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Transmission Lines
Ranga Rodrigo
January 13, 2009
Antennas and Propagation: Transmission Lines 1/46
1 Basic Transmission Line Properties
2 Standing Waves
Antennas and Propagation: Transmission Lines Outline 2/46
Outline
1 Basic Transmission Line Properties
2 Standing Waves
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 3/46
Introduction
A transmission line is a circuit element thattransfers energy in the form of electromagneticwaves from one place to the other.
Transmission Lines
Balanced
E.g., flat twin
Unbalanced
E.g., coaxial cable
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 4/46
The construction of the line will vary dependingon its end use:
Copper wires for low-frequency audio applications.Copper dielectric mixtures for VHF, UHF, andmicrowave.Solid dielectric such as plastic or glass for opticaluse.
By carefully exploiting the properties of a givenline configuration, useful circuit componentssuch as filters and impedance matchingnetworks can be built.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46
The construction of the line will vary dependingon its end use:
Copper wires for low-frequency audio applications.
Copper dielectric mixtures for VHF, UHF, andmicrowave.Solid dielectric such as plastic or glass for opticaluse.
By carefully exploiting the properties of a givenline configuration, useful circuit componentssuch as filters and impedance matchingnetworks can be built.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46
The construction of the line will vary dependingon its end use:
Copper wires for low-frequency audio applications.Copper dielectric mixtures for VHF, UHF, andmicrowave.
Solid dielectric such as plastic or glass for opticaluse.
By carefully exploiting the properties of a givenline configuration, useful circuit componentssuch as filters and impedance matchingnetworks can be built.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46
The construction of the line will vary dependingon its end use:
Copper wires for low-frequency audio applications.Copper dielectric mixtures for VHF, UHF, andmicrowave.Solid dielectric such as plastic or glass for opticaluse.
By carefully exploiting the properties of a givenline configuration, useful circuit componentssuch as filters and impedance matchingnetworks can be built.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46
The construction of the line will vary dependingon its end use:
Copper wires for low-frequency audio applications.Copper dielectric mixtures for VHF, UHF, andmicrowave.Solid dielectric such as plastic or glass for opticaluse.
By carefully exploiting the properties of a givenline configuration, useful circuit componentssuch as filters and impedance matchingnetworks can be built.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46
Distributed Nature of Trans. Lines10 MHz and Below
Wavelengths are long (> 30 m).
Standard components, capacitors, inductors,etc., appear very short.Gives rise to the notion of lumped circuits.
Higher Frequencies
Wavelength is comparable with that ofcomponents.Distributed circuit techniques must be used.Division occurs when the component dimensionis not greater than 1/20 of the wavelength.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46
Distributed Nature of Trans. Lines10 MHz and Below
Wavelengths are long (> 30 m).Standard components, capacitors, inductors,etc., appear very short.
Gives rise to the notion of lumped circuits.
Higher Frequencies
Wavelength is comparable with that ofcomponents.Distributed circuit techniques must be used.Division occurs when the component dimensionis not greater than 1/20 of the wavelength.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46
Distributed Nature of Trans. Lines10 MHz and Below
Wavelengths are long (> 30 m).Standard components, capacitors, inductors,etc., appear very short.Gives rise to the notion of lumped circuits.
Higher Frequencies
Wavelength is comparable with that ofcomponents.Distributed circuit techniques must be used.Division occurs when the component dimensionis not greater than 1/20 of the wavelength.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46
Distributed Nature of Trans. Lines10 MHz and Below
Wavelengths are long (> 30 m).Standard components, capacitors, inductors,etc., appear very short.Gives rise to the notion of lumped circuits.
Higher Frequencies
Wavelength is comparable with that ofcomponents.Distributed circuit techniques must be used.Division occurs when the component dimensionis not greater than 1/20 of the wavelength.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46
Distributed Nature of Trans. Lines10 MHz and Below
Wavelengths are long (> 30 m).Standard components, capacitors, inductors,etc., appear very short.Gives rise to the notion of lumped circuits.
Higher FrequenciesWavelength is comparable with that ofcomponents.
Distributed circuit techniques must be used.Division occurs when the component dimensionis not greater than 1/20 of the wavelength.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46
Distributed Nature of Trans. Lines10 MHz and Below
Wavelengths are long (> 30 m).Standard components, capacitors, inductors,etc., appear very short.Gives rise to the notion of lumped circuits.
Higher FrequenciesWavelength is comparable with that ofcomponents.Distributed circuit techniques must be used.
Division occurs when the component dimensionis not greater than 1/20 of the wavelength.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46
Distributed Nature of Trans. Lines10 MHz and Below
Wavelengths are long (> 30 m).Standard components, capacitors, inductors,etc., appear very short.Gives rise to the notion of lumped circuits.
