Transmission line analysis is used when… The frequency of operation is high The length of the transmission line is long The length of the line is an appreciable fraction of the signal wavelength In general, a line is considered as a transmission line if its length exceeds one sixteenth (1/16) of a wavelength.
Transmission line analysis is used when… The frequency of operation is high The length of the transmission line is long The length of the line is an appreciable fraction of the signal wavelength In general, a line is considered as a transmission line if its length exceeds one sixteenth (1/16) of a wavelength.
A transmission line is a metallic conductor system that is used to transfer electrical energy from one point to another. Electrical power propagates along transmission lines in the form of TEM of TEM waves. waves.
Signal transfer Pulse generator Filters and tuned circuits Provides phase shift and time delay Impedance matching
The velocity of a wave depends on the type of wave and the medium of propagation. In free space, electromagnetic waves travel at the speed of light.
c = 299 792 458 m/s
c = 3 × 108 m/s
The velocity of any wave is given by
v = λf
BALANCED LINE Both conductors carry current; e.g. twin-lead The current in the two conductors are equal in magnitude but travels in opposite direction UNBALANCED LINE One conductor carries the signal while the other is at a ground potential; e.g. coaxial cable
Balanced lines can be connected to unbalanced lines and vice versa using special transformers called BALUNS.
BALANCED LINE Balance is defined in terms of the impedances of the two signal conductors with respect to a reference, which is usually "ground" . UNBALANCED LINE An UNBALANCED input or output connects one of its signal conductors to ground and has a non-zero impedance at the other signal conductor. very popular in consumer electronics, electronic musical instruments, and low cost (often called "semi-pro") audio equipment.
Parallel open-wire line Twin lead Twisted pair Unshielded Twisted Pair (UTP) Shielded pair Coaxial lines Balanced/unbalanced lines
characterized by high radiation losses and is susceptible to noise
conductor distance is between 2 to 6 in.
it is normally operated in the balanced mode
Also known as the ribbon cable The typical conductor separation is 5/16 inch. Commonly used insulators are Teflon and PE.
Common applications include TV and antennas. Typical input impedance is 300 ohms.
Two insulated wires are twisted to form a flexible line without the use of spacers. It is not used for high frequencies. The conductors are twisted together to reduce interference.
computer networking cable capable of handling a 100 MHz bandwidth Most often used in LANs Transmits data up to 100 Mbps for a length of 100 m consists of four color-coded pairs of 22 or 24 gauge wires terminated with an RJ-45 connector
consists of parallel conductors separated and surrounded by a solid dielectric The copper braid isolates the inner conductor from interference and noise.
The conductors are balanced to ground.
consists of two concentric conductors separated by a dielectric outer conductor – copper or aluminum tube or wire braid inner conductor – wire or small tube The dielectric may be air, plastic or ceramic Coaxial cables are used extensively for high frequency applications Limited to unbalanced applications
Rigid or air-filled coaxial line Flexible or solid coaxial line
Both are relatively immune to external radiation and can propagate at higher frequencies than parallel wire line.
The amount of loss in the signal strength as it propagates throughout a transmission line is called Return loss B. SWR C. Attenuation D. Fading A.
The unwanted coupling caused by overlapping electric and magnetic fields Cross talk B. EMI C. Coupling loss D. Noise A.
A measure of the ratio of power transmitted into a cable to the amount of power returned or reflected. Reflection coefficient B. SWR C. attenuation D. Return loss A.
The characteristics of a transmission line are determined by
Electrical properties
Conductivity of the conductor Dielectric constant of the insulator
Physical properties
Wire diameter Conductor separation
Primary Electrical Constants Series dc resistance (R) Shunt capacitance (C) Series inductance (L) Shunt conductance (G)
These are the secondary constants of a transmission line which are determined from primary constants.
Characteristic impedance, Z 0 also called surge impedance Propagation constant, γ It expresses the attenuation and the phase shift per unit length of a transmission line.
It is the ratio of the voltage to current at any point in an infinitely long transmission line.
Z
0
E I
If the line is of infinite length, then there would be no reflection and the impedance is purely resistive.
the input impedance of a finite transmission line whose terminals are shorted.
In terms of physical characteristics d εr
Z
276
0
εr
log
2s
d
s
Z 0
120
εr
ln
2s
d
Z = R + j ωL Y = G + j ωC Characteristic Impedance
Lossy Line
Lossless Line
R j ωL Z G j ωC
R Z G
L Z C
0
0
0
μ
2s
L ln π d
C
επ
ln
2s
d
R 8.34 10
G
πσ
ln
2s
d
8
f r
s = conductor separation d = conductor diameter μ = permeability ε = permittivity σ = conductivity
A transmission line has 2.5 pF of capacitance per foot and 100 nH of inductance per foot. Calculate its characteristic impedance.
