
Transmission
lines 
Coaxial cable gives very good eprotection
against
interference, because of its
completely shielding geometry.
Transmission lines (TL) play
central role
in nuclear physics experiments, where they are used to pipe signals
around
from one place to another within a circuits, and often from the
preamplifier
output to input of the pulse shaper or fast amplifier.
TL for signal of frequencies up to 1GHz
comes
in 2 major types:
 Parallel type of conductor line
 Coaxial line (widely used e.g in
short
lengths with
BNC fitting to carry signal between instruments.)

Example of the
various TL: (a) coaxial
cable, (b) twin lead 
In the domain of ultrahigh frequency
circuitry
there are "stripline" techniques used. In "microwave range" of
frequency
(upwards of 2GHz) conventional TL is replaced by cavity and waveguide
techniques. For nuclear physics applications the familiar coax is
probable
the best choice.
By processing short duration's pulses ,
or fast
rise pulses is observed that speed of electromagnetic wave propagation
=> transmission depends on
geometrical configuration (layout) of
electronics
details.
When we consider a wave with frequency f
propagating in transmission medium with dielectric constant e
with velocity v=c/e
(cvelocity of light in space) then its wavelength can be
calculated l
=v/f~c/f. (For example for
100MHz l =3m.) Electronic
properties
of TL are no more lumped, or focused to 1 point, but (R, L, C, G of
line)
are continually distributed in space, and are depending on distance x
too:
=> u(t,x), i(t,x).
TL
must be considered as a circuit
with distributed
parameters.
Homogenous unit part of the TL
can be characterized
their (R, L, C, G of line) per
meter. The differential
equation leads
to
solution based on superposition of 2 waves:
 main
(proceeding, in advance moving,
shifting),and
 reflected.

Equivalent
unit part of
the TL 

Distribution
of u(t,x),
i(t,x) in unit part of the TL.


Differential
"telegraph"
equation based on distribution u(t,x), i(t,x ) according fig.
above. 

Solution
of the differential
"telegraph" equation based on superposition of 2 waves:
 h  main (proceeding,
shifting)
 o  reflected

In
any section of TL characteristic
impedance
Z_{0}, which well described properties of TL as a ratio of
voltage wave to current wave (equal Z_{0})
For
short part of TL, such line with R=0
and G=infinite
can be assumed as a lossless line and Z_{0}=(L/C)^{0.5
}=>resistive
in range 50 to 100 ohm.
A
part TL of length d can be
used as element
of delay line with delay T =d(LC)^{0.5}.

Time
delaying property
T=d/v of the TL 

Mismatched
transmission lines 
When TL is used in high frequency range
(or pulse
with short rise time) signals, it is important to "match" the load to
the
characteristic impedance of the line. The following facts are
important:
 A transmission line terminated with
a
load equal
to its characteristic impedance (resistance) will transfer an applied
pulse
to termination without reflection. In that case all the power
in
the signal is transferred to the load.
 The impedance looking into such a
terminated line,
at any signal's frequency, is equal to its characteristic impedance.
A mismatched TL has some interesting and
occasionally
useful properties.

Properties
of a mismatched TL. 
The ratio
of reflected wave amplitude to
incident
wave amplitude is given by

r=
A_{o}/A_{h}=(R Z_{0})/(R+ Z_{0}). 
Similarly as coefficient of reflection r
is a coefficient of (voltage) radiation s_{1}
and s_{2 }(current)
defined:
_{. .}

s_{1}=(2Z)/(Z+
Z_{0})=1+r 

s_{2}=(2Z_{0})/(Z+
Z_{0})=1r 
A line terminated in a short circuits
produced
a reflected wave determined by the electrical length of the line. (A
speed
of the wave propagation in coax lines is about twothirds the light
speed
=> 1m thin coax delays ~5ns)

The
reflected wave is
inverted if R<Z_{0} 
In similar manner, an opencircuited
cable
(boundary condition of zero current at the end) produces a noninverted
reflection of amplitude equal to the applied signal.

The
reflected wave is
noninverted if R>Z_{0}. 
Cables terminated with a resistance R
unequal
to characteristic resistance also produce reflections, although of
lesser
amplitude.

The
reflected wave at:
 R=Z_{0}/2
 R=2Z_{0}

This property of shorted cable part is
sometimes
exploited to generate a short pulse from step wave's form. The step
input
is applied the cable input through a resistance equal to Z_{0,}
with the other end of the cable shortened. The pulse wave's form at the
input is then a pulse of width equal to the roundtrip travel time,
since
the reflected step pulse cancels the input step.
In nuclear physics experiments is
dangerous to
forget some opencircuits cable, which can then help to generate a
false
pulses and so make distortion in pulse high amplitude.

Experimental
equipment for
check how is "matched" the load to the characteristic impedance of the
line.
