Here's the continuation of the report: 3.2.1 Parallel Plate Capacitor (continued) As the IV fluid droplets move between the plates of the capacitor, the capacitance increases due to the change in the dielectric constant, resulting in the observation of a peak in capacitance. 3.2.2 Semi-cylindrical Capacitor The semi-cylindrical capacitor consists of two semi-cylindrical conductors (plates) facing each other with a gap between them. The gap between the plates is filled with a dielectric material, typically the IV fluid. When a potential difference is applied across the plates, electric field lines form between them. The dielectric material between the plates enhances the capacitance by reducing the electric field strength and increasing the charge storage capacity. 3.2.3 Cylindrical Cross Capacitor The cylindrical cross capacitor is composed of two cylindrical conductors (rods) intersecting at right angles to form a cross shape. The space between the rods is filled with a dielectric material, such as the IV fluid. When a potential difference is applied between the rods, electric field lines form between them. The dielectric material between the rods enhances the capacitance by reducing the electric field strength and increasing the charge storage capacity, similar to the semi-cylindrical design. 3.3 Advantages of Capacitive Sensing Approach Capacitive sensing for IV fluid monitoring offers several advantages over other automated monitoring methods: 1. Non-invasive operation: The sensors do not require direct contact with the IV fluid, reducing the risk of contamination or disruption to the therapy. 2. High sensitivity: Capacitive sensors can detect minute changes in capacitance, enabling precise tracking of IV fluid droplets. 3. Low cost: The sensors can be constructed using relatively inexpensive materials, making them a cost-effective solution. 4. Low power consumption: Capacitive sensors typically have low power requirements, making them suitable for continuous monitoring applications. 5. Ease of implementation: The sensors can be easily integrated into existing IV setups without significant modifications. 6. Stable measurements: Capacitive sensors can provide stable and repeatable measurements across different IV fluid types. Chapter 4: Experimental Setup and Results 4.1 Description of Experimental Setup To evaluate the performance of capacitive sensors for IV fluid monitoring, an experimental setup was constructed. The setup included various capacitive sensor designs, such as parallel plate, semi-cylindrical, and cylindrical cross capacitors, positioned around an IV drip chamber. The sensors were connected to a capacitance measurement circuit, which recorded the changes in capacitance as IV fluid droplets passed through the sensor's electric field. Multiple experiments were conducted using different IV fluid types and flow rates to assess the sensors' accuracy, repeatability, and sensitivity. 4.2 Measurements with

1. Updated New Syllabus
UNIT III
 CLASS NO: 7
 DATE:19-03-2024

2. First Order System & its Response
Fig 1 : Block diagram of a 1st order system
Input Output Relationship:
𝑪 𝒔
𝑹(𝒔)
=
𝟏
𝑻𝒔 + 𝟏
Objective:
To analyse the system responses to
such inputs as the
 unit-step
 unit-ramp
 unit-impulse functions
The initial conditions are assumed to
be zero.

3. Unit Step Response of First Order System
Input Output Relationship:
𝑪 𝒔
𝑹(𝒔)
=
𝟏
𝑻𝒔 + 𝟏
𝑪(𝒔) =
𝟏
𝑻𝒔 + 𝟏
𝑹(𝒔)
Laplace Transform of Input S
Here it is 𝓛 𝒖 𝒕 =
𝟏
𝒔
𝑪 𝒔 =
𝟏
𝑻𝒔 + 𝟏
𝟏
𝒔
𝑪 𝒔 =
𝟏
𝑻𝒔 + 𝟏
𝟏
𝒔
Using Partial Fraction
Technique:
𝑪 𝒔 =
𝟏
𝒔
−
𝑻
𝑻𝒔 + 𝟏
=
𝟏
𝒔
−
𝟏
𝒔 +
𝟏
𝑻
𝓛−𝟏
𝑪(𝒔)
𝒄 𝒕 = 𝟏 − 𝒆−
𝒕
𝑻, for 𝒕 ≥ 𝟎
Time (𝒕)
𝒄(𝒕)
𝑢 𝑡 = 1, 𝑡 ≥ 0
𝑢 𝑡 = 0, 𝑡 < 0

