Chapter 4 - System Response
Dr. Uche Wejinya
University of Arkansas - Fayetteville
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Overview
System Response
Fourier Series Representation of Signals
Fourier Transform
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System Response
System response: the relationship between the output of a
mechatronic or measurement system and its input.
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System Response
System response: the relationship between the output of a
mechatronic or measurement system and its input.
Use measurement system as an example, which consists of three
parts
• Transducer: converts a physical quantity into a time-varying
voltage (analog signal)
• Signal processor: modify the analog signal
• Recorder: displays or records the signal
Input: physical variable we wish to measure
Output: transducer transforms (converts) the input into a form
that is compatible with the signal processor
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System Response
Goal: reproduced output signal that match the input as closely as
possible unless we want to eliminate certain input (i.e., noise).
The following conditions should be satisfied for a time-varying (i.e.,
sine wave) input
1. Amplitude linearity
2. Adequate bandwidth
3. Phase linearity
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System Response: Amplitude Linearity
Amplitude linearity: Output always changes by the same factor
times (multiplied by) the change
of the input
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System Response: Amplitude Linearity
Amplitude linearity: Output always changes by the same factor
times (multiplied by) the change
of the input
Vout (t) Vout (0) = ↵[Vin (t) Vin (0)]
where ↵ is a constant of proportionality
If this fails, system is not linear with
respect to amplitude, and it becomes
more difficult to interpret the output
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System Response: Amplitude Linearity
Amplitude linearity: Output always changes by the same factor
times (multiplied by) the change
of the input
Vout (t) Vout (0) = ↵[Vin (t) Vin (0)]
where ↵ is a constant of proportionality
If this fails, system is not linear with
respect to amplitude, and it becomes
more difficult to interpret the output
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System Response: Amplitude Linearity
Amplitude Linearity of Ideal measurement system
• Exhibit amplitude linearity for any amplitude or frequency of
the input.
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System Response: Amplitude Linearity
Amplitude Linearity of Ideal measurement system
• Exhibit amplitude linearity for any amplitude or frequency of
the input.
Amplitude Linearity of Actual measurement system
• A measurement system will satisfy amplitude linearity over
only a limited range of input amplitudes
• The system usually responds linearly only when the rate of
change of the input is within certain limits (system
bandwidth)
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System Response: Amplitude Linearity
Amplitude Linearity of Ideal measurement system
• Exhibit amplitude linearity for any amplitude or frequency of
the input.
Amplitude Linearity of Actual measurement system
• A measurement system will satisfy amplitude linearity over
only a limited range of input amplitudes
• The system usually responds linearly only when the rate of
change of the input is within certain limits (system
bandwidth)
To get a better interpretation of Amplitude Linearity , we need to
use a mathematical tools: Fourier Series.
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Fourier Series Representation of Signals
Fourier series: any periodic waveform can be represented as an
infinite series of sine and cosine waveforms of di↵erent amplitudes
and frequencies.
Example: some complicated but periodic waveforms are decomposed
into a series of sine and cosine waveforms
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Dirichlet Conditions
A periodic signal x(t), has a Fourier series if it satisfies the following
conditions
• x(t) is absolutely integrable over any period, namely
Z a+T
a
|x(t)|dt < 1
• x(t) has only a finite number of maxima and minima over
any period
• x(t) has only a finite number of discontinuities over any
period
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Fourier Series Representation of Signals
Complicated periodic waveforms can be decomposed into sine and
cosine waveforms
The fundamental or first harmonic !0 is the lowest frequency component of a periodic waveform
!0 =
2⇡
= 2⇡f0
T
f0 is the fundamental frequency expressed in hertz (Hz).
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Fourier Series Representation of Signals
Complicated periodic waveforms can be decomposed into sine and
cosine waveforms
The fundamental or first harmonic !0 is the lowest frequency component of a periodic waveform
!0 =
2⇡
= 2⇡f0
T
f0 is the fundamental frequency expressed in hertz (Hz).
The other sine and cosine waveforms have frequencies that are integer multiples of the fundamental frequency, such as the second
harmonic is 2!0 and the third harmonic is 3!0 .