Ans: Z0 = 200 ohms
An open-wire line uses wire with a diameter of 2 mm. What should the wire spacing be for an impedance of 150 Ω?
Ans: 3.5 mm
In terms of physical characteristics
D Z log d εr 138
0
D ln Z εr d 60
0
μ D L ln d 2π
C
2επ
D ln d
R 8.34 10
G
2πσ
D ln d
8
1 1 f d D
D = outer conductor dia. d = inner conductor dia. μ = permeability ε = permittivity σ = conductivity
ratio of the speed of an electromagnetic wave in a medium to its speed in vacuum.
v f
v c
v f
1
r
εr of air = 1.0006
εr = relative permittivity or dielectric constant εr of materials commonly used for translines : 1.2 – 2.8
the velocity at which the signal propagates in a medium v
1
LC
v = λ f
c f r
The characteristic impedance of a cable
Increases with length B. Increases with frequency C. Increases with voltage D. None of the above A.
The velocity factor of a cable depends mostly on:
the wire resistance B. the dielectric constant C. The inductance per foot D. All of the above A.
If a coaxial cable uses plastic insulation with a dielectric constant εr = 2.6 , what is the velocity factor for the cable?
Ans: v f = 0.62
If a cable has a velocity factor of 0.8, how long would it take a signal to travel 3000 kilometers along the cable?
Ans: t = 12.5 ms
j R 2 Z
ZY LC
0
v
2
γ = propagation constant
α = attenuation coefficient (Np/L) β = phase shift coefficient (rad/L)
1 Np = 8.686 dB
Compared to a 300-ohm line, the loss of a 50-ohm cable carrying the same power:
Would be less B. Would be more C. Would be the same D. Cannot be compared A.
A line terminated with a resistive load equal to its characteristic impedance is called a nonresonant line, matched line or a flat line. line. ZL = Z0
All the energy travelling down the line is absorbed by the load. The incident current and voltage in a matched line are always in-phase. An infinitely long line is also nonresonant.
A transmission line terminated with a load not equal to its characteristic impedance. Reflection Reflecti on occurs in a resonant line. Reflected Reflecte d power is the portion of the incident power that is not absorbed by the load.
Pr ≤ Pi
A shorted and open lines are both resonant line.
There are two travelling waves in a transmission line. From source to load: incident waves From the load to source: reflected waves The interaction between the incident and the reflected waves due to reflection in a mismatched line creates a pattern of waves that appears to be stationary. This is called standing wave. wave.
λ/2
Voltage Current
λ/4
No phase reversal for reflected voltage 180 ° phase reversal for reflected current. The voltage and current repeats λ/2. The impedance is max. at the open end Impedance inversion occurs λ/4 The sum of Ei and Er is max. at the open end and min. at λ/4 from the open end.
λ/2
Voltage Current
λ/4
180 ° phase reversal for reflected voltage. No phase reversal for reflected current The voltage and current repeats every λ/2. The impedance is min. at the shorted end. Impedance inversion occurs λ/4. The sum of Ii and Ir is max. at the shorted end and min. at λ/4 from the shorted end.
A voltage maximum occurs
At the end of a shorted line B. Quarter wavelength from the end of a shorted line C. Half wavelength from the end of a short line D. Quarter wavelength from the end of an open line A.
It is a vector quantity that represents the ratio of the reflected wave to the incident wave. Γ
E r E i
ZL Z Γ ZL Z
0
0
The maximum (worst case) value is 1. The minimum (ideal condition) value is 0.
Condition ZL Γ open 1 ∞ short 0 –1 matched Z0 0 mismatch ≠ Z0 (0, 1)
Description total reflection w/o phase shift total reflection with phase shift no reflection
partial reflection
A negative reflection coefficient means
The load is greater than the line impedance B. The incident and reflected voltages are in phase at the load C. The terminal of the line is always shorted D. None of the above A.
SWR is a measure of mismatch between the load and the characteristic impedance.
Emax Ei E r SWR Emin Ei E r
SWR
1 Γ 1 Γ
When the load is purely resistive
Z Z L SWR or ZL Z 0
0
A cable has a VSWR of 10. If the minimum voltage along the cable is 20 volts, what is the maximum voltage along the cable?
