# Why Do Cascaded Low Pass Filters Not Simply Multiply Transfer Functions?

• hasan_researc
In summary: It's a simple operational amplifier circuit that helps connect the first filter to the second. It doesn't amplify the input, it just matches the impedance of the first filter.
hasan_researc
I do not understand section 8.6 of my lecture notes (see the attachment).

1. "If two low pass filters are cascaded the output voltage is not simply the product of
two transfer functions." : Why not? Why would we even think of multiplying two transfer functions? Which transfer functions are being referred to? Why?

2. "If you try to analyse such a circuit using phasors and an Argand diagram you will soon realize the shortcomings of that technique. This is because the second filter acts as a load for the first." : Could someone please explain how the second statement leads to the first statement?

3. "However, we shall later that a simple operational amplifier circuit called the unity gain buffer acts as an impedance matcher so that loading of the first filter by the second does not occur. " : What is an impedance matcher? Why does the loading not occur due to an impedance matcher?

Thanks in advance for any help.

Where's the attachment?

Here it is.

#### Attachments

• gtfhhhhhhhhhhhhhhh.pdf
130.2 KB · Views: 1,642
So a low pass works by being a two resistance voltage divider that is frequency dependent, and a low pass has a complex frequency dependent total resistance. If you put one in parallel to the capacitor of another low pass (which is what you do when chaining then) the voltage divider looks different, does it not? Because the second resistance is not formed by a pure capacitor but by a capacitor in parallel to a low pass.

The transfer function tells you how much of the input reaches the output. So it seems natural that you can multiply them, if the first low pass let's 50% through and the second is the same you expect 25% in the output, but this is normally not correct due to the above reasons.

You can imagine a unity gain buffer like a voltmeter that drives a power supply. If it reads 10V on the input, then it supplies 10V on the output. The nice thing is, that the input impedance is so high, that there is almost no current running through it, so the input circuit doesn't see it, and the output doesn't change the voltage when you draw current from it (which is what happens when you draw current from a low pass). In other words it has low output impedance.

Your questions are very elementary, maybe you should talk with other students about them.

0xDEADBEEF, you are totally correct. However, it should be pointed out that it is quite possible to build higher order lowpass filters that doesn't use buffer amplifiers. The buffer is not the only way to do it.

Antiphon said:
0xDEADBEEF, you are totally correct. However, it should be pointed out that it is quite possible to build higher order lowpass filters that doesn't use buffer amplifiers. The buffer is not the only way to do it.

The cascaded filter shown in section 8.6 of the pdf file is already a higher order filter (higher order than a single R-C stage). And, if additional R-C stages are cascaded, the filter order will increase by one for each additional stage added, without buffers.

So, it's already apparent that higher order filters (R-C filters in this case) can be built without buffers; it didn't need to be pointed out.

You must have meant to say something more than that, didn't you? :-)

Yes, I meant that the thread (not necessarily your post) seems to imply that cascading filters is somehow difficult. In fact it's one of the best established ways to make filters. Usually the stages are LC, not RC.

Last edited:
An empedance matcher transforms the maginitude and phase of the voltage and current of a signal such that a desired impedance is presented to a cascaded network. Passive impedance transformers have no power gain of course. For the unity gain buffer amplifier the voltage is not changed but the current is changed, and the buffer op amp will provide the needed current to maintain the same voltage at the output as at the input of this amp (up to a limit in practical amps of course).

## 1. What is a cascaded low pass filter?

A cascaded low pass filter is a type of electronic filter that is made up of multiple low pass filters connected in series. This means that the output of one filter is connected to the input of the next filter, resulting in a gradual reduction of high frequency signals.

## 2. How does a cascaded low pass filter work?

A cascaded low pass filter works by using a combination of resistors, capacitors, and inductors to create a circuit that allows low frequency signals to pass through while attenuating high frequency signals. Each filter in the cascade reduces the amplitude of high frequency signals, resulting in a smoother output signal with fewer high frequency components.

## 3. What are the advantages of using cascaded low pass filters?

One of the main advantages of cascaded low pass filters is that they provide a sharper cutoff for high frequency signals compared to a single low pass filter. This allows for better noise reduction and improved signal quality. Additionally, cascaded filters can be designed to have a steeper roll-off, meaning that they can more effectively attenuate unwanted high frequency signals.

## 4. Are there any limitations to using cascaded low pass filters?

While cascaded low pass filters have many advantages, they also have some limitations. One limitation is that the overall signal attenuation of the cascade is dependent on the individual attenuation of each filter. This means that if one filter has a lower attenuation than the others, it can limit the overall effectiveness of the cascade. Additionally, cascaded filters can introduce phase shifts in the output signal, which can impact the accuracy of the filtered signal.

## 5. What are some common applications of cascaded low pass filters?

Cascaded low pass filters are commonly used in audio and video equipment to remove unwanted high frequency noise from signals. They are also used in communication systems to remove interference and improve signal quality. Additionally, cascaded filters are used in electronic devices such as power supplies and motor controllers to reduce high frequency noise and improve the stability of the system.

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