transmission line theory prp

1
CHAPTER : 2
TRANSMISSION LINE THEORY
Dr. Pravin R. Prajapati

2
Index
 Introduction to Microwaves
 Transmission Line Equations
 The Lossless Line
 Terminated Transmission Lines
 Reflection Coefficient
 VSWR
 Return Loss
 Transmission Lines Impedance Equations
 Special Cases of Terminated Transmission Lines

3
SPECTRUM & WAVELENGTHS
Waves in the electromagnetic spectrum vary in size from
very long radio waves the size of buildings, to very short
gamma-rays smaller than the size of the nucleus of an atom.
Wavelength of a wave is the distance we have to move
along the transmission line for the sinusoidal voltage to
repeat its pattern

4
INTRODUCTION
 Microwave refers to alternating current signals with frequencies between 300
MHz and 300 GHz.
 Figure 1 shows the location of the microwave frequency
Long wave
radio
AM
broad
Castin
g radio
Short
wave
radio
VHF
TV
FM
broad
casting
radio
Microwaves
Far
infrared
infrared Visible
light
3 x105 3 x 106 3 x107 3x 108 3x109 3x1010 3x1011 3x1012 3x 1013 3x1014
103 102 101 1 10-1 10-2 10-3 10-4 10-5 10-6
Typical frequencies
AM broadcast band 535-1605 kHz VHF TV (5-6) 76-88 MHz
Shortwave radio 3-30 MHz UHF TV (7-13) 174-216 MHz
FM broadcast band 88-108 MHz UHF TV (14-83) 470-890 MHz
VHF TV (2-4) 54-72 MHz Microwave ovens 2.45 GHz

5
MICROWAVE BAND DESIGNATION
Frequency
(GHz)
Wavelength (cm) IEEE band
1 - 2 30 - 15 L
2 - 4 15 - 7.5 S
4 - 8 7.5 - 3.75 C
8 - 12 3.75 - 2.5 X
12 - 18 2.5 - 1.67 Ku
18 - 27 1.67 - 1.11 K
27 - 40 1.11 - 0.75 Ka
40 - 300 0.75 - 0.1 mm

6
APPLICATION OF MICROWAVE
ENGINEERING
 Communication systems
 UHF TV
 Microwave Relay
 Satellite Communication
 Mobile Radio
 Telemetry
 Radar system
 Search & rescue
 Airport Traffic Control
 Navigation
 Tracking
 Fire control
 Velocity Measurement
 Microwave Heating
 Industrial Heating
 Home microwave ovens
 Environmental remote
sensing
 Medical system
 Test equipment

7
TYPICAL Rx ARCHITECTURE
Typical receiver (from RF & Microwave Wireless Systems, Wiley)

8
TYPICAL Rx ARCHITECTURE
 When signal arrives at Rx, normally it is amplified
by a Low Noise Amplifier (LNA)
 Mixer then produce a down-converted signal at freq
of fIF+fm OR fIF-fm; fIF<fm
 Signal is then filtered to remove undesired
harmonics & spurious products from mixing
process
 Signal is then amplified by an intermediate freq
(IF) amplifier
 Output signal of amplifier goes to detector for the
recovery of the original message

9
TYPICAL Tx ARCHITECTURE
Typical transmitter architecture (from RF & Microwave Wireless System, Wiley)

10
TYPICAL Tx ARCHITECTURE
 Input baseband signals (video, data, or voice)
is assumed to be bandlimited to a freq fm
 Signal is filtered to remove any components
beyond passband
 Message signal is then mixed with a local
oscillator (LO) to produce modulated carrier
(up-conversion) (fLO+fm OR fLO-fm), fm<fLO
 Modulated carrier can be amplified &
transmitted by the antenna

11
TRANSMISSION LINES
Low frequencies
 wavelengths >> wire length
 current (I) travels down wires easily for efficient power transmission
 measured voltage and current not dependent on position along wire
High frequencies
 wavelength » or << length of transmission medium
 need transmission lines for efficient power transmission
 matching to characteristic impedance (Zo) is very important for low
reflection and maximum power transfer
 measured envelope voltage dependent on position along line
I
+ -

