Transmission
line model
Updated November
24, 2007
Click
here to go to our main page on transmission lines
Click here
to go to our separate page on characteristic impedance
Light,
phase and group velocities (rewritten for December 2007!)
Click
here to go to our page on transmission line attenuation
Click here
to go to our page on characteristic impedance
New page for December 2007!
This page was recently amalgamated from material from propagation
constant, phase velocity and characteristic impedance pages.
The generalized lumpedelement
model of a transmission line can be used to calculate characteristic
impedance, phase velocity, and both parts of the propagation constant
(phase and attenuation). The model uses an infinitesimally small
section of a transmission line with four elements as shown below.
Here the series resistance, series inductance, shunt conductance
and shunt capacitance are all normalized per unit length (denoted
by the "prime" notation).
By the way, the transmission
line model graphic can be downloaded in a Word document along with
lots of other microwave schematic symbols, just visit our download
page.
Let's examine the relationships
between phase constant, frequency, phase velocity and wavelength.
Recall that there are 2
radians in a wavelength, therefore the relationship between phase
constant and wavelength is simply:
Here's phase constant as function
of frequency:
Note that the phase constant
is proportional to frequency. It also turns out that the expression
SQRT(L'C') is the reciprocal of phase velocity of the transmission
line. Here's a separate page on
that topic! But for now, remember that it is always less than or
equal to the speed of light in a vacuum, which is "approximately"2.99792458E+08
meters per second.
For completeness, here's some
expressions for wavelength in terms of phase constant, or frequency:
The series impedance and shunt
admittance of the structure are simply:
The general form for the propagation
constant starts out as this simple expression:
Propagation constant of lossless
transmission line
If the transmission line is lossless,
then R' and G' terms in the propagation constant equation are zero.
For the lossless case the lumped model reduces to:
If R' and G' terms in the propagation
constant equation are zero, the attenuation constant is also zero.
The general equation for propagation constant is neatly simplified:
In the case of a lossless transmission
line, the propagation constant is purely imaginary, and is merely
the phase constant times SQRT(1):
Propagation constant of lowloss
transmission line
The propagation constant equation
does not easily separate into real and imaginary parts for
and
in the case where R' and G' are nonzero terms. But significant
approximations can be made for "lowloss" transmission
lines. For these approximations to hold, these conditions must be
met:
What does lowloss mean here?
Let's assume that the ratios in the above relations are held to
10%. We made a calculation for PTFE coax, 50 ohms, at 10 GHz. At
the conditions described, R'=1500 ohms/meter, and G'=0.6 Siemens/meter.
Both would result in losses of 130 dB/meter (or 0.13 dB/mm). This
is a very lossy cable by lab standards. Although this is slightly
oranges/apples comparison the condition is even lossier than what
you might measure on a MMIC transmission line at Xband. Note that
the condition scales with frequency, Wband signals can have ten
times as much loss and still meet the condition. So the approximation
holds for just about any transmission line, no worries!
Now on to the propagation constant
equation. Approximations are made using the first two terms of a
Taylor series expansion. We refer you to Pozar's
excellent book if you want to study this. Here's the separated
phase and attenuation constants.
Note that the phase constant is
calculated exactly the same from way from capacitance and inductance
per unit length, regardless if the transmission line is lossy or not.
You get a nonzero attenuation
constant if either G' or R' in the transmission line model (above)
are nonzero terms (when G' and R' are zero the transmission line
is lossless and =0).
The approximation of the attenuation constant under these conditions
is calculated as:
In microwave engineering, we
tend to separate the attenuation constant into different components.
The mechanisms of series resistance and shunt conductance can be
separated into two independent loss expressions:
The term alpha1 is actually the
metal loss in a transmission line due to the skin
depth effect. The term alpha2 can be further separated into
loss due to dielectric loss tangent, and loss due to substrate conductivity.
We have a separate page on transmission
lines loss that deals further with the topic of splitting up the
attenuation constant, check it out!
Velocity of light in a transmission
line
The velocity of light in a transmission
line is often called the "phase velocity". We make a distinction
because phase velocity can mean
something very different when we discuss waveguide.
Velocity of light can be derived
from the inductance and capacitance per unit length of a transmission
line. Under the normal (losloss) conditions of:
The velocity of
light in the transmission line is simply:
For a TEM transmission line (coax,
stripline) with air dielectric the velocity of light reduces to
the constant "c" which is the velocity of light in a vacuum
(2.997E8 maters/second).
Transmission line characteristic
impedance
The general expression that defines
characteristic impedance is:
Note that in its general form,
characteristic impedance can be a complex number. Also note that
it only becomes complex if either R' or G' are nonzero, which will
give you a headache if you think about it too long. In practice
we try to achieve nearly lossless transmission lines. For a lowloss
transmission line, the following relationships will occur:
Then for all practical purposes
we can ignore the contributions of R' and G' from the equation and
end up with a nice scalar quantity for characteristic impedance.
For lossless (or near lossless) transmission lines the characteristic
impedance equatin reduces to:
What are L' and C' to the lay
person? L' is the tendency of a transmission line to oppose a change
in current, while C' is the tendency of a transmission line to oppose
a change in voltage. Characteristic impedance is a measure of the
balance between the two. How do we calculate L' and C'? that depends
on what the transmission line is. For example our page on coax
give the coax equations.
Relationship
of L' and C' to Z_{0} and V_{P}
There are many situations where
you need to know inductance per unit length and capacitance per
unit length of a transmission line. Both can be calculated from
the characteristic impedance and the propagation velocity of the
wave in a transmission line. The key to solving these equations
is that the propagation velocity of a transmission line is a very
simple function of its capacitance and inductance per unit length:
Note than when you plug in inductance
in Henries/meter and capacitance in Farads/meter, you are talking
about one speedy wave, limited to be less than or equal to the velocity
of light which is about 3x10E8 meters/second. From this equation
and that for Z_{0} (above) you can arrive at the following
for L' and C':
Now the truth comes out... for
a TEM transmission line such as stripline or coax, and for a given
dielectric material (which would correspond to Keff in the above
equations), the inductance and capacitance per unit length don't
change when you scale the geometry up and down. So all semirigid,
50 ohm PTFEfilled cables (and PTFEfilled stripline!) will have
94.8 pF/meter (28.9 pF/foot) capacitance and 237 nH/meter (72.2
nH/foot) inductance! Let's state that as a Microwaves101
Rule of Thumb:
For coax and stripline 50 ohm transmission lines
that employ PTFE dielectric (or any dielectric material with dielectric
constant=2), the inductance per foot of is approximately 70 nH,
and the capacitance per foot is about 30 pF.
