Transmission lines are nothing but guided conducting structures which are used in power distribution at low frequencies, in communications and computer networks at higher frequencies.
The main aim of this chapter is to provide the overall concepts of Transmission Line theory. They include:
A transmission line is a means of transfer of information from one point to another. Usually it consists of two conductors. It is used to connect a source to a load. The source may be a transmitter and the load may be a receiver.
The various types of transmission lines are
Typical configurations of the above are shown in Fig. 7.1.
(a) Two-wire parallel line
(b) Coaxial line
(c) Twisted pair of lines
(d) Planar line
(e) Wire above conducting plane
(f) Microstrip line
(g) A typical step-index optical fibre
Fig. 7.1 Types of transmission lines
Applications of transmission lines are:
The equivalent circuit of a transmission line is a distributed network. This consists of cascaded sections and each section consists of a series Resistance R, series Induction L, shunt Capacitance C, and shunt conductance G. One section of the equivalent circuit is shown in Fig. 7.2. Here R is expressed in ohm/unit length, L in Henry/unit length, C in Farad/unit length and G in Mho per unit length.
Fig. 7.2 Equivalent circuit of a two-conductor transmission line
Electric and Magnetic Fields in Parallel Plate and Coaxial Lines
In parallel plate transmission, if z is the direction of propagation, the electric and magnetic field distributions are shown in Fig. 7.3.
Fig. 7.3 E and H in a parallel plate transmission line
The E and H fields in a coaxial line are shown in Fig. 7.4.
Fig. 7.4 E and H fields in a coaxial line
The R (Ω /Km), L (H/Km), C (F/Km) and G (mho/Km) are known as primary constants.
Salient aspects of primary constants:
The relations are presented for quick reference.
For parallel wires
where |
σc |
= |
conductivity of conductors |
|
a |
= |
radius of wire |
|
|
|
|
|
d |
= |
spacing between wires |
|
∊d |
= |
dielectric constant of the dielectric material |
|
μr |
= |
relative permeability of the conductor material |
|
|
= |
1 for non-magnetic material |
Internal inductance (Li) is due to internal flux linkages in the conductors.
It is
External inductance (Le) is due to flux linkages with the flux external to the wire.
For coaxial cable
where |
a |
= |
inner radius |
|
b |
= |
outer radius |
|
t |
= |
outer thickness |
For parallel plates
where |
w |
= |
width |
|
d |
= |
separation |
Consider Fig. 7.5 in which a line of length l, voltage and current at source end, Vs and Is are shown. The voltage, VL and current IL at the load end are also shown.
Fig. 7.5 A transmission line with load, ZL
Let V and I be the voltage and current on the line at any arbitrary location. Assume Vf and If are for the forward wave and Vr and Ir are for the reflected wave. Then, we can write
Differentiating and combining, we get
where |
γ2 |
= |
(R + jωL) (G + jωC) |
or, |
γ |
= |
|
|
|
= |
propagation constant |
And the series impedance
The shunt admittance
The solutions of Equations (7.1) and (7.2) are either in exponential form or in hyperbolic function form. In the first form,
These represent the sum of the forward and reflected waves.
The characteristic impedance,
z0 is related to R, L, C and G and is given by
If the line is terminated in a load impedance of zL, then
or, zL Vf − zL Vr = z0 Vf + z0 Vr
Dividing both sides by Vf, we get
or,
or,
The reflection coefficient,
Input impedance zi is defined as the ratio of voltage and current at the sending end.