Ans: v = 200 V
A lossless line has a characteristic impedance of 50 ohms, but is terminated with a 75-ohm resistive load. What SWR do you expect to measure?
Ans: SWR = 1.5
If a cable has an SWR of 1.5, what will be the absolute value of its voltage coefficient of reflection?
Ans: Γ = 0.2
The reflected power as a function of the incident power is given by the equation
Pr Γ Pi 2
SWR
1
Pr Pi
1
Pr Pi
so that Γ
Pr
Pi
condition ideal worst case
Γ
SWR
0
1
1
∞
This is the difference between the incident and the reflected power.
PL Pi 1 Γ PL
2
4 SWR 2
1 SWR
Pi
100 percent of the source power does not reach the load Corona can be produced due to excessive dielectric heating caused by a high value SWR. Reflection and subsequent reflections cause more power loss. Reflection causes ghost image and interference.
A generator matched to a line with a voltage coefficient of reflection equal to 0.2 transmits 100 watts into the line. How much power is actually absorbed by the load?
Ans: PL = 96 W
ZL Z tanh l Zin Z Z ZL tanh l 0
0
0
For lossless lines
ZL jZ tanh l Zin Z Z jZL tanh l 0
0
0
If ZL = Z0, then Zin = Z0
For lossless lines whose l = λ/4 2
Z Z in Z L 0
Length
Input Equivalent Phase shift Description Impedance circuit in degrees
L < λ/4
capacitive, decreases with length
– 90
L = λ/4
series resonant
0
L > λ/4
minimum
purely resistive
Inductive, Increases with length
90
Length
Input Equivalent Phase shift Description Impedance circuit in degrees
L < λ/4
inductive, increases with length
90
L = λ/4
parallel resonant
0
L > λ/4
maximum
purely resistive
capacitive, decreases with length
– 90
Length
OPEN CIRCUIT
SHORT CIRCUIT
L < λ/4
capacitive C decreases with length
inductive L increases with length
L = λ/4
series LC Zi is resistive & minimum
parallel LC Zi is resistive & maximum
L > λ/4
inductive L increases with length
capacitive C decreases with length
The length of a transmission line in wavelengths as opposed to its actual physical length. Lines whose length is not equal to a wavelength produces a phase delay equal to
360L
If a cable has a velocity factor of 0.8, what length of cable is required for a 90° phase shift at 100 MHz?
Ans: l = 0.6 m
Major transmission line losses: Conductor or copper loss Radiation loss Dielectric loss Coupling loss
when a signal passes through a conductor, some energy is lost in the form of heat. This loss is called conductor loss. This is proportional to the square root of the line length and inversely proportional to the impedance. Conductor loss is frequency dependent At high frequencies, I 2R loss is generally due to skin effect.
A phenomenon that occurs at high frequencies where the current flows on the surface of the conductor. This is because of the higher reactance of the conductor at its center. skin effect increases with frequency
The electromagnetic field and the electrostatic field cause the conductors to act like an antenna and radiate energy. Radiation loss is directly proportional to the frequency. Can be minimized by properly shielding the cable
The potential difference between the conductors in a transmission line causes dielectric heating. This loss is called dielectric loss and is proportional to the voltage across the dielectric. This loss is negligible for air dielectrics and it increases with frequency.
How do you call the luminous discharge that occurs between the conductors of a transmission line when the potential difference between them exceeds the breakdown voltage?
Arcing B. Spark C. Corona D. Photodielectric effect A.
This occurs in transmission lines that are connected together Discontinuities tend to heat up, radiate energy and dissipate power.
Whenever there is a mismatch, standing waves are created on the line. A higher value of SWR means a greater degree of mismatch and hence, greater loss. To minimize reflection and losses on the line, the load and the line impedance should be matched. This is called impedance matching. This can be done in two ways: Quarter wavelength transformer Stub matching
Used to match a line to a purely resistive load Z0 ≠ ZL Zin
Z0
ZL
Z0
Z
RL
Z Z RL 0
λ/4
A quarter wavelength transformer is a quarter wavelength section of transmission line that acts like a transformer.
Used when the load is purely reactive or a complex impedance. A transmission line stub is an additional transmission line connected in parallel with the line used to cancel the susceptance of the load. Open or short stubs can be used but shorted stubs are preferred. Half wavelength lines or shorter are used.
A Smith Chart is used to calculate
Transmission line impedances B. SWR and reflection coefficient C. Optimum length of a transmission line D. Transmission line losses A.