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Fourier Series Representation of Signals
Complicated periodic waveforms can be decomposed into sine and
cosine waveforms
The fundamental or first harmonic !0 is the lowest frequency component of a periodic waveform
!0 =
2⇡
= 2⇡f0
T
f0 is the fundamental frequency expressed in hertz (Hz).
The other sine and cosine waveforms have frequencies that are integer multiples of the fundamental frequency, such as the second
harmonic is 2!0 and the third harmonic is 3!0 .
Note: we don’t need entire infinite series because a finite number of
the sine and cosine waveforms can adequately represent the original
signal
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Fourier Series Representation of Signals
The Fourier series representation of an arbitrary periodic waveform
F (t) = C0 +
1
X
An cos(n!0 t) +
n=1
1
X
Bn sin(n!0 t)
n=1
where C0 is the DC component of the signal and the two summations
are infinite series of sine and cosine waveforms. C0 represents the
average value of the waveform over its period
2
C0 =
T
2
An =
T
Z T
0
Z T
f (t)dt =
A0
2
2
f (t) cos(n!0 t)dt; Bn =
T
Z T
0
f (t) sin(n!0 t)dt
0
where f (t) is the waveform being represented and T is the period
of the waveform
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Fourier Series Representation of Signals
The cosine An and sine Bn terms can be combined with a trigonometric identity to create an alternative representation with a single
amplitude and phase
F (t) = C0 +
1
X
Cn cos(n!0 t +
n)
n=1
The total amplitude for each harmonic is given
p
Cn = A2n + Bn2
The phase for each harmonic is given by
✓ ◆
1 Bn
tan
n =
An
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Fourier Series Representation of Signals: Example
Application and meaning of the Fourier Series
F (t) =
2
Bn =
T
(
1
1
Z T /2
0  t < T /2
T /2  t < T
f (t) sin(n!0 t)dt
0
Bn =
2
T
Z T
T /2
✓
f (t) sin(n!0 t)dt
!
1
1
T /2
[cos(n!0 t)]0 +
[cos(n!0 t)]TT /2
n!0
n!0
(
4
n : odd
2
Bn =
[1 cos(n⇡)] = n⇡
n⇡
0
n : even
◆
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Fourier Series Representation of Signals: Example
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Fourier Series Representation of Signals: Example
Figure: f(1,t), ..., f(20,t)
Figure: f(1,t), ..., f(100,t)
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Harmonic Decomposition of Square Wave
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Fourier Series Representation of Signals
As the harmonic frequency increases, the amplitudes of the harmonics decrease
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Fourier Series Representation of Signals
As the harmonic frequency increases, the amplitudes of the harmonics decrease
First, third, and fifth harmonics and add them together, we obtain
a waveform that begins to look similar to a square wave. Higher
order harmonics reduces sharp changes
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Fourier Series Representation of Signals
Top plot: time-domain representation of the signal;
Bottom plot: Fourier series amplitudes vs. frequency, spectrum.
The spectrum is the signal’s frequency-domain representation
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Fourier Series Representation of Signals
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Fourier Transform
Fourier transform (FT) is a mathematical function that transforms
a signal from the time domain, x(t), to the frequency domain, X(f )
X(f ) =
Z +1
x(t)e j2⇡f t dt
1
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Fourier Transform
Fourier transform (FT) is a mathematical function that transforms
a signal from the time domain, x(t), to the frequency domain, X(f )
X(f ) =
Z +1
x(t)e j2⇡f t dt
1
The inverse Fourier transform may be used to convert a signal from
the frequency domain to the time domain as follows
x(t) =
Z +1
X(f )ej2⇡f t df
1
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Fourier Transform
Fourier transform (FT) is a mathematical function that transforms
a signal from the time domain, x(t), to the frequency domain, X(f )
X(f ) =
Z +1
x(t)e j2⇡f t dt
1
The inverse Fourier transform may be used to convert a signal from
the frequency domain to the time domain as follows
x(t) =
Z +1
X(f )ej2⇡f t df
1
When the Fourier transform is to be expressed in terms of the angular
frequency !(rad/s) rather than the frequency f (Hz)
X(!) =
Z +1
x(t)e j!t dt
1
x(t) =
Dr. Uche Wejinya
Lecture 8
1
2⇡
Z +1
1
X(!)e j!t d!
