mw tx lines
DESCRIPTION
Microwave transmission lineTRANSCRIPT
Tx line can be analyzed by: a. Maxwell’s field equations-involves three space variables in addition to time variable
b. Distributed circuit method-involves only one space variable in addition to time variable
Tx Line Equations & Solutions
Transmission Line
Types of TL
Analysis of Tx Line
Analysis of Tx Line
Analysis of Tx Line
Analysis of Tx Line
The dielectric between the conductors has a not zero conductivity, which is responsible of a current flowing from one conductor to the other through the insulator.This phenomenon is accounted for by the conductance G∆z
Analysis of Tx Line
Analysis of Tx Line
If we ignore loss the equation become:
These two equations are called Telegrapher equations or the Transmission line equations
Analysis of Tx Line
WAVE EQUATIONS PHASE VELOCITY and WAVE EQUATION
TRANSMISSION-LINE EQUATIONS
Summation of voltage drop around the central loop
Transmission-Line Equations
(z, t) (z, t)(z, t) i(z, t) R (z, t) (1)
Rearranging, dividing by z and omitting (z,t) we get
- (2)
i vv z L z v z
t z
v iRi L
z t
• The summation of current at point B:
(z , t)i(z, t) (z , t)G (z , t)
(z, t) (z, t)[ (z, t) ]G [ (z, t) ]
i(z, t) (3)
Rearranging, dividing by z, omit
v zv z z C z i z
tv v
v z z C z v zz t z
c z
ting (z,t) and summing
z equal to zero (4)i v
Gv Cz z
Transmission-Line Equations
Differentiating eqn (2) wrt ‘z’ and eqn (4) wrt ‘t’ and combining, the final transmission line equation in voltage form:
Differentiating eqn (2) wrt ‘t’ and eqn (4) wrt ‘z’ and combining, the final transmission line equation in current form:
2 2
2 2(RC LG) (5)
v v vRGv LC
z t t
2 2
2 2(RC LG) (6)
i i iRGi LC
z t t
Transmission-Line Equations
Transmission line equations are applicable to the general transient solution. Voltage and Current on the line are the functions of both position z and time t. The instantaneous line voltage and current are:(z, t) ReV(z)e (7)
i(z, t) ReI(z)e (8)
The phasor give the magnitudes and phases of the
sinusoidal function at each position of z and expressed as:
V(z) V e V e (9)
I(z) e
j t
j t
t t
t
v
I
e (10)
Where, = +j is the propagation constant
tI
Transmission-Line Equations
2 22
2
Substitute j for in equaton (2), (4), (5) and (6)
and divide each equation by e , the Tx equation
in frequency domain
(12) and (13)
(14) and
j t
t
dV dIZI YV
dz dz
d V d IV
dz dz
22
I (15)
Where,
(16) ; Y=G+j C (17)
and = = +j (18)
Z R j L
ZY
Transmission-Line Equations
2 22 2
2 2
For Lossless line, R=G=0 and transmission line equation:
(19) and (20)
(20) and LCI (22)
dV dIj LI j CV
dz dz
d V d ILCV
dz dz
Transmission-Line Equations
Solutions of Transmission-Line Equations
22
2
One possible solution of equation (14) i. e:
is:
V(z) V e V e V e e V e e (23)z z z j z z j z
d VV
dz
V and V- are complex quantities
The term e-jβz , wave travelling in positive z direction and ejβz , wave travelling in negative z direction
The quantity βz is called the electrical length of the line in radians
Solutions of Transmission-Line Equations
22
2
0 0
00
One possible solution of equation (15) i. e:
I is:
(V e V e ) (V e e V e e ) (24)
Characteristic Impedance of the line is defined as:
1
z z z j z z j z
d I
dz
I Y Y
Z R j LZ
Y Y G j C
0 0 (25)R jX
Magnitude of Voltage and Current wave shown in Fig
Solutions of Transmission-Line Equations
Solutions of Transmission-Line Equations
2
At microwave frequencies R L and G
y using Binomial Expansion, the Propagation
Constant can be expressed as:
= (R )(G ) ( ) 1 1
1 11 1
2 2
C
B
R Gj L j C j LC
j L j C
R Gj LC
j L j C
(27)
2
1
1
12
R Gj LC
j L j
jC L
R GL C
LC
C
Solutions of Transmission-Line Equations
The Attenuation and Phase Constant are:
(28)
(29)
1
2
LC
C LR G
L C
Solutions of Transmission-Line Equations
1/2 1/2
