e −2αl , − 0.1 = 10 log e −2αl → e −2αl = 10
−0.1 10
Γin = ΓL 10
= 0.967464848, sin =
1 + Γin
= 60.47
1 + Γin
5- Aşağıdaki toplam kaybı 3 dB olan bir iletim hattında yükte harcanan güç PL = 1 W .
a) Girişteki yansıma katsayısını bulun Γin = ? b) Girişteki net gücü bulun Pin = ?
−3 10
ΓL = 0.99, − 3 = 10 log e −2αl → e −2αl = 10
2
−3 10
, Γin = ΓL e −2αl = 0.99× 10
2
= 0.496
PL = PL+ (1 − ΓL ) → PL+ = 50.25, Pin+ = PL+e 2αl = 100.26, Pin = Pin+ (1 − Γin ) = 75.6W
2
Pin =
e 2αl PL (1 − Γin )
2
(1 − ΓL )
= 75.6W
6- Aşağıdaki iletim hattı için yükte harcanan gücü hesaplayın.
6- What is the power delivered to the load for the lossless transmission line below?
VS = 15 V (rms), Z S = Z 0 = 75 Ω, Z L = 60 − j 40 Ω, l = 0.7 λ .
2
1) ΓL = 0.303∡ − 94° → PL = Pinc (1 − ΓL ) =
V0+
Z0
2
(1 − ΓL ) =
VG
2
2) Z in = 55.4 ∠ 29.5° → PL = Pin = I in Rin =
2
ZG + Z in
Vg
2
4Z 0
2
(1 − ΓL ) = 0.681 W
2
Re {Z in } = 0.681 W
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.
a) λ =
c
f εr
b) Vin = Vg
= 0.6 m, ΓR =
Z in
Z g + Z in
1 + Γin
150 − 75 1
= → Γin = ΓRe −2 αle −2 j βl = 0.211 → Z in = Z 0
= 115.13 Ω
150 + 75 3
1 − Γin
= 42.81 V → I in =
2
c) PL = Pin+e −2αl (1 − ΓR ) =
Pin
1 − Γin
Vg
Z g + Z in
= 371.87 mA → Pin = 12 Re Vin I in∗  = 7.96 W


2
2
e −2αl (1 − ΓR ) = 4.689 W → Pline = Pin − PL = 3.271 W
7- For a transmission line circuit VS = 20 V (rms), Z S = Z 0 = 100 Ω, f = 500 MHz, l = 4 m
Calculate the input power and power delivered to the load when the power is attenuated by
a) 0.0 dB m and Z L = 150 Ω
b) 0.5 dB m and Z L = 150 Ω .
c) 0.5 dB m and Z L = 100 Ω
2
a) ΓL = 0.2, Pin = PL = P + (1 − ΓL ) = 1 (1 − (0.2)2 ) = 9.6 W
b)
P 
Pfinal
0.5
= e −2 αl ⇒ −0.5 = 10 log  fin  = 10 log(e −2α×1m ) ⇒ α =
= 0.0576 Np/m
Pinitial
20 log10 e
 Pinit 
2
Γin = ΓL e −2αl = 0.2 e −8 α = 0.13 ⇒ Pin = Pin+ (1 − Γin ) = 1× (1 − (0.13)2 ) = 0.984 W
2
PL = Pin+ e −2αl (1 − ΓL ) = 1e −2(0.23)(1 − (0.2)2 ) = 0.606 W
c) ΓL = 0, Pin+ = 1 W = 30 dBm, PL+ = 28 dBm
3- A 50 Ω transmission line is matched to a 10 W source, and feeds a load Z L = 100 Ω . If
the line is 2.3 λ long and has an attenuation constant α = 0.5 dB λ , find the power
delivered by the source, lost in the line and delivered to the load.
