Each of the items consists of two statements, one labeled as ‘Statement (I)’ and the other as ‘Statement (II)’. Examine these two statements carefully and select the answers to these items:

Statement (I): Elements with non-minimum phase transfer functions introduce large phase lags with increasing frequency resulting in complex compensation problems.

Statement (II): Transportations lag commonly encountered in process control systems is a non-minimum phase element.

Statement I: Non-minimum phase system introduces large phase lags with increasing frequency resulting in compensation problems.

Justification:

Large phase lag means large transportation delay, and the introduction of transportation delay in a system means making the system non-minimum phase system.

Example:

Consider a system as shown:

\(G\left( s \right) = \frac{1}{{s\left( {s + 1} \right)}}\)

∠G(jω) = -90° – tan^-1(ω)

\(\left[ {\begin{array}{*{20}{c}} {at~\omega = 0:}&{\angle G\left( {j\omega } \right) = - 90^\circ }\\ {at~\omega = \infty :}&{\angle G\left( {j\omega } \right) = - 180^\circ } \end{array}} \right]\)

So, varying ω from 0 to ∞, the phase of the system varies from -90° to -180°.

Now, the transportation delay is introduced in the control system, i.e.

\(G\left( s \right) = \frac{{{e^{ - s{t_d}}}}}{{s\left( {s + 1} \right)}} \simeq \frac{{\left( {1 - s{t_d}} \right)}}{{s\left( {s + 1} \right)}}\)

We observe that the transportation delay has made the system a non-minimum phase system.

Now,

∠G(jω) = -tan^-1 (ωtd) – 90° - tan^-1 (ω)

At ω = 0        ∠G(jω) = -90°

At ω = ∞       ∠G(jω) = -270°

So, now the phase varies from -90° to -290°, as ω is varied from 0 to ∞.

Hence, we can say that transportation lag is a non-minimum phase element and it introduces large phase lags with increasing frequency which results in complex compensation problems.