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8 matching network configurations

Image source: OpenLab at City Tech (New York City College of Technology) - Lesson 04: Matching networks – impedance matching

What would be their resonance frequency ω_0? In an L-sections, Β=ω_0/(Q ). Where Q = the quality factor and bandwidth = B. I've seen that while evaluating ZL is being taken as real in some online materials, should I also follow that for simplicity?

Is there a way to simplify and / or solve equations using a program tool, online? I tried MATLAB, but if the output is shown in subscripts too that'll be great.

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  • \$\begingroup\$ These L-match network operates at one frequency, whereas the terms bandwidth, resonance evokes filter attributes over a range of frequencies. Yes, a matching network Q can be determined - I'd limit Q's usefulness to evaluating component tolerances for the impedance match at the single design frequency. It is likely that the filtering abilities of these L-networks are insufficient for attenuating undesired frequencies. Or try more complex networks if you wish to combine matching with filtering. \$\endgroup\$
    – glen_geek
    Commented Sep 5, 2023 at 14:06
  • \$\begingroup\$ @glen_geek As it turns out, Q here is as representative for bandwidth as it is anywhere else. To impedance-match a bandpass filter, an L-match with Q factor less than the bandpass Q is required. How much, exactly, I don't have the formulas for (or frankly understanding; I would have to work out a lot of math to do that), but to a first order, and assuming input and output impedances are real, this is the case. To the OP, I would suggest using SPICE simulations to show overall response: note these networks contribute poles and zeroes, so affect response of the network they're a part of. \$\endgroup\$
    – Tim Williams
    Commented Sep 5, 2023 at 15:21
  • \$\begingroup\$ Thank you for responding @TimWilliams, math is what I aim for. \$\endgroup\$
    – Praskand
    Commented Sep 5, 2023 at 18:26
  • \$\begingroup\$ Well, books can be (and have been) written on network design, including impedance matching and bandwidth. Generally, we're talking filters. Mathematically, it's in the domain of rational polynomials, with trig added if you need to work with phase as well. Optimization algorithms quickly take over, especially if you need to work multiple variables at once, account for component tolerance etc. There are programs for this, which are well worth it. Or you can plug things into SPICE and just tweak values around. \$\endgroup\$
    – Tim Williams
    Commented Sep 5, 2023 at 18:58
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    \$\begingroup\$ @Praskand - Hi, For the second time, your edit to the question has been reversed (rolled-back). You should not extend or significantly alter a question, after it has received any answers. Otherwise, it makes answers to the original question look wrong or incomplete. (This has been discussed over on Meta.SE using the term "chameleon questions", so you can read more over there.) I realise you are new to the site, so please take this advice - don't change this question again. If you want to ask a new, follow-up question, you can do that & include link to this question for context. Thanks. \$\endgroup\$
    – SamGibson ♦
    Commented Sep 6, 2023 at 7:39

1 Answer 1

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What would be their resonance frequency ω_0?

You are slightly missing the point here. For the CL high-pass and LC low-pass L-pads, true impedance matching is not at the natural resonant frequency of the circuit. True impedance matching can be significantly lower than the natural resonant frequency (low-pass) or, significantly higher for the high-pass.

However, if you are using an L-pad to match (say) 50 Ω to (say) a load of 300 Ω the difference is quite small but, there is still a difference: -

enter image description here

Image from my basic website. The derivation of the formulas are included as well so, when you say this question: -

Is there a way to simplify and / or solve equations using a program tool, online?

The answer is yes for the common examples provided on this page. It's still a work-in-progress by the way. As a double-check I had this set-up in microcap: -

enter image description here

I've plotted input impedance (magnitude and phase) against frequency. As you can see, at 10 MHz, the input impedance (bar the insignificant rounding errors) is resistive at 50 Ω. At the natural resonant frequency, the angle of the input impedance is lagging by nearly 11°. Now this isn't much but, it's important enough to note.

You will also find that the amplification at the true matching frequency equals the Q of the circuit. This is interesting because, at the natural resonant frequency, amplification is also equal to the Q-factor for a 2nd order filter. However, you don't get good impedance matching of resistive impedances at the natural resonant frequency.

I've seen that while evaluating ZL is being taken as real in some online materials, should I also follow that for simplicity?

That's the sensible approach because, complex impedances can easily be made resistive by using another series or parallel reactive component at the impedance matching frequency.

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  • \$\begingroup\$ Thank you for responding, math is what I aim looking for. \$\endgroup\$
    – Praskand
    Commented Sep 5, 2023 at 18:43
  • \$\begingroup\$ @Praskand if we are done here, please take note of this: What should I do when someone answers my question. If you are still confused about something then leave a comment to request further clarification. \$\endgroup\$
    – Andy aka
    Commented Sep 5, 2023 at 19:17
  • \$\begingroup\$ Thanks for replying, I have edited the body, please see! \$\endgroup\$
    – Praskand
    Commented Sep 5, 2023 at 19:54
  • \$\begingroup\$ @Praskand no; once an answer is given, changes you make to the question must not undermine the answer(s) given. If you want to take the search for knowledge in a spin-off direction, raise a new formal question. I'll also add that looking for on-line tools that solve algebra is off-topic on two counts (1) asking for on-line resource recommendations falls under asking shopping questions (off-topic) and (2) asking for tools that solve algebra is not a valid EE question because it's basically a math problem and not specifically EE (i.e. tools that solve equations are general and not EE based). \$\endgroup\$
    – Andy aka
    Commented Sep 5, 2023 at 20:06
  • \$\begingroup\$ I've rolled back your question to how it was when I answered the question. \$\endgroup\$
    – Andy aka
    Commented Sep 5, 2023 at 20:06

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