[SI-LIST] Re: characteristic impedance at DC

  • From: Shawn Zheng <shawn.h.zheng@xxxxxxxxx>
  • To: "haaeri@xxxxxxxxx" <haaeri@xxxxxxxxx>
  • Date: Tue, 10 Apr 2012 13:01:24 +0800

There is no DC characteristic impedance. It is AC concept. In theory it 
approaches infinity for lossy line.  In reality it doesn't exist and no way it 
can be measured unless the line can be infinitely long, which correspond to 
infinite long wavelength.

Sent from my iPhone

On 2012-4-10, at 11:17, mohammad haaeri <haaeri@xxxxxxxxx> wrote:

> Hi Howard,
> Thanks a lot for your explanation. Now I understand its physics not just
> formula.
> 
> Thanks,
> mohammad
> 
> On Mon, Apr 9, 2012 at 7:38 PM, Howard Johnson <howie03@xxxxxxxxxx> wrote:
> 
>> I love this question.
>> 
>> It is best addressed, in my opinion, by the following rather ridiculous
>> tought experiment.
>> 
>> First, stretch a transmission line from here to planet Pluto.  (OK, Pluto
>> is
>> not really called a planet anymore, but you know which rock I'm talking
>> about).  Now, hook up a DC constant-current source and observe the voltage
>> at the input to the line. According to the relationship V = I*Z, the
>> voltage
>> should be proportional to the DC impedance.
>> 
>> REMEMBER that if your signal travels at the speed of light it will take
>> about 5-1/2 hours for the current from your "DC" source to reach the end of
>> the line (*see note).  During that time (if the signal travels at the speed
>> of light), you would observe a linearly increasing voltage as the current
>> traversed an increasingly long segment of the transmission line. At the end
>> of 5-1/2 hours, just before your signal struck the end of the structure, if
>> you were using category-5 UTP having a DC resistance of 0.16 ohms per
>> meter,
>> you would observe an effective impedance at the near end of the cable of
>> approximately (0.16 ohm/meter)(5,869,660,000,000 meters) = 939,145 megohms
>> (essentially, infinity).
>> 
>> Get the idea?  If there is resistance in the line, then the "characteristic
>> impedance" as measured over time scale "T" must take into account the total
>> series resistance encountered during that amount of time.  In 5-1/2 hours,
>> on a cable going to Pluto, that adds up to a LOT of resistance.
>> 
>> As it happens in the real world, a signal travelling to Pluto experiences
>> another phenomenon related to the capacitance of the line that
>> significantly
>> slows the speed of propagation.  You can see this if you imagine how, after
>> a single hour of propagation, the first 170,753 megohms of line resistance
>> tries to drive a signal into the capacitance of several billion km of cable
>> -- the response would be really slow, wouldn't it?  That idea, developed
>> mathematically, is called an "RC" transmission line. The "RC" propagation
>> effect may be observed any time that the line length and measurement
>> interval are made long enough that the accumulated resistance becomes
>> significant. In the case of our Pluto measurement, the distances and times
>> are easily large enough to observe the RC effect. When stimulated with a
>> constant current source, the real-world RC effect causes the measured
>> voltage at the near end of a tranmission line to rise in proportion to the
>> square root of time.  In terms of impedance, we say that the impedance
>> varies inversely with the square root of frequency.
>> 
>> If you look at the formula for characterisitic impedance
>> sqrt((jwL+R)/(jwC+G)), and assume for a good dielectric that G=0 at low
>> frequencies, then as w gets smaller, there must be a point below which jwL
>> is much smaller than R, leaving us with only two significant terms in the
>> equation: sqrt((R)/(jwC)).  This function is what we call the
>> "characteristic impedance of an RC-mode transmission line."  As I hinted
>> earlier, as w tends to zero, this function varies inversely with the square
>> root of frequency.
>> 
>> Does all this mean that you can never drive true DC current into a
>> transmission line? No, not at all. Even though the *characteristic
>> impedance* of the cable goes to infinity at DC, the *input impedance* of a
>> practical, finite-length cable leading to a DC load equals just the DC
>> resistance of the load plus the total series resistance of the transmission
>> line.
>> 
>> Best regards,
>> Dr. Howard Johnson, Signal Consulting Inc.,
>> tel +1 509-997-0750,  howie03@xxxxxxxxxx
>> www.sigcon.com -- High-Speed Digital Design seminars, publications and
>> films
>> 
>> *NOTE: the average radius of Pluto's orbit is 5,869,660,000 km; see
>> http://wiki.answers.com/Q/How_far_does_Pluto_orbit_from_the_Sun
>> 
>> 
>> -----Original Message-----
>> From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
>> On
>> Behalf Of Yuriy Shlepnev
>> Sent: Monday, April 09, 2012 5:15 PM
>> To: haaeri@xxxxxxxxx; si-list@xxxxxxxxxxxxx
>> Subject: [SI-LIST] Re: characteristic impedance at DC
>> 
>> Mohammad,
>> 
>> See my answers below.
>> 
>> Best regards,
>> Yuriy
>> 
>> Yuriy Shlepnev, Ph.D.
>> President, Simberian Inc.
>> 3030 S Torrey Pines Dr. Las Vegas, NV 89146, USA Office +1-702-876-2882Cell
>> +1-206-409-2368
>> Skype: shlepnev
>> www.simberian.com
>> 
>> 
>> 
>> -----Original Message-----
>> From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
>> On
>> Behalf Of mohammad haaeri
>> Sent: Monday, April 09, 2012 3:31 PM
>> To: si-list@xxxxxxxxxxxxx
>> Subject: [SI-LIST] characteristic impedance at DC
>> 
>> Hi,
>> What is the characteristic impedance of a transmission line at DC? If you
>> are saying Z0=sqrt(Rdc/Gdc) at DC, since Gdc=0, and Rdc is not zero,
>> therefore Z0 is infinite. Is it correct?
>> YS: Yes, this is correct for a lossy line that does not have conductive
>> losses in the admittance per unit length (technically in dielectric).
>> Though, there is no waves at DC, for TEM mode we can calculate asymptotes
>> of
>> the impedance and admittance per unit length and the characteristic
>> impedance at DC.
>> 
>> How does behavior of L, R, G, and C (line parameters) change vs. frequency
>> (at low and DC, and at very high frequency)?
>> YS: It obviously depends on a transmission line type. See analysis for a
>> microstrip line in this app note
>> http://www.simberian.com/AppNotes/MicrostripImpedanceAndTDR_2009_04.pdf
>> Impedance grows at lower frequencies if dielectric model has only
>> polarization losses. In reality, there are some conductive losses in
>> dielectric and thus the asymptote of the characteristic impedance ad DC is
>> not infinity. As someone already noted, the low-frequency growth of the
>> impedance has small impact on overall behavior of the line. It should also
>> not be confused with the conductor resistance that is more important to
>> account at DC. For a microstrip line, the impedance also grows at very high
>> frequencies.
>> 
>> Can Z0=sqrt(R+jwl/G+jwc) be used for all frequencies?
>> YS: Yes, as long as the impedance (R+jwL) and admittance (G+iwC) per unit
>> length are appropriately defined. The formula does not have limitations
>> neither at low nor at high frequencies, though this is relatively
>> complicated subject for a short posting.
>> 
>> Thanks,
>> mohammad
>> 
>> 
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> 
> 
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