CHAPTER 1
TRANSMISSION LINE
1.1
INTRODUCTION
A transmission line is a structure used to guide the flow of electromagnetic energy from one point to
another point. This line may be of any physical structure; that is, it may be made of two parallel wires
or two parallel plates or coaxial conductors, or it may be of hollow conductor variety (waveguides).
The general characteristics of electromagnetic wave propagation in these lines are the same. The
preference depends only on the frequency of wave propagation and the use to which these lines are put.
1.2
BASIC TRANSMISSION LINE EQUATIONS
In general, if we examine a transmission line, we will find four parameters, i.e., series resistance (R),
series inductance (L), shunt capacitance (C) and shunt conductance (G), distributed along the whole
length of the line. If R, L, C and G be these primary constants per unit length of the line, then the unit
length of the line may be represented by an equivalent circuit of the type shown in Fig. 1.1. Naturally,
a relatively long piece of line would contain several such identical sections as shown in Fig. 1.2.
R/2
L/2
L/2
G
R/2
C
Fig. 1.1 Equivalent circuit of a unit length of transmission line
1
2
BASIC MICROWAVE TECHNIQUES AND LABORATORY MANUAL
First section
Second section
Third section
Fig. 1.2 A long piece of line as a multi T-section line
The series impedance and shunt admittance per unit length of the line are given by:
Z = R + jωL
Y = G + jωC.
(1.1)
(1.2)
The expressions for voltage and current per unit length are, respectively,
dV
= – (R + jωL)I
(1.3)
dz
dI
= – (G + jωC)V
(1.4)
dz
where negative sign indicates decrease in voltage and current as z increases. The current and voltage are
measured from the receiving end; i.e., at receiving end, z = 0 and line extends in negative z-direction.
Differentiating Eqs. (1.3) and (1.4) and combining them,
d 2V
= γ2 V
dz 2
(1.5)
d2I
= γ2 I
dz 2
These are wave equations of voltage and current respectively propagating on the line; where
and
γ = ZY = ( R + jωL ) ( G + jωC )
(1.6)
(1.7)
is called the propagation constant which is in general a complex quantity and so may be difined as
γ = α + jβ
γ = α2 + β2
(1.8)
α, called the attenuation constant, is the real part of Eq. (1.7), and β, the phase constant is the imaginary part. Thus, propagation constant γ is a measure of the phase shift and attenuation per unit length
along the line. Separating γ into real and imaginary parts, we have,
L (R + ω
α=M
MN
L (R + ω
β=M
MN
2
2
and
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2
2
2
OP
PQ
LC ) O
PP
Q
L2 )(G 2 + ω 2 C 2 ) + ( RG − ω 2 LC )
L2 )(G 2 + ω 2 C 2 ) − ( RG − ω 2
2
1/ 2
(1.9)
1/ 2
(1.10)
3
TRANSMISSION LINE
α is measured in decibels or nepers per unit length of the transmission line (1 neper = 8.686 decibels).
β is the phase shift per unit length of transmission line and is measured in radians per unit length of this
line. Now
(1.11)
β = 2π/λl or λl = 2π/β
where λl is the distance along the line corresponding to a phase change of 2π radians. The phase
velocity Vp = fλl, where f is the signal frequency.
The solutions of voltage and current wave Eqs. (1.5) and (1.6) may be written as
V = V1e–γz + V2e+γz
→+z –z←
I = I1e–γz + I2e+γz.
→+z –z←
(1.12)
(1.13)
These solutions are shown as the sum of two waves; the first term indicates the wave travelling
in positive z-direction, i.e., incident wave, and the second term indicates the wave travelling in the
negative z-direction, i.e., reflected wave.
1.3
CHARACTERISTIC IMPEDANCE
A voltage (rf) applied across the conductors of an infinite line causes a current I to flow. By this
observation, the line looks like an impedance which is denoted by Z0 and is known as characteristic
impedance Z0.
Z0 =
V1
.
I1
The expression for current I, using Eqs. (1.3) and (1.4), is given by
I=
1
∂V
−1
.