Higher FrequenciesWavelength is comparable with that ofcomponents.Distributed circuit techniques must be used.Division occurs when the component dimensionis not greater than 1/20 of the wavelength.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46
Z0
CG
R L
CG
R L
CG
R L
Z0
R: series loss resistance (copper losses).G: shunt loss conductance (losses in dielectric).L: series inductance representing energystorage within the line.C : shunt capacitance representing energystorage within the line.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 7/46
I1
V1
I2
V2
I3
V3
l l
If an infinitely long pair of wires were consideredand the voltage and current were somehowmeasured at uniform spaced points along theline, then
V1
I1= V2
I2= ·· · = Vk
Ik= constant = Z0Ω. (1)
This is termed the characteristic impedance ofthe line and is denoted by Z0.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 8/46
I1
V1
I2
V2
I3
V3
l l
If an infinitely long pair of wires were consideredand the voltage and current were somehowmeasured at uniform spaced points along theline, then
V1
I1= V2
I2= ·· · = Vk
Ik= constant = Z0Ω. (1)
This is termed the characteristic impedance ofthe line and is denoted by Z0.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 8/46
I1
V1
I2
V2
I3
V3
l l
If an infinitely long pair of wires were consideredand the voltage and current were somehowmeasured at uniform spaced points along theline, then
V1
I1= V2
I2= ·· · = Vk
Ik= constant = Z0Ω. (1)
This is termed the characteristic impedance ofthe line and is denoted by Z0.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 8/46
∆Y
Z0
∆Z
Zin = Z0
V
Figure: Characteristic line impedance.
The input impedance Zin for the L-section shownabove is
Zin = Z0 = (Z0 +∆Z )/∆Y
Z0 +∆Z +1/∆Y= Z0 +∆Z
1+Z0∆Y∆Y ,∆Z −→ 0.
(2)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 9/46
∆Y
Z0
∆Z
Zin = Z0
V
Figure: Characteristic line impedance.
The input impedance Zin for the L-section shownabove is
Zin = Z0 = (Z0 +∆Z )/∆Y
Z0 +∆Z +1/∆Y= Z0 +∆Z
1+Z0∆Y∆Y ,∆Z −→ 0.
(2)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 9/46
∆Y
Z0
∆Z
Zin = Z0
V
Figure: Characteristic line impedance.
The input impedance Zin for the L-section shownabove is
Zin = Z0 = (Z0 +∆Z )/∆Y
Z0 +∆Z +1/∆Y= Z0 +∆Z
1+Z0∆Y∆Y ,∆Z −→ 0.
(2)Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 9/46
Solving Equation 2 for Z0 gives
Z0 =(∆Z
∆Y
)1/2
, (3)
where ∆Z = R + jωL and ∆Y =G jωC hence,
Z0 =(
R + jωL
G + jωC
)1/2
Ω . (4)
This expression is important as it relates the lumpedcircuit model for the transmission line to one of theprimary line constants, characteristic impedance.
At very low frequencies:
Z0 =(
R
G
)1/2
Ω. (5)
For high frequencies (ωL À R and ωC ÀG):
Z0 =(
L
C
)1/2
Ω (6)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 10/46
Let’s consider Equation 4,
Z0 =(
R + jωL
G + jωC
)1/2
Ω.
At very low frequencies:
Z0 =(
R
G
)1/2
Ω. (7)
For high frequencies (ωL À R and ωC ÀG):
Z0 =(
L
C
)1/2
Ω (8)
Since transmission line circuit designs are done athigh frequencies, Equation 8 is often used. If lossescannot be neglected, then the line is said to be lossy.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 11/46
ExampleA lossless transmission line has a characteristicimpedance of 50 Ω and s self-inductance of 0.08µH/m. Calculate the capacitance of a 4-meter lengthof line.
Solution
The line is lossless. Therefore, R = 0, G =∞.
Z0 =(
L
C
)1/2
⇒C = L
Z 2o
= 0.08×10−6
50×50= 32 pF/m.
The total capacitance for a 4-meter length is4×32 pF/m= 128 pF/m. This capacitance will limit thehigh frequency response of the cable.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 12/46
ExampleA lossless transmission line has a characteristicimpedance of 50 Ω and s self-inductance of 0.08µH/m. Calculate the capacitance of a 4-meter lengthof line.
Solution
The line is lossless. Therefore, R = 0, G =∞.
Z0 =(
L
C
)1/2
⇒C = L
Z 2o
= 0.08×10−6
50×50= 32 pF/m.
The total capacitance for a 4-meter length is4×32 pF/m= 128 pF/m. This capacitance will limit thehigh frequency response of the cable.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 12/46
Propagation ConstantConsider the lumped equivalent circuit again.
Z0
CG
R L
CG
R L
CG
R L
Z0
The voltage drop ∆V across one lumped section is
∆V =−I (R + jωL)∆x. (9)
here ∆x is the incremental length of the linerepresented by the lumped section.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 13/46
Dividing Equation 9 by ∆x and since ∆x → 0,
dV
d x=−(R + jωL)I . (10)
Similarly we can obtain
d I
d x=−(G + jωC )V. (11)
By differentiating and substitution
d 2V
d x2= (R + jωL)(G + jωC )V. (12)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 14/46
Equation 12 can be written as
d 2V
d x2= γ2V , (13)
where γ=√(R + jωL)(G + jωC ) is termed the
propagation constant. This is usually expressed as
γ=α+ jβ, (14)
where α represents the attenuation per unit lengthand β the phase shift per unit length.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 15/46
Equation 13 has a solution of the form
V (x) =
Ae−γx +Beγx . (15)
This suggests that the line will contain two waves,one traveling in the positive x-direction (e−γx) and theother traveling in the negative x-direction (eγx).To evaluate A, let’s consider an infinite-length lineexcited by sine wave of amplitude Vin. If the linecontains resistive elements, then at x =∞, anyvoltage would have decayed to zero.
