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Transmission line impedance etc

  1. May 28, 2017 #1
    All transmission lines have a characteristic impedance which is different based on their length and voltage and frequency (for AC lines)
    , Also all transformers and their primaries have a different impedance based on the turnsratio, core steel type etc factors.
    Now I read that for any given line to supply the most of its capability with the highest efficiency the load impedance needs to be matched to the line impedance, or the load simply needs to be 100% resistive like a heater element or a large resistor, but most loads are not pure resistors, let's tale the typical mains transformer as an example.
    Since mains transformers come in different sizes and specifications for different devices is it fair to say that some transformers utilize the line power more efficiently while others cause more reactive losses due to the mismatch in impedance between the transformer impedance and the line impedance and so deliver a worse power factor?

    Thank you
  2. jcsd
  3. May 28, 2017 #2


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    For your learning purposes, I think you can assume that a transmission line has a characteristic impedance which is resistive, and it does not depend on length or voltage.
    Ordinary transformers are very efficient and I think you should assume they provide a resistive load.
    For AC power transmission we do not use a load that equals the characteristic impedance of the line. This is acceptable because practical lines are short compared with the wavelength at power frequencies, so we do not notice standing wave effects.
  4. May 28, 2017 #3


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    this is incorrect as tech99 states

    .... or frequency

    all coax or parallel line cable I can buy has a fixed impedance regardless of freq otherwise it would be useless

    some reading up on the subject would be a good idea for you :smile:



    some basics ....
    a twin parallel line .....


    a coax cable .......


    you will have noticed that length and voltage and frequency didn't factor into the equations

  5. May 28, 2017 #4

    jim hardy

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    Radio guys think of wave propagation down a line that's more than a wavelength long and the foregoing discussion applies to that field of electronics.

    Power system guys deal with lines that generally are a lot shorter than a wavelength (3100 miles at 60 hz) so they can be treated like ordinary circuit elements , resistance capacitance and inductance. Most of our lines are short enough that radio wave propagation isn't the way to think about them. A quarter wave is 775 miles.
    Lines more than 150 miles long are considered "Long" and their impedance is calculated using distributed parameters, as in radio work.

    Since the utility is interested in selling power not dissipating it as heat in their lines
    they keep line impedance miniscule compared to load impedance so as to maximize the fraction of power that's NOT dissipated in the distribution system. That's intentional mismatch.

    So that theorem about maximum power conveyance while quite true
    is of far far less use to power system guys than to radio communication or audio amplifier guys.

    For us power plant guys, our impedance as a source really is of interest only for calculating fault currents into a short circuit.

    When a power transmission line becomes long enough to behave as a classic transmission line with significant standing waves it's become too long.
    We don't want it to act like an antenna radiating our hard earned megawatts out into space before they've gone through a customer's KWH meter.
    And that's another reason utilities work hard to generate a clean sine wave - harmonics have shorter wavelengths so will radiate energy from shorter lines.

    I'm not saying that long lines can't show a voltage rise along their length
    it happens when that length becomes a sizeable chunk of a ¼ wavelength..
    That's discussed in this old book that Google turned up
    https://books.google.com/books?id=rdpQAAAAYAAJ&pg=PA278&lpg=PA278&dq=voltage+rise+in+long+utility+transmission+lines&source=bl&ots=03RcH5-zhw&sig=gPgEf-gBFUWVoGuL33PxEpgqb0A&hl=en&sa=X&ved=0ahUKEwj25s_155LUAhXE6IMKHf-GCVgQ6AEIUjAI#v=onepage&q=voltage rise in long utility transmission lines&f=false
    and it's called "Ferranti Effect"
    same self resonance phenomenon as in an antenna. A necessary evil that power system engineers have to contend with.

    Now your point about power factor is well taken.
    Transformers indeed draw magnetizing current hence have a lagging power factor. And there are plenty of them in the distribution system. Utilities scatter capacitor banks around the system to bring power factor back closer to unity. The square boxes here:

    Hope this helps.

    @anorlunda - how'd i do ?

    old jim
  6. May 28, 2017 #5


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    Great answer Jim. I found it very informative.
  7. May 28, 2017 #6

    jim hardy

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    Thanks for the kind words, @phinds
    I was afraid to delve into the math of long transmission lines for the formulas have a most fearsome look about them.- i always preferred Smith Charts.
    I have got too rusty to demonstrate even that straightforward graphical method.
  8. May 28, 2017 #7
    ok, my bad I got mixed up while writing because I was writing just seconds after I watched a video about this stuff , and thanks for the explanations folks, ok so any given line has an impedance which is the result of its length, materials(both conductor and dielectric) and shape I guess?? For example if I take a straight wire and simply make it into a solenoid shaped without any core but air it now has a different impedance than if it was simply straight right? I guess this is how they make car and other antennas shorter than they would otherwise need to be.

