# Ladder LC Filter - why are they minimum phase?

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 BODEs 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and filter synthesis (here I'm only interested in analog filtering with capacitor and selfs (and resistors), a rather outdated science).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.

• Minimal Phase what? shift ? sensitivity, slope? noise? and no they aren't. but can be if , only by good design, Jun 17, 2017 at 22:39
• You only get minimal phase shift in the passband by parameter choice. Jun 17, 2017 at 22:51
• @TonyStewart.EEsince'75 I mean minimal shift, that is to say minimal slope of the phase versus frequency (Nothing to do with sensitivity and noise). One thing too : I'm thinking only of lowpass filters. Hope I'm clear (I have problem with English and it's a very old question in my mind). Do you confirm your "no they aren't" in this new context ? Jun 17, 2017 at 23:42
• I suppose, the questioner means "minimum phase"? Answer: All transfer functions which have no zeros in the right half of the s-plane (RHP) have a phase response that can be derived from the magnitude response (BODEs relationship). These filters are called "mimimum phase filters". This definition applies to the shown LC-circuits, but does NOT apply to ALL LC-configurations.
– LvW
Jun 18, 2017 at 7:31
• @LvW, I would extend your nice comment to RHP poles and delays as well. If a transfer function features one or several RHPP, RHPZ or a delay, then the relationship linking phase and magnitude is lost. The Bode stability criteria no longer works and one must consider Nyquist instead. I have found another interesting discussion on minimum- or non-minimum phase systems here dsp.stackexchange.com/questions/2241/…. By the way, do you have examples of passive LC configurations which are no longer minimum-phase systems? Thanks. Jun 18, 2017 at 16:12

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: $$\small G(s)=\frac{1}{1+(LC)s^2}$$

# 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.

• 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

## Inductor specification comparison document table.

• These generalities have nothing to do with the question, and to much do with Texas Intrument (TI). Jun 18, 2017 at 7:23
• @andre et al failed to give a better answer or contribute anything at all Jun 23, 2018 at 1:02
• @andre the meat of my answer is the intuitive approach for stored energy response times within the passband or the causality of phase shift and significance with its rate of change being envelope group delay. The realization with passive filters must be the lowest Q or minimal liner phase. TI's tool allows these choices. Nyquist Diagrams are a mathematical tool to also design these. Sep 19, 2018 at 19:01
• FWIW, your question has an invalid assumption. LC ladders are not all equal , nor minimum phase. In a minimum phase system, the bulk of the energy in the impulse response of that system is at the beginning of the response. (Minimum group delay.) unlike a tsunami... Sep 19, 2018 at 19:07
• @andre314 These generalities have nothing to do with the question?? Did you read/ understand the beginning of the answer. BTW a Ladder has equal space rungs. and typically means a distributed line of equal LC values but then got modified to be any values of any spaced poles by some. Apr 9, 2019 at 21:37