0 = Ae−γ∞+Beγ∞ ⇒ B = 0. (16)
Therefore, we have
V (x) = Ae−γx . (17)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 16/46
Equation 13 has a solution of the form
V (x) = Ae−γx +Beγx . (15)
This suggests that the line will contain two waves,one traveling in the positive x-direction (e−γx) and theother traveling in the negative x-direction (eγx).To evaluate A, let’s consider an infinite-length lineexcited by sine wave of amplitude Vin. If the linecontains resistive elements, then at x =∞, anyvoltage would have decayed to zero.
0 = Ae−γ∞+Beγ∞ ⇒ B = 0. (16)
Therefore, we have
V (x) = Ae−γx . (17)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 16/46
When we consider that at the driving end of the line(x = 0), the voltage is Vin, we get A =Vin. Therefore,
V (x) =Vine−γx . (18)
Since γ=α+ jβ,
V (x) =Vine−αxe− jβx . (19)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 17/46
e−αx TermLet’s consider the attenuation term, e−αx. For a linecontaining uniform loss per unit length,
V2 = kV1,
V3 = kV2,...= ...
Vn+1 = kVn.
(20)
where k = e−αx is a constant less than one (k = 1 forlossless). Therefore,
Vn+1 =V1kn. (21)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 18/46
Taking the natural logarithm, we have
ln
(Vn+1
V1
)=−nαx . (22)
The term nαx is defined as the total line attenuationand is measured in units called nepers (Np). It canbe shown that 1 Np is equal to 8.686 dB.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 19/46
e− jβx TermNow let’s consider the term e− jβx. Taking the currentI as a reference, we find that the voltage drop acrossa single incremental inductance element L isjωLI∆x, while the voltage across the element Z0 isI Z0.
I Z0
ωLI∆x
∆β
∆β= tan−1
(ωLI∆x
I Z0
). (23)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 20/46
For small angles, tanθ ≈ θ, so
∆β= ωL∆x
Z0. (24)
For a lossless line,
∆β= ωL∆x
(L/C )1/2=ω(LC )1/2∆x. (25)
The quantity ∆β/∆x is called the phase-shift-changeper unit length or the wave number β.
β=ω(LC )1/2. (26)
We have
vp
= f λ= ω
2πλ, β= 2π
λ, vp = ω
β, Z0 = vpL = 1
vpC.
(27)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46
For small angles, tanθ ≈ θ, so
∆β= ωL∆x
Z0. (24)
For a lossless line,
∆β= ωL∆x
(L/C )1/2=ω(LC )1/2∆x. (25)
The quantity ∆β/∆x is called the phase-shift-changeper unit length or the wave number β.
β=ω(LC )1/2. (26)
We have
vp = f λ= ω
2πλ, β
= 2π
λ, vp = ω
β, Z0 = vpL = 1
vpC.
(27)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46
For small angles, tanθ ≈ θ, so
∆β= ωL∆x
Z0. (24)
For a lossless line,
∆β= ωL∆x
(L/C )1/2=ω(LC )1/2∆x. (25)
The quantity ∆β/∆x is called the phase-shift-changeper unit length or the wave number β.
β=ω(LC )1/2. (26)
We have
vp = f λ= ω
2πλ, β= 2π
λ, vp
= ω
β, Z0 = vpL = 1
vpC.
(27)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46
For small angles, tanθ ≈ θ, so
∆β= ωL∆x
Z0. (24)
For a lossless line,
∆β= ωL∆x
(L/C )1/2=ω(LC )1/2∆x. (25)
The quantity ∆β/∆x is called the phase-shift-changeper unit length or the wave number β.
β=ω(LC )1/2. (26)
We have
vp = f λ= ω
2πλ, β= 2π
λ, vp = ω
β, Z0
= vpL = 1
vpC.
(27)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46
For small angles, tanθ ≈ θ, so
∆β= ωL∆x
Z0. (24)
For a lossless line,
∆β= ωL∆x
(L/C )1/2=ω(LC )1/2∆x. (25)
The quantity ∆β/∆x is called the phase-shift-changeper unit length or the wave number β.
β=ω(LC )1/2. (26)
We have
vp = f λ= ω
2πλ, β= 2π
λ, vp = ω
β, Z0 = vpL = 1
vpC.
(27)Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46
Sending End ImpedanceWhen a load is placed at the end of a section oftransmission line the line will produce atransformation effect. The load impedance willappear different when viewed from the sendingend of the line.
Using this effect, the impedance transformationcaused by inserting a section of transmissionline between a given impedance and ameasurement point can be carefully controlledto allow impedance matching.If measurements are to be made on an unknownload impedance, the impedance transformationcaused by inserting a connecting section oftransmission line needs to be taken into account.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 22/46
Sending End ImpedanceWhen a load is placed at the end of a section oftransmission line the line will produce atransformation effect. The load impedance willappear different when viewed from the sendingend of the line.Using this effect, the impedance transformationcaused by inserting a section of transmissionline between a given impedance and ameasurement point can be carefully controlledto allow impedance matching.
If measurements are to be made on an unknownload impedance, the impedance transformationcaused by inserting a connecting section oftransmission line needs to be taken into account.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 22/46
Sending End ImpedanceWhen a load is placed at the end of a section oftransmission line the line will produce atransformation effect. The load impedance willappear different when viewed from the sendingend of the line.Using this effect, the impedance transformationcaused by inserting a section of transmissionline between a given impedance and ameasurement point can be carefully controlledto allow impedance matching.If measurements are to be made on an unknownload impedance, the impedance transformationcaused by inserting a connecting section oftransmission line needs to be taken into account.Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 22/46
Let’s consider Equation 15,
V (x) = Ae−γx +Be+γx .