ΓL = 0.333, α = 0.5 dB / λ → αl = 1.15 dB → αl = 0.1324 Np → Γin = 0.333 e −2(0.1324) = 0.256
Pin =
PL =
2
V0+
2
2Z 0
V0+
(1 − Γin ) = 10(1 − (0.256)2 ) = 9.34 W
2
2Z 0
2
e −2αl (1 − ΓL ) = 10e −2(0.23)(1 − (0.333)2 ) = 6.85 W → Plost = Pin − PL = 2.49 W
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 l = 30 cm and a
distributed attenuation of 0.2 dB/cm? (Hint: 1 Np = 0.0183 dB )
2
α = 0.2 dB cm = 0.023 Np cm, Γ(0) = 0.05 → Γ(0) = 0.224 → Γ(l ) = Γ(0)e −2 γl
Γ(l ) = 0.224 e
−2(0.023 Np cm) (30 cm)
= 0.056 → RL = −20 log | Γ |= 25 dB
4- Aşağıdaki iletim hattı için f = 200 MHz ve u p = 2 × 108 m s ise V (z = −l 2) = ?
4- Consider the lossless transmission line circuit shown below. It is assumed that f = 200
MHz, u p = 2 ×108 m s . Find the input impedane of the transmission line, the input voltage,
the forward travelling the voltage at load and the phasor voltage at z = −l 2 .
λ=
up
f
= 1m → β =
Vg Z in
2π
= 2π rad m → Z in = 46.1 + j 11.5 Ω → Vin =
= 4.07∠8.37° V
λ
Z g + Z in
ZL − Z 0
= 0.227∡ − 31.4°, V (z ) = V0+ (e − j βz + ΓLe j βz ) → V (−l ) = V0+ (e j βl + ΓLe − j βl ) = Vin
ZL + Z 0
V
− j βl 2
j βl 2
) = 5.68∠54.8° V
→ V0+ = j βl in − j βl = 5∠90° V → V (z = −l 2) = V0+ (e
+ ΓLe
(e + ΓLe )
ΓL =
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.
(50)2
Z
= 22.94 + j 6.88 → Z = 1 = 11.45 + j 3.44 Ω
100 − j 30
2
10
10(11.47 + j 3.44)
I in =
= 0.073 − j 0.035 A,VAB = 1
= 0.96 − j 0.15 V
100 + j 50 + 11.47 + j 3.44
100 + j 50 + 11.47 + j 3.44
Z1 = Z 2 =
Pave =
1
P
Re{VAB I in∗ } = 0.0377 W Each antenna gets half the power → Pant = ave = 0.0188 W
2
2
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. Z L = 72 + j 40 Ω, Z 0 = 300 Ω, Z S = 50 Ω, VS = 10∠0°, f = 100 MHz, l = 1.75 m .
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.
Z L = 72 + j 40 Ω, Z 0 = 300 Ω, Z S = 50 Ω, VS = 10∠0°, f = 100 MHz, l = 1.75 m .
ΓL =
ZL − Z 0
= −0.595 + j 0.171, λ = c f = 3 m, βl = 2πl λ = 1.17 π = 3.67 rad
ZL + Z 0
Z L + jZ 0 tan βl
72 + j 40 + j 300 tan(3.67)
= 300
= 111.88 + j 218.79 Ω
Z 0 + jZ L tan βl
300 + j (72 + j 40) tan(3.67)
VZ
Vg
= 22.4 − j 29.9 mA → Pin = PL = 21 Re [Vin I in∗ ] = 0.077 W
Vin = g in = 8.88 + j 1.5 V, I in =
ZG + Z in
ZG + Z in
Z in = Z 0
3- Consider the transmission line. Find the voltage, current and the forward traveling voltage
at input. Z 0 = 50 Ω, Z L = 100 Ω, l = λ 4, Z in = 75 + j 25 Ω, Z S = 25 Ω, VS = 1 ∡ 0°.
Vin =
Vg Z in
Z in + Z S
= 0.76 + j 0.05 V → I in = VinZ in = 9.7∠ − 14 mA, ΓL = 1 3, βL = π 2
V (z ) = V0+e − j βz + V0−e j βz = V0+ (e − j βz + ΓLe j βz ) =V0+e − j βz (1 + ΓLe j 2 βz )
Vin
λ
jπ 2
jπ 2
Vin = V (z = − ) = V0+e (1 + ΓLe − j π ) = jV0+ (1 − ΓL ) → V0+ =
→ Vin+ = V0+e
+
4
jV0 (1 − ΓL )
5- Consider the transmission line with Z 0 = Z S = 50 Ω, Z L = 75 Ω, l = 0.15λ, VS = 100 ∡ 0° V.