12
TRANSMISSION LINE EQUATIONS
 Complex amplitude of a wave may be defined in 3 ways:
 Voltage amplitude
 Current amplitude
 Normalized amplitude whose squared modulus equals
the power conveyed by the wave
 Wave amplitude is represented by a complex phasor:
 length is proportional to the size of the wave
 phase angle tells us the relative phase with respect to
the origin or zero of the time variable

13
 Transmission line is often schematically represented as a
two-wire line.
i(z,t)
V(z,t)
Δz
z
Figure 1: Voltage and current definitions.
The transmission line always have at least two conductors.
Figure 1 can be modeled as a lumped-element circuit, as
shown in Figure 2.

14
 The parameters are expressed in their respective name per
unit length.
RΔz LΔz
GΔz CΔz
i(z,t) i(z + Δz,t)
Δz
v(z + Δz,t)
R = series resistant per unit length, for both conductors, in Ω/m
L = series inductance per unit length, for both conductors, in H/m
G = shunt conductance per unit length, in S/m
C = shunt capacitance per unit length, in F/m
Figure 2: Lumped-element equivalent circuit

15
 The series L represents the total self-inductance of
the two conductors.
 The shunt capacitance C is due to close proximity
of the two conductors.
 The series resistance R represents the resistance
due to the finite conductivity of the conductors.
 The shunt conductance G is due to dielectric loss in
the material between the conductors.
 NOTE: R and G, represent loss.

16
 By using the Kirchoff’s voltage law, the wave
equation for V(z) and I(z) can be written as:
    02
2
2
 zV
dz
zVd
     02
2
2
 zI
dz
zId

  CjGLjRj  
where
γ is the complex propagation constant, which is function of
frequency.
α is the attenuation constant in nepers per unit length, β is
the phase constant in radians per unit length.
[1] [2]
[3]

17
 The traveling wave solution to the equation [2] and [3]
before can be found as:
 
  zz
zz
eIeIzI
eVeVzV






00
00
[4]
[5]
The characteristic impedance, Z0 can be defined as:
CjG
LjRLjR
Z








0
[6]
Note: characteristic impedance (Zo) is the ratio of voltage
to current in a forward travelling wave, assuming there is
no backward wave

18
 Zo determines relationship between voltage and current waves
 Zo is a function of physical dimensions and r
 Zo is usually a real impedance (e.g. 50 or 75 ohms)
characteristic impedance
for coaxial airlines (ohms)
10 20 30 40 50 60 70 80 90 100
1.0
0.8
0.7
0.6
0.5
0.9
1.5
1.4
1.3
1.2
1.1
normalizedvalues
50 ohm standard
attenuation is lowest
at 77 ohms
power handling capacity
peaks at 30 ohms

19
 Voltage waveform can be expressed in time domain as:
      zz
eztVeztVtzv 
 
 coscos, 00
The factors V0
+ and V0
- represent the complex quantities. The Φ±
is the phase angle of V0
±. The quantity βz is called the electrical
length of line and is measured in radians.
Then, the wavelength of the line is:



2

[7]
[8]
and the phase velocity is:
fvp 


 [9]

20
EXAMPLE 1.1
 A transmission line has the following
parameters:
R = 2 Ω/m G = 0.5 mS/m f = 1 GHz
L = 8 nH/m C = 0.23 pF
Calculate:
1. The characteristic impedance.
2. The propagation constant.

23
THE LOSSLESS LINE
 The general transmission line are including loss effect, while the
propagation constant and characteristic impedance are complex.
 On a lossless transmission line the modulus or size of the wave complex
amplitude is independent of position along the line; the wave is neither
growing not attenuating with distance and time
 In many practical cases, the loss of the line is very small and so can be
neglected. R = G = 0
 So, the propagation constant is:
LCjj   [10]
LC 
0
[10a]
[10b]

24
THE LOSSLESS LINE
 For the lossless case, the attenuation constant α is zero.
 Thus, the characteristic impedance of [6] reduces to:
C
L
Z 0
[11]
The wavelength is:
LC




22

and the phase velocity is:
LC
vp
1



[11a]
[11b]