0
0Characteristic Impedance Z :
1The Phase V
(R )Z = 1 1
(G )
1 11 1 (30)
2 2
11
2
(elocity= 31) p
j L L R G
j C C j L j C
L R G
C j L j C
L R G
C j L j
L
C
vLC
C
The product LC is independent of the size and separation of the conductors and depends on only on the Permeability μ and Permittivity ε of the insulating medium
If a lossless transmission line used for microwave frequencies has an air dielectric and contains no ferromagnetic materials, free-space parameters can be assumed
The numerical value of for air insulated conductor is approximately equal to the Velocity of Light in vacuum
Solutions of Transmission-Line Equations
1/ LC
Solutions of Transmission-Line Equations
8
0 0
1 13 10 / sec
When the dielectric of lossy MW Tx line is not air
the Phase Velocity is smaller than the Velocity of
1
phase velocityhe Phase Velocity Factor=
ight in vacum
p
r r
actual
v c x mLC
cL
T
v
of l
ight
1r
r r
veloci
vv
c
ty
REFLECTION COEFFICIENT AND TRANSMISSION COEFFICIENT
The travelling wave along the line contains two components:a. One travelling in the positive z directionb. Other travelling in the negative z direction
If the load impedance is equal to the line characteristic impedance; the reflected travelling wave does not exist
Reflection Coefficient
Reflection Coefficient
A wave experiences partial transmittance and partial reflectance when the medium through which it travels suddenly changes. The reflection coefficient determines the ratio of the reflected wave amplitude to the incident wave amplitude.
Standing Wave FormationStanding Wave Formation
Standing Wave FormationStanding Wave Formation
The animation depicts two waves moving through a medium in opposite directions. The blue wave is moving to the right and the green wave is moving to the left.
As is the case in any situation in which two waves meet while moving along the same medium, interference occurs. The blue wave and the green wave interfere to form a new wave pattern known as the resultant. The resultant in the animation below is shown in black.
The resultant is merely the result of the two individual waves - the blue wave and the green wave - added together in accordance with the principle of superposition.
Standing Wave FormationStanding Wave Formation
The result of the interference of the two waves above is a new wave pattern known as a standing wave pattern. Standing waves are produced whenever two waves of identical frequency interfere with one another while traveling opposite directions along the same medium.
Standing wave patterns are characterized by certain fixed points along the medium which undergo no displacement. These points of no displacement are called nodes (nodes can be remembered as points of no displacement). The nodal positions are labeled by an N in the animation above.
The nodes are always located at the same location along the medium, giving the entire pattern an appearance of standing still (thus the name "standing waves").
Standing Wave FormationStanding Wave Formation
A careful inspection of the above animation will reveal that the nodes are the result of the destructive interference of the two interfering waves. At all times and at all nodal points, the blue wave and the green wave interfere to completely destroy each other, thus producing a node.
Midway between every consecutive nodal point are points which undergo maximum displacement. These points are called antinodes; the anti-nodal nodal positions are labeled by an AN. Antinodes are points along the medium which oscillate back and forth between a large positive displacement and a large negative displacement. A careful inspection of the above animation will reveal that the antinodes are the result of the constructive interference of the two interfering waves.