=
(– γ) (V1 e–γz – V2e+γz)
( R + jωL ) ∂z ( R + jωL )
(1.14)
I = I1e–γz – I2eγz (Phase reversed due to reflection).
For infinite line there are no reflections, that is , V2 and I2 are zero. So we have
I1e–γz =
γV1e − γz
( R + jωL)
or
V1
=
I1
R + jω L
( R + jωL)(G + jωC )
or
Z0 =
R + jωL
= R0 + jX0
G + jωC
(1.15)
Where R0 and X0 are the real and imaginary parts of Z0. R0 should not be mistaken for R while R
is in ohms per metre; R0 is in ohms.
For loss-less line (R and G being zero)
Z0 =
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L /C
and β = ω LC
(1.16)
4
BASIC MICROWAVE TECHNIQUES AND LABORATORY MANUAL
We can see that the finite transmission line terminated by its Z0 [Fig. 1.3(a)] has input impedance also equal to Z0, that is, the finite line of characteristic impedance Z0 has an input impedance Z0
when it is terminated in Z0. A line terminated in its characteristic impedance will absorb all the power
and there will be no reflection and hence it behaves as an infinite line.
Z0
Z in = Z 0
VS
VR
(a)
ZL
IR
(b)
Fig. 1.3 (a) Finite transmission line
terminated by its Z0
Fig. 1.3 (b) Finite transmission line terminated
in an impedance Z L
One can obtain the expression for input impedance of line when it is terminated in an impedance
ZL [Fig. 1.3(b)] located at z = 0 as
Zin =
Vs VR cosh γz + Z0 I R sinh γz
=
V
Is
I R cosh γz + R sinh γz
Z0
Zin = Z0
where
LM Z
NZ
L
0
+ Z0 tanh γz
+ Z L tanh γz
Vs = Voltage at the sending end
Is = Current at the sending end
z = Length of the line
Z0 = Characteristic impedance
VR = Voltage at the receiving end
IR = Current at the receiving end
(1.17)
OP
Q
UV at z = 0.
W
If the line is short-circuited (ZL = 0), we have short-circuited input impedance, Zsc, given by
(VR = 0),
Zsc = Z0 tanh γz
(1.18)
The open-circuited input impedance (ZL = ∞), Z0c, can be found by putting zL = ∞ and IR = 0 in
Eq. (1.17),
Zoc = Z0 coth γz
(1.19)
The product of Eqs. (1.18) and (1.19) gives
Z0 =
1.4
zoc + Z sc
(1.20)
LUMPED CONSTANT DELAY LINE
In laboratory we can study the general characteristics of a transmission line of given primary constants
using a number of lumped T-sections as shown in Fig. 1.2. Such an artificial line is known as lumped
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5
TRANSMISSION LINE
constant delay line. Such lines are made to simulate the actual transmission line and when operated in
the audio-frequency range, they can be made very compact. In actual experiment, 20 to 25 such Tsections are used.
EXPERIMENT 1.1
To determine the characteristic impedance of lumped constant delay line.
EQUIPMENT
A lumped constant delay line board having 20 to 25 T-sections
Audio-oscillator
Resistance box.
From Eq. (1.20) it is clear that the determination of Z0 reduces to the determination of Zsc
and Zoc.
(a) Load-end Short-circuited (ZL= 0)
Let Vsc be the input voltage (can be measured) to the line and VRsc be the voltage across the
series resistance, R then
and
Zsc =
Vsc
I sc
Isc =
VRsc
R
Zsc =
Vsc
R.
VRsc
(1.21)
(b) Load-end Open-circuited (ZL = ∞)
Similarly, if Voc be the input voltage and VRoc be the voltage across the series resistance R in the
case when line is open-circuited, we have
Zoc =
Voc
R
VRoc
(1.22)
From Eqs. (1.21) and (1.22),
Z0 = R
Voc × Vsc
VRoc × VRsc
(1.23)
All the quantities in Eq. (1.23) can be determined experimentally; hence one can determine Z0
and can compare it with the calculated value obtained using Eq. (1.15).