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?
2π × 0.15λ
Z + jZ 0 tan βl
= 0.942 rad, ΓL = 0.2, Z in = Z 0 L
= 41.25 − j 16.35 Ω
λ
Z 0 + jZ L tan βl
Vg
I in =
= 1.08∠10.1° A → Vin = I in Z in = 47.86∠ − 11.46° V → Pin = 12 Re [Vin I in∗ ] = 24 W
ZG + Z in
a) βl =
b) V (l ) = VL+e j βl (1 + ΓLe − j 2 βl ) → Vin = VL+e j βL (1 + ΓLe − j 2 βL ) → VL+ = 50e − j 54° V, VL = 60e − j 54° V
I (l ) =
VL+ j βl
e (1 − ΓLe − j 2 βl ) → I L = 0.8e − j 54° A → PL = 12 Re [VLI L∗ ] = 24 W → PL = Pin (lossless)
Z0
1- Aşağıda verilen iletim hattı devresi için giriş empedansını, girişteki gerilim ve akımı, ve
yükteki ortalama gücü hesaplayın. Z L = 25 Ω, Z 0 = 50 Ω, Z S = 75 Ω, VS = 10∠0°, l = λ
1- A transmission line circuit below has the following parameters:
Z L = 25 Ω, Z 0 = 50 Ω, Z S = 75 Ω, VS = 10∠0°, l = λ
Z in = 25 Ω, Vin =
Vg Z in
Z in + Z S
= 2.5 V, I in =
Vin
= 0.1 A, Pin = PL = 21 Re [Vin I in∗ ] = 0.125 W
Z in
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?
2
Z 02
75 =
→ Z 02 = 150 Ω
300
1- Şekildeki devre için Z 0 = 50 Ω olsun. a) VS = 1V ve Z S = Z 0 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 Z 0 = 50 Ω . a) Determine the
voltage across the load when VS = 1V and Z S = Z 0 . 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.
+ j βl
0
a) V (l ) =V e (1 + ΓLe
− j 2 βl
VS Z in
Z 02
), Vin =
, Z in =
, ΓL = −0.15 − j 0.76
Z S + Z in
ZL
VS Z 02
jπ 2
= V (l = λ4 ) = V0+e (1 + ΓLe − j π ) = jV0+ (1 − ΓL )
Vin = 2
Z 0 + ZS Z L
−jVS
Z 02
V = 2
= −j 0.5 V → VL =V0+ (1 + ΓL ) = −0.38 − j 0.425 = 0.572∡ − 132° V
Z 0 + Z S Z L 1 − ΓL
+
0
b) Zs = Z in∗ = Z 02 Z L∗ = 14.7 − j 58.8 Ω. There are reflections but they add up properly to achieve
maximum power transfer
2- Find the load reflection coefficient, the input impedance and power delivered to the load.
Vg = 2 V (peak), f = 30 GHz, Z g = 50 Ω, Z 0 = 50 Ω, R = 100 Ω, l = 1 cm, u p = 3 × 108 m s
Γ(0) =
V
R − Z0
1
= , l = 0.01, λ = p = 0.01 → βl = 2π ⇒ tan 2π = 0
R + Z0
3
f
2
V
Z + jZ 0 tan βl
2Z in
4
16
Z in = Z 0 L
= Z L = 100 Ω → Vin =
= V → PL = in =
= 0.0089 W
Z 0 + jZ L tan βl
Z in + Z g 3
2Rin 1800
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.
+
L
a) ΓL = 0.57∡ − 80°, P =
+
0
b) V
+
L
= 2V
VL+
2
2Z 0
−
0
= 10 2 V, V
2
2
= 1 W, PL− = ΓL PL+ = 0.325 W → PL = PL+ (1 − ΓL ) = 0.675 W
VL−
5.7
Γ
=
=
V, Γin = L = 0.285∡ − 80°
2
2
2
2
c) P0+ = 2PL+ = 2 W → Pin = P0+ (1 − Γin ) = 1.83755 W → Pline = Pin − PL = 1.16245 W
7- Aşağıdaki iletim hattı için Z S = Z 0 = 50 Ω, l = 6 λ, α = 0.5 dB / λ, Z L = 30 − j 50 Ω .
