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EM Waves and Guide Structures
ECEg 4291
Chapter 3 Part 1
Transmission Lines
Transmission Lines
Introduction
 So far we have seen wave propagation in unbounded media,
media of infinite extent.
 Such wave propagation is said to be unguided in that the
uniform plane wave exists throughout all space
 And EM energy associated with the wave spreads over a wide
area
 Wave propagation in unbounded media is used in radio or TV
broadcasting, where the information being transmitted is
meant for everyone who may be interested.
 Such means of wave propagation will not help in a situation
like telephone conversation, where the information is received
privately by one person.
Transmission Lines
Introduction
 Another means of transmitting power or information is by
guided structures.
 Guided structures serve to guide (or direct) the propagation
of energy from the source to the load.
 Typical examples of such structures are transmission lines
and waveguides.
 Transmission lines are commonly used in power
distribution at lower frequency and in communications at
higher frequency
 A transmission line basically consists of two or more
parallel conductors used to connect a source to a load.
Transmission Lines
Introduction
 The source may be a hydroelectric generator, a transmitter, or
an oscillator; the load may be a factory, an antenna, or an
oscilloscope, respectively
 Examples of transmission lines are shown in next slide
 Coaxial cables are routinely used in electrical laboratories and
in connecting TV sets to TV antennas.
 Micro strip lines are particularly important in integrated
circuits where metallic strips connecting electronic elements
are deposited on dielectric substrates.
Transmission Lines
Introduction
 Examples: (a) coaxial cable (b) two-wire line
(c) planar lines (d) wire above conducting plane
(e) microstrip
Transmission Lines
Transmission Line Parameters
Line Parameters:
1. R – resistance per unit length (series)
- opposition to current flow
2. L – inductance per unit length (series)
- self inductance
3. C – capacitance per unit length (shunt)
- two conductors separated by an insulator
4. G – conductance per unit length (shunt)
- due to dielectric medium separating the conductors
Transmission Lines
Transmission Line Parameters
Line Parameters:
Distributed parameters of a two-conductor transmission line
Transmission Lines
Transmission Line Parameters
Formulas for calculating the values of R, L, C, and G:
Transmission Lines
Transmission Line Parameters
Dimensions: (a) coaxial line (b) two-wire line (c) planar line
Transmission Lines
Transmission Line Parameters
Notes:
1. The line parameters are not discrete or lumped but
distributed. By this we mean that the parameters are
uniformly distributed along the entire length of the line.
2. For each line, the conductors are characterized by σc, μc,
εc=εo and the homogenous dielectric separating the
conductors is characterized by σ, μ, ε.
3. G ≠ 1/R. R is the ac resistance per unit length of the
conductors comprising the line and G is the conductance
per unit length due to the dielectric medium separating the
conductors.
Transmission Lines
Transmission Line Parameters
Notes:
4. The value of L shown in the table is the external
inductance per unit length; that is L = Lext . The effect of
internal inductance Lin (= R/ω) are negligible as high
frequencies at which most communication system operate.
Self inductance (internal) – inductance measured with
the current flowing in the conductor
Mutual inductance (external) – flux linkage due to
nearby current carrying conductor to the conductor current
5. For each line,
LC = με and G/C = σ/ε
Transmission Lines
Transmission Line Parameters
 Let us consider how an EM wave propagates through a two-
conductor transmission line.
 For example, consider the coaxial line connecting the
generator or source to the load as in Figure in the next slide.
 When switch S is closed, the inner conductor is made positive
with respect to the outer one so that the E field is radially
outward
 According to Ampere's law, the H field encircles the current
carrying conductor
 The Poynting vector (E X H) points along the transmission
line.
Transmission Lines
Transmission Line Parameters
 Thus, closing the switch simply establishes a disturbance,
which appears as a transverse electromagnetic (TEM) wave
propagating along the line.
 This wave is a non-uniform plane wave and by means of it
power is transmitted through the line.
Transmission Lines
Transmission Line Equations
 As mentioned a two-conductor transmission line supports a
TEM wave
 An important property of TEM waves is that the fields E
and H are uniquely related to voltage V and current I,
respectively:
 Let us examine an incremental portion of length Δz of a
two-conductor transmission line.
 We intend to find an equivalent circuit for this line and
derive the line equations.
Transmission Lines
Transmission Line Equations
 From Figure in slide 7, we expect the equivalent circuit of
a portion of the line to be as in Figure in next slide
 The model in Figure in the next slide is in terms of the line
parameters R, L, G, and C, and may represent any of the
two-conductor lines of Figure in slide 7
 The model is called the L-type equivalent circuit
 In this model, we assume that the wave propagates along
the +z-direction, from the generator to the load.
Transmission Lines
Transmission Line Equations
L-type equivalent circuit model of a differential length Δz of a two-
conductor transmission line.
Transmission Lines
Transmission Line Equations
 By applying Kirchhoff's voltage law to the outer loop of
the circuit in Figure in previous slide, we obtain
 Taking the limit of eq. (11.3) as Δz -> 0 leads to
 Similarly, applying Kirchoff's current law to the main node
of the circuit in Figure in previous gives
Transmission Lines
Transmission Line Equations
As Δz —> 0, eq. (11.5) becomes
 If we assume harmonic time dependence so that
Transmission Lines
Transmission Line Equations
 where Vs(z) and Is(z) are the phasor forms of V(z, t) and
I(z, t), respectively, eqs. (11.4) and (11.6) become
Transmission Lines
Transmission Line Equations
 In the differential eqs. (11.8) and (11.9), Vs and Is are
coupled.
 To separate them, we take the second derivative of Vs in eq.
(11.8) and employ eq. (11.9) so that we obtain
 Where
Transmission Lines
Transmission Line Equations
 By taking the second derivative of Is in eq. (11.9) and
employing eq. (11.8), we get
 We notice that eqs. (11.10) and (11.12) are, respectively,
the wave equations for voltage and current
 The wavelength and wave velocity u are, respectively,
given by



