# Tool to calculate cascaded but independent single-pole RC hi-pass filters

I frequently use a single op-amp stage for both gain and hi-pass filtering. I normally implement 2 independent single-pole low-pass filters: the input to the op-amp and the ground end of the negative-feedback gain-set resistor. Circuit example follows:

simulate this circuit – Schematic created using CircuitLab

RC network R1, C1 is a high-pass filter set at about 159 Hz.

RC network R2, C2 is another high-pass filter at about 159 Hz, except that the filter flattens out as the frequency drops to the point where the amplifier gain approaches unity.

Cascading these filters in this fashion does NOT result in a -3dB break-point at 159 Hz.

I use this type of circuit on a regular basis but I always wind up iterating component values until I reach the desired break-point frequency.

My question is: is there a technique that I can use to calculate component values that give me a closer approximation to my desired break frequency?

Just to be clear: I'm looking for a tool that allows me to calculate the effect of two cascaded but otherwise independent single-pole filters rather than the standard tools that calculate the component values for a two-pole filter.

This particular project is redoing a design done by someone else who just didn't get it right.

The circuit contains 4 functional blocks: a gain stage, a band-pass filter, a true-RMS detector and a 4-20mA transmitter stage. I have the opportunity to include 3- single pole RC filters within the signal flow: 2 stages exactly as shown above and a 3rd stage between the band-pass filter output and the true-RMS detector input.

I fully understand that cascading multiple single-pole filters like this does not give me the ideal response. However, what they give me is a response that is "Good Enough". Adding these filters takes the design from barely working to working quite well.

I don't mind iterating component values to take me to my desired break-point frequencies. I'm just looking for a tool that gets me there quicker.

As mentioned earlier, this is a trick that I use quite often in my designs because it costs almost nothing to include but can result in radically better performance.

If you use this circuit on a regular basis I guess it would be worth to calculate its transfer function once and for all, so you can just easily evaluate it given the concrete component values. Doing the math you get (assuming an ideal OP)

$$H(\omega)=\frac{j\omega R_1C_1}{1+j\omega R_1C_1}\left(1+\frac{j\omega R_4C_2}{1+j\omega R_2C_2}\right)\tag{1}$$

A plot of (1) in dB with the specified component values looks like this (|H|/db vs f in Hz):

from which you can see that the -3dB point is at about $250\,\text{Hz}$.

If you further assume that

$$R_1C_1=R_2C_2=\tau$$

and if you denote the gain at large frequencies by

$$g=1+\frac{R_4}{R_2}$$

then with a bit more math you get this exact expression for the 3dB cut-off frequency in radians

$$\omega_c=\frac{1}{\tau}\sqrt{1-\frac{1}{g^2}+\sqrt{\left(1-\frac{1}{g^2}\right)^2+1}}\tag{2}$$

which with the given component values gives

$$\omega_c=2\pi\cdot 247.28$$

For a large gain $g\gg 1$, formula (2) is closely approximated by

$$\omega_c=\frac{1}{\tau}\sqrt{1+\sqrt{2}}\approx\frac{1.55}{\tau}\tag{3}$$

There are a billion such tools (well, that's hyperbolic, but there are many). They may not use the exact topology you have there, but they'll all get the job done.

TI and ANALOG DEVICES are just two chip makers that provide good design tools online, and Microchip has a downloadable program

Note that when you cascade two two-pole filters, your -3dB point for the first filter then becomes your -6dB point. If you really want to do the math, just solve for your -1.5dB point $$\left | H\left (j\omega_c \right ) \right | = -1.5dB,$$ where H is the transfer function of your first stage and $\omega_c$ is your target cutoff. Of course, if your caps are 20% tolerance, YMMV on what the transition region actually ends up looking like.

Still, the best way to get an optimal filter is by providing your filter specs and figuring out how many poles you need to get it given a particular filter topology, and then using a pole-placing topology to get it -- e.g., use the filter design tools the way they were meant to be used.

• I've looked at the tools at all 3 of the manufacturers that you mention (and many others) - several times over the past many years. I also have purchased filter calculator programs that are very effective at designing multiple-pole filters. None of them addresses the situation that I've described. – Dwayne Reid Feb 23 '15 at 21:05
• That's because cascading low-pole filters does not produce an optimal high-pole filter. The way to use a good filter design calculator is to spec out the frequency response you need, and let the filter designer give you the appropriate filter with the appropriate number of poles in the appropriate places. – Scott Seidman Feb 23 '15 at 21:12
• you are correct. However, space constraints force me to use a single 8-pin package. That gives me two op-amp stages and one of those is already used. This is the other op-amp in the package and provides the gain needed. Because the gain stage is there and available, I have the ability to include my hi-pass filters at this point. I have my choice: 0, 1, or 2 poles of high-pass filter in this stage. I want to use 2. – Dwayne Reid Feb 23 '15 at 21:25
• Modified my answer. If you want to design that way, you need to calculate for the -1.5dB point for one stage. – Scott Seidman Feb 23 '15 at 21:29

It is not quite clear to me what you are looking for. Are you forced to use the shown topology? I suppose you know that there are second-order highpass filters which are simple to design?

The problem with your circuit is that the second first-order highpass (opamps feedback) is not a "classical" highpass because the gain has a finite value of unity for w=0. Is this topology a requirement? This complicates the calculation of the 3-dB frequency because the numerator of the transfer function has the form N(s)=as+bs² (a=0 for classical functions).

Because you are asking for a "tool". At present, I only can think of a classical circuit simulation program. Performing several ac simulations with parameter stepping (e.g. C2) should give you the desired information.

• I frequently have a number of cascaded stages with op-amps in those stages. These stages are usually gain or filters. Because these stages are AC coupled, I have the freedom to change those coupling components into a hi-pass filter. That makes adding extra poles of filtering extremely inexpensive. The downside is that the break point is "mushy" and not well defined. That's OK - I simply set all the combined break points low enough that I still get the passband response needed. But the extra attenuation at lower frequencies often greatly improves performance. – Dwayne Reid Feb 25 '15 at 4:54
• Nevertheless, a simple 2nd-order highpass filter in Sallen-Key topology is also capacitively coupled - and is easier to design. More than that, it is more flexible because you can implement pole-Q values larger than 0.5 (upper limit for your design). – LvW Feb 25 '15 at 8:05

http://sim.okawa-denshi.jp/en/Fkeisan.htm

You can try this,It's good for filter design.

• Hello Karthik, a link is not sufficient as an answer, please add some summary of its content, in case it gets broken – clabacchio Jul 25 '17 at 13:31