# Frequency Response of LC Filter has a Peak with an Impedance Load

#### Nurahmed Omar

Joined May 12, 2019
3
Hi,

I am designing an LC filter with a cut-off frequency of 20 kHz for the Class-D amplifier. The filter schematic is shown in Fig.1 top right. I understand how to choose the LC values for the critically damped case for the pure resistive load as shown in Fig. 2.

The problem is I want to drive an impedance (400 Ohm + 15 mH) load, where the transfer function becomes a 3rd order, and it has a zero too (Fig.1). The frequency response has a peak for impedance load as shown in Fig.3 & Fig.4. The pole-zero map for impedance load is shown in Fig.5. The transfer function with values for impedance load in MATLAB is also shown in Fig.6.

How can I reduce the quality factor (peak) for this impedance load? How can I find out the quality factor (or damping ratio) for the 3rd order system?
(The problem is the complex poles and the real pole are close together, so I cannot approximate the 3rd order system with the 2nd order system where I can easily write down the damping ratio expression).

Fig.1 LC filter & transfer function

Fig.2 LC values calculation for a critically damped filter for resistive load

Fig.3 Frequency response of LC filter with resistive & impedance load

Fig.4 Frequency response of LC filter with resistive & impedance load (LTspice)

Fig.5 Poles & Pole-Zero map

Fig.6 Transfer function of 3rd order filter with values in MATLAB

#### Papabravo

Joined Feb 24, 2006
20,985
I think you have two choices:
2. Design an active filter so you can isolate the filter from the load

#### Tesla23

Joined May 10, 2009
542
Hi,

I am designing an LC filter with a cut-off frequency of 20 kHz for the Class-D amplifier. The filter schematic is shown in Fig.1 top right. I understand how to choose the LC values for the critically damped case for the pure resistive load as shown in Fig. 2.

The problem is I want to drive an impedance (400 Ohm + 15 mH) load, where the transfer function becomes a 3rd order, and it has a zero too (Fig.1). The frequency response has a peak for impedance load as shown in Fig.3 & Fig.4. The pole-zero map for impedance load is shown in Fig.5. The transfer function with values for impedance load in MATLAB is also shown in Fig.6.
One way that may help is to synthesize a filter with a trailing series L that can absorb the load inductance, your problem is though, that 400ohm and 15mH is too high a Q for 20kHz of bandwidth. Using the tool here: https://rf-tools.com/lc-filter/

if your inductor was 4.5mH or less you could absorb it into the final L. I can get about 6kHz of bandwidth for a load of 15mH and 400 ohms:

I haven't thought much about the right way to synthesise a filter for a class-D amplifier as your source impedance is almost definitely not 400 ohms, I'm sure papers on class D design talk about this.

If the right way to synthesize this filter is as a doubly terminated filter (whch is what the filters above are), then you are running into the Fano matching limit. In that case lookup the many references on broadband matching.

#### Nurahmed Omar

Joined May 12, 2019
3
I think you have two choices:
2. Design an active filter so you can isolate the filter from the load
Since this is part of designing Class-D amplifier (power amplifier), I guess active filter is not suitable for this. Any idea of how to redesign the filter?

#### Nurahmed Omar

Joined May 12, 2019
3
One way that may help is to synthesize a filter with a trailing series L that can absorb the load inductance, your problem is though, that 400ohm and 15mH is too high a Q for 20kHz of bandwidth. Using the tool here: https://rf-tools.com/lc-filter/

View attachment 206844

if your inductor was 4.5mH or less you could absorb it into the final L. I can get about 6kHz of bandwidth for a load of 15mH and 400 ohms:
View attachment 206845
I haven't thought much about the right way to synthesise a filter for a class-D amplifier as your source impedance is almost definitely not 400 ohms, I'm sure papers on class D design talk about this.

If the right way to synthesize this filter is as a doubly terminated filter (whch is what the filters above are), then you are running into the Fano matching limit. In that case lookup the many references on broadband matching.