ZTZ0
ZS
l
x = 0
IincIref
The total current flowing in the line IT is the sum ofcurrent traveling in the forward direction from x = 0 tox = l , I inc, and the amount of current that is reflectedfrom the load termination ZT , Iref.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 23/46
Let’s consider Equation 15,
V (x) = Ae−γx +Be+γx .
ZTZ0
ZS
l
x = 0
IincIref
The total current flowing in the line IT is the sum ofcurrent traveling in the forward direction from x = 0 tox = l , I inc, and the amount of current that is reflectedfrom the load termination ZT , Iref.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 23/46
This reflected current is the portion of the incident(forward) current that is not absorbed by the load.
IT = I inc− Iref, (28)
IT = A
Z0e−γx − B
Z0e+γx . (29)
At position x = l the impedance is ZT .
ZT = VT
IT=
Ae−γl +Be+γl
Ae−γl −Be+γlZ0. (30)
Therefore,B
A= e−2γl ZT −Z0
ZT +Z0. (31)
The term e−2γl is the loss encountered by the signaltraversing from source to termination and back.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 24/46
This reflected current is the portion of the incident(forward) current that is not absorbed by the load.
IT = I inc− Iref, (28)
IT = A
Z0e−γx − B
Z0e+γx . (29)
At position x = l the impedance is ZT .
ZT = VT
IT= Ae−γl +Be+γl
Ae−γl −Be+γlZ0. (30)
Therefore,B
A= e−2γl ZT −Z0
ZT +Z0. (31)
The term e−2γl is the loss encountered by the signaltraversing from source to termination and back.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 24/46
This reflected current is the portion of the incident(forward) current that is not absorbed by the load.
IT = I inc− Iref, (28)
IT = A
Z0e−γx − B
Z0e+γx . (29)
At position x = l the impedance is ZT .
ZT = VT
IT= Ae−γl +Be+γl
Ae−γl −Be+γlZ0. (30)
Therefore,B
A= e−2γl ZT −Z0
ZT +Z0. (31)
The term e−2γl is the loss encountered by the signaltraversing from source to termination and back.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 24/46
Considering the situation at x = 0, we get,
ZS = VS
IS= A+B
A−BZ0. (32)
ZS
Z0= 1+B/A
1−B/A. (33)
Here ZS is the sending end impedance, i.e., theimpedance seen looking into the line toward theload. Substituting,
ZS
Z0=
1+e−2γl ZT −Z0ZT +Z0
1−e−2γl ZT −Z0ZT +Z0
=
ZT (1+e−2γl )+Z0(1−e−2γl )
ZT (1−e−2γl )+Z0(1+e−2γl ). (34)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 25/46
Considering the situation at x = 0, we get,
ZS = VS
IS= A+B
A−BZ0. (32)
ZS
Z0= 1+B/A
1−B/A. (33)
Here ZS is the sending end impedance, i.e., theimpedance seen looking into the line toward theload. Substituting,
ZS
Z0=
1+e−2γl ZT −Z0ZT +Z0
1−e−2γl ZT −Z0ZT +Z0
= ZT (1+e−2γl )+Z0(1−e−2γl )
ZT (1−e−2γl )+Z0(1+e−2γl ). (34)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 25/46
sinhγl = 1
2
[eγl −e−γl] ,
1−e−2γl = e−γl [eγl −e−γl] ,
= 2sinh(γl )e−γl .
1+e−2γl = 2cosh(γl )e−γl .
(35)
ZS
Z0=
ZT cosh(γl )+Z0 sinh(γl )
ZT sinh(γl )+Z0 cosh(γl ). (36)
Equation 36 is the desired result. It enables us toevaluate the sending end impedance ZS, in terms ofthe termination impedance ZT and characteristicimpedance Z0.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 26/46
sinhγl = 1
2
[eγl −e−γl] ,
1−e−2γl = e−γl [eγl −e−γl] ,
= 2sinh(γl )e−γl .
1+e−2γl = 2cosh(γl )e−γl .
(35)
ZS
Z0= ZT cosh(γl )+Z0 sinh(γl )
ZT sinh(γl )+Z0 cosh(γl ). (36)
Equation 36 is the desired result. It enables us toevaluate the sending end impedance ZS, in terms ofthe termination impedance ZT and characteristicimpedance Z0.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 26/46
Sometimes, it is convenient to use the followingequation as the governing equation (be dividing eachterm by Z0 cosh(γl )):
ZS
Z0= ZT cosh(γl )+Z0 sinh(γl )
ZT sinh(γl )+Z0 cosh(γl )
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ). (37)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 27/46
Sometimes, it is convenient to use the followingequation as the governing equation (be dividing eachterm by Z0 cosh(γl )): ZS
Z0= ZT cosh(γl )+Z0 sinh(γl )
ZT sinh(γl )+Z0 cosh(γl )
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ). (37)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 27/46
Sometimes, it is convenient to use the followingequation as the governing equation (be dividing eachterm by Z0 cosh(γl )): ZS
Z0= ZT cosh(γl )+Z0 sinh(γl )
ZT sinh(γl )+Z0 cosh(γl )
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ). (37)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 27/46
From Equation 36, we see that if the load terminationZT = Z0, then the sending end impedance ZS willequal to Z0. In this case, no reflection will occur fromthe load, and the line is said to be matched.
