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Characteristic
impedance
Updated April
25, 2008
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phase and group velocities (rewritten for December 2007!)
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New for February 2007!
We've added some discussion of the equations that convert Z0 and
VP to L' and C'! See below.
Note to readers.. if you want
more discussion on this topic we suggest you pick up a copy of the
latest revision of Pozar's Microwave Engineering, look for
it on our book page.
What is this thing called characteristic
impedance, also known as "Z-zero"? Let's start with a
common misconception. If you have a 50-ohm coax cable (or microstrip
line or any other 50 ohm transmission line) that isn't connected
to anything on either end, and connect an ohmmeter across it, you
might expecting to read 50 ohms if you were new to microwave engineering.
But you'll measure an open circuit (unless something is loading
the opposite end which means you didn't follow our simple instructions...)
nowhere near fifty ohms. What's going on?
But wait, Jens from Denmark wants
you to know that the above statement is not 100% technically correct:
On the new page for characteristic
impedance, you start out by claiming that you cannot measure anything
if you connect an ordinary ohmmeter to a 50 ohm cable or microstrip.
This is simply because the line is too short for the meter to
get the reading. The meter may take about 1 second to get a stable
reading. If you take a loss free transmission of say c x 1.5 (about
4.5 x exp(8) meters!) you will get a reading that lasts about
3 seconds. You may have some practical problems with this experiment
though. The line will wind the equator about 11.25 times. And
getting it lossless will also be difficult.
Thanks, and let us know when
you have that cable in place so we can perform the measurement!
And now an opposing viewpoint
from Raphael...
I believe Jens is somehow
wrongly deriving is conclusion from the wave propagation phenomena.
As I see it, assuming you could make the experiment, you would
never see (let's say) 50 Ohm on your meter because traditional
multimeters derive resistance from measured voltage and current
to display the result of V / I and you would never have any current
flowing between the two conductors.
The definition of resistance
is "the opposition to the flow of electric current"
whereas the definition of impedance is "The opposition to
an alternating current". Meaning that a 50 Ohm "resistance"
behaves the same (offer the same opposition to current) in both
direct or alternating current. A 50 Ohm "impedance"
on the other hand does not necessarily behave the same in direct
or alternating current.
It's not because an impedance
has only a real part that's it's necessarily a "resistance"
... it might just still be an "impedance".
Z = R + jX
Z = R
We're inclined to agree with
Raphael on this, but we look forward to further discussion! And
here comes some more discussion from Mark, who agrees with Jens...
Although a DC measurement
will indeed show no current passing through an unterminated coaxial
cable, in a short term view (think in nanoseconds) this is incorrect.
When you touch your meter leads to the cable, those leads are
at a different electrical potential when compared to the electrical
potential between the center conductor and the shield. In order
to equalize the potential between the test leads and the cable,
current MUST flow. After all, you need charge to have an electrical
potential and you need a current to create that charge. During
the time where the initial energy applied to the cable is still
propagating away from the meter, the meter will indeed read 50
ohms. The problem is, as Jens pointed out, you would either need
a very fast multimeter or a very long cable to actually see 50
ohms on a display.
To look at it another way,
a lossless cable can be approximated as an infinite series of
"L" shaped series inductor, shunt capacitor pairs. When
you think in terms of series inductors and shunt capacitors, you
can easily imagine getting a different reading until all the L's
and C's have stabilized at their steady state values.
To state that the meter
would always read an open, requires information to travel faster
than the speed of light. After all, how can the meter find out
if a cable is terminated until it "queries" it with
electrons? For the first 100 us, a 100 us long cable is going
to look the same to a meter regardless of any open, short, or
loaded conditions on the other end. To do otherwise would require
information to travel faster than the wave propagating along the
cable.
I have trouble with the message
that a meter would take a "look" at the impedance in the
first nanosecond and give a meaningful result. The operator would
wait a few seconds for the measurement, and he would see an open,
then eat another bite of his donut. But you have all given us food
for thought! - Unknown Editor
Characteristic impedance
Now back to the topic at hand...
Any media that can support a
electromagnetic wave has a characteristic impedance associated with
it. Although characteristic impedance units are in Ohms, it is not
a "real" impedance you can measure using direct current
equipment such as a DC Ohmmeter. And although transmission lines
have real loss at microwave frequencies, this isn't what we're talking
about either.
Let's add one caveat to the above
paragraph.... waveguide must be treated
differently from this whole discussion of characteristic impedance,
and we'll get to that eventually. Technically, waveguide isn't really
a transmission line, even though
we often treat it as one!
The best way to think about characteristic
impedance it envision an infinitely long transmission line, which
means that there will be no reflections from the load. Placing an
alternating current voltage Vin(t) will result in a current
Iin(t). The impedance of the transmission line is then:

Sounds simple enough, but unless
you are dealing with "free space", there is no transmission
line that is infinitely long. But that equation is starting to look
like a version of Ohm's law, where R=V/I.
Transmission line lumped element
equivalent circuit
The transmission line model now
has its own page.
Now let's look at the general
equivalent circuit of an infinitesimally small piece of a transmission
line. All circuits elements are normalized to length in transmission
line models; in the metric system the units are Ohms/meter, Farads/meter,
mhos/meter and Henries per meter, we will use the "prime"
notation when we are discussing quantities that are normalized per
unit length.

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.
The T-line model is repeated
infinite times along the length of a real transmission line. Hmm,
this is starting to sound like calculus, which we have pledged to
avoid on Microwaves101. And we will, so we'll stop with this one
section. For microwave engineers, the general expression that defines
characteristic impedance is:

Here R', G', L' and C' are normalized
to length, the same as in the model. 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 non-zero, 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 low-loss
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 transmission lines the transmission line model reduces
to this:

and the more familiar equation
for characteristic impedance is simply:

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 Z0 and VP
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 Z0 (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 PTFE-filled cables (and PTFE-filled 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.
Telegrapher's equations
This solution to the wave equation
for transmission lines was developed by Oliver
Heaviside a long time ago. We'll cover this later. You can look
it up in Wikipedia for now!
Intrinsic
impedance
Characteristic impedance does
not even need a transmission line, there is a characteristic impedance
associated with wave propagation in any uniform medium. In this
case we use the Greek letter eta for impedance.
The intrinsic impedance is a measure of the ratio of the electric
field to the magnetic field.
Intrinsic impedance is calculated
the same way as any transmission line. Assuming there are no "real"
conductances or resistance in the medium, the equation is reduced
to the simpler SQRT(L'/C') form. In this case the inductance per
unit length reduces to the permeability
of of the medium, and the capacitance per unit length reduces to
the permittivity of medium.

Note that the approximation that
0
is related to .
There is no physical reason for this, it just happens that 1/36
gives better than 99.9% accuracy!
Impedance
of free space
In space the terms for relative
permeability and relative permittivity are each equal to unity,
so the intrinsic impedance equation is simplified to the equation
for characteristic impedance of free space:

Here's where the approximation
involving 1/36
for permeability is what gives us that 120
value for free-space impedance (accurate to 99.9%!). Note that permeability
and permittivity of the atmosphere on earth behave very close to
free space (if it ain't raining!), so we use 377 ohms for the characteristic
impedance of free space in most calculations involving atmospheric
propagation.
Please feel free to forward
comments or questions on this page, we want you to understand
it!
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