25
EXAMPLE 1.2
A transmission line has the following per unit length
parameters: R = 5 Ω/m, G = 0.01 S/m, L = 0.2 μH/m
and C = 300 pF. Calculate the characteristic impedance
and propagation constant of this line at 500 MHz.
Recalculate these quantities in the absence of loss
(R=G=0)

29
TERMINATED TRANSMISSION LINES
RF
Incident
Reflected
Transmitted
Lightwave
DUT
• Network analysis is concerned with the accurate measurement
of the ratios of the reflected signal to the incident signal, and
the transmitted signal to the incident signal.
Waves travelling from generator to load have complex amplitudes usually
written V+ (voltage) I+ (current) or a (normalised power amplitude).
Waves travelling from load to generator have complex amplitudes usually
written V- (voltage) I- (current) or b (normalised power amplitude).

30
TERMINATED LOSSLESS
TRANSMISSION LINE
 Most of practical problems involving transmission lines
relate to what happens when the line is terminated
 Figure 3 shows a lossless transmission line terminated
with an arbitrary load impedance ZL
 This will cause the wave reflection on transmission lines.
Figure 3: A transmission line terminated in an arbitrary load ZL

31
TERMINATED LOSSLESS
TRANSMISSION LINE
 Assume that an incident wave of the form V0
+e-jβz is
generated from the source at z < 0.
 The ratio of voltage to current for such a traveling
wave is Z0, the characteristic impedance [6].
 If the line is terminated with an arbitrary load ZL=
Z0 , the ratio of voltage to current at the load must
be ZL.
 The reflected wave must be excited with the
appropriate amplitude to satisfy this condition.

32
TERMINATED LOSSLESS
TRANSMISSION LINE
 The total voltage on the line is the sum of incident and
reflected waves:
  zjzj
eVeVzV  
 00
The total current on the line is describe by:
  zjzj
e
Z
V
e
Z
V
zI 
0
0
0
0




[12]
[13]
The total voltage and current at the load are related by the load
impedance, so at z = 0 must have:
 
  0
00
00
0
0
Z
VV
VV
I
V
ZL 



 [14]

33
TERMINATED LOSSLESS
TRANSMISSION LINE
 Solving for V0
+ from [14] gives:



 0
0
0
0 V
ZZ
ZZ
V
L
L
[15]
The amplitude of the reflected wave normalized to the amplitude
of the incident wave is defined as the voltage reflection
coefficient, Γ:
0
0
0
0
ZZ
ZZ
V
V
L
L


 

[16]
The total voltage and current waves on the line can then be written as:
   zjzj
eeVzV 
 
0
   zjzj
ee
Z
V
zI 
 

0
0
[17]
[18]

34
TERMINATED LOSSLESS
TRANSMISSION LINE
 The time average power flow along the line at the point z:
 2
0
2
0
1
2
1


Z
V
Pav [19]
• [19] shows that the average power flow is constant at any point of
the line.
• The total power delivered to the load (Pav) is equal to the incident
power minus the reflected power
• If |Γ|=0, maximum power is delivered to the load. (ideal case)
• If |Γ|=1, there is no power delivered to the load. (worst case)
• So reflection coefficient will only have values between 0 < |Γ| < 1
0
2
0
2Z
V 
0
22
0
2Z
V 

35
STANDING WAVE RATIO (SWR)
 When the load is mismatched, the presence of a reflected
wave leads to the standing waves where the magnitude
of the voltage on the line is not constant.
  ljzj
eVeVzV  2
0
2
0 11 
 [21]
 lj
eV  2
0 1 

The maximum value occurs when the phase term ej(θ-2βl) =1.
  
10max VV
The minimum value occurs when the phase term ej(θ-2βl) = -1.
  