Standing Wave FormationStanding Wave Formation
• Solve the transmission line problem from the receiving end rather than the sending end, since the voltage to current relationship at the load point is fixed by the load impedance
Reflection Coefficient
voltage and Current travelling along tx line
V V e V e (1)
I e e (2)
z z
z z
Incident
I I
Reflection Coefficient
CIRCUIT DIAGRAM
Transmission line terminated in a load impedance
Reflection Coefficient
0 0
0
Current wave interms of Voltage
V V I e e (3)
and Current at receiving end, line length l
V V e V e (4)
V I e
z z
l ll
ll
Z Z
Voltage
Z
0
0
Ve (5)
V V e V e load Impedance Z (6)
I V e V e
l
l ll
l l ll
Z
The Z
Reflection Coefficient
(7)
The Reflection Coefficient at the receiving end:
Reflected Voltage or CurrentReflection Coefficient =
Voltage or Current
ref ref
inc inc
Incident
V I
V I
V
0
0
(8)l
ll
l
Z Ze
V e Z Z
If the load impedance and characteristics impedances are complex then the Reflection Coefficient is also complex
Reflection Coefficient
=Phase angle between inciden
t and refl
(9)
whe
ected
voltages at the receiv
re, 1 ; ne
ing e
ver greater than unity
generalized reflection coefficien
n
t
i
d
l
l
j
l
l l
The
e
s defined as:
(10) z
z
V e
V e
• Let z=l-d, then the Reflection Coefficient at some point located at a distance d from the receiving end is:
• Very important equation fro determining the reflection coefficient at any point along the line
Reflection Coefficient
(l d)2 2
(l d) = = (11)
Reflection Coefficient at that point can be expressed in
terms of Reflection Coefficeint at the receiving end as:
d
ld d
d l
l
l
V e V ee e
V e V e
e
2 ( 2 )2 2 2 (12)lj dd j d dle e e
For lossy line, both the magnitude and phase of the reflection coefficient are changing in an inward-spiral way:
Reflection Coefficient
Reflection Coefficient for lossy line
For a lossless line α=0, the magnitude of Reflection Coefficient remains constant, only Phase of Г is changing circularly toward the generator with an angle of -2βd.
Reflection Coefficient
Reflection Coefficient for lossless line
It is evident that Гl will zero and there will be no reflection from the receiving end when the terminating impedance is equal to the Characteristic Impedance of the line.
Terminating impedance that differs from the characteristic impedance will create a reflected wave traveling towards source.
The reflection, upon reaching the sending end, will itself be reflected if the source impedance differs from the line characteristic at the sending end.
Reflection Coefficient
TRANSMISSION COEFFICIENT
Transmission line terminated in its Characteristic Impedance is called Properly Terminated line otherwise Improperly Terminated line.
There is a Reflection Coefficient at any point along an improperly terminated line.
Incident power minus Reflected power is equal to the power transmitted to the load
Transmission Coefficient
2 20 transmitted to Load=1- (13)
T=Transmission
Voltage or Current
Voltage or C
Coef
urr
ficient
(1e
4)nt
tr tr
inc in
l
c
l
V ITransmittedT
Incident
ZPower T
Z
V I
Transmission Coefficient
Power Transmission on a line
0 0
0
the travelling wave at the receiving end
V e V e = V e (15)
VV V and e e = e (16)
(16) x Z and substituting in (15)
V e
V e
l l ltr
z z ltr
l
l
ll
l l
Let
Z Z Z
Z Z
Z
0
(17)l Z
Transmission Coefficient
0
0
0
, V e V e = V e
V eV e 1+ =
V e V e
V 1+ = =T
V
(182
T= )
l l ltr
lltr
l l
l tr
l
l
l
Now
Z Z
Z
Z
Z
Z
Z
Transmission Coefficient
2 2
0 0
Power carried by two waves in the side
of the incident and reflected waves
( ) ( )P P -P (19)
2 2
l l
inr inc ref
The
V e V e
Z Z
Transmission Coefficient
2
0
2
2
The power carried to the load by the transmitted waves:
( ) P (20)
2
and using equation (17) and (18)
T (1 )
P =
P
lt
i
rtr
n
l
r
l
l
r t
V e
Z
Putt n
Z
i g
Z
(21)
relation varifies the previous statementThis
Transmission Coefficient
STANDING WAVE AND STANDING WAVE RATIO
Transmission-line equation consist of two waves traveling in opposite directions with unequal amplitude. FIG
Standing Wave
The equation we have derived:
V(z) V e V e V e e V e e
V e cos sin V e cos sin (1)
(V e V e )cos (V e V e )sin
z z z j z z j z
z z
z z z z
z j z z j z
z j z
Standing WaveStanding Wave Pattern/Amplitude of Voltage Wave
0
2 20
2 2 1/2
no loss line we can assume V e and V e real
voltage equation V V e (2)
Where,
V {(V e V e ) cos
(V e V e ) sin }
z z
js
z z
z z
For
The
z
z
(3)Phase Pattern of Standing Wave
V e V earctan tan
V e V (4)
e
z z
z zz
Standing WaveThe Maximum Amplitude
maxV V e V e V e (1 ) (5)
occurs at z=n where, n=0, 1, 2...
