# Matching impedance network for an LC ladder network

Let's say I have a 5 stage L-C ladder network with L= 4.7 micro Henry and C= 60 pF. The characteristic impedance of the network becomes 280 ohms (square root(L/C)). However, from circuit analysis, the impedance of the circuit becomes j*561 ohms at 25 MHz. If I want to drive a load of about 50 ohms which will be placed at the end of the L-C ladder network what impedance of the ladder network will come into consideration for matching? Will it be 280 ohms or j*561 ohms for maximum power transfer? The source is a 50 ohm AC source. I want to pass a signal of about 9 MHz through this ladder network.

• Rather what will this undefined filter do, which can be easily bode plotted in Falsad’s site , what do you need? Define s11,s22,s21 Commented Oct 24, 2018 at 19:01
• why use a mismatched filter? Commented Oct 24, 2018 at 19:07
• The whole LC-chain-as-transmission-line approximation only works below the cutoff frequency. Around and above the cutoff frequency it looks like a low-pass filter. 1/(2*pi*sqrt(L*C)) is around 9.5MHz -- so it's really no astonishment at all that the thing looks purely reactive to the generator. Commented Oct 24, 2018 at 19:13
• Thanks for the comments. Actually, I am interested to know what will happen below 9.5 MHz for the mentioned value of inductor and capacitors. If it is a single L-C low pass filter I can understand it will attenuate high-frequency components over 9.5 MHz. However, I am not clear about how the chain L-C network will work below 9.5 MHz in this case. If I need to pass a signal from generator with a frequency below 9.5 MHz to a 50-ohm antenna what parameters should I consider to modify? I need to keep the L-C ladder network which is a constraint for me while ensuring signal transfer to load/antenna Commented Oct 24, 2018 at 20:15
• @TimWescott you are forgetting a transmission also needs a distributed R, so it has large ripples in the decade below cutoff. Commented Oct 25, 2018 at 10:10

Well, I have no idea why you would want to, but the impedance and transfer looks like this:

Gain at 9MHz isn't very good. Needless to say, there's nothing left at 25MHz (at least, assuming the source and load are actually 50Ω, as given, and the components are ideal). The passband ripple is very high because of the mismatch. This is why we don't pick arbitrary values and try to force them into working just anywhere.

Input impedance, zoomed 8-10MHz:

Cursors showing approx. ±20% bounds on Rin, centered on 400Ω. In the same range, Xin varies 727-826Ω.

If you wanted to match it at 9MHz, the simplest network is an L-match. We can use a calculator such as this,
https://www.eeweb.com/tools/l-match/
which would get this transfer curve:

The overall schematic looks like so:

The general effect of the additional LC (of very particular values) is to shift around the ripple/peaks in the passband, while also adding one more ripple overall, as it is itself a lowpass section. Essentially the same filter profile (as original) can be had by simply adjusting one pair of elements (almost any pair, really) in the original 5L-5C section to achieve the same condition.

Real results, using lossy components, or including parasitics for that matter, or component tolerances, are left as an exercise for the student.

Since each LC is loaded by smaller reactive load, each pole shifts after each successive “step” in the LC ladder. This results in a 6dB swing in the transfer function or +/-3dB from pole to pole till you get to the output.

I chose Falstad’s filter simulator since LC Ladder is a defined circuit, then stretched the y axis spectrum slider right THEN the frequency response so that I got near your values.

You can see the results here.

What you really want is a smooth low Q, flat group delay, Bessel flat response or a steep Chebychev maximally flat response. These 2 types which can be adjusted with more specs like 3dB flat or 6dB flat and degree of flatness unlike the LC ladder which has large de-tuned ripples. Chebychev staggers the peaks too but in a way that the ripple is minimized to any amount like 1dB or 0.1dB which trades off steepness of the skirt. There are many more options like Cauer Elliptical, raised cosine, Gaussian linear phase, etc.