ExampleA transmission line of length 100 m is operated at10.0 MHz, and has an attenuation per unit length of0.002 nepers/m. The phase velocity of the line is2.7×108 m/s, and the line has a characteristicimpedance of 50 Ω. What is the value of the terminalload impedance, if the input impedance looking intothe line is measured to be 30− j 10Ω?
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 28/46
From Equation 36, we see that if the load terminationZT = Z0, then the sending end impedance ZS willequal to Z0. In this case, no reflection will occur fromthe load, and the line is said to be matched.
ExampleA transmission line of length 100 m is operated at10.0 MHz, and has an attenuation per unit length of0.002 nepers/m. The phase velocity of the line is2.7×108 m/s, and the line has a characteristicimpedance of 50 Ω. What is the value of the terminalload impedance, if the input impedance looking intothe line is measured to be 30− j 10Ω?
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 28/46
Solution
Following relations are useful:
tanh( jβl ) = j tan(βl ).
tanh(γl ) = tanh(αl + jβl ),
= tanh(αl )+ j tan(βl )
1− tanh(αl ) tan(βl )
tanh(α± jβl ) = sinh(2αl )± j sin(2βl )
cosh(2αl )+cos(2βl ).
(38)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 29/46
We need to find ZT , given ZS and line parameters.From Equation 36,
ZS
Z0= ZT cosh(γl )+Z0 sinh(γl )
ZT sinh(γl )+Z0 cosh(γl ),
we obtainZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ). (39)
Rearranging,
ZT
ZS=
ZSZ0
− tanh(γl )
1− ZSZ0
tanh(γl ). (40)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 30/46
We need to find ZT , given ZS and line parameters.From Equation 36,
ZS
Z0= ZT cosh(γl )+Z0 sinh(γl )
ZT sinh(γl )+Z0 cosh(γl ),
we obtainZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ). (39)
Rearranging,
ZT
ZS=
ZSZ0
− tanh(γl )
1− ZSZ0
tanh(γl ). (40)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 30/46
Equation 40 is written for traveling from load togenerator. However, as we need to find ZT , we needto travel toward the load. Therefore, we need to use−l in place of +l . We note that tanh(−γl ) =− tanh(γl ).
ZT = quantity to be found,
ZS = 30− j 10Ω,
Z0 = 50+ j 0Ω,
l = 100 m,
α= 0.002 nepers/m,
vp = 2.7×108 m/s.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 31/46
ZS
Z0= 0.6−0.2 j ,
αl = 0.002×100 = 0.2 nepers,
∴ 2αl = 0.4 nepers,
βl = ωl
vp= 2π×10×106 ×100
2.7×108= 23.27 radians,
∴ 2βl = 46.54 radians,
tanh(γl ) = tanh(αl + jβl ),
= tanh(0.2+ j 23.27).
tanh(γl ) = tanh(0.2+ j 23.27),
= sinh0.4+ j sin46.54
cosh0.4+cos46.54.
sinh0.4 = e0.4 −e−0.4
2= 0.411.
cosh0.4 = 1.081.
sin46.54 = 0.5513.
cos46.54 =−0.8343.
tanh(γl ) = 0.411+ j 0.5513
1.081−0.8343.
tanh(γl ) = 0.411+ j 0.5513
1.081−0.8343,
= 1.666+ j 2.2347.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 32/46
ZS
Z0= 0.6−0.2 j ,
αl = 0.002×100 = 0.2 nepers,
∴ 2αl = 0.4 nepers,
βl = ωl
vp= 2π×10×106 ×100
2.7×108= 23.27 radians,
∴ 2βl = 46.54 radians,
tanh(γl ) = tanh(αl + jβl ),
= tanh(0.2+ j 23.27).
tanh(γl ) = tanh(0.2+ j 23.27),
= sinh0.4+ j sin46.54
cosh0.4+cos46.54.
sinh0.4 = e0.4 −e−0.4
2= 0.411.
cosh0.4 = 1.081.
sin46.54 = 0.5513.
cos46.54 =−0.8343.
tanh(γl ) = 0.411+ j 0.5513
1.081−0.8343.
tanh(γl ) = 0.411+ j 0.5513
1.081−0.8343,
= 1.666+ j 2.2347.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 32/46
ZS
Z0= 0.6−0.2 j ,
αl = 0.002×100 = 0.2 nepers,
∴ 2αl = 0.4 nepers,
βl = ωl
vp= 2π×10×106 ×100
2.7×108= 23.27 radians,
∴ 2βl = 46.54 radians,
tanh(γl ) = tanh(αl + jβl ),
= tanh(0.2+ j 23.27).
tanh(γl ) = tanh(0.2+ j 23.27),
= sinh0.4+ j sin46.54
cosh0.4+cos46.54.
sinh0.4 = e0.4 −e−0.4
2= 0.411.
cosh0.4 = 1.081.
sin46.54 = 0.5513.
cos46.54 =−0.8343.
tanh(γl ) = 0.411+ j 0.5513
1.081−0.8343.
tanh(γl ) = 0.411+ j 0.5513
1.081−0.8343,
= 1.666+ j 2.2347.Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 32/46
Substituting this into the governing equation gives
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZT
Z0= 0.6− j 0.2− (1.666+ j 2.2347)
1− (0.6− j 0.2)(1.666+ j 2.2347),
= 4.55− j 0.7063
3.0.