In hyperbolic function form, the solutions of Equations (7.1) and (7.2) are given by
The constants, A1, A2, B1 and B2 are found using the boundary conditions, that is,
Then
zL is chosen at x = 0 and if l = −x, we have
Here, l is measured from the load end. From Equations (7.3) and (7.4), we have
If the line is short circuited at the receiving end, zL = 0 and VL = 0. Then the input impedance is
Now if zL = ∞, IL = 0, then the input impedance is
or,
The secondary constants are
Propagation constant, γ and
Characteristic impedance, z0
Propagation constant
Definition 1
Definition 2
Definition 3
Definition 4 γ ≡ α + jβ
where |
α |
= |
attenuation constant, dB/m |
|
β |
= |
phase constant, rad/m |
It may be noted that
Consider
From Equations (7.5) and (7.6), we have
or,
From Equations (7.6) and (7.7), we get
or,
Characteristic impedance, z0
Definition 1 The characteristic impedance, z0 of a line is defined as the ratio of the forward voltage wave to the forward current wave at any point on the line,
that is,
Definition 2 z0 is also defined as the ratio of the square root of series impedance to the square root of shunt admittance, or,
Definition 3 z0 is defined as the minus of the ratio of the reflected voltage wave to the reflected current wave at any point on the line, or,
A transmission line is said to be lossless if the conductors of the line are perfect, or, σc = ∞ and the dielectric medium between the lines is lossless, or, σd = 0. Also, a line is said to be lossless, if
For lossless line,
As R = 0, G = 0
or,
As R = 0 = G
The velocity of propagation in lossless line
A transmission line is said to be distortionless when the attenuation constant, α is frequency-independent and the phase constant, β is linearly dependent on the frequency or when
Consider
If
Consider
The velocity of propagation for distortionless line is
The overall transmission line characteristics are shown in Table 7.1.
Table 7.1 Propagation Parameters for Different Types of Lines
Energy is propagated along a transmission line in the form of Transverse Electromagnetic wave (TEM wave). The phase velocity, υp for TEM wave is
For a transmission line, μr = 1, but ∊r may be different. Then
For a lossless and distortionless transmission line,
Individual waves propagate with the same phase velocity if β is proportional to ω. If β is not proportional to ω and if the wave components travel with different velocities, the envelope of the wave travels with a velocity, known as group velocity, υg, that is,
This is also the velocity at which energy is propagated along the line.
Introduction of inductance in series with the line is called loading and such lines are called loaded lines.
Effect of loading
This is shown in Fig. 7.6.
Fig. 7.6 Effect of loading on the cable
Types of loading
For lossless line,
and as tanh jβl = j tan βl and z0 = R0 , zi becomes
For shorted line, zL = 0
For open circuited line, zL = ∞
and
Input impedances of a transmission line for different cases are given in Table 7.2.
Table 7.2 Input Impedance for Different Loads
Type of line | Input impedance, zi |
---|---|
Lossy line |
|
Lossless line |
|
Lossy line with shorted load (zL = 0) |
z0 tanh β l |
Lossy line with open circuited load (zL = ∞) |
z0 coth βl |
Lossy line with matched load (zL = z0) |
z0 |
Lossless line with shorted load |
j z0 tan βl |
Lossless line with open circuited load |
− j z0 cot βl |
Lossless line with matched load |
z0 |
The variation of input impedance of lossless line when shorted and open circuited are shown in Fig. 7.7.
(a) Shorted load
(b) Open circuited load
Fig. 7.7 Input impedance variation
At radio frequencies,
|
ωL >> R |
|
ωC >> G |
Then, |
Z = R + jωL ≈ jωL |
|
Y = G + jωC ≈ jωC |
|
|
and |
|
|
As γ = α + jβ, α ≈ 0, , it is not enough if α is small compared to β. Hence let us consider,
For RF lines, the input impedance is
where |
ρ |
= |
reflection coefficient |
|
zL |
= |
load impedance |
|
z0 |
= |
characteristic impedance |
Proof For lossless RF lines, we have
Similarly,
At the load, (l = 0)
I at any point from load is
Load current, IL
But by definition, the reflection coefficient, ρ is
and
Dividing both sides by Vi, we get
If |
zL |
= |
z0, ρ = 0 |
|
zL |
= |
0, ρ = −1 |
|
zL |
= |
∞, ρ = 1 |
|
zL |
= |
purely reactive, |ρ|= 1 |
VSWR is defined as
Proof We can write
Similarly,
We can also write |ρ| as
When the line is terminated by purely resistive load,
By selecting a terminated line of suitable length, it is possible to produce the equivalent of a pure resistance, inductance and capacitance or any desired combination thereof.
Equivalent circuits for shorted and open lines are shown in Table 7.3.
Table 7.3 Equivalent Circuits for Shorted and Open Lines
Length of line | Equivalent circuit element | |
---|---|---|
Shorted line | Open line | |
Inductor |
Capacitor |
|
Tank circuit |
Series-resonant circuit |
|
Capacitor |
Inductor |
|
Series-resonant circuit |
Tank circuit |
In Table 7.4, lines of different length with short and open ends along with their equivalent circuits are shown.
