1 ekt 441 microwave communications chapter 1: transmission line theory
TRANSCRIPT
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EKT 441
MICROWAVE COMMUNICATIONS
CHAPTER 1:
TRANSMISSION LINE THEORY
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OUR MENU (PART 1)
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
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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
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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
Casting radio
Short
wave
radio
VHF
TV
FM
broad
casting
radio
MicrowavesFar
infraredinfrared 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
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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
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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
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TYPICAL Rx ARCHITECTURE
Typical receiver (from RF & Microwave Wireless Systems, Wiley)
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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
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TYPICAL Tx ARCHITECTURE
Typical transmitter architecture (from RF & Microwave Wireless System, Wiley)
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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
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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+ -
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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
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TRANSMISSION LINE EQUATIONS
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.
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TRANSMISSION LINE EQUATIONS
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/mL = series inductance per unit length, for both conductors, in H/mG = shunt conductance per unit length, in S/mC = shunt capacitance per unit length, in F/m
Figure 2: Lumped-element equivalent circuit
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TRANSMISSION LINE EQUATIONS
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.
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TRANSMISSION LINE EQUATIONS
By using the Kirchoff’s voltage law, the wave equation for V(z) and I(z) can be written as:
022
2
zVdz
zVd 022
2
zIdz
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]
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TRANSMISSION LINE EQUATIONS
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
LjRLjRZ
0[6]
Note: characteristic impedance (Zo) is the ratio of voltage to current in a forward travelling wave, assuming there is no backward wave
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TRANSMISSION LINE EQUATIONS
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 impedancefor 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
norm
aliz
ed v
alue
s 50 ohm standard
attenuation is lowest at 77 ohms
power handling capacity peaks at 30 ohms
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TRANSMISSION LINE EQUATIONS
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]
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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.
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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]
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THE LOSSLESS LINE
For the lossless case, the attenuation constant α is zero.
Thus, the characteristic impedance of [6] reduces to:
C
LZ 0
[11]
The wavelength is:
LC
22
and the phase velocity is:
LCvp
1
[11a]
[11b]
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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)
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TERMINATED TRANSMISSION LINES
RF
Incident
Reflected
Transmitted
LightwaveDUT
• 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).
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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
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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.
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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 eZ
Ve
Z
VzI
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
0Z
VV
VV
I
VZL
[14]
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TERMINATED LOSSLESS TRANSMISSION LINE
Solving for V0+ from [14] gives:
00
00 V
ZZ
ZZV
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 eeZ
VzI
0
0
[17]
[18]
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TERMINATED LOSSLESS TRANSMISSION LINE
The time average power flow along the line at the point z:
2
0
2
01
2
1
Z
VPav [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
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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 20
20 11 [21]
ljeV 20 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]
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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
VSWR [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
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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 <
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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)
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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
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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
1SWR
)20/(10 RL
1
1
SWR
SWR
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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
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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):
zjzj
zj
eV
V
eV
eV
zincidentV
zreflectedVz
2
0
0
0
0
)(
)(
[12]
Defining or ΓL as the reflection coefficient at the load;
)0(0
0
V
VL
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TRANSMISSION LINE IMPEDANCE EQUATION
Substituting ΓL into eq [14 and 15], the impedance along the line is given as:
ljlj
ljlj
ee
eeZ
zI
zVzZ
0)(
At x=0, Z(x) = ZL. Therefore;
L
LL ZZ
1
10
0
00 ZZ
ZZZe
L
LjLL
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TRANSMISSION LINE IMPEDANCE EQUATION
At a distance l = -z from the load, the input impedance seen looking towards the load is:
0
0
0 ZeeV
eeV
lI
lVZ
ljlj
ljlj
in
02
2
1
1Z
e
elj
lj
[25a]
When Γ in [16] is used: lj
Llj
L
ljL
ljL
in eZZeZZ
eZZeZZZZ
00
000
ljZlZ
ljZlZZ
L
L
sincos
sincos
0
00
ljZZ
ljZZZ
L
L
tan
tan
0
00
[26a]
[26b]
[26c]
[25b]
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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.
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SOLUTION TO EXAMPLE 1.4
It can be shown as:
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(i) The reflection coefficient,
076
0
0
242.0502550
502550 j
L
LL
ej
j
ZZ
ZZ
(ii) VSWR
64.11
1
L
LVSWR
SOLUTION TO EXAMPLE 1.4 (Cont’d)
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(iii) The input impedance seen by the source, Zin
8.38.30
255050
50255050
tan
tan
0
00
j
j
jj
jZZ
jZZZZ
L
Lin
48
2 1
4tan
Need to calculate
Therefore,
SOLUTION TO EXAMPLE 1.4 (Cont’d)
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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
z0-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
1SWR
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SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
Referred to Figure 4, equation [17] and [18] the voltage and current on the line are:
zjVeeVzV zjzj sin2 00 [27]
zZ
Vee
Z
VzI zjzj cos
2
0
0
0
0
[28]
From [26c], the ratio V(-l) / I(-l), the input impedance is:
ljZZ in 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.
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SPECIAL CASE OF LOSSLESS 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
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52
SPECIAL CASE OF LOSSLESS 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
z0-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
1SWR
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53
SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
For an open circuit I = 0, while the voltage is a maximum. The input impedance is:
ljZZ in 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:
Lin Z
ZZ
20 [32]
Note: also known as quarter wave transformer.
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SPECIAL CASE OF LOSSLESS 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.
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SPECIAL CASE OF LOSSLESS TRANSMISSION LINES