The locations of the poles of in the s-plane determines the system behavior.
Here we consider the pole locations and the corresponding system response as
changes from to . We assume
below.

The inverse Laplace transform of is the impulse response of this system

where

This second order system can also be considered as the product (i.e., cascade) of two first order systems:

and in the Bode diagram of this system, both the magnitude and the phase plot of can be obtained by adding the corresponding plots of and . (For example, the sum of two curves with 20 dB/decade drop at high frequency is a curve with 40 dB/decade drop.)

The frequency response at any frequency is determined by the product of the two vectors and in the denominator of , and the behavior of the system as the frequency changes can be qualitatively determined by observing the locations of the poles in the s-plane, which vary as a function of the parameter . We first list the two poles for different values of :

comments | |||

real poles | |||

repeated real poles | |||

complex conjugate pair | |||

imaginary poles | |||

complex conjugate pair | |||

repeated real poles | |||

real poles | |||

**Note 1:**
When , the poles are a complex conjugate pair located along a circle
with center at the origin and radius equal to . The phase angles of
the poles are
. In other words, different values
are represented by concentric circles of different radius, and different
values are represented by rays from the origin of different angles.

**Note 2:**
When increases from to , each of the two poles moves
in the s-plane along a path called the *root locus*.

**Motion of **:

**Motion of **:

Here we discuss the pole locations corresponding to various values, and the corresponding system behaviors:

- , both roots and are negative real and we have:

The impulse response is the difference of two exponentially decaying functions, which decays to zero in time. In the s-plane, the two roots are on the horizontal axis ( on the right close to 0 and on the left). The middle point between the two roots is , the distance between the two roots is . When , will be very close to the origin while is far from it on the left. At low frequency, the behavior of the system is dominated by corresponding to the pole close to 0; - ,
are two repeated
roots. We have

and

This corresponds to a critically damped system. - , the two roots are a complex conjugate pair in the
2nd and 3rd quadrants, respectively:

and

This is an exponentially decaying sinusoid with frequency . In the s-plane, the magnitude of the two pole vectors (from the origin to the complex poles) is

and the angle between the pole vectors and the horizontal axis is

The two poles and are on a circle with radius centered about the origin. - , the two poles and
are imaginary,
and is

The system becomes unstable. In particular, when , i.e., the frequency of the input is the same as the system's natural frequency, we have

and resonance occurs. (Structures like bridges can be damaged!) - , the two poles are a complex conjugate pair in the 1st
and 4th quadrants. The expression for is the same as in the case of
:

However, as , the exponent is positive and is an exponentially growing sinusoid, i.e., the system is unstable. Also note that the geometry of the two poles in the s-plane is symmetric to that of the case , i.e., the poles are on the circle with radius and centered about the origin. - , and
are two repeated
roots. We have

and

The impulse response grows monotonically. - , and both roots and are positive real on the
horizontal axis of the s-plane. As the real parts of both and are
positive, the terms and in

and their difference grow exponentially as and the system is unstable.

Summarizing the above, we can trace the movement of the two roots (root locus) in the s-plane when changes its value in the range , and find the corresponding system behaviors, as shown in the table below ( ).

Comments on | |||

exponential growth | |||

exponential growth | |||

exponentially growing sinusoid | |||

constant sinusoid | |||

exponentially decaying sinusoid | |||

critically damped | |||

exponential decay |