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Fourier Transform: sine wave
v(t) = sin(!t)
Z +1
V (!) =
sin(!t)e
j!t
dt
1
V (!) =
V (!) =
Z +1
1
j
2
ej!0 t
1
Z 1 ⇣
e
e
2j
j!0 t
ej!t dt
j(!+!0 )t
j(! !0 )t
e
1
2⇡ (! + !0 ) = e
j(!+!0 )t
2⇡ (!
j(! !0 )t
!0 ) = e
V (!) = ⇡j[ (! + !0 )
(!
⌘
dt
!0 )]
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Fourier Transform: Unit Impulse Function
(!) =
Z +1
=e
(t
t0 )e j!t dt
1
j!0 t
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Fourier Transform: Processing Sound Signal
Voltage signal as a function of time corresponding to the sound of
the middle A note of a piano
Transform the signal to the frequency domain via FFT
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Fourier Transform: Example
The signal is measured from oscilloscope, it contains a cos component signal and two sin components signal
Question:
• is there any DC components in the signal
• write down the mathematical representation of this signal
• what’s the source of 60Hz
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Fourier Transform: Example
Signal containing 60 Hz ”noise” and its amplitude spectrum. The
composite signal is given by
x(t) = 1 + cos(1000⇡t) + 2 sin(600⇡t) + sin(120⇡t)
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Chapter 4 CONTND: System Response
Dr. Uche Wejinya
University of Arkansas - Fayetteville
1/27
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Lecture 9
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Overview
Bandwidth
Phase Linearity
Distortion of Signals
Dynamic Characteristics of Systems
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Good Measurement System
Goal: reproduced output signal match the input as closely as
possible unless we want to eliminate certain input (i.e., noise).
Three conditions for a measurement system
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Good Measurement System
Goal: reproduced output signal match the input as closely as
possible unless we want to eliminate certain input (i.e., noise).
Three conditions for a measurement system
1. Amplitude linearity
2. Adequate bandwidth
3. Phase linearity
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Bandwidth
Ideal system should replicate all frequency components of a signal.
Real systems have limitations in their ability to reproduce all frequencies.
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Bandwidth
Ideal system should replicate all frequency components of a signal.
Real systems have limitations in their ability to reproduce all frequencies.
We typically use decibel scale to measure the degree of fidelity of
a measurement system’s reproduction at di↵erent frequencies.
dB = 20 log10
✓
Aout
Ain
◆
Ain is the input amplitude and Aout is the output amplitude of a
particular harmonic frequency.
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Bandwidth
Frequency response curve, which is also called Bode plot of a
system. It is a plot of the amplitude ratio, AAout
, vs. the input
in
frequency.
It characterizes how components of an input signal are amplified or
attenuated by the system.
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Bandwidth
The term bandwidth is used to quantify the range of frequencies a
system can adequately reproduce.
The bandwidth of a system is defined as the range of frequencies
where the input is not attenuated by more than -3 dB
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Bandwidth
A system usually has two frequencies at which the attenuation of
the system is -3 dB.
They are defined as the low and high corner or cuto↵ frequencies
!L and !H . These two frequencies define the bandwidth of the
system: !L to !H
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Bandwidth
The -3 dB cuto↵ is the decibel value when the power of the output
signal (Pout ) is attenuated to half of its input value (Pin ):
Pout
1
=
Pin
2
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Bandwidth
The -3 dB cuto↵ is the decibel value when the power of the output
signal (Pout ) is attenuated to half of its input value (Pin ):
Pout
1
=
Pin
2
The power of a sinusoidal signal is proportional to the square of the
signal’s amplitude, thus at the cuto↵ value
r
r
Aout
Pout
1
=
=
⇡ 0.707
Ain
Pin
2
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Bandwidth
The -3 dB cuto↵ is the decibel value when the power of the output
signal (Pout ) is attenuated to half of its input value (Pin ):
Pout
1
=
Pin
2
The power of a sinusoidal signal is proportional to the square of the
signal’s amplitude, thus at the cuto↵ value
r
r
Aout
Pout
1
=
=
⇡ 0.707
Ain
Pin
2
At the cuto↵ frequencies, the amplitude of the signal is attenuated
by 29.3% (to 70.7% of its original value), which is approximately -3
dB:
r
1
dB = 20 log10
⇡ 3 dB
2
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Bandwidth
Measurement systems often exhibit no attenuation at low frequencies (i.e., !L ⇡ 0), and only degrades only at high frequencies. For
these systems, the bandwidth extends from 0 to !H .