Table 7.4 Equivalent Circuit and Impedance
Transmission line | Equivalent circuit | Input impedance |
---|---|---|
zi = + jz0 tan βl |
||
zi = + jz0 cot βl |
||
zi = + jz0 tan βl |
||
zi = + jz0 cot βl |
||
Losses in transmission lines are of three types:
Copper Loss
These losses occur because of the following reasons:
Dielectric Losses
These losses exist due to improper characteristics of dielectric.
Salient features:
Radiation Losses
Salient features:
Problem 7.1 A transmission line with air as dielectric has z0 = 50 Ω and a phase constant of 3.0 rad/m at 10 MHz. Find the inductance and capacitance of the line.
Solution A line with air dielectric is lossless as σ = 0.
and
or,
As
Problem 7.2 A lossy cable which has R = 2.25 Ω/m, L = 1.0 μH/m, C = 1pF/m, and G =0 operates at f = 0.5 GHz. Find the attenuation constant of the line.
Solution The propagation constant is given by
Problem 7.3 A transmission line in which no distortion is present has the following parameters: z0 = 50 Ω, α = 0.020 m-1, υ = 0.6υ0. Determine R, L, G, C and wavelength at 0.1 GHz.
Solution For a distortionless line the condition is
and
and hence
Problem 7.4 For a transmission line which is terminated in a normalised impedance zn, VSWR = 2. Find the normalised impedance magnitude.
Solution Normalised impedance, zn
or,
We have
Problem 7.5 A lossless transmission line used in a TV receiver has a capacitance of 50 pF/m and an inductance of 200 nH/m. Find the characteristic impedance for sections of a line 10 metre long and 500 metre long.
Solution The characteristic impedance of a lossless transmission line is
The inductance, L of the line
For 10 m line,
|
L |
= |
200 × 10−9 × 10 = 2000 × 10−9 |
|
|
= |
2 × 10−6 H |
|
C |
= |
50 pF/m |
For 10 m line,
The characteristic impedance, z0
The inductance of 500 m line
|
L |
= |
200 × 10−9 × 500 |
|
|
= |
10,0000 × 10−9 |
|
L |
= |
10−4 H |
The capacitance of 500 m line
C | = | 50 × 10−12 × 500 | |
= | 25000 × 10−12 = 25 × 10−9 F |
The characteristic impedance, z0 of 500 m line
Problem 7.6 A two-wire open air line, whose diameter is 2.588 mm, is used in several applications. The wires are spaced at 290 mm between the centres. Find out the characteristic impedance of the line.
Solution Radius of the wire
Spacing between the wires is
The characteristic impedance of the two-wire open air line is
Problem 7.7 A copper coaxial line has an outside tubing of thickness 1.8 mm and its outside diameter is 30 mm. The thickness of the inner tubing is 1.0 mm and its outside diameter is 8 mm. Find the characteristic impedance of the line.
Solution Diameter of the outside conductor is
Diameter of the inner conductor is
For a coaxial cable, Z0 is
Problem 7.8 If a signal of 30 MHz is transmitted through a coaxial cable which has a capacitance of 30 pF/m and an inductance of 500 nH/m. (a) Find the time delay for a cable 1 m long, (b) propagation velocity, and (c) propagation delay over a cable length of 10 m.
Solution
The capacitance of 10 metre cable is
The time delay,
Problem 7.9 The attenuation coefficient of a transmission line is 0.2mNp/m. Find the attenuation coefficient in (a) dB/m (b) dB/mile.
Solution
Attenuation coefficient, in dB/m is
The attenuation coefficient, α in dB/mile is
Problem 7.10 A lossless transmission line is terminated in a load impedance of 30 – j 23 Ω. Find the phase constant and the reflection coefficient of a line of length 50 m. Characteristic impedance, z0 = 50Ω. Wavelength on the line = 0.45 m.
Solution
|
zL |
= |
(30 − j23) Ω |
|
z0 |
= |
50 Ω |
|
λ |
= |
0.45 m |
|
l |
= |
50 m |
Phase constant,
Reflection coefficient,
Problem 7.11 A coaxial cable has z0 of 75 Ω and a capacitance of 70 pF/m. Find its inductance per metre. If the radius of the inner conductor is 0.292 mm and the relative permittivity of the dielectric is 2.3, determine the radius of the outer conductor.