f
u 




2

Transmission Lines
Transmission Line Equations
 The solutions of the linear homogeneous differential
equations (11.10) and (11.12) are
 Thus, we obtain the instantaneous expression for voltage
as
Transmission Lines
Transmission Line Equations
 The characteristic impedance Zo of the line is the ratio of
positively traveling voltage wave to current wave at any
point on the line.
 By substituting eqs. (11.15) and (11.16) into eqs. (11.8)
and (11.9) and equating coefficients of terms and ,
we obtain
Transmission Lines
Transmission Line Equations
 Ro should not be mistaken for R— while R is in ohms per
meter; Ro is in ohms.
 The propagation constant and the characteristic impedance Zo
are important properties of the line because they both depend
on the line parameters R, L, G, and C and the frequency of
operation.
 The reciprocal of Zo is the characteristic admittance Yo, that
is, Yo = 1/Zo.
 The transmission line considered so far is the lossy type in
that the conductors comprising the line are imperfect and the
dielectric in which the conductors are embedded is lossy
Transmission Lines
Transmission Line Equations
 We may now consider two special cases of lossless
transmission line and distortionless line.
Case for Lossless Transmission Line
 A transmission line is said to be lossless if the conductors
of the line are perfect (σc ≈ ∞) and the dielectric
medium separating them is lossless (σ ≈ 0).
 For such a line, it is evident that when:
σc ≈ ∞ and σ ≈ 0,
This is necessary for a line to be lossless.
G
R 
 0
Transmission Lines
Transmission Line Equations
Case for Lossless Transmission Line
Thus, for such a line,
,
0

 LC
j
j 

 




f
LC
u 


1
,
0

O
X
C
L
R
Z O
O 

Transmission Lines
Transmission Line Equations
Case for Distortionless Transmission Line
 A signal consists of band of frequencies; wave amplitudes
of different frequency components will be attenuated
differently in a lossy line as α is frequency dependent.
This results in distortion.
 A distortionless line is one in which the attenuation
constant α is frequency independent while the phase
constant β is linearly dependent on frequency.
 From the general expression for α and β, a distortion line
results if the line parameters are such that
C
G
L
R

