transmission line theory - delta univ · 2016-03-22 · transmission line problems are usually...
TRANSCRIPT
Transmission Line Theory
Dr. M.A.Motawea
Introduction:
In an electronic system, the delivery of power requires the connection of
two wires between the source and the load. At low frequencies, power is
considered to be delivered to the load through the wire.
In the microwave frequency region, power is considered to be in electric
and magnetic fields that are guided from place to place by some physical structure.
Any physical structure that will guide an electromagnetic wave place is called a
Transmission Line.
Transmission lines are used in power distribution (at low frequencies), and
in communications (at high frequencies).
A transmission line consists of two or more parallel conductors used to
connect a source to a load., the source may be a generator, a transmitter, or
an oscillator and the load may be a factory, an antenna, or an oscilloscope,
respectively.
Transmission lines include coaxial cable, a two wire line, a parallel plate or
planar line, a wire above the conducting plane, and a micro-strip line
Cross sectional views of these lines consists of two conductors in figure
,each of these lines consists of two conductors in parallel
Coaxial cables are used in electrical laboratories and in connecting T.V sets
to T.V antennas
Micro-strip lines are important in integrated circuits where metallic strips
connecting electronic elements are deposited on dielectric substrates.
There are different types of modes propagatr between the two conductorsof
transmission line as:
- TE, transverse electric, i.e. 0,0 ZZ HE
- TM, transverse magnetic, i.e. 0,0 ZZ EH
- TEM, transverse electro-magnetic i.e. 0 ZZ EH
- Propagate in Z-direction, ZZ EH ,
Transmission line problems are usually solved using EM field theory and electric
theory, the two major theories on which electrical engineering is based, we use
circuit theory because it is easier to deal with mathematics.
Our analysis of transmission lines will include the derivation of transmission line
equations and characteristic quantities, the use of Smith chart , various practical
applications of transmission lines, and transients on transmission lines.
a b
c d e
Cross sectional view of transmission lines:
a-coaxial line b-two wire line c- planar line d- wire above conducting plane
e- microstrip line
RL
S IRS
Vg
coaxial line
loadgenerator
E
k
E & H fields in the coaxial line
For conductor ( 0,, CCC ), and for dielectric ( ),, .
Transmission Lines Parameters:
We must describe a transmission line in terms of its line parameters.
R[Ω/m] ….conductivity of conductor
L[H/m]….self inductance of wire
G[Ω-1
/m]….dielectric between two conductors
C[F/m]….proximity between two conductors.
We have : LC , ,CG
to generator
i(z,t) i(z+ z, t )
+
v(z,t)v(z+ z,t)
z
R z L z
C zG z
to load
+
- -
z z+ z
I
1- The line parameters R,L,G, and C are not discrete or lumped but distributed as
shown .by this we mean that the parameters are uniformly distributed along the
entire length.
2- For each line, the conductors are characterized by C , C , 0 C , and the
homogeneous dielectric separating the conductors is characterized by , ,
3- R
G1
; R is the ac resistance per unit length of the conductors comprising the line
and G is the conductance per length due to the dielectric medium separating the
conductors.
4- The external inductance per unit length; that is, extLL . The effects of internal
inductance )( RlLin are negligible at high frequencies at which most
communication systems operate.
5- For each line, LC and
C
G
Transmission Line Equations:
As mentioned above , two conductor transmission line supports TEM wave; the
electric and magnetic fields on the line are transverse to the direction of wave
propagation . an important property of TEM waves is that the fields E and H are
uniquely related to voltage V and current I respectively:
dlEV . dlHI .
In view of this , we will use circuit equations V and I in solving the transmission
line problems instead of solving field quantities E and H ( i.e solving Maxwell’s
equations and B.C.), the circuit model is simpler and more convenient.
Let us examine an incremental portion of length z of a two conductor transmission
line . we intend to find an equivalent circuit for this line and drive the line
equations. From the figure of distributed element model of transmission line, we
assume that the wave propagates along +z direction , from the generator to the load.
