{"id":387575,"date":"2024-06-29T07:24:07","date_gmt":"2024-06-29T07:24:07","guid":{"rendered":"http:\/\/savepearlharbor.com\/?p=387575"},"modified":"-0001-11-30T00:00:00","modified_gmt":"-0001-11-29T21:00:00","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=387575","title":{"rendered":"About LC ladders<\/span>"},"content":{"rendered":"
<\/div>\n
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LC ladder component values can have different values for the same transfer function. Published tables have only one set. Why?<\/p>\n

Computing LC ladders<\/h2>\n

If we write a transfer function of an LC ladder, we find that the same result can be achieved using different LC values.<\/p>\n

Short example for a 2nd order series LC ladder with parameters:
a cutoff frequency is 1 rad (0.159 Hz), quality factor is 0.636 (Bessel), source is 1 Ohm, termination is 1 Ohm.<\/p>\n

The transfer function is:<\/p>\n

                A H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        (1 + s\/(Q \u03c9) + s\u00b2\/\u03c9\u00b2)            Rt                              1 H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        Rs + Rt 1 + s (L + C Rs Rt) \/ (Rs + Rt) + s\u00b2 L C Rt \/ (Rs + Rt)<\/code><\/pre>\n
Solving the equations using a math software, obtain the values:<\/p>\n
L = 0.885 H, C = 2.26 F L = 2.26 H, C = 0.885 F<\/code><\/pre>\n
\"2nd
2nd order LC ladders<\/figcaption><\/figure>\n
The transfer functions match.<\/p>\n
Also, for higher order ladders several solutions can be found.<\/p>\n
For high frequency circuits input impedance of a ladder is important. One of known ways to compute LC ladder values to match impedances is:<\/p>\n
    \n
  1. \n
    Compute poles of a transfer function.<\/p>\n<\/li>\n
  2. \n
    Compute a polynomial from the poles to get H(s).<\/p>\n<\/li>\n
  3. \n
    Compute product of reflection functions p(s)*p(-s).<\/p>\n<\/li>\n
  4. \n
    Find input reflection p(s).<\/p>\n<\/li>\n
  5. \n
    Compute input impedance Zin from p(s).<\/p>\n<\/li>\n
  6. \n
    Compute ladder component values by continued fraction expansion.<\/p>\n<\/li>\n<\/ol>\n
    The most complex part is extracting p(s) from p(s)*p(-s).<\/p>\n
    Books say that to find it, zeroes of the p(s)*p(-s) polynomial should be computed. They also say that there are no restrictions for used zero group positions. They also say that a ladder structure is defined by numerator and denominator orders.<\/p>\n
    Comparison of results<\/h2>\n
    Let\u2019s compute variants for 5th order Butterworth LC ladder. A cutoff frequency is 1 rad (0.159 Hz) and to make it more interesting, a source is 1 Ohm, a termination is 10 Ohm.<\/p>\n
    The roots of the numerator of p(s)*p(-s) in this case:<\/p>\n
    -0.9607 + 0i, +0.9607 + 0i -0.7772 + 0.5647i, -0.7772 - 0.5647i +0.7772 + 0.5647i, +0.7772 - 0.5647i -0.2969 + 0.9136i, -0.2969 - 0.9136i +0.2969 + 0.9136i, +0.2969 - 0.9136i<\/code><\/pre>\n
    There are 3 groups of the roots:<\/p>\n
    \"Groups
    Groups of complex zeroes of p(s)p(-s)<\/figcaption><\/figure>\n
    7 different solutions with positive values were found with the same polynomial orders:<\/p>\n
      Zin     2.0 s^5 + 6.345 s^4 + 10.07 s^3 + 9.878 s^2 + 5.992 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       0.1273 s^4 + 0.4039 s^3 + 0.594 s^2 + 0.48 s + 0.1818  Z1 = 15.71, Z2 = 0.1727, Z3 = 14.09, Z4 = 0.0912, Z5 = 3.152    Zin     2.0 s^5 + 4.424 s^4 + 5.941 s^3 + 4.559 s^2 + 2.183 s + 0.1818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       2.049 s^4  + 4.531 s^3  + 5.913 s^2  + 4.289 s+ 1.818  Z1 = 0.976, Z2 = 12.19, Z3 = 0.1794, Z4 = 11.4, Z5 = 0.0452    Zin     2.0 s^5 + 3.236 s^4 + 5.236 s^3 + 5.236 s^2 + 3.236 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       3.236 s^4 + 5.236 s^3 + 5.236 s^2 + 3.236 