z z z
This
The Minimum Amplitude
minV V e V e V e (1 ) (6)
occurs at z=(2n-1) / 2 where, n=0, 1, 2...
z z z
This
Standing Wave Pattern in a Lossy Line
Voltage Standing Wave Pattern in a Lossless Line
Standing Wave
1
distance between and two successive
maxima and minina is one-half wave length
occurs at z=n where, n=0, 1, 2...
n nz (7)
2
there is n
/ 2 2
Note o zeros i that, n the minimum
The
This
n z
Distance Between Maxima and Minima
Standing Wave
max
min
,
e e e (1 ) (8)
e e e (1 ) (9)
z z z
z z z
Similarly
I I I I
I I I I
Standing Wave Positive and Negative wave have equal
amplitude: V e V e i.e. Magnetude
of Reflection Coefficient is unity, the Standing
Wave with zero Phase is:
V 2V e cos
z z
zs
When
z
0
Pure Standing Wave for Cu
(12)
I 2 e sin (13)
Called Pure Stand
rren
ing W v
t
a e
zs j Y V z
• The Voltage and Current Standing Waves are 900 out of phase along the line. The point of zero current are called the current nodes
• Voltage and Current nodes are interlaced and a quarter wave apart
Standing Wave
• The voltage and current may be expressed as real functions of time and space:
Standing Wave
0
(14)
(15)
the amplitudes of these two e
The voltage is maximu
(z, t) Re[V (z)e ] 2V e cos
quations
m at the
vary sinu
(z, t) Re[I (z)e ] 2
soidally
with ti
instan
me;
cos
t w
V e sin s
h
n
en
i
j t z
z
s
j ts s
sv
i Y z t
z t
the current
is zero and vice versa. Fig.