By applying Kirchoff’s voltage law to the outer loop of figure above , we obtain:
KVL: 0V
),(),(),(*),( tZZVtZIt
ZLtZIZRtZV
),(),(),(),(
tZIt
LtZRIZ
tZVtZZV
By taking the limit of this equation, as 0Z , leads to
),(),(),( tZIt
LtZRItZVZ
(1)
KCL: 0I at the main node of the circuit
ItZZItZI ),(),(
t
tZZVZCtZZZVGtZZItZI
),(),(),(),(
t
tZZVCtZZGV
Z
tZItZZI
),(),(
),(),(
By taking the limit of this equation, as 0Z , leads to
t
tZVCtZGV
Z
tZI
),(),(
),( (2)
If we assume harmonic time dependence so that :
])(Re[),( tj
s eZVtZV ])(Re[),( tj
s eZItZI
Where Vs(Z) and Is(Z) are the phasor forms of V(Z,t) and I(Z,t) respectively
Then eqn (1) becomes
),(),(),( tZIt
LtZRItZVZ
tj
s
tj
s
tj
s eZIt
LeZIReZVZ
)(Re)(Re)(Re
tj
s
tj
s
tj
s eZIt
LeZIReZVZ
)(Re)(Re)(Re
tj
s
tj
s
tj
s eZLIjeZRIeZVZ
)()()(
)()()()()( ZILjRZLIjZRIZVZ
ssss
)()()( ZILjRZVZ
ss
(3)
Also,
)()()( ZVCjGZIdZ
dss (4)
By taking d/dt of 3: )()()(2
2
ZIdZ
dLjRZV
dZ
dss
Substitute in 4: )())(()(2
2
ZVCjGLjRZVdZ
dss
0)())(()(2
2
ZVCjGLjRZVdZ
dss
0)()( 2
2
2
ZVZVdZ
dss Wave eq
n for voltage (5)
Also,
0)()( 2
2
2
ZIZIdZ
dss Wave eq
n for current (6)
These are wave equations for voltage and current similar in form to the wave
equations obtained for plane waves in previous chapter.
jCjGLjR ))(( (7)
is the propagation constant.
is attenuation factor, [Np/m, dB/m]. is phase constant [rad/sec].
2 is wavelength, [m].
fu is wave velocity, [m/s].
Solution of wave equation:
The solutions of the linear homogeneous differential equations, 5 and 6 are:
ZZ
s eVeVZV 00)( (8) and ZZ
s eIeIZI 00)( (9)
+z -z +z -z
Where :
,, 00
VV
00 , II are wave amplitudes; + and – sign respectively denote wave
traveling along +Z and –Z directions, as is also indicated by the arrows.
Thus we obtain the instantaneous expression for voltage as :
)cos()cos(])(Re[),( 00 zteVzteVeZVtzV ZZtj
s (10)
We define the C/C impedance Z0 of the line as the ratio of positively traveling
voltage wave to current wave at any point on the line. Z0 is analogous to η
(intrinsic impedance of the medium of wave propagation).
By substituting eqs, (8) and (9) into eqs. (5) , (6) and equating coefficients of terms
Zeand
Ze as:
We have:
ZZ
s eVeVZV 00)( , )()()( ZILjRZVZ
ss
ZZZZ eILjReILjReVeV 0000 )()(
By equating coefficient of the exponential , then:
00 )( ILjRV ,
)(
0
0 LjR
I
V
(11)
00 )( ILjRV ,
)(
0
0 LjR
I
V
(12)
CjG
LjR
CjGLjR
LjR
I
V
I
VZ
))((
)(
0
0
0
00 (13)
CjG
LjRZ
0 C/C impedance of the line
We have also:
ZZ
s eIeIZI 00)( )()()( ZVCjGZIdZ
dss
))(( 0000
ZZZZ eVeVCjGeIeI
By equating coft of the exponential , then:
00 )( VCjGI CjGI
V
0
0
(13)
00 )( VCjGI CjGI
V
0
0
(14)
CjG
LjR
CjG
CjGLjR
CjGI
V
I
VZ
))((
0
0
0
0
0 (15)
000 jXRCjG
LjRZ
Where R0 [Ω] is real part of Z0 , X0 [Ω] is imaginary part of Z0 . The propagation
constant and the C/C impedance Z0 are important properties of the line because
they both depend on the line parameters R,l,G, and C and the frequency of
operation. the reciprocal of z0 is the C/c admittance Y0, that is Y0 = 1/Z0.
We may now consider two special cases of lossless transmission line and
distortion-less line.
A- Lossless line:
A transmission line is said to be lossless if the conductors of the line are
perfect ( C ) and the dielectric medium separating them is lossless ( )0 . For
such line : GR 0 necessary condition for a line to be lossless.
Then: jCjGLjR ))((
jCjLj ))((
0 , LC ,
f
LCu
1
Also, C
LjXR
CjG
LjRZ
000
00 X , C
LRZ 00
It is desirable for power transmission,
B- Distortionless Line :
A signal normally consists of a band of frequencies; wave amplitudes of
different frequency components will be attenuated differently in a lossy line as
α is frequency dependent. This results in distortion. A distortion-less line is one
in which the attenuation constant α is frequency independent while the phase
constant β is linearly dependent on frequency. From the general expression for
α and β . it is evident that a distortion-less line results if the line parameters are
such that :
C
G
L
R
jG
CjRG
G
Cj
R
LjRGCjGLjR )1()1)(1())((
RG , LCC
LC
G
RC
G
CRG
f
LCu
1
Showing that α deos not depend on frequency, whereas β is a linear function of
frequency.