Transmission Lines
Transmission Line Equations
Case for Distortionless Transmission Line
Thus, for a distortionless line,
or
This shows that α does not depend on frequency
whereas β is a linear function of frequency















G
C
j
R
L
j
RG


 1
1



j
G
C
j
RG 








 1
,
RG

 LC

 
Transmission Lines
Transmission Line Equations
Case for Distortionless Transmission Line
Also,
or
and
 
  O
O
O jX
R
C
L
G
R
G
C
j
G
R
L
j
R
Z 








1
1
,
C
L
G
R
RO 
 0

O
X



f
LC
u 


1
Transmission Lines
Transmission Line Equations
Summary
Transmission Lines
Transmission Line Equations
Note:
1. The phase velocity is independent of frequency because
the phase constant β linearly depends on frequency. We
have shape distortion of signals unless α and u are
independent of frequency.
2. u and ZO remain the same as for the lossless lines.
3. A lossless line is also distortionless line, but a
distortionless line is not necessarily lossless. Although
lossless lines are desirable in power transmission,
telephone lines are required to be distortionless.
Transmission Lines
Transmission Line Equations
Example 1:
An air line has characteristic impedance of 70 Ω and phase
constant of 3 rad/m at 100 MHz. Calculate the inductance per
meter and the capacitance per meter of the line. Note: air line
can be regarded as a lossless line.
Transmission Lines
Transmission Line Equations
Dividing eq. (11.1.1) by eq. (11.1.2) yields
Transmission Lines
Transmission Line Equations
From eq. (11.1.1),
Example 2: A distortionless line has Zo = 60 Ω, α = 20 mNp/m,
u = 0.6c (c = speed of light). Find R, L, G, C and λ at 100 MHz.
Hence,
Transmission Lines
 Transmission Line Equations
But
From eq. (11.2.2b),
Dividing eq. (11.2.1) by eq. (11.2.3) results in
Transmission Lines
Transmission Line Equations
From eq. (11.2.2a),
Multiplying eqs. (11.2.1) and (11.2.3) together gives
Transmission Lines
Input impedance, SWR and power
 Consider a transmission line of
length l, characterized by γ and
Zo, connected to a load ZL as
shown below.
 Looking into the line, the
generator sees the line with the
load as input impedance Zin.
Transmission Lines
Input impedance, SWR and power
 Let the transmission line extend from z = 0 at the generator
to z = l at the load.
 First of all, we need the voltage and current waves, that is
 To find V0
+ and V0
- , the terminal conditions must be given.
For example, if we are given the conditions at the input, say
Transmission Lines
Input impedance, SWR and power
 Substituting these into the wave and current waves
equations results in
 If the input impedance at the input terminals is Zin, the input
voltage V0 and the input current I0 are easily obtained from
the figure.
Transmission Lines
Input impedance, SWR and power
 On the other hand, if we are given the conditions at the load,
say
 Substituting these into the wave and current wave’s
equations gives
Transmission Lines
Input impedance, SWR and power
 Next, we determine the input impedance Zin = Vs(z)/Is(z) at
any point on the line.
 At the generator, for example, voltage and current wave’s
equations yield
 Substituting the equations of Vo
+ and Vo
- into Zin and
utilizing some useful identities and properties
(Lossy Line)
Transmission Lines
Input impedance, SWR and power
Useful Identities:
Transmission Lines
Input impedance, SWR and power
 For a lossless line, γ = jβ, tanh j βl = jtan βl , and Zo = Ro,
so Zin becomes
(Lossless line)
 Showing that the input impedance varies periodically with
distance from the load.
 The quantity βl is usually referred to as the electrical length
of the line and can be expressed in degrees or radians.
Transmission Lines

Transmission Lines

Transmission Lines
Input impedance, SWR and power
 The standing wave ratio s (otherwise denoted by SWR) as
 As mentioned at the beginning of this chapter, a transmission
is used in transferring power from the source to the load
 The average input power at a distance l from the load is given
by
 Since the last two terms are purely imaginary, we have
Transmission Lines
Input impedance, SWR and power
 Since the last two terms are purely imaginary, we have
 The first term is the incident power Pi while the second term is
the reflected power Pr. Thus eq. above may be written as
Pt = Pi – Pr
 where Pt is the input or transmitted power and the negative
sign is due to the negative going wave since we take the
reference direction as that of the voltage/current traveling
toward the right.
Transmission Lines