Pure Standing Wave of Voltage and Current
STANDING WAVE RATIO
Standing Wave Ratio
Standing results from the simultaneous presence of waves traveling in opposite directions on a transmission line. The ratio of the maximum of the standing wave pattern to the minimum is defined as the standing wave ratio ρ
Standing Wave Ratio
max max
min min
voltage or currenttanding Wave Ratio=
voltage or current
(16)
MaximumS
Mnimum
V I
V I
• Standing wave results from the fact that two traveling wave component add in phase at some points and subtract at other points
• The Standing wave ratio of a pure traveling wave is unity and that of a pure standing wave is infinite
• Standing wave ration of Voltage and current are identical
• When the standing wave ratio is unity, there is no reflected wave and the line called a FLAT LINE
Standing Wave Ratio
The standing wave ratio can not ne defined on a lossy line because the standing wave pattern changes markedly from one position to another
Low loss line the ration remains fairly constant and it may be defined for some region
For a lossless line, the ration remains same throughout the line
Standing Wave Ratio
o Since the reflected wave is defined as the product of and incident wave and its reflection coefficient, the standing wave ratio is related to the reflection coefficient by :
o FIGo The standing wave ratio is a positive real number
and never less than unity. o The magnitude of the reflection coefficient is
never greater than unity
Standing Wave Ratio
1 (17)
1
1 and vice versa (18)
1
1 1
LINE IMPEDANCE AND ADMITTANCE
Line ImpedanceFIG
0
(z) Impedance of Tx Line Z= (1)
I(z)
V=V +V V e +V e (2)
I=I +I (V e -V e ) (3)
At the sending end z=o (2) and (3)
V
z zinc ref
z zinc ref
s s
VThe
Y
I Z
0
0
0
+V (4)
V -V (5)
V ( ) (6)2
V ( ) (7)2
s
ss
ss
I Z
IZ Z
IZ Z
Line Impedance
0 0
0 00
0 00
0
V and V in (2) and (3)
V= [( )e ( )e ] (8)2
I= [( )e ( )e ] (9)2
The line Impedance at point z from sending end
( )e ( )e
( )e (
z zss s
z zss s
z zs s
zs
Substituting
IZ Z Z Z
IZ Z Z Z
Z
Z Z Z ZZ Z
Z Z Z
0 )e zs Z
Line Impedance
0
0 00
0 0
0
The line Impedance at point z=l from Receiving
end interms of and
( )e ( )e (11)
( )e ( )e
The line Impedance can be expressed
interms of and ; At z=l,
s
l ls s
r l ls s
l
Z Z
Z Z Z ZZ Z
Z Z Z Z
Z Z
0
V
V e +V e (12)
V e -V e (13)
r l l
l ll l
l ll
I Z
I Z
I Z
Line Impedance
0
0
V e +V e (12)
V e -V e (13)
Solving these two equation for V and V
V ( )e (14)2
V
l ll l
l ll
lll
l
I Z
I Z
IZ Z
I
0( )e (15)2
llZ Z
Line Impedance
0 0
0 00
these results in (2) and (3)
and letting z=l-d
[( )e ( )e ] (16)2
[( )e ( )e ] (17)2
d dll l
d dll l
Substituting
IV Z Z Z Z
II Z Z Z Z
Z
Line Impedance
0
0 00
0 0
00
The line Impedance any from Receiving
end interms of and
( )e ( )e (18)
( )e ( )e
line impedance at the sending end can
be found from (18). Let, l=d
(
l
d dl l
r d dl l
ls
Z Z
Z Z Z ZZ Z
Z Z Z Z
The
Z ZZ Z
0
0 0
)e ( )e (19)
( )e ( )e
l ll
i ll l
Z Z
Z Z Z Z
These equations can be simplified by replacing the exponential by Hyperbolic or Circular functions
Line Impedance
00
0
00
0
Hyperbolic function obtained from
e cosh( ) sinh( )
At any point from the sending end in terms of
Hyperbolic function
Z cosh( ) Z sinh( )Z= Z
Z cosh( ) Z sinh( )
Z Z tanh( )= Z
Z Z tanh( )
z
s
s
s
s
The
z z
z z
z z
z
z
(21)
Line Impedance
0 00
0 0
00
0
00
0
( )e ( )e (18)
( )e ( )e
, from (18), from the Receiving End
in terms of Hyperbolic function
Z cosh( ) Z sinh( )Z= Z
Z cosh( ) Z sinh( )
Z +Z tanh( )= Z
Z +Z
d dl l
r d dl l
l
l
l
Z Z Z ZZ Z
Z Z Z Z
Similarly
d d
d d
d
(22)tanh( )l d
Line Impedance0 0
00
0
00
0
0
