How to implement a transfer function with discrete components

[garvind25]

Hi,

I need to implement a second order transfer function in 's' domain (having 2 poles, 1 zero and 2.5 DC gain) using transistors and discrete components. Since I have not done so previously, can anyone guide me how to do so? I did a search with the keywords ‘realise a second order transfer function, bjt’ but ended up with papers on filters etc.

If possible, pls. point to me to any document / app note/ website demonstrating the transfer function realization with an example.

I am looking forward to your comments.

Thanks,
Arvind Gupta.

[Akanimo]

Hi,

A common-emitter amplifier should do, I think.
What are the zero and poles frequencies?

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Add:

What type of zero is it? Inverted or noninverted? Write out your transfer function too.

[LvW]

............ but ended up with papers on filters etc.

Thanks,
Arvind Gupta.

A transfer function with 2 poles and one zero IS A FILTER !!

[garvind25]

Thanks for the response.

The TF = 2.5 (s - 1)/ {(s+5) (s+60)}

Regards,
Arvind Gupta

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A transfer function with 2 poles and one zero IS A FILTER !!

Thanks for the clarification. BTW is there any general technique for implementing TFs, viz. one pole TFs, 2 pole TFs, 3 pole TFs and so on.

Regards,
Arvind Gupta

[asdf44]

As said, yes this is a filter and you need to be familiar with basic filter building blocks. There is no magic way to create an arbitrary system but there are some basic building blocks:

First order low pass (pole)
First order high pass (zero)
Integrator (pole)
Differentiator (zero)

And then more advanced filter topological which include multiple poles and zeros (and the possibility of complex poles as zeros) such as sallen-key or biquad (2 pole, 2 zero).

It's possible to create 3rd order and higher but, for various reasons, quite common to build those instead by cascading multiple first or second order blocks in series.

Note that everything mentioned above is equally applicable to analog or digital (see IIR) systems.


In the specific case of your transfer function I built it in LTSpice with RC->Differentiator (with opamp)->RC to create the two poles and one zero.

[garvind25]

As said, yes this is a filter and you need to be familiar with basic filter building blocks. There is no magic way to create an arbitrary system but there are some basic building blocks:

First order low pass (pole)
First order high pass (zero)
Integrator (pole)
Differentiator (zero)

And then more advanced filter topological which include multiple poles and zeros (and the possibility of complex poles as zeros) such as sallen-key or biquad (2 pole, 2 zero).

It's possible to create 3rd order and higher but, for various reasons, quite common to build those instead by cascading multiple first or second order blocks in series.

Note that everything mentioned above is equally applicable to analog or digital (see IIR) systems.


In the specific case of your transfer function I built it in LTSpice with RC->Differentiator (with opamp)->RC to create the two poles and one zero.

Thanks for the information. How did you provide a DC gain of 2.5 in your implementation pls? Can I ask you to share the ckt diagram / LTSpice file for a better understanding. I have not implemented TFs before and hence asking.

Regards,

Arvind Gupta

[Akanimo]

...
How did you provide a DC gain of 2.5 in your implementation pls? Can I ask you to share the ckt diagram / LTSpice file for a better understanding. I have not implemented TFs before and hence asking.
...
Arvind Gupta

asdf44 mentioned that he did it with an OpAmp, so basically, it would be Rf/R1=2.5 for a DC gain of 2.5.

You haven't implemented transfer functions before? Why then are you going for transistor-based implementation? Why not prefer opamp-based implementation which would be far easier?

Your transfer function is not in normalised form. Repost it's normalised form here.

[garvind25]

asdf44 mentioned that he did it with an OpAmp, so basically, it would be Rf/R1=2.5 for a DC gain of 2.5.

You haven't implemented transfer functions before? Why then are you going for transistor-based implementation? Why not prefer opamp-based implementation which would be far easier?

Your transfer function is not in normalised form. Repost it's normalised form here.

OK. I was asked to look for an inexpensive way to implement the transfer function. So I guessed that I would have to use transistors.

The normalized form of the TF is:

TF= 0.0083 (jw-1) / {-0.0033w^2 + 0.2166jw + 1}

Regards,
Arvind Gupta

[Akanimo]

OK. I was asked to look for an inexpensive way to implement the transfer function. So I guessed that I would have to use transistors.

Opamps are inexpensive components. If you use them together with resistors and capacitors, transfer functions can be implemented at very low cost.

The normalized form of the TF is:

TF= 0.0083 (jw-1) / {-0.0033w^2 + 0.2166jw + 1}

Okay, let's see.

[Akanimo]

I was trying to normalise the TF but I encountered some challenge. Here is where I got stuck:

TF(s) = (2.5/300)*(s-1)/(1+(s/(300/65))+((s^2)/300))

I am thinking of ways to normalise the numerator.

How did you arrive at this transfer function? Where did you get it from?

[Akanimo]

I have been thinking about this transfer function.

(s-1) in the numerator. Isn't that a RHPZ?

And the frequencies are very low. For instance, if the zero (s-1) is compared to a standard form (s-w_z) then w_z = 1 = 2*pi*f_z.
Therefore zero frequency f_z = 1/(2*pi) = 1/6.284 Hz.

The transfer function doesn't seem like it is for a compensator. It seems like it is the transfer function of an uncompensated loop gain that you need to compensate. That's why I ask how you arrived at it. Maybe you should clarify.

[FvM]

(s-1) in the numerator. Isn't that a RHPZ?

Obviously. Can be implemented with basic active circuit.

[Akanimo]

FvM,
Please review. The DC gain should be 2.5*1/(5*60) = 2.5/300.

But not for the RHPZ, it could be implemented with a single Opamp.

[FvM]

The DC gain should be 2.5*1/(5*60) = 2.5/300.

Yes, that's what the circuit implements.

[vivekroy]

There is a classic book on this topic by Kendall Su. It is really an exhaustive book on filter design using passive components and/or active components.

[asdf44]

I admit when I said I had modeled it above I was wrong which is why I never posted it. I had matched the gain and thought the phase mismatch I was seeing was a simulation artifact but in-fact was because I wasn't handling the RHP zero correctly. Thanks FvM for posting the correct circuit.

[Akanimo]

Now, I'm wondering what application we would need to implement a RHPZ for. Why would we want to increase gain slope by +20dB/decade but lose phase margin by 45 degrees?

[garvind25]

Could I ask how you arrived at this solution?

Regards,
Arvind Gupta

[FvM]

1. Know that a RHPZ + single pole combo with unequal pole/zero frequencies can be implemented by modification of a standard active all-pass
2. Add a second pole respectively first order low-pass
3. Adjust the parameters by comparison of coefficients

Used Sapwin to validate the transfer function