Transmission Lines
Input impedance, SWR and power
We now consider special cases when the line is connected to
the load , ZL = 0, ZL = ∞, ZL = Zo
A. Shorted Line (ZL = 0)
From the equation of input impedance of lossless line, at ZL
= 0, it becomes:
Also,
Г = -1, SWR = ∞
Transmission Lines
Input impedance, SWR and power
A. Shorted Line (ZL = 0) .....cont...
We notice that Zin is a pure reactance, which could be
capacitive or inductive depending on the value of l.
B. Open-Circuited Line (ZL = ∞)
In this case, Zin becomes
Also,
Г = 1, SWR = ∞
Transmission Lines
Input impedance, SWR and power
C. Matched Line (ZL = Zo)
 This is the most desired case from the practical point of
view. For this case, equation of Zin reduces to
Also,
Г = 0, SWR = 1
 The incident power is fully absorbed by the load.
 Thus, maximum power transfer is possible when a
transmission line is matched to the load.
Transmission Lines
Input impedance, SWR and power
Input Impedance of a lossless line when Shorted (ZL = 0)
Transmission Lines
Input impedance, SWR and power
Input Impedance of a lossless line when Shorted (ZL = ∞)
Transmission Lines
Input impedance, SWR and power
Example 1: A certain transmission line operating at ω = 106
rad/s has α = 8 dB/m, β = 1 rad/m, and ZO = 60 + j40 Ω and
is 2 m long. If the line is connected to a source of 10 /0o V,
Zg = 40 Ω and terminated by a load of 20 + j50 Ω,
determine: (a) Zin (b) the sending-end current (c) the
current at the middle of the line
Solution
Transmission Lines
Transmission Lines
Transmission Lines
Example 2:
Transmission Lines
Solution
Transmission Lines
Transmission Lines
Example 3: A 50-Ω coaxial cable feeds a 75 + j20-Ω dipole
antenna. Find Γ and s.
Solution
Transmission Lines
Example 4 :
Transmission Lines
Transmission Lines
Transmission Lines
Example 5: Refer to the lossless transmission line shown in
Figure in example 4 slide. (a) Find Γ and s. (b) Determine Zin at
the generator.