4.  Interesting
Observations:
• 𝒄 𝒕 𝒕=𝟎 = 𝒄 𝟎 = 𝟎 and 𝒄 𝒕 𝒕=∞ = 𝒄 ∞ =
𝟏
• The curve behaviour is
exponential
• At 𝒕 = 𝑻, 𝒄 𝑻 = 𝟏 − 𝒆−𝟏 = 𝟎. 𝟔𝟑𝟐
• The smaller the Time-Constant 𝑻 ,
the faster the system response.
• The Slope of the Tangent Line at
𝒕 = 𝟎 is
𝟏
𝑻
since
𝒅𝒄(𝒕)
𝒅𝒕 𝒕=𝟎 =
𝟏
𝑻
𝒆−
𝒕
𝑻
𝒕=𝟎 =
𝟏
𝑻
?

5. Unit-Ramp Response of First Order System
Input Output Relationship:
𝑪 𝒔
𝑹(𝒔)
=
𝟏
𝑻𝒔 + 𝟏
𝑪(𝒔) =
𝟏
𝑻𝒔 + 𝟏
𝑹(𝒔)
Laplace Transform of Input Si
Here it is 𝓛 𝒓 𝒕 =
𝟏
𝒔𝟐
𝑪 𝒔 =
𝟏
𝑻𝒔 + 𝟏
𝟏
𝒔𝟐
𝑪 𝒔 =
𝟏
𝑻𝒔 + 𝟏
𝟏
𝒔𝟐
Using Partial Fraction
Technique:
𝑪 𝒔 =
𝟏
𝒔𝟐 −
𝑻
𝒔
+
𝑻𝟐
𝑻𝒔 + 𝟏
𝓛−𝟏 𝑪(𝒔)
𝒄 𝒕 = 𝒕 − 𝑻 + 𝑻𝒆−
𝒕
𝑻, for 𝒕 ≥
𝟎
Time (𝒕)
𝒄(𝒕)

6.  Interesting
Observations:
The error
signal 𝒆 𝒕
𝒆 𝒕 = 𝒓 𝒕 − 𝒄 𝒕 = 𝑻 𝟏 − 𝒆−
𝒕
𝑻
𝒆 ∞ = 𝑻 𝟏 − 𝒆−
∞
𝑻 = 𝑻
*** The error signal 𝒆 𝒕 approaches to 𝑻
at steady state

7. Unit-Impulse Response of First Order System
Input Output Relationship:
𝑪 𝒔
𝑹(𝒔)
=
𝟏
𝑻𝒔 + 𝟏
𝑪(𝒔) =
𝟏
𝑻𝒔 + 𝟏
𝑹(𝒔)
Laplace Transform of Input Si
Here it is 𝓛 𝜹 𝒕 = 𝟏
𝑪 𝒔 =
𝟏
𝑻𝒔 + 𝟏
𝑪 𝒔 =
𝟏
𝑻𝒔 + 𝟏
𝓛−𝟏
𝑪(𝒔)
𝒄 𝒕 =
𝟏
𝑻
𝒆−
𝒕
𝑻 for 𝒕 ≥ 𝟎
Time (𝒕)
𝒄(𝒕)
𝟏
𝑻
= 𝟐

8. An Important Property of Linear Time-
Invariant (LTI) Systems
𝒄 𝒕 = 𝒕 − 𝑻 + 𝑻𝒆−
𝒕
𝑻, for 𝒕 ≥
𝟎
𝒄 𝒕 = 𝟏 − 𝒆−
𝒕
𝑻, for 𝒕 ≥ 𝟎
𝒄 𝒕 =
𝟏
𝑻
𝒆−
𝒕
𝑻 for 𝒕 ≥ 𝟎
Unit-Ramp
Response
Unit-Step
Response
Unit-Impulse
Response
 Interesting
Observations:
d (Unit-Ramp Response) / dt = Unit-Step
Response
d (Unit-Step Response ) / dt = Unit-Impulse
Response
 The response to the integral of the original signal can be obtained by
integrating the response of the system to the original signal and by determining
the integration constant from the zero-output initial condition. This is a
property of LTI systems.
 Linear time-varying (LTV) systems and nonlinear systems do not possess this
Remark