10min VV
[22]
[23]

36
STANDING WAVE RATIO (SWR)
 As |Γ| increases, the ratio of Vmax to Vmin increases, so the
measure of the mismatch of a line is called standing wave ratio
(SWR) can be define as:



1
1
min
max
V
V
SWR [24]
• This quantity is also known as the voltage standing wave ratio,
and sometimes identified as VSWR.
• SWR is a real number such that 1 ≤ SWR ≤
• SWR=1 implies a matched load


37
RETURN LOSS
 When the load is mismatched, not all the of the available power
from the generator is delivered to the load.
 This “loss” is called return loss (RL), and is defined (in dB) as:
 log20RL [20]
• If there is a matched load |Γ|=0, the return loss is dB (no
reflected power).
• If the total reflection |Γ|=1, the return loss is 0 dB (all incident
power is reflected).
•So return loss will have only values between 0 < RL <



38
Attenuation loss
 It is measure of power loss due to signal absorption in the
device.
 This “loss” is called return loss (RL), and is defined (in dB) as:
•
Attenuation (dB) =
10log(i/p energy- reflected energy)/(transmitted energy to the load)
  l
rs
rs
eVV
VV
222
22
log10 




39
Reflection loss
 It is measure of power loss due to transmission during
reflection of the signal as a result of impedance mismatch.
•
Reflection loss (dB) = 10log(i/p energy)/(i/p energy- reflected energy)
S
S
L
4
)1(
log10
1
1
log10
2
2





40
Transmission Loss
 It is measure of power loss due to transmission through the line
or device.
•
Transmission loss (dB) = 10log(i/p energy)/(transmitted energy)
= Attenuation loss + Reflection loss
  l
rs
s
eVV
V
222
2
log10 



41
Insertion Loss
 It is measure of power loss due to transmission through the line
or device computed to direct delivery of energy without line or
device.
•
2
1
log10
P
P

Where P1= power received by load when connected directly to source.
P2= power received by load when device is inserted between source
and load with input power constant.

42
SUMMARY
 Three parameters to measure the ‘goodness’
or ‘perfectness’ of the termination of a
transmission line are:
1. Reflection coefficient (Γ)
2. Standing Wave Ratio (SWR)
3. Return loss (RL)

43
EXAMPLE 1.3
Calculate the SWR, reflection coefficient magnitude, |Γ| and
return loss values to complete the entries in the following
table:
SWR |Γ| RL (dB)
1.00 0.00
1.01
0.01
30.0
2.50


44
EXAMPLE 1.3
The formulas that should be used in this calculation are as
follow:
[20]
[24]
mod from [20]
mod from [24]
 log20RL



1
1
SWR
)20/(
10 RL

1
1



SWR
SWR

45
EXAMPLE 1.3
Calculate the SWR, reflection coefficient magnitude, |Γ| and
return loss values to complete the entries in the following
table:
SWR |Γ| RL (dB)
1.00 0.00
1.01 0.005 46.0
1.02 0.01 40.0
1.07 0.0316 30.0
2.50 0.429 7.4


46
TERMINATED LOSSLESS
TRANSMISSION LINE
 Since we know that total voltage on the line is
  zjzj
eVeVzV  
 00
And the reflection coefficient along the line is defined as Γ(z):
  zj
zj
zj
e
V
V
eV
eV
zincidentV
zreflectedV
z 


2
0
0
0
0
)(
)(





[24 a]
Defining or ΓL as the reflection coefficient at the load;
)0(
0
0
 

V
V
L

47
TRANSMISSION LINE IMPEDANCE
EQUATION
 Substituting ΓL into eq [14 and 15], the impedance along the line
is given as:
 
  ljlj
ljlj
ee
ee
Z
zI
zV
zZ 





 0)(
At x=0, Z(x) = ZL. Therefore;
L
L
L ZZ



1
1
0
0
0
0
ZZ
ZZ
Ze
L
Lj
LL


 

48
TRANSMISSION LINE IMPEDANCE
EQUATION
 At a distance l = -z from the load, the input impedance seen
looking towards the load is:
 
 
 
  0
0
0
Z
eeV
eeV
lI
lV
Z ljlj
ljlj
in 









02
2
1
1
Z
e
e
lj
lj







[25a]
When Γ in [16] is used:
   
    lj
L
lj
L
lj
L
lj
L
in
eZZeZZ
eZZeZZ
ZZ 






00
00
0
ljZlZ
ljZlZ
Z
L
L


sincos
sincos
0
0
0



ljZZ
ljZZ
Z
L
L


tan
tan
0
0
0



[26a]
[26b]
[26c]
[25b]

49
EXAMPLE 1.4
A source with 50  source impedance drives a
50  transmission line that is 1/8 of wavelength
long, terminated in a load ZL = 50 – j25 .
Calculate:
(i) The reflection coefficient, ГL
(ii) VSWR
(iii) The input impedance seen by the source.