Solution Radius of the inner conductor,
We have
or,
For a coaxial cable, z0 is also given by
or,
where r0 = radius of the outer conductor.
Problem 7.12 A lossless transmission line of length 100 m has an inductance of 28μH and a capacitance of 20 nF. Find (a) propagation velocity (b) phase constant at an operating frequency of 100 kHz (c) characteristic impedance of the line.
Solution Length of transmission line,
Inductance of the line
Inductance/metre
Capacitance of the line = 20 nF
Capacitance per metre
Problem 7.13 The dielectric material between two conductors of a lossless coaxial cable has ∈r = 4 and μr = 1. Diameter of the inner conductor is 2 mm. Characteristic impedance of the 10 m long cable is 50Ω. Determine the diameter of the outer conductor of the coaxial cable.
Solution The expression for inductance and capacitance of a coaxial cable is
and
where |
μ |
= |
permeability = μ0 μr |
|
∈ |
= |
permittivity = ∈0∈r |
|
d0 |
= |
diameter of outer conductor |
|
di |
= |
diameter of inner conductor |
The expression for the characteristic impedance, z0 is
Here
Problem 7.14 A transmission line is lossless and is 25 m long. It is terminated in a load of zL = 4o + j30Ω at a frequency of 10 MHz. The inductance and capacitance of the line are L = 300 nH/m, C = 40 pF/m. Find the input impedance at the source and at the mid-point of the line.
Solution The length of transmission line = 25 m.
Load impedance,
Inductance,
Capacitance,
Characteristic impedance,
Phase constant, β
Input impedance at the source end is
Similarly, input impedance at 12.5 m from source end is
Smith chart is a polar plot of the reflection coefficient in terms of normalised impedance, r + jx. In other words, it is a graphical plot of normalised resistance and reactance in the reflection coefficient plane.
It was designed by Phillip H. Smith in 1939.
Construction of Smith Chart
It is constructed within a circle of unit radius (| ρ | ≤ 1) as in Fig. 7.8.
Fig. 7.8 Construction of Smith chart on unit circle
Smith chart provides the relation between reflection coefficient, ρ load, zL and characteristic impedance, z0. Here, the impedances are always normalised with respect to characteristic impedance,
|
|
or, |
ρ = |ρ| ∠θρ = ρr + jρi |
Smith charts are constructed in terms of normalised impedances (zL / z0) to avoid construction of one chart for each z0. Normalised load impedance is
or,
or,
and
The Equations (7.8) and (7.9) are the equations of circles like
Equation (7.8) is known as r-circle and Equation (7.9) is known as x-circle.
The resistance circle has centre at and radius of as in Fig. 7.9.
Fig. 7.9 r-circles
The reactance circle has centre at and radius as shown in Fig. 7.10.
Fig. 7.10 The reactance circles
Salient features of r and x-circles
values of normalised resistance along the line.
values of normalised line reactance,
Applications of Smith Chart
It can be used to:
find the parameters of mismatched transmission lines
find normalised admittance from normalised impedance or vice-versa
find VSWR for a given load impedance
design stubs for impedance matchings
locate a voltage maximum on the line
find the input impedance of a transmission line.
A stub is a piece of transmission line. It can be short circuited at the far end or open circuited. It has a pure reactance or susceptance. It is used to cancel out reactance or susceptance of a transmission line. In other words, it is used for impedance matching.
In general, shorted stubs are more frequently used since open ended stubs tend to radiate. The design parameters of stubs are (1) stub length and (2) stub distance from the load. The matching of transmission lines is done by the design of a single stub or a double stub.
Design of Single Stub Matching
The design consists of the following steps:
A typical stub is shown in Fig. 7.11.
Fig. 7.11 Shorted stub connected to a transmission line
Problem 7.15 Find the input impedance of 75Ω lossless transmission line of length 0.1λ when the load is a short.
Solution Load impedance,
Characteristic impedance,
Length of the line = 0.1λ
The normalised load impedance is
Steps Involved
Fig. 7.12 Lossless transmission line where load is short
Analytical method The expression for zi (for lossless line) is
If
Problem 7.16 Find the input impedance of a 75Ω lossless transmission line of length (0.1λ) if it is terminated in open circuit.