s + 0.1818  Z1 = 0.618, Z2 = 1.618, Z3 = 11.0, Z4 = 0.1618, Z5 = 6.18    Zin     2.0 s^5 + 1.315 s^4 + 7.082 s^3 + 3.463 s^2 + 4.94 s + 0.1818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource         5.157 s^4 + 3.39 s^3 + 7.01 s^2 + 1.533 s + 1.818  Z1 = 0.3878, Z2 = 1.182, Z3 = 2.177, Z4 = 7.248, Z5 = 0.1521    Zin     2.0 s^5 + 5.157 s^4 + 7.082 s^3 + 7.01 s^2 + 4.939 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       1.315 s^4 + 3.39 s^3 + 3.463 s^2 + 1.533 s + 0.1818  Z1 = 1.521, Z2 = 0.7248, Z3 = 21.77, Z4 = 0.1182, Z5 = 3.878    Zin     2.0 s^5 + 2.048 s^4 + 5.941 s^3 + 5.913 s^2 + 2.183 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       4.424 s^4 + 4.531 s^3 + 4.559 s^2 + 4.289 s + 0.1818  Z1 = 0.4521, Z2 = 1.14, Z3 = 1.794, Z4 = 1.219, Z5 = 9.763    Zin     2.0 s^5 + 0.1273 s^4 + 10.07 s^3 + 0.594 s^2 + 5.992 s + 0.1818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource         6.345 s^4 + 0.404 s^3 + 9.878 s^2 + 0.48 s + 1.818  Z1 = 0.3152, Z2 = 0.9123, Z3 = 1.409, Z4 = 1.727, Z5 = 1.571<\/code><\/pre>\n
    Use a simulator to verify the results.<\/p>\n
    Since all the polynomials have numerator orders higher than their denominator, they all should work in series structure. But it is not!<\/p>\n
    \"5th
    5th order LC ladders<\/figcaption><\/figure>\n
    \"Transfer
    Transfer functions<\/figcaption><\/figure>\n
    Frequency responses match for sets 1, 3, 5, 6 in series structure and 2, 4, 7 in shunt structure. There must be some other sign of the structure.<\/p>\n
    Now compare their input impedances. Amplitude shows impedance module. 0 dB is 1 Ohm in the chart, +20 dB is 10 Ohm. Phase shows image part of a value. 0 degrees means that impedance is real.<\/p>\n
    \"Input
    Input impedances<\/figcaption><\/figure>\n
    Only 1st circuit has input impedance closest to a real value (green lines) in the pass-band. It is the values which can be found in tables in books. Now it is clear why.<\/p>\n
    The values of the 1st circuit are computed using only left half plane zeroes of the p(s)*p(-s).<\/p>\n
    Conclusion<\/h2>\n
    LC ladders with the same transfer functions can be built using different component values. There is no big difference if it is a low frequency circuit. For high frequency circuits input impedance is important. Use either values from tables or compute them using only left half plane zeroes of a p(s)*p(-s).<\/p>\n
    Unfortunately, some books have tables with typos. Use [1].<\/p>\n
    Verify what you read.<\/p>\n
    There are no solutions for even-order Chebyshev filters when a source is equal to termination. The higher ripple the higher a termination to source ratio should be to get a solution. Some tools compute ladder values using iterative equations [2] and can compute values at any source and termination for even-order Chebyshev filters. Be careful with such tools. [3] offers a way to find solutions for band-pass filters.<\/p>\n
    If you are interested to see a code for old computers, see [4].<\/p>\n
    References<\/h2>\n
      \n
    1. \n
      Anatol I. Zverev, \u201cHandbook of Filter Synthesis\u201d.<\/p>\n<\/li>\n
    2. \n
      Lars Wanhammar, \u201cAnalog Filters Using MATLAB\u201d.<\/p>\n<\/li>\n
    3. \n
      Songbai Zhang, Lei Zhu, \u201cGeneral synthesis method for symmetrical even-order Chebyshev bandpass filter\u201d.<\/p>\n<\/li>\n
    4. \n
      Thomtavanit Hatayodom, \u201cLadder network transfer function characteristics\u201d. Link<\/a>.<\/p>\n<\/li>\n
    5. \n
      «idealCircuit», a simulator. Link<\/a>.<\/p>\n<\/li>\n
    6. \n
      Nuhertz, Filter Solutions. Link<\/a>.<\/p>\n<\/li>\n
    7. \n
      «Circuit Calculator», an electronics circuit design tool. Link<\/a>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
      <\/div>\n