50
SOLUTION TO EXAMPLE 1.4
It can be shown as:

51
(i) The reflection coefficient,
 
 
0
76
0
0
242.0
502550
502550 j
L
L
L
e
j
j
ZZ
ZZ








(ii) VSWR
64.1
1
1




L
L
VSWR
SOLUTION TO EXAMPLE 1.4 (Cont’d)

52
(iii) The input impedance seen by the source, Zin







8.38.30
255050
502550
50
tan
tan
0
0
0
j
j
jj
jZZ
jZZ
ZZ
L
L
in




48
2 


  1
4
tan 

Need to calculate
Therefore,
SOLUTION TO EXAMPLE 1.4
(Cont’d)

53
SPECIAL CASE OF LOSSLESS
TRANSMISSION LINES
 For the transmission line shown in Figure 4, a line is terminated
with a short circuit, ZL=0.
 From [16] it can be seen that the reflection coefficient Γ= -1.
 Then, from [24], the standing wave ratio is infinite.
VL=0
V(z),I(z)
Z0, β
z
0-l
IL=0
ZL=0
Figure 4: A transmission line terminated with short circuit
0
0
0
0
ZZ
ZZ
V
V
L
L


 




1
1
SWR

54
TRANSMISSION LINES
 Referred to Figure 4, equation [17] and [18] the voltage and
current on the line are:
    zjVeeVzV zjzj

sin2 00

 [27]
    z
Z
V
ee
Z
V
zI zjzj

cos
2
0
0
0
0



 [28]
From [26c], the ratio V(-l) / I(-l), the input impedance is:
ljZZin tan0 [29]
When l = 0 we have Zin=0, but for l = λ/4 we have Zin = ∞ (open circuit)
Equation [29] also shows that the impedance is periodic in l.

55
TRANSMISSION LINES
V(z)/2jV0
+
z
-
λ/
4
-
λ/
2
-
3λ/
4
-λ
1
-1
(a)
I(z)Z0/2V0
+
z
-
λ/
4
-
λ/
2
-
3λ/
4
-λ
1
-1
(b)
Xin/Z0
z
-
λ/
4
-
λ/
2
-
3λ/
4
-λ
1
-1(c)
Figure 5: (a) Voltage (b) Current (c) impedance (Rin=0 or ∞)
variation along a short circuited transmission line

56
TRANSMISSION LINES
 For the open circuit as shown in Figure 6, ZL=∞
 The reflection coefficient is Γ=1.
 The standing wave is infinite.
VL=0
V(z),I(z)
Z0, β
z
0-l
IL=0
ZL=∞
Figure 6: A transmission line terminated in an open circuit.
0
0
0
0
ZZ
ZZ
V
V
L
L


 




1
1
SWR

57
TRANSMISSION LINES
 For an open circuit I = 0, while the voltage is a maximum.
 The input impedance is:
ljZZin cot0 [30]
When the transmission line are terminated with some special
lengths such as l = λ/2,
Lin ZZ  [31]
For l = λ/4 + nλ/2, and n = 1, 2, 3, … The input impedance [26c]
is given by:
L
in
Z
Z
Z
2
0
 [32]
Note: also known as quarter wave transformer.

58
TRANSMISSION LINES
(a)
(b)
(c)
V(z)/2V0
+
z
-
λ/
4
-
λ/
2
-
3λ/
4
-λ
1
-
1
I(z)Z0/-2jV0
+
z
-
λ/
4
-
λ/
2
-
3λ/
4
-λ
1
-
1
Xin/Z0
z
-
λ/
4
-
λ/
2
-
3λ/
4
-λ
1
-
1
Figure 7: (a) Voltage (b) Current (c) impedance (R = 0 or ∞)
variation along an open circuit transmission line.

59
TRANSMISSION LINES

transmission line theory prp

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