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
and finally, the desired result
ZT = 76− j 12Ω.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 33/46
Substituting this into the governing equation gives
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZT
Z0= 0.6− j 0.2− (1.666+ j 2.2347)
1− (0.6− j 0.2)(1.666+ j 2.2347),
= 4.55− j 0.7063
3.0.
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
and finally, the desired result
ZT = 76− j 12Ω.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 33/46
Substituting this into the governing equation gives
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZT
Z0= 0.6− j 0.2− (1.666+ j 2.2347)
1− (0.6− j 0.2)(1.666+ j 2.2347),
= 4.55− j 0.7063
3.0.
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
and finally, the desired result
ZT = 76− j 12Ω.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 33/46
Short Circuit Termination
If a short circuit is used as the load termination, thenthe sending-end impedance becomes,
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZS|SC = Z0 tanh(γl ), (41)
and, neglecting line losses for short lengths we get
ZS|SC = Z0 tanh( jβl ) = j Z0 tan(βl ). (42)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 34/46
Short Circuit Termination
If a short circuit is used as the load termination, thenthe sending-end impedance becomes,
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZS|SC = Z0 tanh(γl ), (41)
and, neglecting line losses for short lengths we get
ZS|SC = Z0 tanh( jβl ) = j Z0 tan(βl ). (42)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 34/46
Short Circuit Termination
If a short circuit is used as the load termination, thenthe sending-end impedance becomes,
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZS|SC = Z0 tanh(γl ), (41)
and, neglecting line losses for short lengths we get
ZS|SC = Z0 tanh( jβl ) = j Z0 tan(βl ). (42)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 34/46
Open Circuit Termination
When the termination is open circuit, then thesending-end impedance becomes,
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZS|OC = Z0
tanh(γl ), (43)
and, neglecting line losses for short lengths we get
ZS|OC = − j Z0
tanh( jβl )=− j Z0 cot(βl ). (44)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 35/46
Open Circuit Termination
When the termination is open circuit, then thesending-end impedance becomes,
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZS|OC = Z0
tanh(γl ), (43)
and, neglecting line losses for short lengths we get
ZS|OC = − j Z0
tanh( jβl )=− j Z0 cot(βl ). (44)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 35/46
Open Circuit Termination
When the termination is open circuit, then thesending-end impedance becomes,
ZS
Z0=
ZTZ0
+ tanh(γl )
1+ ZTZ0
tanh(γl ).
ZS|OC = Z0
tanh(γl ), (43)
and, neglecting line losses for short lengths we get
ZS|OC = − j Z0
tanh( jβl )=− j Z0 cot(βl ). (44)
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 35/46
RemarksEquations 42 and 44 indicate that, by correctlychoosing l , we can make the line (stub) to havethe sending end impedance equivalent to alumped capacitor or an inductor.
When a transmission line is terminated in aperfect open or a short circuit, the incidentenergy is reflected back into the line.Product of Equations 42 and 44 gives
ZS|OC×ZS|SC = Z 20 (45)
This is a useful method of measuring thecharacteristic impedance of a transmission line.For this the line length selected should be avalue close to an odd number of λ/8.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 36/46
RemarksEquations 42 and 44 indicate that, by correctlychoosing l , we can make the line (stub) to havethe sending end impedance equivalent to alumped capacitor or an inductor.When a transmission line is terminated in aperfect open or a short circuit, the incidentenergy is reflected back into the line.
Product of Equations 42 and 44 gives
ZS|OC×ZS|SC = Z 20 (45)
This is a useful method of measuring thecharacteristic impedance of a transmission line.For this the line length selected should be avalue close to an odd number of λ/8.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 36/46
RemarksEquations 42 and 44 indicate that, by correctlychoosing l , we can make the line (stub) to havethe sending end impedance equivalent to alumped capacitor or an inductor.When a transmission line is terminated in aperfect open or a short circuit, the incidentenergy is reflected back into the line.Product of Equations 42 and 44 gives
ZS|OC×ZS|SC = Z 20 (45)
This is a useful method of measuring thecharacteristic impedance of a transmission line.For this the line length selected should be avalue close to an odd number of λ/8.Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 36/46
βl = 2πlλ
− tan(βl )
π
2
π 3π
2
2π 5π
2
−6
−4
−2
0
2
4
6
Normalized input resistance versus βl for ashort-circuited, lossless, transmission line.Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 37/46
βl = 2πlλ
−cot(βl )
π
2
π 3π
2
2π 5π
2
−6
−4
−2
0
2
4
6
Normalized input resistance versus βl for aopen-circuited, lossless, transmission line.Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 38/46
ExampleThe design of a microwave amplifier requires aninductor to series resonate a capacitive reactance of15,000 Ω. Calculate the residual resistance if theinductor consists of a short-circuited line withZ0 = 75Ω, α= 0.002 dB/cm, and λ= 10 cm.
Solution
To series resonate the − j 15000Ω, the inputimpedance of the shorted line must be + j 15000Ω. Ifthe line were lossless, Zin = j 15000 = j 75tan(βl ), andtherefore
βl = tan−1 200 ≈ π
2− 1
200rad.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 39/46
ExampleThe design of a microwave amplifier requires aninductor to series resonate a capacitive reactance of15,000 Ω. Calculate the residual resistance if theinductor consists of a short-circuited line withZ0 = 75Ω, α= 0.002 dB/cm, and λ= 10 cm.