       \u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u043e\u0440\u0438\u0433\u0438\u043d\u0430\u043b \u0441\u0442\u0430\u0442\u044c\u0438        https:\/\/habr.com\/ru\/articles\/693606\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
      <\/div>\n
      \n
      \n
      \n
      \n
      LC ladder component values can have different values for the same transfer function. Published tables have only one set. Why?<\/p>\n
      Computing LC ladders<\/h2>\n
      If we write a transfer function of an LC ladder, we find that the same result can be achieved using different LC values.<\/p>\n
      Short example for a 2nd order series LC ladder with parameters:
      a cutoff frequency is 1 rad (0.159 Hz), quality factor is 0.636 (Bessel), source is 1 Ohm, termination is 1 Ohm.<\/p>\n
      The transfer function is:<\/p>\n
                      A H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        (1 + s\/(Q \u03c9) + s\u00b2\/\u03c9\u00b2)            Rt                              1 H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        Rs + Rt 1 + s (L + C Rs Rt) \/ (Rs + Rt) + s\u00b2 L C Rt \/ (Rs + Rt)<\/code><\/pre>\n
      Solving the equations using a math software, obtain the values:<\/p>\n
      L = 0.885 H, C = 2.26 F L = 2.26 H, C = 0.885 F<\/code><\/pre>\n
      2nd order LC ladders<\/figcaption><\/figure>\n
      The transfer functions match.<\/p>\n
      Also, for higher order ladders several solutions can be found.<\/p>\n
      For high frequency circuits input impedance of a ladder is important. One of known ways to compute LC ladder values to match impedances is:<\/p>\n
        \n
      1. \n
        Compute poles of a transfer function.<\/p>\n<\/li>\n
      2. \n
        Compute a polynomial from the poles to get H(s).<\/p>\n<\/li>\n
      3. \n
        Compute product of reflection functions p(s)*p(-s).<\/p>\n<\/li>\n
      4. \n
        Find input reflection p(s).<\/p>\n<\/li>\n
      5. \n
        Compute input impedance Zin from p(s).<\/p>\n<\/li>\n
      6. \n
        Compute ladder component values by continued fraction expansion.<\/p>\n<\/li>\n<\/ol>\n
        The most complex part is extracting p(s) from p(s)*p(-s).<\/p>\n
        Books say that to find it, zeroes of the p(s)*p(-s) polynomial should be computed. They also say that there are no restrictions for used zero group positions. They also say that a ladder structure is defined by numerator and denominator orders.<\/p>\n
        Comparison of results<\/h2>\n
        Let\u2019s compute variants for 5th order Butterworth LC ladder. A cutoff frequency is 1 rad (0.159 Hz) and to make it more interesting, a source is 1 Ohm, a termination is 10 Ohm.<\/p>\n
        The roots of the numerator of p(s)*p(-s) in this case:<\/p>\n
        -0.9607 + 0i, +0.9607 + 0i -0.7772 + 0.5647i, -0.7772 - 0.5647i +0.7772 + 0.5647i, +0.7772 - 0.5647i -0.2969 + 0.9136i, -0.2969 - 0.9136i +0.2969 + 0.9136i, +0.2969 - 0.9136i<\/code><\/pre>\n
        There are 3 groups of the roots:<\/p>\n
        Groups of complex zeroes of p(s)p(-s)<\/figcaption><\/figure>\n
        7 different solutions with positive values were found with the same polynomial orders:<\/p>\n
          Zin     2.0 s^5 + 6.345 s^4 + 10.07 s^3 + 9.878 s^2 + 5.992 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       0.1273 s^4 + 0.4039 s^3 + 0.594 s^2 + 0.48 s + 0.1818  Z1 = 15.71, Z2 = 0.1727, Z3 = 14.09, Z4 = 0.0912, Z5 = 3.152    Zin     2.0 s^5 + 4.424 s^4 + 5.941 s^3 + 4.559 s^2 + 2.183 s + 0.1818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       2.049 s^4  + 4.531 s^3  + 5.913 s^2  + 4.289 s+ 1.818  Z1 = 0.976, Z2 = 12.19, Z3 = 0.1794, Z4 = 11.4, Z5 = 0.0452    Zin     2.0 s^5 + 3.236 s^4 + 5.236 s^3 + 5.236 s^2 + 3.236 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       3.236 s^4 + 5.236 s^3 + 5.236 s^2 + 3.236 s + 0.1818  Z1 = 0.618, Z2 = 1.618, Z3 = 11.0, Z4 = 0.1618, Z5 = 6.18    Zin     2.0 s^5 + 1.315 s^4 + 7.082 s^3 + 3.463 s^2 + 4.94 s + 