The impedance of a lossless line Z
be expressed in terms of circular function
Z cos( z) sin( z)Z= R
cos( z) jZ sin( z)
Z -jR tan( z)= R (25)
-jZ tan( z)
ZZ= R
s
s
s
s
R
can
jR
R
R
0
0
00
0
cos( ) sin( )
cos( ) jZ sin( )
Z +jR tan( )= R (26)
+jZ tan( )
l
l
l
l
d jR d
R d d
d
R d
Impedance in Terms of Reflection Coefficient
0
0
2
0 2
-ZRe (18) and substituting =
Z
line impedance looking from Receiving end
1 Z=Z (27)
1
The Reflection Coefficient at a distance d
from receiving en
ll
l
dl
dl
Zarranging
Z
e
e
( 2 )2 2
0
d,
(29)
Then the simple equation at a distance d from
1load, Z=Z (30)
1
lj dd dl le e e
Impedance in Terms of Reflection Coefficient
2
0
is a complex quantity (31)
, 2
Impedance variation along the lossless line
1 1 (cos sin )Z(d)=
1 1 (cos sin )
(d) jX(d) Z(
jl
dl l
j
j
e
e d
The
e jR
e j
R
d) (32)dje
Impedance in Terms of Reflection Coefficient
2
0 2
2
0 2
0 2
2
1 2 coswhere, Z(d) (33)
1 2 cos
1 R(d)= (34)
1 2 cos
2 sin X(d)= (35)
1 2 cos
2 sin(d)=arctan arctan (
1
R
R
R
X
R
36)
Impedance in Terms of Reflection Coefficient
, = 2 , then = 2 if,
However,cos( 2 ) cos and
sin( 2 ) sin
, Z(d)=Z(d+ )=Z(d+ ) (37)2
, impedance along the lossless line
will be repe
l l
l l
l l
Since d d
then
Hence
ated for every interval at
a half-wavelength
Impedance in Terms of Reflection Coefficient
Since, the magnitude of a reflection coefficient is
related to the standing wave ratio as
1 1 and (38)
1 1
The line impedance at any location from
the receiving end can be written
0
( 1) ( 1)eZ=R (39)
( 1) ( 1)e
j
j
Determination of Characteristic Impedance
Determination of Characteristic Impedance and Propagation Constant of a given transmission line - take two measurementa. Measure the sending end impedance with the receiving end short-circuited and record the result:b. Measure the sending end impedance with the receiving end open circuited and record the result:
0 tanh( l) (40)scZ Z
0 coth( l) (41)ocZ Z
• Then the Characteristic Impedance of the measured transmission line is:
• And the propagation constant of the line can be computed from:
Determination of Characteristic Impedance
0 (42)oc scZ Z Z
1arctan (43)sc
oc
Zj h
l Z
NORMALIZED IMPEDANCE The Normalized Impedance is defined as:
Lower case letters are commonly designated for normalized quantities in describing distributed transmission line circuits
{Equations (39), (40) and (44)}Normalized impedance fro a lossless line has the following significant features:
0
1 (44)
1
Zz r jx
Z
1. Maximum Normalized Impedance
Normalized Impedance
maxmaxmax
0 0 min
max
1 (45)
1
Where is the positive real value and it is
equal to Standing Wave Ratio the location
of any max voltage on the line
VZz
R R I
z
at
2. Minimum Normalized Impedance
Normalized Impedance
minminmin
0 0 max
min
1 1 (46)
1
Here, is a positive real number but
reciprocal of Standing Wave Ratio at the
location of any min voltage on the line
VZz
R R I
z
Normalized Impedance
max min
max max
min min
3. For every interval of half-wavelength
distance along the line z or z is repeated
z (z) z (z / 2) (47)
z (z) z (z / 2) (48)
Normalized Impedance
max min
max
min
maxmin
4. Since, V and V are separated by a
quarter-wavelength, z is equal to reciprocal
of z for any /4 separation:
1z (z /4) (49)
z (z)
LINE ADMITTANCE• When a transmission line is branched, it is better
to solve the line equations for the line voltage, current and transmitted power in terms of admittance rather than impedance.
• The characteristic admittance and the generalized admittance are defined as:
0 0 00
0 0
1 (50)
1 (51)
Admittance can be written:
1y= (52)
Y G jBZ
Y G jBZ
Normalized
Y Zg jb
Y Z z