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chapter 3 part 1.ppt

  • 1. EM Waves and Guide Structures ECEg 4291 Chapter 3 Part 1 Transmission Lines
  • 2. Transmission Lines Introduction  So far we have seen wave propagation in unbounded media, media of infinite extent.  Such wave propagation is said to be unguided in that the uniform plane wave exists throughout all space  And EM energy associated with the wave spreads over a wide area  Wave propagation in unbounded media is used in radio or TV broadcasting, where the information being transmitted is meant for everyone who may be interested.  Such means of wave propagation will not help in a situation like telephone conversation, where the information is received privately by one person.
  • 3. Transmission Lines Introduction  Another means of transmitting power or information is by guided structures.  Guided structures serve to guide (or direct) the propagation of energy from the source to the load.  Typical examples of such structures are transmission lines and waveguides.  Transmission lines are commonly used in power distribution at lower frequency and in communications at higher frequency  A transmission line basically consists of two or more parallel conductors used to connect a source to a load.
  • 4. Transmission Lines Introduction  The source may be a hydroelectric generator, a transmitter, or an oscillator; the load may be a factory, an antenna, or an oscilloscope, respectively  Examples of transmission lines are shown in next slide  Coaxial cables are routinely used in electrical laboratories and in connecting TV sets to TV antennas.  Micro strip lines are particularly important in integrated circuits where metallic strips connecting electronic elements are deposited on dielectric substrates.
  • 5. Transmission Lines Introduction  Examples: (a) coaxial cable (b) two-wire line (c) planar lines (d) wire above conducting plane (e) microstrip
  • 6. Transmission Lines Transmission Line Parameters Line Parameters: 1. R – resistance per unit length (series) - opposition to current flow 2. L – inductance per unit length (series) - self inductance 3. C – capacitance per unit length (shunt) - two conductors separated by an insulator 4. G – conductance per unit length (shunt) - due to dielectric medium separating the conductors
  • 7. Transmission Lines Transmission Line Parameters Line Parameters: Distributed parameters of a two-conductor transmission line
  • 8. Transmission Lines Transmission Line Parameters Formulas for calculating the values of R, L, C, and G:
  • 9. Transmission Lines Transmission Line Parameters Dimensions: (a) coaxial line (b) two-wire line (c) planar line
  • 10. Transmission Lines Transmission Line Parameters Notes: 1. The line parameters are not discrete or lumped but distributed. By this we mean that the parameters are uniformly distributed along the entire length of the line. 2. For each line, the conductors are characterized by σc, μc, εc=εo and the homogenous dielectric separating the conductors is characterized by σ, μ, ε. 3. G ≠ 1/R. R is the ac resistance per unit length of the conductors comprising the line and G is the conductance per unit length due to the dielectric medium separating the conductors.
  • 11. Transmission Lines Transmission Line Parameters Notes: 4. The value of L shown in the table is the external inductance per unit length; that is L = Lext . The effect of internal inductance Lin (= R/ω) are negligible as high frequencies at which most communication system operate. Self inductance (internal) – inductance measured with the current flowing in the conductor Mutual inductance (external) – flux linkage due to nearby current carrying conductor to the conductor current 5. For each line, LC = με and G/C = σ/ε
  • 12. Transmission Lines Transmission Line Parameters  Let us consider how an EM wave propagates through a two- conductor transmission line.  For example, consider the coaxial line connecting the generator or source to the load as in Figure in the next slide.  When switch S is closed, the inner conductor is made positive with respect to the outer one so that the E field is radially outward  According to Ampere's law, the H field encircles the current carrying conductor  The Poynting vector (E X H) points along the transmission line.
  • 13. Transmission Lines Transmission Line Parameters  Thus, closing the switch simply establishes a disturbance, which appears as a transverse electromagnetic (TEM) wave propagating along the line.  This wave is a non-uniform plane wave and by means of it power is transmitted through the line.
  • 14. Transmission Lines Transmission Line Equations  As mentioned a two-conductor transmission line supports a TEM wave  An important property of TEM waves is that the fields E and H are uniquely related to voltage V and current I, respectively:  Let us examine an incremental portion of length Δz of a two-conductor transmission line.  We intend to find an equivalent circuit for this line and derive the line equations.
