Given: Transfer function of the system is 1/Ts 1.
The system is subjected to a step input and takes 2 seconds to reach 50% of step height.

To find: The value of t.

Solution:
We know that the transfer function of a first-order system is given by:

G(s) = K/(1+Ts)

where K is the gain and T is the time constant.

Here, the transfer function is 1/Ts 1. So,

K = 1
T = s

The response of the system to a step input is given by:

C(s) = G(s) * R(s)

where C(s) is the output (response), R(s) is the input (step), and G(s) is the transfer function.

For a step input, R(s) = 1/s.

Therefore,

C(s) = G(s) * R(s)
= 1/(s^2 + sT)

The time-domain response is obtained by taking the inverse Laplace transform of C(s). The Laplace transform of 50% of step height is 0.5/s. So, we need to find the time when C(t) = 0.5/s.

C(s) = 0.5/s
1/(s^2 + sT) = 0.5/s
sT + 1 = 2s
sT = 2s - 1
T = (2s - 1)/s

Substituting s = jw (where j is the imaginary unit) to get the frequency response:

T(jw) = (2jw - 1)/(jw)
= 2 - (1/jw)

The magnitude of the frequency response at w = 0 (DC gain) is:

|T(j0)| = |2 - (1/j0)| = 2

This means that the system has a gain of 2 at DC.

The magnitude of the frequency response at w = infinity (high-frequency gain) is:

|T(jinf)| = |2 - (1/jinf)| = 2

This means that the system has a gain of 2 at high frequencies.

The magnitude of the frequency response at w = 1/T (crossover frequency) is:

|T(j1/T)| = |2 - (1/j1/T)| = |2 - jT| = sqrt(2^2 + T^2)

Since the system takes 2 seconds to reach 50% of step height, the time constant T is:

T = 2/1.5 = 1.333 seconds

Therefore, the crossover frequency is:

w = 1/T = 0.75 rad/s

At this frequency,

|T(j0.75)| = sqrt(2^2 + 1.333^2) = 2.564

This means that the system has a gain of 2.564 at the crossover frequency.

The phase angle at the crossover frequency can be found as:

tan(phi) = Im(T(j0.75))/Re(T(j0.75))
= -1.333/1.564
phi = -39.9 degrees

Since the phase angle is negative, the system is a type 1 system (no phase lag).

Now, to find the time when the response reaches 50% of step height,