Yükte harcanan güç PL = 0.675 W ise hattın girişindeki net gücü hesaplayın.
7- For the given transmission line Z S = Z 0 = 50 Ω, l = 6 λ, α = 0.5 dB / λ, Z L = 30 − j 50 Ω .
The power delivered to load is PL = 0.675 W, Calculate the input power.
ΓL = 0.57∡ − 80°, α = 0.5 dB/λ → αl = 3 dB → αl = 0.35 Np
2
Γin = ΓL e −2αl = 0.57 e −2(0.35) = 0.285 → PL = PL+ (1 − ΓL ) = 0.675 W → PL+ = 1 W
2
Pin+ = PL+e 2 αl = 2 W → Pin = Pin+ (1 − Γin ) = 1.84 W
7- Aşağıdaki iletim hattı için Z S = Z 0 = 50 Ω, l = 6 λ, α = 0.5 dB / λ, Z L = 30 − j 50 Ω .
Voltaj kaynağının gücü 2 W ise yükte harcanan gücü ve hatta harcanan gücü bulun.
7- For the given transmission line Z S = Z 0 = 50 Ω, l = 6 λ, α = 0.5 dB / λ, Z L = 30 − j 50 Ω .
Find the power lost in the line and at the load if the line is connected to a 2 W source.
ΓL = 0.57∡ − 80°, α = 0.5 dB / λ → αl = 3 dB → αl = 0.35 Np
2
Γin = ΓL e −2 αl = 0.57 e −2(0.35) = 0.285, Pin+ = 2W → Pin = Pin+ (1 − Γin ) = 1.84 W
2
PL+ = 1 W → PL = PL+ (1 − ΓL ) = 0.675 W → Pline = Pin+ − PL+ = 2 − 1 = 1 W
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.
f = 1.5 GHz → λ =
Z mid
c
λ
π
= 0.2 m, 2.5 cm = → βl = ⇒ tan βl = 1
f
8
4
Z L + j 30
λ
(150)2
= 30
= j 150 Ω ⇒ Z L = j 20 Ω, 5 cm = @1.5 GHz, Z in =
= −j 150Ω
30 + jZ L
4
j 150
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.
2
VG
max power ⇒ Z in = Z = 50 + j 50 Ω → since Z R = Z in → PL = Pin =
= 0.25 W
8 Re {ZG }
∗
G
4 π 3λ 

−j
Vin
1 + ΓRe λ 2  = −V0+ (1 + ΓR ) → V0+ = −


(1 + ΓR )


V Z
→ VR = V0+ (1 + ΓR ) = −Vin = − G in = −(5 + j 5) V
ZG + Z in
Vin = V (l = 1.5λ) = V0+e
j
2 π 3λ
λ 2
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.
2
VG
Z2
max power ⇒ Z in = Z = 50 + j 50 Ω → Z R = 0 = 25 − j 25 Ω → PL = Pin =
= 0.25 W
Z in
8 Re {ZG }
∗
G
2π λ 

2j
jVin

Vin = V (z = − λ 4) = V e
1 + ΓRe λ 4  = jV0+ (1 − ΓR ) → V0+ = −


(1 − ΓR )


(1 + ΓR )
Z
→ VR = V0+ (1 + ΓR ) = −jVin
= −jVin L = −j 5 V
(1 − ΓR )
Z0
+
0
j
2π λ
λ 4
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.
Z L + jZ 0 tan βl
2π λ
π
, Z L = 0 → Z insc = jZ 0 tan βl, Z L = ∞ → Z inoc = − jZ 0 cot βl, βl =
=
Z 0 + jZ L tan βl
λ 8
4
Z − 50
j −2
= j 50, Z inoc = j 50 → Zeq = j 25 → Z in = j 25 → ΓL = eq
=
→ ΓL = 1 → PL = 0
Zeq + 50
j +2
Z in = Z 0
Z insc
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.