    So what I guess I wanted to say is that in any given impedance line depending on the frequency there will either be no noticeable effects (low frequencies) or standing waves will occur (high frequencies) and then as one increases it further or lowers it they will get bigger or smaller. This was the most interesting fact to me, about which i would like to state further questions hopefully with such great answers as i have already got.
    How about voltage since voltage is the amplitude of the signal (say sine wave for example) does amplitude for a given frequency affect the standing wave characteristic (assuming we are already in the frequency range for a given conductor that standing waves have arisen) ? Or is it simply affected by frequency and impedance?

    The thing why I ask this is , let's assume we have either a long enough line or a high enough frequency (in the 250khz+ range) that in our line standing waves are occurring, lets say that our standing wave amplitude is the same as our signal amplitude, now first of all would that mean that as I measured between or at certain spots along the line I would get zero volts because those are the sports where the signal cancels out? I take this from a mechanical analog example used in an old Bell labs video where the teacher used a mechanical analog and it could be seen that for a given frequency there are certain spots where the mechanical arms stand still while at others they cycle up and down rapidly, so I assume the still place would represents close to or zero voltage conditions on an electrical line, what happens with the current at these places is the current also zero or is it not? Something in my intuition tells me the current shouldn't be zero , although since we are talking about AC signals I assume the current is also zero at those spots when the sine wave reaches is upper and lower peaks because at other time as it rises and falls current is flowing through the line either back or forth. Although since the current is probably no in phase that could mean these conditions which I just described are a bit different, I would be happy if you would help me out with some clarification here.
  9. May 28, 2017 #8

    jim hardy

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    I do not have a short intuitive explanation for transmission line equations. A google search turns up plenty but the math is more than i could type even with Latex.
    Try these three links

    One can appreciate them without college level differential equations, though. I recommend ARRL Antenna Handbook for a practical text. It's what we used in my high school.
    That Bell Labs video is most excellent for painting a mental picture.
    inductance is the electrical analogy to inertia and capacitance is the analogy to 'springiness ' or elasticity.

    Yes the voltage and current waves are out of phase by a quarter wavelength as you surmised.

    The first step toward understanding transmission lines it to appreciate the concept of resonance in a straight piece of wire.
    Thought Experiment in two steps:
    1. Imagine in space an electromagnetic field that is alternating at high frequency. Let us use for our example the field produced by radio station WLS of Chicago , which is 890 khz. I used to get that station on my car radio in Miami on clear winter nights so i know it's a real field.
    Its wavelength would be c/890,000 = 337 meters.

    2. Now add to your mental picture a lone wire one half wavelength long, 168.5 meters, oriented so the electromagnetic waves encounter it crosswise.
    Observe that as a wave encounters the wire it begins pushing charge toward one end and away from the other end.
    At the center of the wire charge flows along freely filling one end, and emptying the other.
    At the ends of the wire charge cannot exit so accumulates there.
    As the electromagnetic wave passes the wire and reverses its polarity, current in the wire reverses . The ends acquire charge of opposite polarity and current in the middle reverses direction.
    Aha ! that's an AC voltage peak at each end and a current peak in the middle.

    halfwave antenna1.jpg

    When your wire is a half wave long its inherent capacitance to rest of the universe and its inherent self inductance make it resonant, just another quirk of the universe. Resonance implies more current and voltage than you'd expect, and indeed that's what you get.

    That's the principle behind TV "Rabbit Ear" antennas with adjustable length 'ears' - a straight wire that's resonant . They take current from the middle and feed it into the TV tuner. Since we're dealing with microwatts impedance matching becomes important.

    A transmission line is two wires in proximity so there's mutual inductance and capacitance between them. That changes things from the free space example just described but not fundamentally. Just instead of permittivity and permeability of free space we use the properties of whatever is in between the wires. That just changes velocity of propagation down the line, instead of c it's a property of the line about 2/3 c for most everyday cables.
    And, since current can flow out the end of the transmission line into a load, one can flatten out his standing waves by attention to impedance matching.. That's the "Reflection coefficient" in those equations.

    Perhaps there's a HAM aboard who can make the next thought step ? Or correct me?

    old jim
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