Solution
|
l |
= |
0.1λ |
|
zL |
= |
∞ (open) |
|
z0 |
= |
75 Ω |
The expression for input impedance is
If the load is open circuit,
Smith chart method
Fig. 7.13 Lossless transmission line terminated in open circuit
Problem 7.17 A transmission line of length 0.40λ has a characteristic impedance of 100Ω and is terminated in a load impedance of 200 + j180Ω. Find the
Solution The data is
(a)
Hence the reflection coefficient is
(b) The reflection coefficient |ρ| = 0.591 circle meets the positive real axis opoc at r = 4, that is,
Fig. 7.14
(c) To determine zi,
Normalised input impedance
Input impedance
Problem 7.18 Design a stub to match a transmission line which is connected to a load impedance of zL = (450 – j 600) Ω. The characteristic impedance of the line is 300 Ω. The operating frequency is 20 MHz.
Solution
The stub distance |
= |
(0.181 0.053) λ = 0.128λ |
|
= |
0.128 × 15 = 1.92 m |
Stub length |
= 0.0842λ |
|
= 0.0842 × 15 = 1.263 m |
Fig. 7.15
The designed stub parameters are:
For the design of any device, it is convenient to have more parameters in designer’s control for more freedom. For this purpose, to match the load with the transmission line, a second stub of adjustable position is included. A typical double stub is shown in Fig. 7.16.
Fig. 7.16 Double stub transmission line
It is also possible to use a triple stub tuner for more design convenience.
In single stub matching the stub is placed on the line at a specified point. Its location varies with zL and frequency. This creates some difficulties as the specified point may occur at an undesirable location. In such cases, double stubs are used. Here the distance between them is fixed such as and so on and the lengths of the two stubs are adjusted to match the load.
Design Methodology
Step 1 |
Fix the distance between the two stubs and keep stub 1 at the location of the load. |
Step 2 |
Draw the circle corresponding to the normalised conductance, g = 1. |
Step 3 |
Obtain the normalised distance of . |
Step 4 |
Rotate the circle in anticlockwise direction by wavelengths towards the load and draw. The point which represents y1 is located here. |
Step 5 |
Locate yL = gL + jbL. |
Step 6 |
Draw g = gL circle. This intersects the rotated g = 1 circle at one or two points where yL = gL + jb1. |
Step 7 |
Locate the corresponding y2 points on the g = 1 circle. y2 = 1 + jb2. |
Step 8 |
Find the stub length l1 between the points representing yl and yL. |
Step 9 |
Find the stub length l2 from the angle between the point representing –jb2 and psc. |
The stub distances from the load need not be found as d is fixed.
In Table 7.5 given on the next page, propagation characteristics of EM waves in free space, in waveguides and in transmission lines are compared.
Table 7.5 Comparison between the Propagation Characteristics of EM Waves in Free Space, Waveguides and Transmission Lines
Fig. 7.17
1. Yes |
2. Yes |
3. Yes |
4. Yes |
5. Yes |
6. Yes |
7. Yes |
8. Yes |
9. No |
10. No |
11. Yes |
12. Yes |
13. Yes |
14. Yes |
15. Yes |
16. No |
17. Yes |
18. Yes |
19. Yes |
20. TEM wave |
21. Two or more conductors
22. If the conductors are separated by the same dielectric, if they have same cross-sectional area along the length of the line and if they have same dimensions
23. A few mm
24. Smith chart
25. If the conductors have σ = ∞ and if the dielectric between the conductors has σ = 0
26. z0 tanh rl
27.
28. zi = zL
29. z0 changes at the point
30.
31.
32. 1 + ρ
33.
34. A piece of transmission lined
35. One
36.
37. The ratio
38. The time taken for the wave to travel from one end to the other
39. l = length of line, u is the velocity of propagation of the wave
40. 1/3
41.
42.
43.
44. Lattice diagram
45. A time-distance diagram
46. PCBs
47.
48.
49.
50.
51. α + jβ
1. (a) |
2. (a) |
3. (a) |
4. (b) |
5. (a) |
6. (b) |
7. (b) |
8. (a) |
9. (b) |
10. (c) |
11. (b) |
12. (c) |
13. (a) |
14. (b) |
15. (a) |
16. (c) |
17. (b) |
18. (a) |
19. (c) |
20. (a) |
Determine the characteristic impedance if the operating frequency is 1.59 kHz.
Find the attenuation constant, phase constant and phase velocity.