Z 0 2Z 0
E1Z 0
E
+
= Z 0 → ΓL = 0 → Z in = Z 0 → V1 =
= 1
3
3
Z 0 + 2Z 0
3
E
E
E
V (l ) = V +e j βl (1 + ΓLe −2 j βl ) = V +e j βl = 1 → V + = 1 e − j βl → V2 = V (l = 0) = 1 e − j βl
3
3
3
2Z 0
V
V0 = 2Z 0 2 3 Z 0 = 23 V2 → V0 = 29 E1e − j βl
3 + 3
ZL =
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 2 →
λ
2
= 50 cm → λ = 1 m, s = 2 → ΓL =
s −1 1
2π
= , θ = 2βz max = 2 z max = 0.8π rad
s −1 3
λ
1
1 + ΓL
ΓL = e j 0.8 π → Z L = Z 0
= 26.93 + j 11.87 Ω, tan βl2 = tan(1.4π) → Z in 1 = 100 Ω
3
1 − ΓL
50−70
1
line 1 → Z L = 100 100 = 50 Ω, ΓL = 50
+70 = − 6 , l1 = 3.75 m = 3.75λ → Z in 2 =
V (l ) = V +e j βl (1 + ΓLe −2 j βl ) → Vin = 10 9898
+70 =
V (l = 0) = V + (1 + ΓL ) = j 5(1 − 61 ) = j
25
6
35
6
2
Z 01
(70)2
= 98 Ω
=
ZL
50
= V +e j βl1 (1 − 16 e −2 j βl1 ) = −jV + (1 + 16 ) → V + = 5 j
V
3- A l = 3.2 m long transmission line has Z 0 = 75 + j 0.94 Ω and γ = 0.224 + j 17.614 m−1 . It
is driven by a generator, operating at f = 105.2 MHz , with VS = 16∠0° V and Z S = 75 Ω .
Using a network analyzer, the input impedance is measured to be Z in = 88.6 + j 20.3 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.
Z in − Z 0
VG Z in
= 0.14341∠47.5°, Vin =
= 8.82∠5.8° V
Z in + Z 0
Z in + ZG
Vin
b) Vin = Vin+ (1 + Γin ) ⇒ Vin+ =
= 8.01∠0.3° V
(1 + Γin )
1 + ΓL
c) Γin = ΓLe −2 γl ⇒ ΓL = Γine 2 γl = 0.6∠26.4°, Z L = Z 0
= 166.36 + j 143.22 Ω
1 − ΓL
a) Γin =
d) VL = Vin+e −γl (1 + ΓL ) = 5.71 + j 2.155 = 6.10∠20.7° V
2- Aşağıdaki kayıpsız iletim hattında faz hızı Vp = 2 ×108 m s ise dalga boyunu, yükteki
yansıma katsayısını, giriş empedansını, girişteki gerilimi ve yükte harcanan gücü bulun.
Z S = Z 0 = 150 Ω, l = 6 m, Z L = 150 − j 50 Ω,VS (t ) = 5 cos(8π × 107 t ) Volt
2- A l = 6 m long lossless TEM transmission line having Z 0 = 150 Ω is driven by a source
with Vg (t ) = 5 cos(8π × 107 t ) Volt and Z g = Z 0 . If the line has a relative permittivity of 2.25
(corresponding to a phase velocity of Vp = 2 × 108 m s ) and is terminated in a load
Z L = 150 − j 50 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) ω = 8π × 107 → f = 40 MHz, β =
b) ΓL =
ω
2π
= 0.4π rad m → λ =
= 5m
Vp
β
ZL − Z0
= 0.027 − j 0.162 = 0.164∠ − 80.5°
ZL + Z 0
Z L + jZ 0 tan βl
150 − j 50 + j (150)(3.078)
= 150
= 116 + j 27 = 119∠13.3° Ω
Z 0 + jZ L tan βl
150 + j (150 − j 50)(3.078)
Vg Z in
5(119∠13°)
= 2.23∠8.5° V
d)Vin =
=
Z in + Z g
266 + j 27
c) Z in = Z 0
2
2
Vg
1 Vin
e) Pav is when Z in = Z g = Z 0, i.e. when Vin = Vg 2 → Pav =
Rin =
= 20.8 mW
2 Z in
8Z 0
2
1 Vin
2
f) PL = Pin → PL =
Rin = 20.3 mW check PL = (1 − ΓL )Pav = 20.3 mW
2 Z in
2- 0.725λ uzunluğunda kayıpsız bir iletim hattında VS (t ) = 100 cos(ωt ) V , Z 0 = 50 Ω ,
Z S = 10 + j 10 Ω Z L = 40 + j 30 Ω ise yükteki yansıma katsayısını, girişteki empedansı ve
gerilimi, ve yükte harcanan gücü hesaplayın.