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Bandwidth
Measurement systems often exhibit no attenuation at low frequencies (i.e., !L ⇡ 0), and only degrades only at high frequencies. For
these systems, the bandwidth extends from 0 to !H .
Ideal measurement system has infinite bandwidth
• Amplitude ratio of 1 extending from 0 to infinite frequency.
• Reproduce all harmonics in a signal without amplification or
attenuation
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Bandwidth
Measurement systems often exhibit no attenuation at low frequencies (i.e., !L ⇡ 0), and only degrades only at high frequencies. For
these systems, the bandwidth extends from 0 to !H .
Ideal measurement system has infinite bandwidth
• Amplitude ratio of 1 extending from 0 to infinite frequency.
• Reproduce all harmonics in a signal without amplification or
attenuation
Real measurement system has limited bandwidth, a↵ected by
• Capacitance, inductance, and resistance in electrical systems
• Mass, sti↵ness, and damping in mechanical systems
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Bandwidth: Example
Vin (t) = A1 sin(!0 t) + A2 sin(2!0 t)
+ A3 sin(3!0 t) + ...
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Bandwidth: Example
Vin (t) = A1 sin(!0 t) + A2 sin(2!0 t)
+ A3 sin(3!0 t) + ...
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Bandwidth: Example
Vin (t) = A1 sin(!0 t) + A2 sin(2!0 t)
+ A3 sin(3!0 t) + ...
0
Ai = (Aout /Ain )i ⇥ Ai
0
A2 = (Aout /Ain )2 ⇥ A2
Vout (t) = 0.25A2 sin(2!0 t)+A3 sin(3!0 t)+
... + A9 sin(9!0 t) + 0.5A10 sin(10!0 t)
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Bandwidth: some notes
When designing or choosing a measurement system for an application, it is important that the bandwidth of the system be large
enough to adequately reproduce the important frequency components present in the input signal
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Bandwidth: some notes
When designing or choosing a measurement system for an application, it is important that the bandwidth of the system be large
enough to adequately reproduce the important frequency components present in the input signal
A measurement system that cannot accurately reproduce signals
that have rapid changes associated with them.
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Bandwidth: some notes
When designing or choosing a measurement system for an application, it is important that the bandwidth of the system be large
enough to adequately reproduce the important frequency components present in the input signal
A measurement system that cannot accurately reproduce signals
that have rapid changes associated with them.
To experimentally determine the bandwidth of a system, it is necessary to systematically apply pure sinusoidal inputs and to determine
the output-to-input amplitude ratio for the desired range of frequencies.
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Bandwidth of Low Pass Filter
Using the voltage divider rule:
Vout =
1
j!C
Vin
1
j!C + R
Vout
1
=
Vin
j!RC + 1
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Bandwidth of Low Pass Filter
Vout
1
=p
Vin
1 + (!RC)2
Cuto↵ Frequency for the circuit is
Using the voltage divider rule:
Vout =
1
j!C
Vin
1
j!C + R
1
RC
1
= p = 0.707
2
!c =
Vout
Vin
Vout
1
=
Vin
j!RC + 1
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Bandwidth of Low Pass Filter
Using !c , Vout and Vin relationship
can be found as
Vout
1
=p
! 2
Vin
1 + ( !c
)
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Bandwidth of Low Pass Filter
Using !c , Vout and Vin relationship
can be found as
Vout
1
=p
! 2
Vin
1 + ( !c
)
Low-pass filter: lower frequencies are “passed” to the output
with little attenuation and higher
frequencies are significantly attenuated
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Bandwidth of High Pass Filter
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Bandwidth of High Pass Filter
High-pass filter: higher frequencies are “passed” to the output
with little attenuation and lower
frequencies are significantly attenuated
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Notch Filter (Band Stop Filter)
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Notch Filter (Band Stop Filter)
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Band Pass Filter
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Band Pass Filter
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Phase Linearity
Phase Linearity: expresses how well a system preserves the phase
relationship.