C
L
G
R
GCjG
RLjRjXR
CjG
LjRZ
)/1(
)/1(000
00 X , C
LRZ 00
Note that:
1. The phase velocity is independent of frequency because the phase
constant linearly depends on frequency. We have shape distortion of
signals unless u and are independent on frequency.
2. u and 0Z remain the same as for lossless line.
3. A lossless line is also distortion-less line,but a distortion-less line is not
necessarily lossless. Although lossless lines are desirable in power
transmission, Telephone lines are required to be distortion-less.
Microwave Engineering
Sheet #3-a
Transmission line
Q1: Define T.L.?
Q2: State types of mode which propagate in T.L.?
Q3: Deduce the formula of propagation constant , ?
Q4: Deduce the formula of C/C impedance, Z0 ?
Q5: Sketch E/H field in coaxial cable?
Q6: An air line has Z0 =70 Ω, and phase constant =3 rad/m at f= 100MHz
Calculate: - the inductance/m - the capacitance/m
Q7: A transmission line operating at f=500 MHz has Z0 =80 Ω , mN p /04.0
mrad /5.1 , Find the line parameters R,L,G, and C.
Q8: A distortion-less line has Z0= 60 Ω , mmN p /20 , u=0.6 c(velocity of light in vacuum).
Find: R,l,G,C and at f=100 MHz?
Q9: A telephone line has R= 30 Ω/km, L= 100 mH/km, G=0, and C= 20 µF/km at f= 1 kHz,
obtain: a- C/C impedance of the line
b- propagation constant
c- the phase velocity
Dr. M.A.Motawea
2- Input Impedance, Standing Wave Ratio, and Power:
Consider a transmission line of length l, characterized by and 0Z , connected to a
load LZ as shown in figure:
Zg I0
Vg V0
+
_
+
_
VLZL
Z=0 Z=l
ZinZin
Z l' = l-Z
IL
a- Input impedance due to a line terminated by a load
I0
Zin
Zg
Vg V0
+
_
b- Equivalent circuit for finding V0 and I0 in terms of Zin at the input
Looking into the line, the generator sees the line with the load as an input
impedance inpZ . It is our intention in this section to determine the input impedance,
the standing wave ratio (SWR), and power flow on the line.
Let the transmission line extend from z = 0 at the generator to z = l at the
load. First of all, we need 0V and 0I , as mention above,
ZZ
s eVeVzV 00)( (1)
ZZ
s eIeIzI 00)( ,
ZZ
s eZ
Ve
Z
VzI
0
0
0
0)( (2)
Where eqn of Z0 has been incorporated,
CjG
LjR
CjG
LjR
I
V
I
VZ
)(
0
0
0
00 (3)
To find
0V and
0V the terminal conditions must be given.
e.g. at the input , (at z = 0) :
),0(0 zVV (4)
),0(0 zII (5)
Substitute (4), (5) into (1) ,(2) results in :
)(2
10000 IZVV
(6)
)(2
10000 IZVV
(7)
If the input impedance at the input terminals is inpZ , the input voltage V0 and the
input current I0 are easily obtained from figure as:
g
gin
in VZZ
ZV
0 ,
gin
g
ZZ
VI
0 (8)
(at z = l ) :
On the other hand, if we are given the conditions at the load, say:
)( lzVVL , )( lzII L (9)
Substitute this into eqn (1), (2) gives:
l
LL eIZVV )(2
100
(10)
l
LL eIZVV )(2
100 (11)
at any point on the line:
Next we determine input impedance )(/)( zIzVZ ssin at any point on the line.
at the generator, for example,
00
000 )(
)(
)(
VV
VVZ
zI
zVZ
s
sin (12)
Substituting eqn (1), (11) into (12) yields:
)2
()2
(
)2
()2
(
2
)(
2
)(
2
)(
2
)(
)(
)(
)(
0
0
0
00
00
0
00
000
LL
L
LL
L
LL
L
LL
L
l
LL
l
LL
l
LL
l
LL
s
s
inee
IZee
V
eeIZ
eeV
ZeIZVeIZV
eIZVeIZV
ZVV
VVZ
zI
zVZ
lZZ
lZZZ
ZlZ
lZZZ
lZlZ
lZlZZ
lZlI
V
lZlI
V
Z
lZlI
V
lZlI
V
ZlIZlV
lIZlVZ
L
L
L
L
L
L
L
L
L
L
L
L
L
L
LL
LL
tanh
)tanh
tanh
)tanh
coshsinh
sinhcosh
coshsinh
sinhcosh
]
coshsinh
sinhcosh
[coshsinh
sinhcosh
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(13)
lZZ
lZZZZ
L
L
in
tanh
)tanh
0
0
0 * for lossy line (14)
ljZZ
ljZZZZ
L
Lin
tanh
)tanh
0
00 * for lossless line (15)
This indicates that the input impedance varies periodically with distance l from
the load. The quantity βl is usually referred to as the electrical length of the line
and can be expressed in degree or radians.
