Exp 2 Instro

FACULTY OF ENGINEERINGMECHANICAL
ENGINEERING DEPARTMENT
[ME472-INSTRUMENTATION AND DYNAMIC LAB]
[Exp#2: System Response]
Name ‫ﻋﺒﺪاﻟﺮﺣﻤﻦ اﻟﻤﺄﻣﻮن ﻣﺤﻤﺪ‬

‫ﻗﺪور‬
ID # 135066
Sec # 1
ENG Mohammad Rawashdh
Date 10 – Mar – 2021
Introduction:
 SYSTEM RESPONSE: The relationship between the desired

output of a mechatronic or measurement system and its actual
output is the basis of system response analysis .
 Black box Model: No prior model is available. Most system
identification algorithms are of this type. In science, computing,
and engineering, a black box is a device, system or object which
can be viewed in terms of its inputs and outputs (or transfer
characteristics), without any knowledge of its internal workings. Its
implementation is "opaque" (black). Almost .
Figure (1): Black Box
 first order system response:

A first order control system is defined as a type of control system
whose input-output relationship (also known as a transfer function)
is a first-order differential equation. A first-order differential
equation contains a first-order derivative, but no derivative higher
than the first order. The order of a differential equation is the order
of the highest order derivative present in the equation.
 The standard form of a first-order differential with a step

input of amplitude A is given by:
dx
τ + x=kA
dt
 So far, what techniques do we have for finding the system
parameters, in particular the time constant ?
We can deduct from plots of the free or step responses of a

first-order system.
Figure (2): Experimental step response, x exp(t), for the charging capacitor showing the 1τ ,2 τ , and 3 τ estimates of
the time constant.
1. RC step response
The system in this experiment is a resistor-capacitor (RC)
circuit, as illustrated in Figure(3). The capacitor in the circuit is
charged when it is connected to a power supply at point A and
discharges when the power is turned off (i.e., when it is
connected to point B). To avoid inconsistencies in our data due
to internal power supply dynamics, we will use a switch in the
circuit to control the input voltage while the power supply is
on. The voltage across the capacitor is the quantity we will
measure and is the dependent variable in the mathematical
model of the system.
Figure (3): Resistor-capacitor circuit

−t
v=v iv (1−e ) τ where τ =RC
And we have other examples of first order system:
Figure(4): Time constants of some typical first-order systems.
2. Second-Order System Transient Response

Second-order state determined systems are described in
terms of two state variables. Physical second-order
system models contain two independent energy storage
elements which exchange stored energy and may
contain additional dissipative elements; such models are
often used to represent the exchange of energy between
mass and stiffness elements in mechanical systems,
between capacitors and inductors in electrical systems,
and between fluid inertance and capacitance elements in
hydraulic systems.
Figure(5):examples of second order system modeling.
MATERIALS AND PROCEDURES:
 Oscilloscope: Electrical instrument used to represent voltage

signals as a function of time.
 AC function generator: The device that is capable of supplying
variable power and frequency to a load.
 capacitor box: The box which contains the capacitance of

different values for estimating and comparing the capacitance is
known as the capacitance box.
 resistance box: The box which contains the resistors of different
values for estimating and comparing the resistance is known as
the resistance box.
 inductor box: The box which contains the inductance of different
values for estimating and comparing the inductance is known as
the inductance box.
 PNC cable: The BNC connector is a miniature quick
connect/disconnect radio frequency connector used for coaxial
cable.
Figure(6):oscilloscope . figure(7):Ac function generator.
Figure(8):capacitor box figure(9):resistance box
Figure(10):inductance box figure(11):PNC cable

Data and result:
1. Fires order system:
 low-pass filter
is a filter that passes signals with a frequency lower than a selected
cutoff frequency and attenuates signals with frequencies higher
than the cutoff frequency. The exact frequency response of
the filter depends on the filter design.
−t
Figure(12):low pass filter V out =V ¿ (1−e )τ
Figure(13): schematic diagram for first order system used to measure time constant.
Figure(14): first order response
If R=12 k ohm, C= 24 n F
From the figure(14) we can find:

1. Theoretical time constant=RC =12∗103∗24∗10−9=288∗10−6 sec
2. Experimental time constant=245∗10−6 sec
¿ |288−245|
3. Percent error=¿ true value−exp value∨ ∗100 %= ≈ 15 % ¿
true value 288
Table 1: cutoff frequency
Input freq. (Hz) VPP IN VPP OUT Ar

60.6 10 9.8 0.98
100.8 10 9.6 0.96
152.4 10 9.2 0.92
200.9 10 9 0.9
261.3 10 8.6 0.86
315.6 10 8 0.8
412.9 10 7.6 0.76
490.9 10 7.2 0.72
560.8 10 6.8 0.68
616.4 10 6.2 0.62
680.6 10 5.6 0.56
560.8+490.9
1. f c exp ≈ 2
≈525.85 HZ
1 1
2. f c theo= 2 πRC = =552.62 HZ
2 π∗12∗103∗24∗10−6
1.2
3.
1
0.8
Ar #
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700 800
input ferq (HZ)
¿ |552.62−525.85|
Percent error=¿ true value−exp value∨ ∗100 %= ≈ 4.84 % ¿
true value 552.62
Cutoff frequency at Ar=.707# , f=525.85 Hz

Figure(15): curve between gain input frequency vs AR.
2. Second order system:
 Low pass filter

