Ladder LC Filter - why are they minimum phase?

Question

Ladder LC filters are well-known to be minimum phase filters.

^{simulate this circuit – Schematic created using CircuitLab}

Above: The ladder LC filter I'm thinking of, though ladder filters are more general, and so far I know, the question stays valid for general ladder filters.

Is there a explanation/demonstration or simply a intuitive explanation of this property of ladder filters?

Bonus question for those interested: What I find strange is that the analysis of lossless delay-line equation (telegraph equation) is done by "approximating" the delay-line by a LC ladder, that is to say, a minimum phase filter, though delay-line are the archetyp of the no minimum phase filter.

EDIT (18 June 2017) I restrict the question to lowpass filters only.

EDIT (19 June 2017) This question is in standby for a few days because I'm invesgating the problem. Lastest news :

• What I mean by "minimum phase filter" is a filter where the phase response can be derived from the magnitude response according to the BODE`s relationship. That's a concept that arises in Automatics and filter synthesis (here I'm only interested in analog filtering with capacitor and selfs (and resistors), a rather outdated science).

• Allpass filter are evidently not minimum phase filter (because they have all the same magnitude response : 0dB whatever the frequency, so we can't deduce the phase from the magnitude response)

• For rational transfert functions, minimum phase filters are the ones that have their zeros in the left half complex plan.

• It seems that the ladders such as the one represented above don't have any zeros (whatever the values of the components, and assuming the driving impedance and the load impedance are purely resistive). In that case it is evident that it's a minimum phase filter.

• Next problem : finally what is the general accepted definition of a ladder filter.

Answer 1

If we model a lossless transmission line as a pure time delay, its transfer function would be \$\small G(s)=e^{-sT}\$. To be minimum phase, the inverse would need to be causal, and \$e^{sT}\$ is non-causal, hence the transmission line is non-minimum phase with this model.

If we model the line by an L-C ladder, then we are doing something akin to using a Pade approximant, and we can make as accurate a model as we wish by increasing the order of the model.

In its simplest form for the problem to hand, the (0, 2) Pade approximant of \$\small e^{-sT}\$ is: $$ \small G(s)=e^{-sT}\rightarrow \small\frac{2}{2+2Ts+T^2\small s^2}$$

which is causal and minimum phase as the inverse can be realised by a differentiator and a double-differentiator.

This compares, for example, with a single-stage L-C ladder TF model: $$\small G(s)=\frac{1}{1+(LC)s^2}$$

Answer 2

There are 2 exceptions to your generalization:

A minimum phase filter and a zero phase filter with the same amplitude response.

It is also possible to have an LC ladder filter with a zero gain amplitude response but shift phase in the region of interest over two decades

Intuitive explanation

Minimum phase means the energy is front loaded or causal like a step input with a fast response and there is no energy before time=0 It can be applied to impulses steps and wavelets.

The minimal phase filter means the the energy rises quickly

Zero Phase means there is maximum energy at relative time at t=0 Maximum phase means the energy is back loaded like a Tsunami.

You cannot For smoothest linear phase shift or flattest group delay, always choose a Bessel filter. THis has low Q =0.5 For higher order filter ALLPASS filters with high group delay, the number of stages must be increased proportional to the delays. Q's are adjusted for each stage so that interstage effects result in a desired group delay passband response or in LC filters, impedance ratios in a ladder results in a flat group delay.

All pass Filter ( delay line)

LC All Pass filters are used to create delay lines for HF pulses or data or analog scope signals so triggers can be viewed.

LC Filters are low loss passive filters of any order and many topologies suitable usually for >100kHz where Op Amp gains are diminished or for passive DC power SMPS filters at lower ranges using the driving point impedance.

If you understand Active filters then you can understand phase and amplitude responses of LC filters. They can share similar characteristics but have unique advantages. With free software you can chose a design an active 7th order Chebyshev low pass filter and have a complete Schematic and Bill of Materials (BOM ) in less than a minute than it takes to read this answer, but with limitations of gain-bandwidths. RLC filters also exist for DC power LPF's used in Class "D" power Amps and SMPS ripple filters.

Additional info

• For LC filters you must choose driving and load impedance.
• As in all design, first you choose the specs then and for filters you later choose topology and then realize it then repeat until specs match results with component tolerances, suitable impedance ranges, noise DC error, and cost.
• For active topology Sallen & Keys or multiple feedback or full differential

• TI software allows you to later change to any standard tolerance values with a RC tolerance dropdown menu

  This TI version does not balance input offset voltages for matched Rs from input bias current or allow easy scaling of RC, but the advanced user will know how to do this anyways. THe WEBENCH version is best to use as it is updated regularly.

Important reading on Filter types

http://www.ti.com/lit/an/sbfa002/sbfa002.pdf for a background on "some" of many kinds of filter properties used in both active RC and passive LC filters. In active filters impedance inversion is possible for negative resistance gain and negative reactance Capacitors to emulate capacitors but are limited by the gain bandwidth product.

For online login to www.ti.com ( free ) there are now hundreds of resources and free design software such as active filters http://www.ti.com/lsds/ti/analog/webench/webench-filters.page ( with TI logon)

You can also download the offline version but it is no longer updated http://www.ti.com/filterpro-dt to design the same Butterworth, Elliptical, Chebyshev

The TI software guides you easily to make any Active filter.

As always you must learn to define the Design Specs 1st with characteristics; (lowpass, highpass, allpass, bandpass, bandstop, (LPF,HPF,APF,BPF,BSF) ; parameters for gain, f1 breakpoint, passband gain ripple (error), f2 bandstop reference and attentuation

Class D amp LC filter design software

Inductor specification comparison document table.