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Calculating total impedence....

Discussion in 'Electronic Design' started by Paul Burridge, Oct 28, 2003.

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  1. Hi,

    It's midnight here and I'm suffering from brain failure.
    Can anyone come to my rescue and show me how to work out the impudence
    of this circuit fragment? (I know, but everyone's brain's entitled to
    not function once in a while and I'm getting my -js and +js all mixed
    up). :-(

    Sine wave input to left of 50R resistor (representing Rgen) at
    frequency of 10Mhz.
    Need to know total impedence from point A through to ground. Thanks.


    50R 33R 10uH
    ___ A ___ ___
    Sig ---------|___|---------|___|----------UUU--+
    input |
    | |
    | .-.
    --- | |
    --- | |
    25p | 50k'-'
    | |
    | |
    | |

    created by Andy´s ASCII-Circuit v1.24.140803 Beta
  2. Mike

    Mike Guest

    I say forget the j's, and stick with s.

    Looking in at point A, toward the 33R resistor:

    Z(s) = R33 + sL + R50k/(sR50kC + 1)

    R33 == 33R
    R50k == 50k
    C == 25pf
    L == 10uH

    If you insist on combining everything together:

    Z(s) = ((sL + R33)(sR50kC + 1) + R50k)/(sR50kC + 1)
    = (s^2(LCR50k) + s(R50kR33C + L) + R33 + R50k)/(sR50kC + 1)
    = ((R33 + R50k)(s^2(LCR50k/(R33 + R50k) +
    s(R50kR33C + L)/(R33 + R50k) + 1)/(sR50kC + 1)

    Now you can substitute s = jw:

    Z(jw) = ((R33 + R50k)(1 - w^2(LCR50k/(R33 + R50k) +
    jw(R50kR33C + L)/(R33 + R50k))/(1 + jwR50kC)

    Is that what you were looking for?

    -- Mike --
  3. "s"?? What's "s"??
    To me, Z(s) means source resistance. I already know that. It's 50
    And "s" is....what???
    Listen, Mike, I'm sure you're trying to be helpful, but none of the
    above makes the slightest bit of sense to me. Thanks for your efforts
    but I'm still none the wiser.
    Can anyone else explain it in a simpler way? It's hardly rocket
    science after all. For a start, is it okay to work out the impedence
    of the series elements seperately from the parallel elements and just
    add the two afterwards? I end up with 629 ohms for the series part and
    625 ohms for the parallel. But since I expect Z(total) for the two
    combined to be around the same as Z(source) this seems way off beam.
    Any ideas?
  4. "S" is the circuit expresd via Laplace transforms.
    Once you undertsnd the laplace bit, you plug in s=jw.

    You then need to know complex math, i.e. how to handle (a+jb)/(c+jd)
    If you cant understand how to handle (a+jb)/(c+jd), we are not going to
    get any further with this problem. Its a must. Do you understand this or

    Kevin Aylward
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.
  5. I read in that Paul Burridge
    >) about 'Calculating total impedence....', on Tue, 28 Oct
    That's the trouble with offering too-sophisticated solutions. If Paul
    can't recall how to work out a rather simple impedance, what's the point
    of introducing a more advanced technique without any explanation?

    's' is the complex frequency variable s = [sigma] + j[omega].
    'Z(s)' is 'Z as a function of s'. The source impedance would have the
    symbol 'Zs', with s maybe as a suffix.
    Yes, it is, but the damage has been done.
    Yes, provided you do it correctly. You have to take the phases into
    You haven't taken the phases into account. You can do it two ways,
    either expressing the impedances as magnitude and phase OR as real and
    imaginary parts. Since you mentioned -j and +j, I suppose you are
    reasonably happy with real and imaginary parts.

    25 pF at 10 MHz is 637 ohms, so you can probably forget the 50 kohms for
    practical purposes. That makes it a lot easier. 10 uH at 10 MHz is 628
    ohms, so the circuit is nearly series-resonant. You need to be careful
    here. The residual reactance is 628 - 637 = -9 ohms, which is a
    capacitive reactance, and the capacitance value is 1.77 nF (1770 pF).
    Yes, it's a LARGE value, because the residual reactance is SMALL.

    We don't know the resistance of the 10 uH coil so we can only assume
    that it's low compared with 33 ohms. So the impedance at A is 33-j9

    If you really wanted to take the 50 kohms into account, you would best
    switch to admittances. The conductance of 50 kohms is 20 uS
    (microsiemens) and the susceptance of the capacitor is 1/637 S = 1570
    uS. So the total impedance is 1/(20 + j1570) Mohms.

