A system has two real poles and one real zero where p1<z1<ρ2<0. (a) a branch of the root locus l

step 1 Here, let's assume z equals to x plus i y equals to x plus i y equals to z squared divided by z

A system has two real poles and one real zero where...

A system has two real poles and one real zero where $p_1<z_1<\rho_2<0$. (a) a branch of the root locus lies on the real axis between $z_1$ and $p_2$ (b) a branch of the root locus lies on the real axis between $z_1$ and $p_1$ (c) a branch of the root locus lies on the real axis between $p_2$ and $\infty$ (d) has no branch on the real axis

A system has two real poles and one real zero where $p_1<z_1<\rho_2<0$.
(a) a branch of the root locus lies on the real axis between $z_1$ and $p_2$
(b) a branch of the root locus lies on the real axis between $z_1$ and $p_1$
(c) a branch of the root locus lies on the real axis between $p_2$ and $\infty$
(d) has no branch on the real axis

Key Concepts

Real-Axis Segment Rule

The real-axis segment rule is a criterion used in constructing the root locus. It states that any point on the real axis is part of the root locus if and only if the total number of poles and zeros to its right is odd. This rule is fundamental in determining which segments of the real axis are included in the root locus and is key when assessing the behavior of system branches between poles and zeros.

Zeros

Zeros are the values of the complex frequency variable that cause the transfer function to become zero. They influence the system's frequency response and contribute to shaping the root locus, as their presence can attract or repel the paths of the poles.

Poles

Poles are specific values of the complex frequency variable that make the system's transfer function approach infinity. They are critical in determining the stability and dynamic behavior of a system, and their location in the s-plane (complex plane) often dictates the response characteristics of the control system.

Root Locus

The root locus is a graphical method used in control systems to analyze how the roots of the characteristic equation (or the closed-loop poles) change as a single parameter, typically the system gain, is varied. It provides insight into system stability and transient performance by illustrating the paths that poles take in the complex plane.

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