Solution
To series resonate the − j 15000Ω, the inputimpedance of the shorted line must be + j 15000Ω. Ifthe line were lossless, Zin = j 15000 = j 75tan(βl ), andtherefore
βl =
tan−1 200 ≈ π
2− 1
200rad.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 39/46
ExampleThe design of a microwave amplifier requires aninductor to series resonate a capacitive reactance of15,000 Ω. Calculate the residual resistance if theinductor consists of a short-circuited line withZ0 = 75Ω, α= 0.002 dB/cm, and λ= 10 cm.
Solution
To series resonate the − j 15000Ω, the inputimpedance of the shorted line must be + j 15000Ω. Ifthe line were lossless, Zin = j 15000 = j 75tan(βl ), andtherefore
βl = tan−1 200 ≈ π
2− 1
200rad.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 39/46
Since β= 2π/λ,
l =
2.5
(π
2− 1
100π
)cm.
Thus a shorted lossless line about 0.3% less than aquarter wavelength (2.5 cm) would provide thedesired reactance without adding any resistance.However, in this line is not lossless and αl = 0.005 dBor 5.76×10−4 Np (1 Np = 8.686 dB). Sincetanh(αl ) ≈αl , substituting Z0 = 75Ω and tan(βl ) = 200in to Equation 37 yields
Zin = 755.76×10−4 + j 200
1+ j 0.115= 1700+ j 14800Ω.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 40/46
Since β= 2π/λ,
l = 2.5
(π
2− 1
100π
)cm.
Thus a shorted lossless line about 0.3% less than aquarter wavelength (2.5 cm) would provide thedesired reactance without adding any resistance.However, in this line is not lossless and αl = 0.005 dBor 5.76×10−4 Np (1 Np = 8.686 dB). Sincetanh(αl ) ≈αl , substituting Z0 = 75Ω and tan(βl ) = 200in to Equation 37 yields
Zin =
755.76×10−4 + j 200
1+ j 0.115= 1700+ j 14800Ω.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 40/46
Since β= 2π/λ,
l = 2.5
(π
2− 1
100π
)cm.
Thus a shorted lossless line about 0.3% less than aquarter wavelength (2.5 cm) would provide thedesired reactance without adding any resistance.However, in this line is not lossless and αl = 0.005 dBor 5.76×10−4 Np (1 Np = 8.686 dB). Sincetanh(αl ) ≈αl , substituting Z0 = 75Ω and tan(βl ) = 200in to Equation 37 yields
Zin = 755.76×10−4 + j 200
1+ j 0.115= 1700+ j 14800Ω.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 40/46
Note the effect of line loss. The slight loss in thereactive portion can be adjusted by slightlyincreasing the line length. The resistive componentis significant. If we adjust the length, the residualresistance will actually be 1748 Ω.
Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 41/46
Outline
1 Basic Transmission Line Properties
2 Standing Waves
Antennas and Propagation: Transmission Lines Standing Waves 42/46
Standing WavesFor any transmission line, a sinusoidal signal ofappropriate frequency introduced at the sendingend by a generator will propagate along thelength of the transmission line.
If the line has infinite length, then the signalnever reaches the end of the line.If the signal is viewed at some distance downthe line away from the generator, then it willappear to have the same frequency, but willexhibit smaller peak to peak voltage swing thanat the generator.The signal, therefore, has been attenuated bythe line losses due to the conductor resistanceand dielectric imperfections.
Antennas and Propagation: Transmission Lines Standing Waves 43/46
Standing WavesFor any transmission line, a sinusoidal signal ofappropriate frequency introduced at the sendingend by a generator will propagate along thelength of the transmission line.If the line has infinite length, then the signalnever reaches the end of the line.
If the signal is viewed at some distance downthe line away from the generator, then it willappear to have the same frequency, but willexhibit smaller peak to peak voltage swing thanat the generator.The signal, therefore, has been attenuated bythe line losses due to the conductor resistanceand dielectric imperfections.
Antennas and Propagation: Transmission Lines Standing Waves 43/46
Standing WavesFor any transmission line, a sinusoidal signal ofappropriate frequency introduced at the sendingend by a generator will propagate along thelength of the transmission line.If the line has infinite length, then the signalnever reaches the end of the line.If the signal is viewed at some distance downthe line away from the generator, then it willappear to have the same frequency, but willexhibit smaller peak to peak voltage swing thanat the generator.
The signal, therefore, has been attenuated bythe line losses due to the conductor resistanceand dielectric imperfections.
Antennas and Propagation: Transmission Lines Standing Waves 43/46
Standing WavesFor any transmission line, a sinusoidal signal ofappropriate frequency introduced at the sendingend by a generator will propagate along thelength of the transmission line.If the line has infinite length, then the signalnever reaches the end of the line.If the signal is viewed at some distance downthe line away from the generator, then it willappear to have the same frequency, but willexhibit smaller peak to peak voltage swing thanat the generator.The signal, therefore, has been attenuated bythe line losses due to the conductor resistanceand dielectric imperfections.Antennas and Propagation: Transmission Lines Standing Waves 43/46
ReflectionIf the line is lossless, then the signal viewed atsome remote point will be identical to that at thegenerator but time delayed by an amountdependant on the position.
