# Second order passive low pass filter design

#### LvW

Joined Jun 13, 2013
1,586
(Ro*s)/(Ro*s^2*C*L+s*L+Ro), {this is the transfer function}
This is the simplified transfer function of a bandpass (with only one single resistor).
Are we talking here about lowpass or bandpass?

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#### MrAl

Joined Jun 17, 2014
9,622
This is the simplified transfer function of a bandpass (with only one single resistor).
Are we talking here about lowpass or bandpass?
Hi there,

It does not matter because it was an exercise in the mathematics not how transfer functions relate to the class of filter. The sole reason for the writing was to show how to get the amplitude from a transfer function, using the most basic mathematical functions and procedures. That's what i think every student should know before using math software.
Also, i did not want to do a transfer function that was the same as the original poster just yet to give them a chance to do it first.

#### Ian0

Joined Aug 7, 2020
6,660
So, let’s go way back to post #124, which has the circuit diagram.
What is the nature of the voltage to be measured, and why is its source impedance 15Ω?
Is the A/D a microcontroller peripheral which has a limit of 0 to 3.3V, or is it some other sort of A/D with a larger input range. If it is the MCU then circuitry will be required to prevent the ±15V output from the op-amp doing some damage.
And if the source impedance is as low as 15Ω why is it necessary to use the op-amp as a buffer?
And if a second order filter really is required, then wouldn’t it be far more sense to use the op-amp in a Sallen & Key filter structure?

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• ericgibbs

#### LvW

Joined Jun 13, 2013
1,586
Question to Ian0: Do you know what the questioner (pinkyponky) really wants?
In this thread, I have seen many different circuits - with and without opamp.

#### Ian0

Joined Aug 7, 2020
6,660
Question to Ian0: Do you know what the questioner (pinkyponky) really wants?
In this thread, I have seen many different circuits - with and without opamp.
He wants an input filter (either antialiasing or interference removing) for an A/D. It took 124 posts to establish that.
I think that now we have established that a practical brick-wall filter cannot be made with a single inductor, so progress is being made towards some sort of second-order filter with the advice of some very patient people.

#### The Electrician

Joined Oct 9, 2007
2,914
Hi MrAl!

I already said the application. This filter will be used to filter the input signal of the ADC and output of the DAC.

You asked me to derive the transfer function. Eventually, Done!. Ok. How can I plot the response from this transfer function?.

How to design the R, L, and C values?. These values will be used to design a filter that attenuate the signal above 100kHz.

S=jw, w=2*pi*f, and j2=-1. After substitution of these values, the final Transfer Function is derived and attached below. Please have a look and tell me how to plot the response from the below transfer function.
PP, the transfer function you have derived here is the transfer function of the circuit in post #150 if you let R=R1, L=L1, C=C1, and Rload =R2+R3.

The response in post #150 is the response of your transfer function; that's what LTSpice does--it plots transfer function responses, among other things.

Are you required to plot the response from the mathematical expression of the transfer function, or will the plot from LTSpice be enough?

#### pinkyponky

Joined Nov 28, 2019
318
PP, the transfer function you have derived here is the transfer function of the circuit in post #150 if you let R=R1, L=L1, C=C1, and Rload =R2+R3.

The response in post #150 is the response of your transfer function; that's what LTSpice does--it plots transfer function responses, among other things.

Are you required to plot the response from the mathematical expression of the transfer function, or will the plot from LTSpice be enough?
Hello Electrician!

As you see that I was derived the transfer function in the post #174, I want to design a RLC values from that derivation, to design RLC filter.

#### pinkyponky

Joined Nov 28, 2019
318
Hi pinkyponky,
as I have told you already before, it is not necessary (and very involved and time consuming) to do all the math with complex figures.
Nobody is doing this (because you would start at "zero".)
This was already done long time ago - and the results are summarized in tables which allow to design a second order filter (or even higher order) for the desired approximation.

In post#177 I gave you the transfer function for your second-order RLC filter.
For your convenience, I repeat it again: The part names are according to your hand written figure in your post#174.
The parallel connection is Rp=R||RL.

H(s)=Ao / {1 + s[(RpC + L/(R+RL)] + s²AoLC}

Writing the transfer function in this form (so called "standard form") has the advantage that we immediately can identify the three main parameters which define the lowpass response:
* DC gain is Ao
* Pole frequency is wp=SQRT(1/AoLC)
* Quality factor (pole-Q) can be found by 1/Qp=wp[(RpC + L/(R+RL)]

* For DC the gain Ao can immediately found to be RL/(R+RL). You are free to select R undf RL according to your requirements.
* The pole frequency wp is in rather vicinity to the wanted 3dB-cutoff frequency wc (for a maximum flat magnitude, Butterworth approximation, it is identical wp=wc). Because Ao now is known you can compute the values for L and C together with the required pole-Q (next point). This results in a set of two equations for the two unknown values L and C.
* The required Q-values can be found in corresponding tables: Q=1/SQRT(2)=0.7071 for max. flat (Butterworth) response. Larger Q-values give a Chebyshev response with a slight "peak" in the vicinity of the pole frequency (tabulated for a peak ("ripple") of 0.05dB, 0.1dB, 0.3dB.........3dB).