PROCEDURE
1. Make the connections as shown in Fig. 1.4(a).
2. Set audio frequency (af) oscillator at 1.5 kHz and its output voltage at a suitable level, say 2 V.
3. Measure the voltage across resistance box (RB) and at the input of lumped delay line as shown
in Fig. 1.4 when the load-end is (i) open-circuited and (ii) short-circuited.
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6
BASIC MICROWAVE TECHNIQUES AND LABORATORY MANUAL
V.T.V.M.
V.T.V.M.
Resistance
box
Single
generator
Transmission
line
To load end
(a)
VR
VC
R
C
V ZC
ZC
VZ
Output
end
Transmission
line
(b)
VR
R
V
p/2
ZC
VC
VZ
p/2
(j L ¢ )
Reactive
(Inductive
in present
case)
(R ¢ ) Resistive
Z SC (or Z OC ) = (R ¢ + j L ¢ ) V R /R
(c)
Fig. 1.4 (a) Circuit arrangement for measuring input impedance.
(b) Circuit arrangement for measuring input impedance when complex.
(c) Vector plot of measured voltages for measuring complex input
impedance.
4. Vary the resistance R and repeat step 3.
5. Record observation in Table 1.1.
TABLE 1.1
S.
No.
Load-end
OPEN-CIRCUIT
Resistance
Load-end
SHORT-CIRCUIT
from
RB
Voltage
across RB
Voltage at input
of delay line
Voltage
across RB
Voltage at input
delay line
R
(VRoc)
(Voc)
(VRsc)
(Vsc)
1.
2.
3.
4.
5.
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7
TRANSMISSION LINE
CALCULATIONS
Z0 = R
Voc × Vsc
VRoc × VRsc
ohms (Ω).
(See eqn. 1.23)
However, Z0 is in general complex and can be measured using a circuit of Fig. 1.4(b). Voltages
across condenser (Vc) and resister (VR) are π/2 out of phase. Assuming no reflections, VR, Vc and line
voltage Vz follow the vector plot shown in Fig. 1.4(c). Hence they allow the determination of reactive
and resistive components of Zoc, when line is open-circuited, or of Zsc when short-circuited.
The procedure shall be as follows:
1. Make the connections as shown in Fig. 1.4 (b) with line open-circuited.
2. Measure voltages Vc , VR , Vzc and Vz with audio-oscillator set for 1.5 kHz and suitable
voltage level, say 2V.
3. Draw a vector plot shown in Fig. 1.4(c) and measure resistive and reactive components of
voltage Vz. Divide Vz with the current (VR/R) to get Zoc.
4. Repeat steps 1 to 3 for short-circuited line to get Zsc.
5. Find out Zo using Eq. (1.20).
6. Repeat observations, for various values of R and C.
In calculations, of course, complex algebra has to be used.
Note: If af or rf milliammeter is available, it can be in series arm. Then resistance box is not necessary.
One can get directly
Zoc =
and
Z0 =
Voc
,
I oc
Zsc =
Vsc
I sc
Z oc × Z sc
EXPERIMENT 1.2
To study voltage distribution along a lumped constant delay line in the cases when it is (i) opencircuited, (ii) short-circuited and (iii) terminated in Z0 and hence determine α, β, γ and λl.
EQUIPMENT
A lumped constant delay line having 20 to 25 sections
Audio-frequency oscillator
VTVM.
Any voltage wave travelling down the line is continuously attenuated if the line is terminated in
Z0, the characteristic impedance of the line. The voltage at nth section, Vn, is given by
Vn = Vse–γn
where Vs is the voltage at the sending end
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8
BASIC MICROWAVE TECHNIQUES AND LABORATORY MANUAL
− αn − jβn
.
Vn = Vs e e
or
Voltages, in the preceding equation are the amplitudes or peak values of sinusoidally varying
functions.
Vn = Vse–αn
or
giving us
α = 2.3026 log10
Vs
Vn
(1.24)
Measuring Vs, Vn and n, we can determine α, the attenuation per section.