    This requires a bit of complex number lore to change to (20-
    j1570)/(20^2 + 1570^2)**. This then evaluates to 8 + j 637 ohms. OK, 8
    ohms isn't all that negligible compared with 33 ohms, but you now know
    how to allow for it anyway. The Q of the coil matters as well: it may
    well not be satisfactory to assume that its series resistance at 10 MHz
    is the same as the d.c. resistance, and there may be losses that are
    best expressed as a parallel resistance as well. But we can only use the
    data we are given.

    There is also a loss resistance associated with the 25 pF, and expressed
    as parallel resistance it might not be negligible compared with the 50
    kohms (or, expressed as a series resistance, it might not be negligible
    with the 8 ohms). Again, we don't have data on that.

    ** We need to make the denominator real, so we do this:

    1/(x + jy) = (x-jy)/{(x - jy)(x + jy)} = (x - jy)/(x^2 + y^2)
  6. Right, thanks for the link. I'll check it out. I've often heard the
    term 'Laplace' bandied about but have no knowledge of it as yet.
    I don't have any problem with algebra, Kev. I'm just wondering though,
    why this Laplace business seems to be the preferred way of solving the
    problem. What advantage does it confer over the '+j/-j' way of
    approaching such problems?
  7. Fair enough. I'd never heard of it.
    Yes, I did remember that much but seem to have forgotten *how* to take
    the phases into account. ISTR something about adding the different
    phases vectorially; i.e, the correct outcome being given by the square
    root of the sum of the squares. Does that not apply in this instance?
    Thanks, John. Obviously I'm going to need a while to absorb this lot
    and look into the background of this Laplace lark as well. It could
    take a while. :-/
  8. Kais Badami

    Kais Badami Guest

    impudence? seems like a well behaved network to me :)
    25p at 10MHz => -636.6j
    10u at 10MHz => 628.3j

    First 25p // 50k

    take reciprocal of both and add them to find total admittance

    1/(-636.6j) = 1.57 X 10E-3 j
    1/(50,000) = 0.02 X 10E-3

    add and find reciprocal to get impedence (i assume you have a
    scientific calculator if not, multiply numerator and denominator by
    complex conjugate
    of denominator)

    8.11 - 636.8j

    add the inductance

    8.11 - 8.3j

    add resistance 33 ohms

    41.1 - 8.3j

    I havent checked this - hopefully the answer is right. But the method
    is certainly right.

    If you dont have a sci calculator, you might want to use a Smith Chart
    for stuff like this to save a lot of calculations. However it is
    efficient only if the resistance / impedence values are not too widely
    separated from one another (as in this case), otherwise selection of a
    normalizing impedence may be difficult.

    - Kais
  9. Hi,

    If you haven't got it already, do a search for AppCad 3.0.2 on
    the Agilent site. In the engineering tools section there is a RPN
    calculator with a parallel-series-parallel conversion function
    that should help you to resolve that network. Their home page is
    offering V2.5.1 but if you poke around you will find the later

    Cheers - Joe
  10. Tim Shoppa

    Tim Shoppa Guest

    The impedance of the 25pF capacitor at 10MHz is 636 ohms (times -i).
    This means that the parallel resistance of 50K is almost negligible.

    The impedance of the 10 uH inductor at 10 MHz is 628 ohms (times i).

    You are very near resonance at 10 MHz... as a result the combination of
    the L and the C gives a near-zero impedance. The 33 ohm resistor then
    dominates. So effectively at resonance (which is I assume why you're
    asking about 10MHz) the stuff to the right of "A" is almost exactly like
    a 33 ohm resistor to ground.

    Does this help any?

  11. Because the j way is messy on paper. Work out all the stuff with sL,
    1/sc and R, then plug in the s=jw at the end.

    Kevin Aylward
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.
  12. Fred Bloggs

    Fred Bloggs Guest

    This is not how it's done in practice. First off, change that 50K to
    60K, as I told him that the DC bias is to be set at 2.0VDC with a
    100K+150K. Then you consider the parallel combination of 25p||60K- and
    its equivalent Qc of Y/G=60KWC at frequency W. This is an impedance of
    60K/(1+jWC60K)=60k/(1+(60KWC)^2)- j*(WC60K^2)/(1+(60KWC)^2) which is
    recognized as a series combination of equivalent Rc and Cc, where
    Rc=60k/(1+(60KWC)^2) and Cc=C*(1+(60KWC)^2)/(WC60k)^2. Substituting Qc,
    these are now: Rc=60K/Qc^2 and Cc=C*(1+Qc^2)/Qc^2. Then at 10MHz, Qc=95,
    so that Rc=7 ohms, and Cc=C*1.0001. Taking Cc=C for this high-Qc results
    in less than 0.0005 fractional error in Wo or 0.05%- well go ahead and
    use 25.0025p if you want. You will have the same effect with L and its
    Q. The main thing is that these components Q's be an order of magnitude
    in square ratio larger than the circuit Q at resonance.
  13. Genome

    Genome Guest

    Bear in mind that the necro asked for the impedance from point A to ground.
    You missed out the R50 from the siggy gen.

