# Bandpass Filter Question

#### Pleorqy

Joined Mar 23, 2020
7
Design a two-stage "bandpass" RC filter, in which the first stage is highpass with a breakpoint of 100Hz, and the second stage is lowpass with a breakpoint of 10kHz. Assume the input signal source has an impedance of 100Ω. What is the worst-case output impedance of your filter, and therefore what is the minimum recommended load impedance?

Thank you for all your supports.

#### ericgibbs

Joined Jan 29, 2010
9,806
hi,
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#### Pleorqy

Joined Mar 23, 2020
7
hi,
E
I tried to value capacitors and resistors in the highpass and the lowpass filter by applying the formula f = 1/2πRC. But I do not know what to do with the input signal source impedance.

#### ericgibbs

Joined Jan 29, 2010
9,806
hi,
One way of starting to solve it, would be to draw out a circuit diagram, showing the HP and LP filters etc.
Show the maths for the two sections.
We can then work from the same diagram
E

#### Pleorqy

Joined Mar 23, 2020
7
hi,
One way of starting to solve it, would be to draw out a circuit diagram, showing the HP and LP filters etc.
Show the maths for the two sections.
We can then work from the same diagram
E

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#### ericgibbs

Joined Jan 29, 2010
9,806
hi P,
This is your circuit design in simulation... do you see what changes are required to have a BW =100Hz thru 10kHz.. ?
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#### LvW

Joined Jun 13, 2013
975
Design a two-stage "bandpass" RC filter, in which the first stage is highpass with a breakpoint of 100Hz, and the second stage is lowpass with a breakpoint of 10kHz.
You are required to design a "two-stage" filter. Speaking of "two stages" we normally assume that in this case we have two stages which are separated from each other - which means: The 2nd stage does not load the first stage.
However, your circuit diagram shows two RC sections which are NOT isolated from each other...

#### Pleorqy

Joined Mar 23, 2020
7
You are required to design a "two-stage" filter. Speaking of "two stages" we normally assume that in this case we have two stages which are separated from each other - which means: The 2nd stage does not load the first stage.
However, your circuit diagram shows two RC sections which are NOT isolated from each other...
I don't think so. The question wants me to construct a bandpass filter, doesn't it? Please correct me if I am mistaken.

#### LvW

Joined Jun 13, 2013
975
I don't think so. The question wants me to construct a bandpass filter, doesn't it? Please correct me if I am mistaken.
Yes - of course. Did I put the bandpass into question?
But - for my understanding - a bandpass is required that consists of two isolated stages.
That means: Put an isolation amplifier (buffer, unity gain amplifier) between the 1st and the 2nd RC section.
In this case, both sections are decoupled - and the calcualtion is much more simpler (if compared with your diagram).
Only in this case (decoupling), both sections (high pass, lowpass) may be designed separately!!

#### WBahn

Joined Mar 31, 2012
25,556
While I can't be sure, I suspect that the question doesn't envision an isolation amplifier between the stages. Otherwise you just throw a couple more at it and eliminate the issues with input and output impedance.

I'm pretty sure one of the objectives of the problem is to explore and see the impact that one stage loading another stage has on the performance.

@Pleorqy : What is the criteria that you have been told to use for establishing the minimum load impedance?

#### JoeJester

Joined Apr 26, 2005
4,259
But - for my understanding - a bandpass is required that consists of two isolated stages.
That means: Put an isolation amplifier (buffer, unity gain amplifier) between the 1st and the 2nd RC section.
In this case, both sections are decoupled - and the calcualtion is much more simpler (if compared with your diagram).
Only in this case (decoupling), both sections (high pass, lowpass) may be designed separately!!
Active filters require an active device. A two stage passive filter doesn't. The question does not differentiate.

see Passive Bandpass

#### MrAl

Joined Jun 17, 2014
7,487
Hello,

To turn a two stage directly connected passive filter into a two stage passive filter with low interaction between stages (and hence less loading of the first stage by the second stage) simply do an impedance scaling on the second stage. The scale factor should really be 10 or greater but you be the judge.
This is pretty easy. If you use 1k on the first stage then use 10k on the second stage, and that means the second stage capacitor will be one tenth (1/10) of the first cap, so if the first cap was 0.1uf then the second stage cap becomes 0.01uf. So in this example we went from 1k to 10k and from 0.1uf to 0.01uf.
If you calculated 1k for the first stage and 2k for the second stage then the new value for the second stage is 20k The cap again scales inversely.