To determine β, one has to plot (Fig. 1.6) voltage at each section of the line in the cases when the
line is open-circuited, short-circuited and terminated with Z0. The distance (No. of sections) between
first and third minima, second and fourth minima, third and fifth minima... gives λ1 in each case, then
β may be computed using Eq. (1.11) and γ from Eq. (1.8).
PROCEDURE
1. Make connections as shown in Fig. 1.5.
2. Set of oscillator at 1.5 kHz and its output at a suitable level, say 2V.
Z0
V
T
V
M
V
T
V
M
Open circuited
end
ZL = ¥
Short circuited
end
ZL = 0
Fig. 1.5 Measurement of parameters on a transmission line terminated in various loads
3. Measure voltage across each section of the line by connecting VTVM when
(a) line terminated in Z0
(b) load-end is open-circuited (ZL = ∞)
(c) load-end is short-circuited (ZL = 0).
Typical graphs are shown in Fig. 1.6.
4. Record observations in Table 1.2.
5. Plot voltages versus number of T-sections for the cases: (a), (b) and (c) on the same graph.
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TRANSMISSION LINE
1.2
1.0
Voltage in volts
0.8
Load end short circuited
0.6
0.4
Terminated
in Z 0
0.2
Load end open circuited
0
4
8
12
16
20
No. of sections of lumped constant delay line
Fig. 1.6 Voltage distribution on a line when short-circuited, open-circuited case
and terminated in characteristic impedance (calculated value)
CALCULATIONS
(i) α =
=
1
2.3026 (log10 Vs – log10 Vn) nepers/section
n
2.3026
(log
n
10
Vs – log10 Vn) × 8.686 decibels/section.
(ii) λ1 = Distance (No. of sections on the line) between alternate minima.
For a given line, sections may be converted into equivalent length.
(iii) β =
(iv) γ =
2π
radians per section.
λl
α 2 + β 2 (no units).
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BASIC MICROWAVE TECHNIQUES AND LABORATORY MANUAL
TABLE 1.2
S.
Section
Voltage across sections of lumped delay line in volts
No.
No.
when load-end is
1
2
Open-circuited
Short-circuited
Terminated with Z0
(calculated value)
3
4
5
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
...
...
...
20.
RESULTS
S.
No.
Quantity
Experimental
value
Theoretical
value*
percent
error
1.
α
...
...
...
2.
β
...
...
...
3.
γ
...
...
...
4.
λl
...
...
...
*These quantities are to be calculated from the various equations defining the respective quantities in terms of primary constants R, L, G and C.
CRITICISM
1.
2.
3.
4.
5.
6.
Why loops are not symmetric?
Why experimentally observed value of Z0 does not agree with the calculated value?
Comment on ‘loss-lessness’ of a practical line.
Comment on design considerations of a lumped delay line.
What should be the shape of a UHF line? and why?
Why minima of voltage along the sections of the line are preferred to the maxima in the calculation of secondary constants?
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TRANSMISSION LINE
Note: The typical graph shown in Fig. 1.6 was obtained in our laboratory by designing an artificial constant delay line having primary constants as:
L = 4mH
R = 10 ohms
C = 0.47 µF
G = 10–4 mhos
at frequency 1.5 kHz.
REFERENCES
JORDEN, R.C., Electromagnetic Waves and Radiating Systems, Prentice-Hall of India (1976).
LAGERELAETTA, Microwave Measurement and Techniques, Artech House Inc., 610 Washington Street,
Dedham, Massachusetts (1976).
RAGAN, G.L., Microwave Transmission Circuit, McGraw-Hill Book Company (1948).
SINNEMA, W., Electronic Transmission Technology in Waves and Antennas, Prentice-Hall Inc., Englewood
Cliffs, NJE (1979).
SISODIA, M.L., AND RAGHUVANSI, G.S., Microwave Circuits and Passive Device, Wiley Eastern
Limited, New Delhi (1987).
SLATER, J.C., Microwave Transmission, McGraw-Hill Book Company (1942).
WHEELER, G.J, Introduction to Microwaves, Prentice-Hall of India (1978).
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