Signal 2 lags signal 1 because it
occurs later on the time axis, with
time displacement td = T /4
=
360td
2⇡td
degrees =
radians
T
T
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Phase Linearity
Phase Linearity: expresses how well a system preserves the phase
relationship.
Phase Linearity: phase angle must
be linear with frequency for equal
time displacement of frequency components.
Signal 2 lags signal 1 because it
occurs later on the time axis, with
time displacement td = T /4
360td
2⇡td
=
degrees =
radians
T
T
= 360f ·td degrees = 2⇡f ·td radians
where we can also use
=k·f
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Distortion of Signals
System does not exhibit amplitude linearity, the amplitudes of
the output frequency components are attenuated.
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Distortion of Signals
System does not exhibit phase linearity, the output frequency
components may be of the proper amplitude but are displaced in
time with respect to one another
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Dynamic Characteristics of Systems
Dynamic systems can be modeled as linear ordinary di↵erential equations with constant coefficients
N
X
n=0
M
X
dn Xout
dm Xin
An
=
B
m
dtn
dtm
m=0
where Xout is output variable, Xin is input variable, An and Bm are
the coefficients. N is the Order of system, N = 0, 1, or 2, M = 0.
Dynamic equations: Newton’s laws, Lagrange equations, and KVL
and KCL equations.
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Zero-Order System
Zero-order System: M = N = 0
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Zero-Order System
Zero-order System: M = N = 0
A0 Xout = B0 Xin
Xout =
B0
Xin = KXin
A0
K is constant referred to as the gain or sensitivity of the system
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Zero-Order System
Zero-order System: M = N = 0
A0 Xout = B0 Xin
Xout =
B0
Xin = KXin
A0
K is constant referred to as the gain or sensitivity of the system
Example: potentiometer used to measure displacement.
x
Vout = R
Rp V s =
Vs
L
Xin
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Zero-Order System
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First-Order System
N
X
An
n=0
N = 1 and M = 0
M
X
dn Xout
dm Xin
=
B
m
dtn
dtm
m=0
dXout
+ A0 Xout = B0 Xin
dt
A1 dXout
B0
+ Xout =
Xin
A0 dt
A0
A1
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First-Order System
N
X
An
n=0
N = 1 and M = 0
M
X
dn Xout
dm Xin
=
B
m
dtn
dtm
m=0
dXout
+ A0 Xout = B0 Xin
dt
A1 dXout
B0
+ Xout =
Xin
A0 dt
A0
The coefficient ratio on the right-hand side is called the sensitivity or
static sensitivity, defined as
A1
K=
B0
A0
Coefficient ratio on the left side is called time constant
A1
⌧=
A0
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First-Order System
The first-order system can be written as
⌧
dXout
+ Xout = KXin
dt
Step input
Xin =
(
0
Ain
t<0
t 0
The output of the system in response to this input is called the
step response of the system
Xout (t) = KAin (1
e t/⌧ )
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First-Order System
Xout (t) = KAin (1
e t/⌧ )
Xout (⌧ ) = KAin (1
e 1 ) = 0.632KAin
Xout (4⌧ ) = KAin (1
e 4 ) = 0.982KAin
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Chapter 4 CONTND: System Response
Sections 4.9 - 4.11
Dr. Uche Wejinya
University of Arkansas - Fayetteville
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Overview
Dynamic Characteristics of Systems: zero- and first-order
Dynamic Characteristics of Systems: Second-order
System Modeling and Analogies
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Dynamic Characteristics of Systems
Dynamic systems can be modeled as linear ordinary di↵erential equations with constant coefficients
N
X
n=0
M
X
dn Xout
dm Xin
An
=
B
m
dtn
dtm
m=0
where Xout is output variable, Xin is input variable, An and Bm are
the coefficients.
