Phase Lead Compensating Network: Pole-Zero Conditions

A phase lead compensating network is a crucial component in control systems, primarily used to improve transient response and increase system bandwidth by adding phase lead to the open-loop transfer function. This type of network is typically constructed using passive components like resistors and capacitors, forming an RC network.

Essential Conditions for Pole and Zero Placement

For a phase lead compensating network to be stable and physically realizable using only capacitors and resistors, two fundamental conditions regarding the locations of its pole (\(P_c\)) and zero (\(Z_c\)) in the s-plane must be satisfied:

1. Location in the s-plane for Realizability and Stability:

   For any passive and stable electrical network, all its poles and zeros must be located in the Left Half Plane (LHS) of the s-plane. This is a fundamental requirement for physical realizability and stability. If a pole were located in the Right Half Plane (RHS), the network itself would be unstable. Similarly, a zero in the RHS would result in a non-minimum phase system, which is generally avoided in simple passive compensators. Therefore, both the pole (\(P_c\)) and the zero (\(Z_c\)) of a phase lead compensating network must be in the LHS.

2. Relative Position of Pole and Zero for Phase Lead:

   The defining characteristic of a phase lead compensator is its ability to contribute a positive phase shift (lead) to the system. This phase lead is achieved through a specific relative placement of the pole (\(P_c\)) and zero (\(Z_c\)) on the negative real axis of the s-plane. To introduce phase lead, the pole (\(P_c\)) must be positioned such that its value is greater than the zero's value (\(Z_c\)) when considering their locations on the negative real axis. That is, \(P_c > Z_c\).

   This configuration means that the pole (\(P_c\)) is closer to the origin than the zero (\(Z_c\)) on the negative real axis. For instance, if \(P_c = -2\) and \(Z_c = -5\), then \(P_c > Z_c\).

   In terms of the transfer function, a phase lead network has the form:

   \(G_c(s) = \frac{s - Z_c}{s - P_c}\)

   where both \(Z_c\) and \(P_c\) are negative real numbers in the LHS, and the condition \(P_c > Z_c\) ensures the lead characteristic.

Conclusion

Based on these conditions, for a phase lead compensating network composed of only capacitors and resistors, both its pole (\(P_c\)) and zero (\(Z_c\)) must be located in the Left Half Plane (LHS), and the pole's value must be greater than the zero's value (\(P_c > Z_c\)). This matches the condition stated in the second option.