Of course you have to use the R's and C's to suite your frequency requirements.
To understand the effects of output resistance you might get away with just modifying the second stage and doing an analysis, and to understand the effects of input resistance you might get away with just doing an analysis on the first stage. You should really do the analysis on the entire circuit though but i am not sure how much you have progressed so far in your studies.

If you must account for the output resistance then you will be limited on the choice of R and C for the second stage. This means the first stage will scale by a factor of 1/10 for the resistor and 10 for the cap. This means the input resistance can be lower but there will be limitations before you end up having to go active rather than passive.

At some point we could look at the difference between two active stages vs two passive stages with impedance scaling. The results are interesting.

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#### JoeJester

Joined Apr 26, 2005
4,259
MrAl,

That would be the interesting assignment. Design both then contrast and compare. Then again, I would restrict the resistors and capacitors to some E-xx standard values. You might as well use the math and choose wisely.

I don't expect the professors to do that.

#### LvW

Joined Jun 13, 2013
975
While I can't be sure, I suspect that the question doesn't envision an isolation amplifier between the stages. Otherwise you just throw a couple more at it and eliminate the issues with input and output impedance.
Yes - perhaps you are right and I am wrong. On the other hand, it is a bad situation that the task description is not detailed enough. To me - a "two-stage bandpass RC filter" contains two "stages/blocks" which can be designed separately.
Without an isolation amplifier it would not be necessary to explicitely mention "two stages" .
However, I know - it is a matter of interpretation.......(not a good situation)

#### WBahn

Joined Mar 31, 2012
25,556
Yes - perhaps you are right and I am wrong. On the other hand, it is a bad situation that the task description is not detailed enough. To me - a "two-stage bandpass RC filter" contains two "stages/blocks" which can be designed separately.
Without an isolation amplifier it would not be necessary to explicitely mention "two stages" .
However, I know - it is a matter of interpretation.......(not a good situation)
Keep in mind that assignments are given in the context of the course material being covered.

#### MrAl

Joined Jun 17, 2014
7,487
I did some quick testing and found some interesting results.
The passive impedance scaled version is very much like the active isolated version.
In one case the only difference is the denominator 's' term which was:
s*(RC+1/A)
for the passive type, where A is the impedance scale factor with A=inf for the active type.

I will look at this more tomorrow.

#### LvW

Joined Jun 13, 2013
975
Keep in mind that assignments are given in the context of the course material being covered.
Yes - but this does not help me at all. I am not participating at this course......therefore, the questioner is the only one who could know some background details (if any).......and he should inform us!

Again - here is another reasoning for my position:
* The task description explicitely requires two corner frequencies (and not a certain center frequency with a corresponding bandwidth). This is a rather simple and straightforward task to realize using two isolated RC sections (highpass, lowpass).

* Following the same approach - without such a buffer amplifier - you must try to select a pretty high impedance niveau for the last section. However, you cannot avoid loading effects for the first section - and the result will be not as correct if compared with the first approach (with a buffer).
More than that, the mentioned high impedance niveau of the last section imposes severe problems as far as the second requirement is concerned: Allowable minimum load impedance (for an errror, not yet specified.)

* Of course, in principle it is possible to use a 4-element circuit as shown previously (without a buffer) and to dermine the parts values so that the bandpass exactly meets both corner frequencies (like the version with buffering) - however, the calculation is a bit more involved.....

* My recommendation to Pleorqy: Try to answer the question based on both approaches - with and without such a buffering between both sections.

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#### MrAl

Joined Jun 17, 2014
7,487
I went over the two versions.
When the impedance scaling is 10x the two responses are nearly the same with the passive set midband gain just 10 percent lower than the active set.
When the impedance scaling is 100x though the two responses overlap so perfectly it is impossible to tell the difference between the two.
The difference in the transfer functions is that the passive has an extra time constant associated with 's' in the denominator, so the damping factor is a bit higher. If the impedance scaling factor is large enough then this second term is small compared to the first term so the first term dominates. In the active set this first term is the only one associated with 's' so that's how the two become so close for large scale factors.
This result is intuitive also because if the second stage does not load the first stage too much then the filter function would be the same as the active set with just a minor difference because in the active set the buffer just prevents the first stage from being loaded by the second stage.