N is the Order of system, N = 0, 1, or 2, M = 0.
Dynamic equations: Newton’s laws, Lagrange equations, and KVL
and KCL equations.
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Zero-order and First-order System
Zero-order System: M = N = 0
A0 Xout = B0 Xin
Xout =
B0
Xin = KXin
A0
K is constant referred to as the gain or sensitivity of the system
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Zero-order and First-order System
Zero-order System: M = N = 0
A0 Xout = B0 Xin
Xout =
B0
Xin = KXin
A0
K is constant referred to as the gain or sensitivity of the system
First-order System: N = 1 and M = 0
A1
dXout
+ A0 Xout = B0 Xin
dt
A1 dXout
B0
+ Xout =
Xin
A0 dt
A0
0
K=B
A0 is called the sensitivity or static sensitivity
A1
⌧ = A0 is called time constant
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First-Order System
The first-order system can be written as
⌧
dXout
+ Xout = KXin
dt
Step input
Xin =
(
0
Ain
t<0
t 0
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First-Order System
The first-order system can be written as
⌧
dXout
+ Xout = KXin
dt
Step input
Xin =
(
0
Ain
t<0
t 0
The output of the system in response to this input is called the
step response of the system
Xout (t) = KAin (1
e t/⌧ )
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First-Order System
Xout (t) = KAin (1
e t/⌧ )
Xout (⌧ ) = KAin (1
e 1 ) = 0.632KAin
Xout (4⌧ ) = KAin (1
e 4 ) = 0.982KAin
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First-Order System: Example
An important example of a first-order system is an RC circuit,
which is very common in timing, filter, and other applications
C
dv0 (t) v0 (t) Vs
+
=0
dt
R
⇣
⌘
t
v0 (t) = Vs 1 e RC
Therefore, the time constant for RC circuit is ⌧ = RC.
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First-Order System: Identification
Experimentally calibrate time constant ⌧ and the static
sensitivity K.
⌧
dXout
+ Xout = KXin
dt
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First-Order System: Identification
Experimentally calibrate time constant ⌧ and the static
sensitivity K.
dXout
+ Xout = KXin
dt
K may be obtained by static calibration, where a known static input
⌧
is applied, and the output is observed.
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First-Order System: Identification
Experimentally calibrate time constant ⌧ and the static
sensitivity K.
dXout
+ Xout = KXin
dt
K may be obtained by static calibration, where a known static input
⌧
is applied, and the output is observed.
A common method to determine the time constant ⌧ is to apply a
step input to the system and determine the time for the output to
reach 63.2% of its final value
Xout = KAin (1
e 1 ) = 0.632KAin
Limitation: time constant ⌧ relies on ONE measurement sample.
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First-Order System: Identification
Alternative method to measure ⌧
Xout (t) = KAin (1
Xout KAin
=
KAin
e t/⌧ )
e⌧
t
t
Xout
=e⌧
KAin
✓
◆
Xout
t
ln 1
=
KAin
⌧
1
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First-Order System: Identification
Alternative method to measure ⌧
Xout (t) = KAin (1
Xout KAin
=
KAin
e t/⌧ )
e⌧
t
Z=
Z
=
t
t
⌧
1
⌧
t
Xout
=e⌧
KAin
✓
◆
Xout
t
ln 1
=
KAin
⌧
1
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Second-Order System
N
X
n=0
M
An
X
dn Xout
dm Xin
=
B
m
dtn
dtm
m=0
Second-Order System: N = 2 and M = 0
Typical Mass-Spring-Damper System
Using Newton’s second Law:
m
d2 x
dx
+b
+ kx = Fext (t)
2
dt
dt
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Second-Order System: Characteristic Equation
Characteristic Equation
ms2 + bs + k = 0
Quadratic equation has two roots
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Second-Order System: Characteristic Equation
Characteristic Equation
ms2 + bs + k = 0
Quadratic equation has two roots
s✓
◆
b
b 2
s1 =
+
2m
2m
s✓
◆
b
b 2
s2 =
2m
2m
k
m
k
m
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Second-Order System: Undamped Motion
If there were no damping in the system, b = 0
q
q
k
k
s1 = j m
s2 = j m
Corresponding homogeneous solution would be
r !