Voltage reflection coefficient ( L ):
We now defined L as the voltage reflection coefficient (at the load) .
L is the ratio of the voltage reflection wave to the incident wave at the load:
l
l
LeV
eV
0
0
,
by eqn 10, 11 gives:
0
0
ZZ
ZZ
L
LL
(16)
In general , the voltage reflection coefficient at any point on the line can be
defined as the ratio of the magnitude of the reflected voltage wave to that of the
incident wave , that is as.
z
l
l
L eV
V
eV
eVz
2
0
0
0
0)(
But 'llz , substituting and combining with eqn l
l
LeV
eV
0
0
, we get:
''
222
0
0)( l
L
ll
L eeeV
Vz
(17)
current reflection coefficient
The current reflection coefficient at any point on the line is negative of the voltage
reflection coefficient at that point. Thus , the current reflection coefficient at the
load is : L
ll eIeI 00 /
as we did in plane waves, we define the standing wave ratio (s) :
L
L
I
I
V
Vs
1
1
min
max
min
max
(18)
It is easy to show that 0maxmax / ZVI and 0minmin / ZVI .
The input impedance inpZ in eqn (14) has maxima and minima that occur
respectively at the maxima and minima of the voltage and current standing wave.
It is easily shown that:
0
min
max
maxsZ
I
VZ in (19)
And s
Z
I
VZ in
0
max
min
min
(20)
As a way of demonstrating these concepts, consider a lossless line with
characteristic of 500Z . For the sake of simplicity, we assume that the line is
terminated in a pure resistive load 100LZ and the voltage at the load is 100 V
(rms). The conditions on the line repeat themselves every half wavelength.
The average input power at a distance z from the load is given by an
equation similar to eqn of Pav,
)]()(Re[2
1 * zIzVP ssav (21)
voltage ¤t wave patterns on a lossless line terminated by a resistive load.
We now consider special cases when the line is connected to load 0LZ ,
LZ , and 0ZZL . These special cases can easily be derived from the general
case.
A. Shorted Line (ZL =0)
For this case , eq. (15) becomes:
ljZZZ
LZinsc tan00
(22)
Also, ,1L s
This impedance is pure reactance, which could be capacitive or inductive
depending on the value of l . The variation of Zin with l is shown in
figure (a).
Input impedance of a lossless line: (a) when shorted, (b) when open
B. Open Circuited Line ( LZ )
In this case , eq (15) becomes
ljZ
lj
ZZZ inzoc
L
cottan
lim 00 (23)
,1L s
The variation of Zin with l is shown in figure (b) . notice from eqs (22),(23) that:
2
0ZZZ scoc (24)
C. Matched Line ( )0ZZ in
This is the most desired case from the practical point of view. For this case, eq(15)
reduces to:
0ZZ in (25)
And
,0L 1s
That is , 00 V , the whole wave is transmitted and there is no reflection . the
incident power is fully absorbed by the load. Thus maximum power transfer is
possible when the transmission line is matched to the load.
Example:
A 30-m long lossless transmission line with Z0 =50 Ω operating at 2 MHz is
terminated with a load 4060 jZL . If cu 6.0 on the line. Find:
a- The reflection coefficient
b- The standing wave ratio s.
c- The input impedance.
0
0
0
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)40110)(4010(
40110
4010
504060
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anglejj
j
jj
jj
j
j
j
j
ZZ
ZZ
L
LL
06.197.
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03.1
03.1
1
1
min
max
min
max
L
L
I
I
V
Vs
ljZZ
ljZZZZ
L
Lin
tanh
)tanh
0
00
3-The Smith Chart
It is a tool to solve problems of transmission lines , it is commonly used of the
graphical techniques. It is basically a graphical indication of a transmission line as
one moves along the line. We calculate L ,s, and Zin and so on.
Example :
a 100+j150 Ω load is connected to a 75 Ω lossless line . find :
a-
b- S
c- The load admittance LY
d- 4.0atZ in from the load
e- The location of Vmax and Vmin w.r.t. the load if the line is 0.6 λ long
f- Zin at the generator.
Smith Chart
Sheet #3-b
Transmission line
Input impedance, reflection coefficient & standing wave ratio
Q1: a transmission line has c/c impedance Z0=50 Ω is connected with a generator has frequency
f=25 MHz and terminated by a load impedance ZL, at l=25 m from the load Zin=50-j100 .
Required: ZL, L