Transmission Line Eq Fixed
-
Upload
erwin-yusuf -
Category
Documents
-
view
223 -
download
0
Transcript of Transmission Line Eq Fixed
-
8/2/2019 Transmission Line Eq Fixed
1/17
1
TRANSMISSION LINE EQUATION
-
8/2/2019 Transmission Line Eq Fixed
2/17
2
Quotes
No mathematical purist could ever dothe work involved in Maxwells
treatise. He might have all the
mathematic, and much more, but it
would be to no purpose, as he could
not put it together without the
physical guidance. This is in no way
to his discredit, but only illustrates
different ways of thought.
Oliver Heaviside
-
8/2/2019 Transmission Line Eq Fixed
3/17
3
Introduction
A transmission line basically consists of two or moreparallel conductors used to connect a source to a load.
The source may be a hydroelectric generator, a
transmitter, or an oscillator; the load may be a factory, an
antenna, or an oscilloscope, respectively. Typical transmission lines include coaxial cable, a two-wire
line, a parallel-plate or planar line, a wire above the
conducting plane, and a microstrip line.
Transmission line problems are usually solved using EM fieldtheory and electric circuit theory, the two major theories
on which electrical engineering is based.
Basically, all engineering problem can be solved employing
field theory equations
-
8/2/2019 Transmission Line Eq Fixed
4/17
4
-
8/2/2019 Transmission Line Eq Fixed
5/17
5
Key point about transmission line
operation
The major deviation from circuit theory with transmission line,
distributed networks is this positional dependence of voltage and
current! Must think in terms of position and time to understand
transmission line behavior
This positional dependence is added when the assumption of
the size of the circuit being small compared to the signaling
wavelength
Voltage and current on a transmission line isa function of both time and position.
V1 V2
dz
I2I1
tz,fI
tz,fV
-
8/2/2019 Transmission Line Eq Fixed
6/17
6
Presence of Electric and Magnetic
Fields
V
I
I
E
+
-
+
-
+
-
+
-
V + DV
I + DI
I + DIV
IH
IH
V + DV
I + DI
I + DI
Both Electric and Magnetic fields are present in the transmissionlines
These fields are perpendicular to each other and to the direction of wavepropagation for TEM mode waves, which is the simplest mode.
Electric field is established by a potential difference between twoconductors.
Implies equivalent circuit model must contain capacitor.
Magnetic field induced by current flowing on the line
Implies equivalent circuit model must contain inductor.
dlEV . dlHI .
-
8/2/2019 Transmission Line Eq Fixed
7/177
-
8/2/2019 Transmission Line Eq Fixed
8/178
-
8/2/2019 Transmission Line Eq Fixed
9/179
Kirchhoffs Law
Current Law I=0
Dividing by x, we obtain:
.(2)
-
8/2/2019 Transmission Line Eq Fixed
10/1710
-
8/2/2019 Transmission Line Eq Fixed
11/1711
Transmission Line Equation
(f-Domain)
dxdxandjtxItxixVtxv ///);(),();(),( Substituting
.(4)
.(3)
IGRIGLCRjIjCLdx
Id
VGRVGLCRjVjCLdx
Vd
'')''''()(''
'')''''()(''
2
2
2
2
2
2
Become
.(6)
.(5)
-
8/2/2019 Transmission Line Eq Fixed
12/1712
The Solution of Transmission Line
Equation
Idx
VdICjRLjR
dx
Id
Vdx
VdVCjRLjR
dx
Vd
2
2
2
2
2
2
2
2
2
2
or)'')(''(
or)'')(''(
xx
eAeAxV
21)(
VeAeAxdx
Vd xx 22
2
1
2
2
2
)(
)'')(''( CjRLjR
General exponential solution ofdAlembert
-
8/2/2019 Transmission Line Eq Fixed
13/1713
The Solution of Transmission Line
Equation
ILIRdx
Vd
j
''
0''
''''Z
CjG
LjRLjR
General current Solution
The Ratio
Characteristic impedance of
the transmission line
is the attenuation (loss) factor
is the phase (velocity) factor
)(''
)( 21xx
eAeALjR
xI
-
8/2/2019 Transmission Line Eq Fixed
14/17
14
-
8/2/2019 Transmission Line Eq Fixed
15/17
15
The distinction between electrically
long and electrically short
Definition in frequency domain (time harmonic quantities)
c
x Definition in frequency domain (time harmonic quantities)
(x)VV
r
c
vC
c/fcT
-
8/2/2019 Transmission Line Eq Fixed
16/17
16
The distinction between electrically
long and electrically short
Definition in time domin),()( txvtv
xconsttvtv )()(
v
ac
lT
Electrically short if the voltage at a given instant
possesses the same value everywhere along a line
i.e. If the rise time of the travelling wave is very
large compared with the propagation time or delaytime
-
8/2/2019 Transmission Line Eq Fixed
17/17
17
fin