# what is the difference in u-controller processor sampling frequency and the z-domain conversion sampling frequency?

I have observed a bit confusing behavior of my system response (or may be i am missing something).

I have a transfer function in S domain converted to Z domain with a 1kHz sampling frequency at the time of conversion using matlab, When I embed this discrete version of the transfer function to my system which is also sampling on the same frequency of 1kHz. The system works the way as expected (i.e. the step response is the same as that of the s-domain analogue controller).

But if I increase the sampling frequency of my system while using the SAME discrete transfer function that i just converted from s to z domain with a SAME conversion sampling frequency of 1kHz , the step response gets further faster.

My question is that, why the discrete system gets faster response than the analogue one, despite the transfer functions of the analogue controller and the discrete controller are the same.

What I understand, the step response of any transfer function should remain the same in either case (i.e. either the function is in s-domain or in z-domain) the response should be the same ?

Does this mean the digital controllers have the ability to fast the response of the same transfer function by changing the sampling frequency of the system?

It is important, NOT to confuse the system sampling frequency of my u-controller at which the u-controller is collecting the samples from ADC, with the sampling frequency that I used as a parameter required to convert the s-domain transfer function to z-domain transfer function.

I thank you all for your time.

OK just read your bold words and realize you know what you are doing but, be absolutely sure that what you say in bold is absolutely true. Anyway, my previous answer was: -

But if I increase the sampling frequency of my system while using the SAME discrete transfer function that i just converted from s to z domain with a SAME conversion sampling frequency of 1kHz , the step response gets further faster.

Of course it does - the rate at which you sample is totally part of the transfer function. If you work out your transfer function at one sample rate and then change the rate, then the transfer function changes.

Basically, when you increased the rate of sampling and thought you were using the same discrete transfer function, you weren't.

If you were mimicking an integrator by using an accumulator, the speed at which you accumulate (sampling frequency) determines the rate at which the integrator integrates.

To expand to Andy's response remember that the z domain and s domain are related by z = e^sT where T is the sampling rate, so if you change your sampling rate in your digital domain by the relation above you are modifying your s plane transfer function. Maybe this quick article will better the relation of the Z-transform to the Laplace transform and show why changing the sampling in the digital domain will change the response.

When you convert a filter from from analogue to digital you have to make some assumptions.

One of the common ways of doing this is to use the 'tustin transform'

$$s = \dfrac{2}{T}\cdot\dfrac{1 - z^{-1}}{1 + z^{-1}}$$

Where $T$ is the time between samples. If you sample faster the response of the system will be faster unless you recalculate your digital filter for the new, faster, rate.