2- A 0.725λ long lossless transmission line having Z 0 = 50 Ω , Z S = 10 + j 10 Ω
Z L = 40 + j 30 Ω is driven by a source with VS (t ) = 100 cos(ωt ) V . Compute
ΓL , Z in , I in , Vin , PL .
a) ΓL =
ZL − Z0
1
= ∠90°
ZL + Z0
3
Z L + jZ 0 tan(1.45π)
= 49.10 − j 35.03 = 60.31∠ − 35.5° Ω
Z 0 + jZ L tan(1.45π)
VS
VS Z in
= 1.56∠22.95° A, Vin =
= 91.72 − j 20.41 V = 93.97∠ − 12.55° V
c) I in =
Z in + Z S
Z in + Z S
b) Z in = Z 0
2
1
1 Vin
Rin = 59.6 W
d) PL = Pin = Re {VI ∗ } =
2
2 Z in
5- 0.725λ uzunluğunda kayıpsız bir iletim hattında VS (t ) = 100 cos(ωt ) V , Z 0 = 50 Ω ,
Z S = 10 + j 10 Ω Z L = 30 + j 40 Ω ise yükteki yansıma katsayısını, girişteki empedansı, akım
ve gerilimi, ve yükte harcanan gücü hesaplayın.
5- Consider the transmission line circuit below with VS (t ) = 100 cos(ωt ) , Z S = 10 + j 10 Ω ,
Z 0 = 50 Ω Z L = 30 + j 40 Ω . Compute ΓL , Z in , I in , Vin , PL .
a) ΓL = 0.5∡90°
b) βl = 1.45π → Z in = 64.36∠ − 51.74 Ω
c) I in = Vs (Z s + Z in ) = 1.55∠39.11 A
d)Vin (−0.725λ, t ) = 101.165 cos(ωt − 12.63°) V
e) PL = 12 Re {VI ∗ } = 48.26W
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.
−2 j βl
50−60
1
Z in = 50 Ω, ΓL = 50
= − 111 , Vin = Vin+ (1 + Γin ) =
+60 = − 11 , Γin = ΓLe
+
in
V
VG Z in
, ΓG =
ZG + Z in
2
2
 Z
 V (1 − Γ )
Vin+
VG
VG
11
1 − ΓG
+

in
G
G
 =

=
= V → Pin =
=
(1 + Γin )  Z g + Z in  2(1 − ΓG Γin )
6
2Z 01
8Z 01 1 − ΓG Γin
2
ZG −Z 0
ZG +Z 0
= 0.25
2
= 28.01 mW
2
Pin− = Pin+ Γ = 0.232 mW, Pin = Pin+ (1 − Γ ) = 27.78 mW
5- Aşağıdaki kayıpsız iletim hattında girişteki ve yükteki gerilimi hesaplayın.
5- Determine the input voltage and load voltage for the following lossless transmission line.
Z S = Z 0 = 75 Ω, l = 30 km, Z L = 100 + j 200 Ω,VS (t ) = 15 cos(8000πt ) V, u p = 2.5 × 108 m s
ω = 8000π → β =
ω
= 1.005 × 10−4 rad m → βl = 3.0159, ΓL = 0.628 + j 0.425 = 0.758∠34°,
up
Z L + jZ 0 tan βl
= 43.6 + j 136 Ω, V (z ) = VL+ (e − j βz + ΓLe j βz ), z = 0(load)
Z 0 + jZ L tan βl
VS Z in
Vin =
= 10.9 + j 4.7° V = 11.87∠23.3° V = VL+ (e j βL + ΓLe − j βL ) → VL+ = −7.28 − j 1.26 V
Z in + Z g
Z in = Z 0
VL = −11.32 − j 5.15° V = 12.4∠ − 155.6° V