Understanding the Control System Transfer Function

The question asks us to identify the type of control network represented by the following transfer function:

$$ G(s) = \frac{{1 + 0.5s}}{{1 + s}} $$

We need to analyze the structure of this transfer function to determine if it corresponds to a lag network, lead network, lag-lead network, or a proportional controller.

Analyzing the Transfer Function Structure

The given transfer function is in the standard first-order form:

$$ G(s) = K \frac{{1 + \tau_z s}}{{1 + \tau_p s}} $$

By comparing the given function $$ G(s) = \frac{{1 + 0.5s}}{{1 + s}} $$ with the standard form, we can extract the following parameters:

• The gain \( K = 1 \).
• The time constant related to the zero, \( \tau_z \), is found from the term \( 1 + 0.5s \). So, \( \tau_z = 0.5 \).
• The time constant related to the pole, \( \tau_p \), is found from the term \( 1 + s \). So, \( \tau_p = 1 \).

Identifying the Network Type Based on Time Constants

The nature of the network (lag or lead) is determined by comparing the time constants \( \tau_z \) and \( \tau_p \).

• For a Lag network, the pole's time constant is typically larger than the zero's time constant, i.e., \( \tau_p > \tau_z \). This configuration causes the system's response to lag, particularly at higher frequencies.
• For a Lead network, the zero's time constant is typically larger than the pole's time constant, i.e., \( \tau_z > \tau_p \). This configuration tends to speed up the system's response and provide a phase lead.

In our case, we have:

• \( \tau_z = 0.5 \)
• \( \tau_p = 1 \)

Comparing these values, we find that \( \tau_p (1) > \tau_z (0.5) \). This condition, \( \tau_p > \tau_z \), is characteristic of a lag network.

Alternative Analysis: Pole-Zero Locations

We can also determine the network type by examining the locations of the poles and zeros on the s-plane.

The zero is located where the numerator is zero:

$$ 1 + 0.5s = 0 \implies s = -\frac{1}{0.5} = -2 $$

The pole is located where the denominator is zero:

$$ 1 + s = 0 \implies s = -1 $$

The zero is at \( s = -2 \) and the pole is at \( s = -1 \). In terms of frequency, the zero is at a higher frequency (\( |-2| \)) than the pole (\( |-1| \)). A pole at a lower frequency than the zero signifies a lag network.

Conclusion

Based on the analysis of time constants (\( \tau_p > \tau_z \)) and pole-zero locations (pole at \( s=-1 \), zero at \( s=-2 \)), the transfer function $$ G(s) = \frac{{1 + 0.5s}}{{1 + s}} $$ represents a Lag network.