Based on these information you immediately can design your filter.
I hope this helps.
Hello LvW,

May I know how the pole frequency and quality factor of equations are derived?

However, I have calculated the parameters which you mentioned in the post #191

According to my requirement, R=1Kohm and Rp=15ohm, RL=7Kohm

The chosen values are R=1kohm, L=100nH, C=2nF

Therefore, The calculated above three parameters are:

Ao=0.875, wp=75Mhz, Qp=0.4

If those are wrong, then what are the best valued to be calculated for Ao, wp, Qp?

#### Ian0

Joined Aug 7, 2020
6,660
IF I remember correctly, this filter precedes an ADC, so therefore must be an anti-aliasing filter.
Therefore the cutoff frequency must be less than half the sampling frequency.
The Q is a matter of choice, depending on many factors, such as:
1) how much the amplitude response can vary within the passband. If it is important, choose Butterworth (Q=0.7)
2) how important it is to have constant group delay (complex signals retain their shape). If this is important, choose Bessel (Q=0.577)
3) if there is energy in the signal close to the sampling frequency. If so, choose Chebyshev.

#### pinkyponky

Joined Nov 28, 2019
318
IF I remember correctly, this filter precedes an ADC, so therefore must be an anti-aliasing filter.
Therefore the cutoff frequency must be less than half the sampling frequency.
The Q is a matter of choice, depending on many factors, such as:
1) how much the amplitude response can vary within the passband. If it is important, choose Butterworth (Q=0.7)
2) how important it is to have constant group delay (complex signals retain their shape). If this is important, choose Bessel (Q=0.577)
3) if there is energy in the signal close to the sampling frequency. If so, choose Chebyshev.
Hi Ian0,

I have chosen the butterworth filter and used the table (which is in post #22) to calculate the RLC value, but the problem is here is that I have calculated high inductance in mH.

So, then, I have followed the instruction which is given by MrAl. He asked me to derive the transfer function from the electrical circuit. After that I don't how to go further to design RLC filter.

People are helping me but not in the same direction. that is the problem.

Please could you help me from the post #181?.

#### Ian0

Joined Aug 7, 2020
6,660
A value of inductance in the mH region is correct.

#### pinkyponky

Joined Nov 28, 2019
318
A value of inductance in the mH region is correct.
I need small size components since we have physical dimension problem on the PCB. If I go with high inductor value then it has high physical dimensions.

In order to fulfill above requirement what I have to do.

#### Ian0

Joined Aug 7, 2020
6,660
Abandon the idea of a filter that uses inductors.

#### pinkyponky

Joined Nov 28, 2019
318
Abandon the idea of a filter that uses inductors.
Why? what happen?

The experts like you its not a difficult design. It's very simple for you. I have given all my requirements and also I put my effort lot to design RLC fiter, such as deriving the tranfer function and also I have calculated the L and C values as per the post #22. Even-though I'm not able to complete this work. I'm requesting you that how to design a RLC filter with low inductance values.

#### Ian0

Joined Aug 7, 2020
6,660
Why? what happen?

The experts like you its not a difficult design. It's very simple for you. I have given all my requirements and also I put my effort lot to design RLC fiter, such as deriving the tranfer function and also I have calculated the L and C values as per the post #22. Even-though I'm not able to complete this work. I'm requesting you that how to design a RLC filter with low inductance values.
Not possible Unless you reduce the load resistance.

#### Ian0

Joined Aug 7, 2020
6,660
You have a load resistance of 3.3k, and a source resistance of 15Ω, and need an attenuation of 3:1, you could use a load resistance of 7.5Ω, which would reduce the inductance by a factor of 100. Your inductance will then be in the 10uH to 100uH range, but can your source drive an impedance that low?

#### BobTPH

Joined Jun 5, 2013
6,041
What would your response be if I said I was having trouble designing a 5 passenger car that can go from 0-60 in 5 sec, has a 500 mile range and the fuel tank can hold only 1 gallon?

• Ian0

#### pinkyponky

Joined Nov 28, 2019
318
What would your response be if I said I was having trouble designing a 5 passenger car that can go from 0-60 in 5 sec, has a 500 mile range and the fuel tank can hold only 1 gallon?
Hi Bob,

I gave the full details such as input resistance, load resistance, application of the filter circuit design, and also I have derive the transfer function. I struck here that reduce the inductance value.

#### Ian0

Joined Aug 7, 2020
6,660
Repeat after me (in a badly faked Scottish accent) “You cannae change the laws of physics, captain”.
If you want a smaller inductance, you must have a higher cutoff frequency, or a smaller load impedance.
Do you really think that I would be using an inductor this big and expensive (it’s 4mH @ 90A) if I could design a smaller and cheaper one? Even when you do make a RLC filter, it won’t be very good, due to parasitic capacitance in the inductor.
If you had listened to what the experts said 3 months ago - use an RCRC filter or a Sallen & Key filter - your project would be up and running by now.
(Even a Sallen &Key filter has problems at high frequencies due to the phase shift in the op-amp.)