This is really interesting. You showed the possibility of using the inductance load as part of the LC filter, so I can get rid of the true inductor from the traditional LC filter.

I don't know how to choose source impedance either. I tried the filter design tool link that you gave above, when I choose source impedance that different from load impedance it shows error (cannot synthesis the filter).

#### Papabravo

Joined Feb 24, 2006
20,985
I suppose you could go get a textbook and learn the basics of filter design. You might or might not have time for that. You do know that passive filters have an insertion loss that needs to be made up somewhere. If you make the active filter early in the process maybe you can avoid putting it just in front of the load,

For a maximally flat, 3 pole Butterworth response you start with a prototype lowpass filter for either 1 Hz or 1 rad/s. You select the component values from the table. I recommend the LCL topology. You then scale the whole thing up in magnitude and frequency. If the source and load impedances are equal, the insertion loss of the passive lowpass filter will be -6 dB. You need to make that up someplace else especially if you are directly driving the output device.

Given the output requirements of 400 Ω and 15 mH there might not be any way to drive that load at 20 kHz.

EDIT: I cannot drive that load with a low impedance voltage source. With the L first and then the R, I'm down 13.6 dB at 20 kHz. If I switch the components, I get a highpass response with -55 dB at low frequencies. I don't think either of those is what you want.

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#### MrAl

Joined Jun 17, 2014
11,250
Hi,

I am designing an LC filter with a cut-off frequency of 20 kHz for the Class-D amplifier. The filter schematic is shown in Fig.1 top right. I understand how to choose the LC values for the critically damped case for the pure resistive load as shown in Fig. 2.

The problem is I want to drive an impedance (400 Ohm + 15 mH) load, where the transfer function becomes a 3rd order, and it has a zero too (Fig.1). The frequency response has a peak for impedance load as shown in Fig.3 & Fig.4. The pole-zero map for impedance load is shown in Fig.5. The transfer function with values for impedance load in MATLAB is also shown in Fig.6.

How can I reduce the quality factor (peak) for this impedance load? How can I find out the quality factor (or damping ratio) for the 3rd order system?
(The problem is the complex poles and the real pole are close together, so I cannot approximate the 3rd order system with the 2nd order system where I can easily write down the damping ratio expression).

View attachment 206826
Fig.1 LC filter & transfer function

View attachment 206827
Fig.2 LC values calculation for a critically damped filter for resistive load

View attachment 206828
Fig.3 Frequency response of LC filter with resistive & impedance load

View attachment 206829
Fig.4 Frequency response of LC filter with resistive & impedance load (LTspice)

View attachment 206830
Fig.5 Poles & Pole-Zero map

View attachment 206831
Fig.6 Transfer function of 3rd order filter with values in MATLAB

Hello,

Sometimes you have to add a small resistance in series with an inductor in order to damp the response. That's beacuse reactive components store energy and dont really dissipate any.
Another way is to design the filter so that it has a peak out of the frequency band of interest.

I guess you are using an LC filter with resistive load or with a load that is made from a series combination of another L and a load resistor, and the filter has to pass up to 20kHz so it is a low pass filter.

I took another look at this and found that it takes a resistance value too high in series with the two series inductors to get the damping to a place where the peak starts to drop significantly. Adding capacitance across the output helps, but it also changes the cutoff frequency and to compensate for that the two series inductor values have to be lowered, which in turn would have the undesirable effect of increasing current from the input.
Perhaps a post LC filter would help in addition to what is already there.

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#### MrAl

Joined Jun 17, 2014
11,250
Hello again,

One of the things you did not mention was what components are allowed to be changed. Can any components be changed?
If so, increasing inductance and increasing the capacitance across the load seem sot help reduce the peak amplitude response quite a bit. For a quick example i tried 10mH for each of the two series inductors and added 0.04uf across the load. The peak response went down a lot.
If that kind of change is acceptable we can look at optimizing the values.