    Whilst we're on the subject, what's all that knobrot about s?

    Taking your sum, plus your nice notation...., ignoring R50, and bending it a

    XA = R33 + XL + XC//R50k

    Or, including the 50 ohm sig gen impedance....

    XA = (R33 + XL + XC//R50k)//R50

    OK, so XC//R50K is,

    [R50K/jwC]/[R50k + 1/jwC]

    You multiply the top and bottom by jwC. What's that then? some sort of
    hidden trick. Whoopsy.... that's multiple negative points for not showing
    your method....... and end up with,


    Oooh impessive.

    XL is jwL and R33 is R33 so, if you wanted to get your original sum

    XA = R33 + jwL + R50k/[jwCR50k + 1]

    Oh dear, I now have to substitute s for jw

    XA = R33 + sL + R50k/[sCR50k + 1]

    What the **** changed, should I also convert XA to Z(jw) and then into Z(s)?

    That's..... impedance as a function of jw or impedance as a function of s.
    Excuse me,

    No clothes before..... substituted for...... no clothes afterwards. And then
    you still have to go through the shitty algebra with a net benefit of no
    extra clothes.

    Now, don't get me wrong. Before I switched off as some mathematician went
    through the concept of Laplace transforms I managed to hear something about
    it converting things from the 'have to divide and multiply in the frequency
    domain' to 'only need to add and subtract in the time domain'. Which really
    sounds super spiffy wonderful, like log tables, but I never understood the
    point of it.........

    And, in this case, I still don't. Seriously it's got to be as much use as
    farting in an upside down bucket. Unless, of course, you do so with your
    left trouser leg rolled up above your kneecap.

  14. Genome

    Genome Guest

    So Mr French bloke was just saving ink?


  15. Bill Sloman

    Bill Sloman Guest

    Your circuit includes a capacitor and an inductor. To calculate the
    impedance oof the whole circuit you have to recognise the fact that
    the impedance of a circuit including reactive elements has to be
    represented by a complex number


    where "i" is the square root of minus one.

    The impedance of a capacitor at a particular frequency f Z(c)= -i/wC

    where w is the frequency in radians per second, which is two pi times
    the frequency in Hz, and c is the capacitance in farads.

    The impedance of an inductor is similarly Z(L)= iwL

    Complex impedances, so defined, can be added in series and in
    parallel, just like simple real resistances.

    "s" is just electrical engineers shorthand for iw or 2.i.pi.f.

    Hope this helps. But I guess that what you really needed was a good
    night's sleep.
  16. Use full impedances for all components (Zr=R; Zc=1/(jwC); Zl=jwL).
    Then combine the full impedances in usual manner, i.e.
    Zpar=(Z1*Z2)/(Z1 + Z2);
    Zseries= Z1 + Z2. Use standard complex number arithmetics.
    This is what the previous poster has suggested.
    In your example:
    - work out impedance of parallel combination of resistor and
    - add impedance of series inductor;
    - add impedance of series resistor. This is your impedance from point
    A toward the right. If you want to include the generator's impedance
    (50 Ohm), it would be parallel to the whole thing as impedance of the
    voltage source is zero.

  17. Genome

    Genome Guest

    My apologies, of course what you meant to say was

    Substitute s for jw to begin with.... do the sums and then.....

    1/s^n becomes -j/w^n when n is odd.
    1/s^n becomes 1/w^n when n is even.
    1/s^n is -1 when n is 0
    s^n becomes -jw^n for n is odd.
    s^n becomes w^n for n is even.
    s^n is 1 when n is 0


    It's so bloody simple I'm surprised I ever managed without it. All I have to
    do is remember the transformation bits and I will be saved. Thankyou,
    thankyou, thankyou!!!!

  18. Genome

    Genome Guest

    to do is remember the transformation bits and I will be saved. >Thankyou,
    thankyou, thankyou!!!!
    Oh shit, did I miss a minus sign somewhere?

  19. I read in that Paul Burridge
    >) about 'Calculating total impedence....', on Tue, 28 Oct

    [big snip]

    I mean you were already confused.
    Sort of, but if you are happy with using 'j', do it that way.
    You don't need to bother with Laplace unless you are going to study what
    happens to LCR circuits with non-sinusoidal signals, such as steps,
    impulses and rectangular waves. While you can use Laplace with sine-
    waves, you don't need to, especially not if all you use it for is
    writing 's' instead of 'j[omega]'.
  20. I read in that Fred Bloggs <>
    It's how I do it in practice. YMMV.
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