  • 15. Transmission Lines Transmission Line Equations  From Figure in slide 7, we expect the equivalent circuit of a portion of the line to be as in Figure in next slide  The model in Figure in the next slide is in terms of the line parameters R, L, G, and C, and may represent any of the two-conductor lines of Figure in slide 7  The model is called the L-type equivalent circuit  In this model, we assume that the wave propagates along the +z-direction, from the generator to the load.
  • 16. Transmission Lines Transmission Line Equations L-type equivalent circuit model of a differential length Δz of a two- conductor transmission line.
  • 17. Transmission Lines Transmission Line Equations  By applying Kirchhoff's voltage law to the outer loop of the circuit in Figure in previous slide, we obtain  Taking the limit of eq. (11.3) as Δz -> 0 leads to  Similarly, applying Kirchoff's current law to the main node of the circuit in Figure in previous gives
  • 18. Transmission Lines Transmission Line Equations As Δz —> 0, eq. (11.5) becomes  If we assume harmonic time dependence so that
  • 19. Transmission Lines Transmission Line Equations  where Vs(z) and Is(z) are the phasor forms of V(z, t) and I(z, t), respectively, eqs. (11.4) and (11.6) become
  • 20. Transmission Lines Transmission Line Equations  In the differential eqs. (11.8) and (11.9), Vs and Is are coupled.  To separate them, we take the second derivative of Vs in eq. (11.8) and employ eq. (11.9) so that we obtain  Where
  • 21. Transmission Lines Transmission Line Equations  By taking the second derivative of Is in eq. (11.9) and employing eq. (11.8), we get  We notice that eqs. (11.10) and (11.12) are, respectively, the wave equations for voltage and current  The wavelength and wave velocity u are, respectively, given by    f u      2 
  • 22. Transmission Lines Transmission Line Equations  The solutions of the linear homogeneous differential equations (11.10) and (11.12) are  Thus, we obtain the instantaneous expression for voltage as
  • 23. Transmission Lines Transmission Line Equations  The characteristic impedance Zo of the line is the ratio of positively traveling voltage wave to current wave at any point on the line.  By substituting eqs. (11.15) and (11.16) into eqs. (11.8) and (11.9) and equating coefficients of terms and , we obtain
  • 24. Transmission Lines Transmission Line Equations  Ro should not be mistaken for R— while R is in ohms per meter; Ro is in ohms.  The propagation constant and the characteristic impedance Zo are important properties of the line because they both depend on the line parameters R, L, G, and C and the frequency of operation.  The reciprocal of Zo is the characteristic admittance Yo, that is, Yo = 1/Zo.  The transmission line considered so far is the lossy type in that the conductors comprising the line are imperfect and the dielectric in which the conductors are embedded is lossy
  • 25. Transmission Lines Transmission Line Equations  We may now consider two special cases of lossless transmission line and distortionless line. Case for Lossless Transmission Line  A transmission line is said to be lossless if the conductors of the line are perfect (σc ≈ ∞) and the dielectric medium separating them is lossless (σ ≈ 0).  For such a line, it is evident that when: σc ≈ ∞ and σ ≈ 0, This is necessary for a line to be lossless. G R   0
  • 26. Transmission Lines Transmission Line Equations Case for Lossless Transmission Line Thus, for such a line, , 0   LC j j         f LC u    1 , 0  O X C L R Z O O  
  • 27. Transmission Lines Transmission Line Equations Case for Distortionless Transmission Line  A signal consists of band of frequencies; wave amplitudes of different frequency components will be attenuated differently in a lossy line as α is frequency dependent. This results in distortion.  A distortionless line is one in which the attenuation constant α is frequency independent while the phase constant β is linearly dependent on frequency.  From the general expression for α and β, a distortion line results if the line parameters are such that C G L R 
  • 28. Transmission Lines Transmission Line Equations Case for Distortionless Transmission Line Thus, for a distortionless line, or This shows that α does not depend on frequency whereas β is a linear function of frequency                G C j R L j RG    1 1    j G C j RG           1 , RG   LC   
  • 29. Transmission Lines Transmission Line Equations Case for Distortionless Transmission Line Also, or and     O O O jX R C L G R G C j G R L j R Z          1 1 , C L G R RO   0  O X    f LC u    1
  • 31. Transmission Lines Transmission Line Equations Note: 1. The phase velocity is independent of frequency because the phase constant β linearly depends on frequency. We have shape distortion of signals unless α and u are independent of frequency. 2. u and ZO remain the same as for the lossless lines. 3. A lossless line is also distortionless line, but a distortionless line is not necessarily lossless. Although lossless lines are desirable in power transmission, telephone lines are required to be distortionless.
