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22..55..55:: SSeeccoonndd OOrrddeerr SSyysstteemm SStteepp RReessppoonnssee
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Overview
In this chapter, we address the case in which the input to a
second order system consists of thesudden application of a constant
voltage or current to the circuit; this type of input can be
modeled asa step function. The response of a system to this type of
input is called the step response of thesystem.
The material presented in this chapter will emphasize the
development of qualitative relationshipbetween the damping ratio
and natural frequency of a system and the system’s
time-domainresponse. We will also see that we can quantitatively
relate several specific response parameters tothe system’s damping
ratio and natural frequency. This approach allows us to infer a
great deal aboutthe expected system response directly from the
damping ratio and natural frequency of the system,without
explicitly solving the differential equation governing the
system.
Before beginning this chapter, you shouldbe able to:
After completing this chapter, you should beable to:
• Define damping ratio and naturalfrequency from the
coefficients of asecond order differential equation(Chapter
2.5.1)
• Write the form of the natural response ofa second order
system (Chapter 2.5.2)
• Classify overdamped , underdamped ,and
critically damped systems
according to their damping ratio(Chapter 2.5.4)
• Identify the expected shape of thenatural response of
over-, under-, andcritically damped systems (Chapter2.5.4)
• State from memory the definition of anunderdamped second
order system’sovershoot, rise time, and steady-stateresponse
• Use the coefficients of a second ordersystem’s governing
equation to estimate thesystem’s overshoot, rise time, and
steady-state response
This chapter requires:
• N/A
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2.5.5: Second Order System Step Response
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owners.
In chapter 2.5.1, we wrote a general differential equation
governing a second order system as:
)t ( f )t ( ydt
)t (dy
dt
)t ( yd nn =++
2
2
2
2 ω ςω (1)
where y(t) is any system parameter of interest (for
example, a voltage or current in an electrical
circuit), nω and ζ are the undamped natural
frequency and the damping ratio of the
system,
respectively, and f(t) is a forcing function applied to the
system.
In this chapter, we restrict our attention to the specific case
in which )t ( f is a step function. Thus,
the forcing function to the system can be written as
>
<==
0
00
0t , A
t ,)t ( Au)t ( f
(2)
Thus, the differential equation governing the system
becomes:
)t ( Au)t ( ydt
)t (dy
dt
)t ( yd nn 0
2
2
2
2 =++ ω ςω (3)
In addition to the above restriction on the forcing function, we
will assume that the initial conditions areall zero (we sometimes
say that the system is initially relaxed ). Thus, for the
second-order systemabove, our initial conditions will be
0
00
0
=
==
=t dt )t (dy
)t ( y
(4)
Solving equation (3) with the initial conditions provided in
equations (4) results in the step response ofthe
system.
As in our discussion of forced first order system responses in
chapter 2.5.1, we write the overallsolution of the differential
equation of equation (3) as the sum of a particular solution and
ahomogeneous solution. Thus,
)t ( y)t ( y)t ( y ph
+=
The homogeneous solution of second order differential equations
has been discussed in chapters2.5.1 and 2.5.4 and will not be
repeated here. The particular solution of the differential equation
(4)can be obtained by examining the solution to the equation after
the homogeneous solution has died
out. Letting t →∞ in equation (3) and noting
that the forcing function is a constant as
t →∞ allows us to
set 02
2
=∞→
=∞→
dt
)t (dy
dt
)t ( yd and thus,
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2.5.5: Second Order System Step Response
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owners.
2
2
n
PPn
A)t ( y A)t ( y
ω ω =⇒= (5)
Combining the particular and homogeneous solutions, assuming the
system is underdamped ( 1
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2.5.5: Second Order System Step Response
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owners.
Time (sec)
A m p l i t u d e
Figure 1. Underdamped second order system step response.