The RLC filter: RLC circuit is described as a second-order circuit,
meaning that any voltage or current in the circuit can be described
by a second-order differential equation in circuit analysis. The
three circuit elements, R, L and C, can be combined in a number of
different circuits .
Figure(16):RLC low pass filter .
R ' 1 1
I '' (t )+ I ( t )+ I ( t )=0 , ω 02 =
L LC LC
R/L
Ϛ= , ωd =ω 0 √ 1−Ϛ 2
2 √1/ LC
I (t)=C∗e−Ϛ ω t sin (ω d t+φ)

0
 Ϛ <1:underdamped
 Ϛ=1 :critcaldamped
 Ϛ >1: overdamped
Figure(17): time vs x(t) or I(t) with change Ϛ
Figure(18): Features of an underdamped step response.
 Rise time (Tr): is the time required for the response to rise from 0
to 90% of the final value.
 Settling time (Ts): is the time required for the response to reach
and stay within a specified tolerance band (2% or 5%) of its final
value.
 Delay time (Td): is the time required for the response to reach 50%
of the final value.
 Peak time (Tp): is the time required for the underdamped step
response to reach the peak of time response (Y p) or the peak
overshoot .
 Percent overshoot (OS%): is the normalized difference between
the response peak value and the steady value (this characteristic is
not found in a first order system and found in higher one for the
underdamped step response
Figure (19): second response

From the figure (19) we can find:
1. Rising time=10 μ sec
2. Settlingtime=125 μ sec
v pp out−v ¿ 14.8−10.2
3. Percentage overshoot = ∗100 %= =31.1 %
v¿ 10.2
4. Peak time ≈ 20 μs
5. Delay time=5 μ sec
Discussion:
 The time constant is very important to indicate how long

the system response takes to provide accurate reading.
 Figure #8 shows that there is a cut-off frequency between
(500-550)Hz.
 In the first order system, when we applied the step input to
the RC circuit, we got a graph from the chart recorder that
graph is called the "Time Response" of the system, which
helps us to find the time constant that is an important
parameter to analyze any first order system.
 the response of a second-order system can be described to a
designer without the need for sketching the response. We
define two physically meaningful specifications for second-
order systems. These quantities can be used to describe the
characteristics of the second-order transient response just as
time constants describe the first-order system response.
 The natural frequency is a time-axis scale factor and does
not affect the nature of the response other than to scale it in
time.
Conclusion:
 after one time constant, the output reaches 63.2% of its
final value.
 First order systems are characterized by a time constant
only, which is a measure of how fast the system responds to
the force function.
 The RC Circuit can be used as low-bass or high-bass filters
depending on the configuration of the resistance and the
capacitor in the circuit.
 Second order systems are characterized by delay time,
rising time, set time, peak time, and maximum overshoot.
 Step Response of higher order systems:

The step responses of higher order systems are harder to generalize
about. For example, a third order system can have:
 three distinct real roots,

 one pair of repeated real roots, and one other distinct real root,
 a single root repeated three times, or
 a complex conjugate pair of roots and a real root.
For more information about the higher order system, see this link:
The Dominant Pole Approximation (swarthmore.edu)

 band-pass filter: is a device that
passes frequencies within a certain range and rejects
(attenuates) frequencies outside that range.
Figure(20):band pass filter
 band-stop filter: is a filter that passes

most frequencies unaltered, but attenuates those in a
specific range to very low levels. It is the opposite of
a band-pass filter. A notch filter is a band-stop filter with a
narrow stopband (high Q factor).
Figure(21):band stop filter.

REFERENCES:
 Instrumentation and Dynamic Systems Lab Manual.

 David G. Alciatore, Michael B. Histand - Introduction to
Mechatronics and Measurement Systems, Fourth Edition
 Instrumentation for Engineering Measurements, James W. Dally,

Second Edition.
 Step response - Wikipedia
 Electrical4U: Learn Electrical & Electronics Engineering (For

Free)

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FACULTY OF ENGINEERINGMECHANICAL

[ME472-INSTRUMENTATION AND DYNAMIC LAB]

[Exp#2: System Response]

Name ‫ﻋﺒﺪاﻟﺮﺣﻤﻦ اﻟﻤﺄﻣﻮن ﻣﺤﻤﺪ‬

 SYSTEM RESPONSE: The relationship between the desired

Figure (1): Black Box

 first order system response:

 The standard form of a first-order differential with a step

We can deduct from plots of the free or step responses of a

Figure (3): Resistor-capacitor circuit

And we have other examples of first order system:

Figure(4): Time constants of some typical first-order systems.

2. Second-Order System Transient Response

MATERIALS AND PROCEDURES:

 Oscilloscope: Electrical instrument used to represent voltage

 capacitor box: The box which contains the capacitance of

Figure(8):capacitor box figure(9):resistance box

Figure(10):inductance box figure(11):PNC cable

From the figure(14) we can find:

2. Experimental time constant=245∗10−6 sec

Input freq. (Hz) VPP IN VPP OUT Ar

Cutoff frequency at Ar=.707# , f=525.85 Hz

2. Second order system:

 Low pass filter

Figure(16):RLC low pass filter .

I (t)=C∗e−Ϛ ω t sin (ω d t+φ)

Figure(18): Features of an underdamped step response.

Figure (19): second response

 The time constant is very important to indicate how long

 Step Response of higher order systems:

 three distinct real roots,

The Dominant Pole Approximation (swarthmore.edu)

Figure(20):band pass filter

 band-stop filter: is a filter that passes

Figure(21):band stop filter.

 Instrumentation and Dynamic Systems Lab Manual.

 Instrumentation for Engineering Measurements, James W. Dally,

 Step response - Wikipedia

 Electrical4U: Learn Electrical & Electronics Engineering (For

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