r !
k
k
xh (t) = A cos
t + B sin
t
m
m
Pure undamped oscillatory motion with angular frequency (aka
natural frequency)
r
k
!n =
m
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Second-Order System: Critically Damped
If there is damping in the system, but the radicand is 0
xh (t) = (A + Bt)e !n t
This represents an exponentially decaying motion.
A system with this behavior is said to be critically damped,
because it is just on the verge of damped oscillatory motion.
The damping constant that results in critical damping is called the
critical damping constant bc .
p
bc = 2 km = 2m!c
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Second-Order System: Critically Damped
The damping ratio ⇣ (zeta) for a damped system is defined as
⇣=
b
b
= p
bc
2 km
It is a measure of the proximity to critical damping. A critically
damped system has a damping ratio of 1
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Second-Order System: Underdamped
With the definitions of natural frequency and damping ratio, the
roots of the characteristic equations can be written as
s✓
◆
p
b
b 2
k
s1,2 =
±
s1 = ⇣!n + !n ⇣ 2 1
2m
2m
m
r
p
b
b
k
⇣=
= p
, !n =
s2 = ⇣!n !n ⇣ 2 1
bc
m
2 km
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Second-Order System: Underdamped
With the definitions of natural frequency and damping ratio, the
roots of the characteristic equations can be written as
s✓
◆
p
b
b 2
k
s1,2 =
±
s1 = ⇣!n + !n ⇣ 2 1
2m
2m
m
r
p
b
b
k
⇣=
= p
, !n =
s2 = ⇣!n !n ⇣ 2 1
bc
m
2 km
If there is damping in the system and the radicand is negative
(⇣ < 1), the roots are complex conjugates:
p
p
xh (t) = e ⇣!n t [A cos(!n 1 ⇣ 2 t) + B sin(!n 1 ⇣ 2 t)]
This motion represents damped oscillation consisting of sinusoidal
motion with exponentially decaying amplitude
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Second-Order System: Underdamped
A system with these characteristics is said to be underdamped
(⇣ < 1)
The damped natural frequency of the system is defined as
p
!d = !n 1 ⇣ 2
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Second-Order System: Underdamped
A system with these characteristics is said to be underdamped
(⇣ < 1)
The damped natural frequency of the system is defined as
p
!d = !n 1 ⇣ 2
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Second-Order System: Overdamped
If there is damping in the system and the radicand is positive (⇣ >
1), the roots are distinct real roots:
p
s1 = ⇣!n + !n ⇣ 2 1
p
s2 = ⇣!n !n ⇣ 2 1
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Second-Order System: Overdamped
If there is damping in the system and the radicand is positive (⇣ >
1), the roots are distinct real roots:
p
s1 = ⇣!n + !n ⇣ 2 1
p
s2 = ⇣!n !n ⇣ 2 1
The resulting transient homogeneous solution is
p
p
2
2
xh (t) = Ae( ⇣+ ⇣ 1)!n t + Be( ⇣ ⇣ 1)!n t
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Second-Order System: Overdamped
If there is damping in the system and the radicand is positive (⇣ >
1), the roots are distinct real roots:
p
s1 = ⇣!n + !n ⇣ 2 1
p
s2 = ⇣!n !n ⇣ 2 1
The resulting transient homogeneous solution is
p
p
2
2
xh (t) = Ae( ⇣+ ⇣ 1)!n t + Be( ⇣ ⇣ 1)!n t
This represents an exponentially decaying output. A system with
these characteristics is said to be overdamped, because its damping
exceeds critical damping as (⇣ > 1)
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System Response for all three cases
The curves represent unforced motion of a second-order system
with di↵erent amounts of damping
Initial State:
dx
(0) = 0,
dt
x(0) = 1
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Second-Order System: Step Response
The step response is a good measure of how fast and smoothly a
system responds to abrupt changes in input
(
0 t<0
Fext (t) =
Fi t 0
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Second-Order System: Step Response
The step response is a good measure of how fast and smoothly a
system responds to abrupt changes in input
(
0 t<0
Fext (t) =
Fi t 0
The step response consists of two parts
• a transient homogeneous solution xh (t), which has been
discussed above
• a steady state particular solution xp (t), which is the result of
the forcing function: xp (t) = Fki
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Second-Order System: Step Response
Fext (t) =
(
0
Fi
t<0
t 0
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Second-Order System: Step Response
Question: underdamped, overdamped,
critically damped
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Second-Order System: Step Response
Steady state value: the value the system
reaches after all transients dissipate.
