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

[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.

 

[The Electrician]

Where's the attachment?

 

[hasan_researc]

Here it is.

 

[0xDEADBEEF]

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.

 

[Antiphon]

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 Electrician]

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? :-)

 

[Antiphon]

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.

 

[ottmarkw]

As the pdf explains, the loading impedance of the cascaded filter changes the effective transfer function of the first filter stage compared to when it is calculated (or measured) without the cascaded filter (and the cascaded filter transfer function is also changed by the impedance of the filter at its input. This effect generally occurs when cascading reactive filters. For filter designs that involve cascading of passive reactive filters, the loading effects of cascaded stages need to be taken into account in the design. There are common methods and tables that help with the design of LC ladder filters. If an ideal buffer amplier is inserted between two cascaded reactive filter stages, the overall tranfer function would then be the product of the individual transfer functions. Sometimes a buffer amplifier is still useful at the final output of a reactive filter in order to ensure the desired filter response is preserved when the impedance at the output of the filter is variable or not known.

 

[ottmarkw]

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).