Control.com - Nerds in Control

I have a basic control system question to the experts of this forum.

What is a transfer function - Is it just a relation between output/input OR is it the 'time response' relation between the output & input.

To make my point clear I am considering the following two examples:

1) I/P converter:
The input is 4 - 20mA current (I) and the output is 3 - 15PSI Pressure (P). Here the Output/Input relation would be a equation of straight line. so P/I = 0.75, the slope. Is this the Transfer function?

The other way, as suggested in some texts, is that the Output (P) responds to a Change in input (I) as a first order equation with a time constant T. Hence Output/Input can be approximated as a first order equation with a finite Time Constant value. Hence the equation is likely to be P/I = K/1+Ts.

Which of the above two is the correct picture of a transfer function?

2) Control Valve:
A similar case can be made out for a Control Valve also. The stem position (X) can be approximated as linear function of Pressure (P) on the diaphragm. So X/P can again be approximated as a constant (= slope). (0 - 100% P will result in 0 - 100 change in X)

But some texts describe the Control valve with a Mass-Spring analogy and determine a more complex equation like X/P = 1/(Ms^2 + fs + K).

Again the question is which of the above two approaches reflects the correct transfer function of this system too?

Regards
Ritika

It all depends on how fast you need to operate. Transfer functions
are usually expressed without regard for time. That doesn't mean you can ignore time, just that the function describes behavior well within the speed capability of the device. Once you are speed or time
limited, the picture changes and often is not characterized.

Regards
cww

Moin Ritika, moin all,

Time and frequency are not independent.

Many dynamic systems can be described by linear time-invariant differential equations or difference equations with constant coefficient. For such systems the Laplace-Transform (or Z-Transform in the discrete case) convert from time to frequency domain and vs.

A transfer function is the transformation of a differential equations to the frequency domain.

Simple example for a first order system PT1
Let u be the input signal, y the output signal.
differential equation
y(t) + a dy(t)/dt = b u(t)
Laplace-Transform
Y(s) + a s Y(s) = b U(s)
Transfer Function
G(s) = Y(s) / U(s) = b / (1 + as)

From a transfer function you can directly get a Bode-Diagram or the steady state for a constant input or else. Multiplying the Transfer Function by 1/s and solving the inverse Laplace-Transform gets you the step response.

Read a text book for Laplace-Transforms and Z-Transforms for more information.

Best regards
Friedrich Haase

Ritika,

I'll try and be brief.

Yes a transfer function is basically a (linear) way of describing a relationship between an input & output. Transfer functions are a well established convention for articulating the system response but their are other ways (differential equations, z tranforms). Which is appropriate can be a matter of preference, tools available or the purpose and context of the model you are developing. Transfer functions encapsulate information about frequency response and time response given certain input signals. With experience you can pick out characteristics.

For your 2 examples which is the appropriate model depends upon the properties of the system being modelled.

In some cases (process time constant much greater than I/P converter time constant) you may be able to approximate the I/P converter as a linear gain and lose little in accuracy. Same for the control valve. In other cases z-transforms may be necessary, or non-linear models. Use of engineering judgement is required and the simplest model that achieves your aims is usually the best. Don't get carried away with model accuracy. So both your answers may be correct or incorrect!

Hope that helps. It sounds like you need more support or reading and experimenting time.

DaveMH