Question: underdamped, overdamped,
critically damped
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Lecture 10
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Second-Order System: Step Response
Steady state value: the value the system
reaches after all transients dissipate.
Rise time: time required for the system to
go from 10% to 90% of the steady state value
Question: underdamped, overdamped,
critically damped
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Lecture 10
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Second-Order System: Step Response
Steady state value: the value the system
reaches after all transients dissipate.
Rise time: time required for the system to
go from 10% to 90% of the steady state value
Overshoot: the measure of the maximum
amount the output exceeds the steady state
value before settling, usually specified as a
percentage of the steady state value
Question: underdamped, overdamped,
critically damped
22/26
Dr. Uche Wejinya
Lecture 10
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Second-Order System: Step Response
Steady state value: the value the system
reaches after all transients dissipate.
Rise time: time required for the system to
go from 10% to 90% of the steady state value
Overshoot: the measure of the maximum
amount the output exceeds the steady state
value before settling, usually specified as a
percentage of the steady state value
Settling time: time required for the sysQuestion: underdamped, overdamped, tem to settle to within an amplitude band,
whose height is a specified ± percentage of
critically damped
the steady state value
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Lecture 10
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Second-Order System: Step Response
Some notes about second order system with
step response:
• An underdamped system exhibits a
fast rise time, but it overshoots and
may take a while to settle
• With the right amount of damping, it
can settle faster than critically
damped or overdamped systems.
• For cases where no overshoot is
allowed or desired, a critically damped
system has the fastest rise and settle
times.
• An overdamped system also has no
overshoot, but has slower rise and
settle times.
1. identify the order of a system, along
with the frequency response amplitude and phase plots
2. If the system is second order, we can
learn whether it is underdamped,
overdamped, or critically damped.
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System Modeling and Analogies
Any system, whether it be mechanical, electrical, hydraulic, thermal,
or some combination—can be modeled by ordinary linear di↵erential
equations that relate the output responses of the system to the
inputs.
24/26
Dr. Uche Wejinya
Lecture 10
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System Modeling and Analogies
Any system, whether it be mechanical, electrical, hydraulic, thermal,
or some combination—can be modeled by ordinary linear di↵erential
equations that relate the output responses of the system to the
inputs.
These di↵erential equations are all very similar mathematically and
di↵er only in the constants that appear in front of the derivative
terms
24/26
Dr. Uche Wejinya
Lecture 10
UARK
System Modeling and Analogies
Any system, whether it be mechanical, electrical, hydraulic, thermal,
or some combination—can be modeled by ordinary linear di↵erential
equations that relate the output responses of the system to the
inputs.
These di↵erential equations are all very similar mathematically and
di↵er only in the constants that appear in front of the derivative
terms
These constants represent the physical parameters of the system,
and there are analogies for these parameters among the di↵erent
system
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System Modeling and Analogies
A resistor in an electrical system is analogous to a damper in a
mechanical system or to a valve or flow restriction in a hydraulic
system
Mass or inertia in a mechanical system is analogous to inductance
in an electrical system and fluid inertance in a hydraulic system
The generic terms used to describe the analogous system parameters and variables are e↵ort, flow, displacement, momentum,
resistance, capacitance, and inertia.
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System Modeling and Analogies
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