I'm stuck on this problem and am unsure how to tackle it.
It is known that an industrial process system is accurately modelled by the standard first order differential equation:
Tx' + x = Ky
i) Given the following results of experimental measurement, deduce the parameters T and K,
When an input Y(t) of 5 units was applied, it was found that the system output x(t) ultimately settled with a value of 20 units.
When a sinusoidal input was applied, it was found that, at an input frequency of 10 radians, the output lagged behind the input by exactly -45 degrees.
ii) If y(t) was a step change drive, at what time would x(t) reach half of its final steady state value (using the values of K and T already deduced)?
I've got as far as:
Tx' + x = K5 when x=20
=> 20 = K5
K = 4
Can anyone show me where to go from here?
I've found a passage in my notes which says "Another useful response of the first order system is the one we observe when the input is a sinusoid. As we saw earlier, all we need do is replace the derivative with jw (I'm using 'w' as omega as I can't find how to change it on my PC) as follows:
Tjwx + x = Ky
How does it work using this formula?