  • 32. Transmission Lines Transmission Line Equations Example 1: An air line has characteristic impedance of 70 Ω and phase constant of 3 rad/m at 100 MHz. Calculate the inductance per meter and the capacitance per meter of the line. Note: air line can be regarded as a lossless line.
  • 33. Transmission Lines Transmission Line Equations Dividing eq. (11.1.1) by eq. (11.1.2) yields
  • 34. Transmission Lines Transmission Line Equations From eq. (11.1.1), Example 2: A distortionless line has Zo = 60 Ω, α = 20 mNp/m, u = 0.6c (c = speed of light). Find R, L, G, C and λ at 100 MHz. Hence,
  • 35. Transmission Lines  Transmission Line Equations But From eq. (11.2.2b), Dividing eq. (11.2.1) by eq. (11.2.3) results in
  • 36. Transmission Lines Transmission Line Equations From eq. (11.2.2a), Multiplying eqs. (11.2.1) and (11.2.3) together gives
  • 37. Transmission Lines Input impedance, SWR and power  Consider a transmission line of length l, characterized by γ and Zo, connected to a load ZL as shown below.  Looking into the line, the generator sees the line with the load as input impedance Zin.
  • 38. Transmission Lines Input impedance, SWR and power  Let the transmission line extend from z = 0 at the generator to z = l at the load.  First of all, we need the voltage and current waves, that is  To find V0 + and V0 - , the terminal conditions must be given. For example, if we are given the conditions at the input, say
  • 39. Transmission Lines Input impedance, SWR and power  Substituting these into the wave and current waves equations results in  If the input impedance at the input terminals is Zin, the input voltage V0 and the input current I0 are easily obtained from the figure.
  • 40. Transmission Lines Input impedance, SWR and power  On the other hand, if we are given the conditions at the load, say  Substituting these into the wave and current wave’s equations gives
  • 41. Transmission Lines Input impedance, SWR and power  Next, we determine the input impedance Zin = Vs(z)/Is(z) at any point on the line.  At the generator, for example, voltage and current wave’s equations yield  Substituting the equations of Vo + and Vo - into Zin and utilizing some useful identities and properties (Lossy Line)
  • 42. Transmission Lines Input impedance, SWR and power Useful Identities:
  • 43. Transmission Lines Input impedance, SWR and power  For a lossless line, γ = jβ, tanh j βl = jtan βl , and Zo = Ro, so Zin becomes (Lossless line)  Showing that the input impedance varies periodically with distance from the load.  The quantity βl is usually referred to as the electrical length of the line and can be expressed in degrees or radians.
  • 46. Transmission Lines Input impedance, SWR and power  The standing wave ratio s (otherwise denoted by SWR) as  As mentioned at the beginning of this chapter, a transmission is used in transferring power from the source to the load  The average input power at a distance l from the load is given by  Since the last two terms are purely imaginary, we have
  • 47. Transmission Lines Input impedance, SWR and power  Since the last two terms are purely imaginary, we have  The first term is the incident power Pi while the second term is the reflected power Pr. Thus eq. above may be written as Pt = Pi – Pr  where Pt is the input or transmitted power and the negative sign is due to the negative going wave since we take the reference direction as that of the voltage/current traveling toward the right.
  • 49. Transmission Lines Input impedance, SWR and power We now consider special cases when the line is connected to the load , ZL = 0, ZL = ∞, ZL = Zo A. Shorted Line (ZL = 0) From the equation of input impedance of lossless line, at ZL = 0, it becomes: Also, Г = -1, SWR = ∞
  • 50. Transmission Lines Input impedance, SWR and power A. Shorted Line (ZL = 0) .....cont... We notice that Zin is a pure reactance, which could be capacitive or inductive depending on the value of l. B. Open-Circuited Line (ZL = ∞) In this case, Zin becomes Also, Г = 1, SWR = ∞
  • 51. Transmission Lines Input impedance, SWR and power C. Matched Line (ZL = Zo)  This is the most desired case from the practical point of view. For this case, equation of Zin reduces to Also, Г = 0, SWR = 1  The incident power is fully absorbed by the load.  Thus, maximum power transfer is possible when a transmission line is matched to the load.
  • 52. Transmission Lines Input impedance, SWR and power Input Impedance of a lossless line when Shorted (ZL = 0)
  • 53. Transmission Lines Input impedance, SWR and power Input Impedance of a lossless line when Shorted (ZL = ∞)
  • 54. Transmission Lines Input impedance, SWR and power Example 1: A certain transmission line operating at ω = 106 rad/s has α = 8 dB/m, β = 1 rad/m, and ZO = 60 + j40 Ω and is 2 m long. If the line is connected to a source of 10 /0o V, Zg = 40 Ω and terminated by a load of 20 + j50 Ω, determine: (a) Zin (b) the sending-end current (c) the current at the middle of the line Solution
  • 60. Transmission Lines Example 3: A 50-Ω coaxial cable feeds a 75 + j20-Ω dipole antenna. Find Γ and s. Solution
  • 64. Transmission Lines Example 5: Refer to the lossless transmission line shown in Figure in example 4 slide. (a) Find Γ and s. (b) Determine Zin at the generator.