When a sinusoidal signal reaches the open endof a section of a lossless transmission line, itcan dissipate no energy.This means that all the energy propagatingalong the line in the forward direction (incident)will be reflected completely, on reaching theopen circuit termination.The reflected wave (backward wave) must besuch that the total current at the open circuit iszero.
Antennas and Propagation: Transmission Lines Standing Waves 44/46
ReflectionIf the line is lossless, then the signal viewed atsome remote point will be identical to that at thegenerator but time delayed by an amountdependant on the position.When a sinusoidal signal reaches the open endof a section of a lossless transmission line, itcan dissipate no energy.
This means that all the energy propagatingalong the line in the forward direction (incident)will be reflected completely, on reaching theopen circuit termination.The reflected wave (backward wave) must besuch that the total current at the open circuit iszero.
Antennas and Propagation: Transmission Lines Standing Waves 44/46
ReflectionIf the line is lossless, then the signal viewed atsome remote point will be identical to that at thegenerator but time delayed by an amountdependant on the position.When a sinusoidal signal reaches the open endof a section of a lossless transmission line, itcan dissipate no energy.This means that all the energy propagatingalong the line in the forward direction (incident)will be reflected completely, on reaching theopen circuit termination.
The reflected wave (backward wave) must besuch that the total current at the open circuit iszero.
Antennas and Propagation: Transmission Lines Standing Waves 44/46
ReflectionIf the line is lossless, then the signal viewed atsome remote point will be identical to that at thegenerator but time delayed by an amountdependant on the position.When a sinusoidal signal reaches the open endof a section of a lossless transmission line, itcan dissipate no energy.This means that all the energy propagatingalong the line in the forward direction (incident)will be reflected completely, on reaching theopen circuit termination.The reflected wave (backward wave) must besuch that the total current at the open circuit iszero.Antennas and Propagation: Transmission Lines Standing Waves 44/46
ReflectionAs the reflected signal travels back along theline toward the generator, it reinforces theincident waveform at certain points formingmaxima (nodes).
Similarly, it can cancel the incident waveform atcertain other points producing minima(antinodes).In an open circuit line, node voltage points willoccur at the same position as the antinodecurrent points.These waves do not represent traveling waves.They are standing waves, implying that there isno net power flow from generator to the(open-circuit) load.
Antennas and Propagation: Transmission Lines Standing Waves 45/46
ReflectionAs the reflected signal travels back along theline toward the generator, it reinforces theincident waveform at certain points formingmaxima (nodes).Similarly, it can cancel the incident waveform atcertain other points producing minima(antinodes).
In an open circuit line, node voltage points willoccur at the same position as the antinodecurrent points.These waves do not represent traveling waves.They are standing waves, implying that there isno net power flow from generator to the(open-circuit) load.
Antennas and Propagation: Transmission Lines Standing Waves 45/46
ReflectionAs the reflected signal travels back along theline toward the generator, it reinforces theincident waveform at certain points formingmaxima (nodes).Similarly, it can cancel the incident waveform atcertain other points producing minima(antinodes).In an open circuit line, node voltage points willoccur at the same position as the antinodecurrent points.
These waves do not represent traveling waves.They are standing waves, implying that there isno net power flow from generator to the(open-circuit) load.
Antennas and Propagation: Transmission Lines Standing Waves 45/46
ReflectionAs the reflected signal travels back along theline toward the generator, it reinforces theincident waveform at certain points formingmaxima (nodes).Similarly, it can cancel the incident waveform atcertain other points producing minima(antinodes).In an open circuit line, node voltage points willoccur at the same position as the antinodecurrent points.These waves do not represent traveling waves.They are standing waves, implying that there isno net power flow from generator to the(open-circuit) load.Antennas and Propagation: Transmission Lines Standing Waves 45/46
A point one-quarter wavelength away from ashort circuit will have voltage and currentmagnitudes equivalent to those obtained for anopen circuit.
For a lossless line the peak value of the standingwave envelope is twice that of the incident wave.There are nulls as well due to the completecancelation.For lossy (practical) line, the peak will be lessthan twice that of the incident wave, andcomplete cancelation rarely results.
Antennas and Propagation: Transmission Lines Standing Waves 46/46
A point one-quarter wavelength away from ashort circuit will have voltage and currentmagnitudes equivalent to those obtained for anopen circuit.For a lossless line the peak value of the standingwave envelope is twice that of the incident wave.
There are nulls as well due to the completecancelation.For lossy (practical) line, the peak will be lessthan twice that of the incident wave, andcomplete cancelation rarely results.
Antennas and Propagation: Transmission Lines Standing Waves 46/46
A point one-quarter wavelength away from ashort circuit will have voltage and currentmagnitudes equivalent to those obtained for anopen circuit.For a lossless line the peak value of the standingwave envelope is twice that of the incident wave.There are nulls as well due to the completecancelation.
For lossy (practical) line, the peak will be lessthan twice that of the incident wave, andcomplete cancelation rarely results.
Antennas and Propagation: Transmission Lines Standing Waves 46/46
A point one-quarter wavelength away from ashort circuit will have voltage and currentmagnitudes equivalent to those obtained for anopen circuit.For a lossless line the peak value of the standingwave envelope is twice that of the incident wave.There are nulls as well due to the completecancelation.For lossy (practical) line, the peak will be lessthan twice that of the incident wave, andcomplete cancelation rarely results.
Antennas and Propagation: Transmission Lines Standing Waves 46/46