0.1818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource         5.157 s^4 + 3.39 s^3 + 7.01 s^2 + 1.533 s + 1.818  Z1 = 0.3878, Z2 = 1.182, Z3 = 2.177, Z4 = 7.248, Z5 = 0.1521    Zin     2.0 s^5 + 5.157 s^4 + 7.082 s^3 + 7.01 s^2 + 4.939 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       1.315 s^4 + 3.39 s^3 + 3.463 s^2 + 1.533 s + 0.1818  Z1 = 1.521, Z2 = 0.7248, Z3 = 21.77, Z4 = 0.1182, Z5 = 3.878    Zin     2.0 s^5 + 2.048 s^4 + 5.941 s^3 + 5.913 s^2 + 2.183 s + 1.818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource       4.424 s^4 + 4.531 s^3 + 4.559 s^2 + 4.289 s + 0.1818  Z1 = 0.4521, Z2 = 1.14, Z3 = 1.794, Z4 = 1.219, Z5 = 9.763    Zin     2.0 s^5 + 0.1273 s^4 + 10.07 s^3 + 0.594 s^2 + 5.992 s + 0.1818 \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Rsource         6.345 s^4 + 0.404 s^3 + 9.878 s^2 + 0.48 s + 1.818  Z1 = 0.3152, Z2 = 0.9123, Z3 = 1.409, Z4 = 1.727, Z5 = 1.571<\/code><\/pre>\n
        Use a simulator to verify the results.<\/p>\n
        Since all the polynomials have numerator orders higher than their denominator, they all should work in series structure. But it is not!<\/p>\n
        5th order LC ladders<\/figcaption><\/figure>\n
        Transfer functions<\/figcaption><\/figure>\n
        Frequency responses match for sets 1, 3, 5, 6 in series structure and 2, 4, 7 in shunt structure. There must be some other sign of the structure.<\/p>\n
        Now compare their input impedances. Amplitude shows impedance module. 0 dB is 1 Ohm in the chart, +20 dB is 10 Ohm. Phase shows image part of a value. 0 degrees means that impedance is real.<\/p>\n
        Input impedances<\/figcaption><\/figure>\n
        Only 1st circuit has input impedance closest to a real value (green lines) in the pass-band. It is the values which can be found in tables in books. Now it is clear why.<\/p>\n
        The values of the 1st circuit are computed using only left half plane zeroes of the p(s)*p(-s).<\/p>\n
        Conclusion<\/h2>\n
        LC ladders with the same transfer functions can be built using different component values. There is no big difference if it is a low frequency circuit. For high frequency circuits input impedance is important. Use either values from tables or compute them using only left half plane zeroes of a p(s)*p(-s).<\/p>\n
        Unfortunately, some books have tables with typos. Use [1].<\/p>\n
        Verify what you read.<\/p>\n
        There are no solutions for even-order Chebyshev filters when a source is equal to termination. The higher ripple the higher a termination to source ratio should be to get a solution. Some tools compute ladder values using iterative equations [2] and can compute values at any source and termination for even-order Chebyshev filters. Be careful with such tools. [3] offers a way to find solutions for band-pass filters.<\/p>\n
        If you are interested to see a code for old computers, see [4].<\/p>\n
        References<\/h2>\n
          \n
        1. \n
          Anatol I. Zverev, \u201cHandbook of Filter Synthesis\u201d.<\/p>\n<\/li>\n
        2. \n
          Lars Wanhammar, \u201cAnalog Filters Using MATLAB\u201d.<\/p>\n<\/li>\n
        3. \n
          Songbai Zhang, Lei Zhu, \u201cGeneral synthesis method for symmetrical even-order Chebyshev bandpass filter\u201d.<\/p>\n<\/li>\n
        4. \n
          Thomtavanit Hatayodom, \u201cLadder network transfer function characteristics\u201d. Link<\/a>.<\/p>\n<\/li>\n
        5. \n
          «idealCircuit», a simulator. Link<\/a>.<\/p>\n<\/li>\n
        6. \n
          Nuhertz, Filter Solutions. Link<\/a>.<\/p>\n<\/li>\n
        7. \n
          «Circuit Calculator», an electronics circuit design tool. Link<\/a>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
          <\/div>\n

           \u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u043e\u0440\u0438\u0433\u0438\u043d\u0430\u043b \u0441\u0442\u0430\u0442\u044c\u0438            https:\/\/habr.com\/ru\/articles\/693606\/<\/a>
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