9. Second Order System and Its Response
Fig 2 : Servo system
Simplified Block Diagram
Plant

10. Relevant Equations related to Servo system ||Transfer
Function of Servo system
The equation for the load
elements is
𝑱𝒄 + 𝑩𝒄 = 𝑻 • 𝑱 & 𝑩 are inertia and viscous-
friction elements
• 𝑻 denotes the Torque produced by
the proportional controller whose
gain is 𝑲
Taking Laplace Transform of both side and
considering initial conditions to be
zero:
𝑱𝒔𝟐𝑪(𝒔) + 𝑩𝒔𝑪(𝒔) = 𝑻(𝒔)
𝑪 𝒔
𝑻(𝒔)
=
𝟏
𝒔(𝑱𝒔 + 𝑩)
𝑪 𝒔
𝑹(𝒔)
=
𝑲
𝑱𝒔𝟐 + 𝑩𝒔 + 𝑲
=
𝑲
𝑱
𝒔𝟐 +
𝑩
𝑱
𝒔 +
𝑲
𝑱
2nd Order system
Closed-Loop Transfer Function
∃ 2 Closed-Loop
Poles

11. Step Response of Second Order System
𝑪 𝒔
𝑹(𝒔)
=
𝑲
𝑱𝒔𝟐 + 𝑩𝒔 + 𝑲
=
𝑲
𝑱
𝒔𝟐 +
𝑩
𝑱
𝒔 +
𝑲
𝑱
Closed-Loop TF
=
𝑲
𝑱
𝒔 +
𝑩
𝟐𝑱
+
𝑩
𝟐𝑱
𝟐
−
𝑲
𝑱
𝒔 +
𝑩
𝟐𝑱
−
𝑩
𝟐𝑱
𝟐
−
𝑲
𝑱
Closed-Loop Poles will be COMPLEX CONJUGATES if 𝑩𝟐
− 𝟒𝑱𝑲
Closed-Loop Poles will be REAL if 𝑩𝟐 − 𝟒𝑱𝑲 ≥ 𝟎
Remark:
For Transient Response analysis, it is convenient to write
𝑲
𝑱
= 𝒘𝒏
𝟐
𝑩
𝑱
= 𝟐𝜻𝒘𝒏 = 𝟐𝝈
Key Terms
 𝝈 : Attenuation
 𝒘𝒏 : Undamped Natural
Frequency
 𝜻 : Damping Ratio
 𝑩 : Actual Damping
 𝑩𝒄 : Critical Ratio
 𝜻 =
𝑩
𝑩𝒄
=
𝑩
𝟐 𝑱𝑲

12. 𝑮 𝒔 =
𝒘𝒏
𝟐
𝒔(𝒔 + 𝟐𝜻𝒘𝒏)
𝐶(𝑠)
𝑅(𝑠)
Fig 3 : Standard Second-order feedback system
Open Loop Transfer Function
𝐸 𝑠 = 𝑅 𝑠 − 𝐶(𝑠)
𝑪 𝒔
𝑹(𝒔)
=
𝒘𝒏
𝟐
𝒔𝟐 + 𝟐𝜻𝒘𝒏 + 𝒘𝒏
𝟐
3 Important Cases To
Study
Case 1: Underdamped || 𝟎 <
𝜻 < 𝟏
Case 2: Critically Damped
|| 𝜻 = 𝟏
Case 3: Overdamped ||
𝜻 > 𝟏
𝑪 𝒔
𝑹(𝒔)
=
𝒘𝒏
𝟐
(𝒔 + 𝜻𝒘𝒏 + 𝒋𝒘𝒅)(𝒔 + 𝜻𝒘𝒏 − 𝒋𝒘𝒅)
Where, 𝑤𝑑 = 𝑤𝑛 1 − 𝜁2
𝑪(𝒔) =
𝒘𝒏
𝟐
𝒔 + 𝜻𝒘𝒏 + 𝒋𝒘𝒅 𝒔 + 𝜻𝒘𝒏 − 𝒋𝒘𝒅
𝟏
𝐬
𝑹 𝒔 =
𝟏
𝒔
Unit Step
Function
𝑪 𝒔
𝑹(𝒔)
=
𝒘𝒏
𝟐
𝒔 + 𝒘𝒏
𝟐
Where, 𝑤𝑑 = 𝑤𝑛 1 − 𝜁2 = 0
𝑪(𝒔) =
𝒘𝒏
𝟐
𝒔 + 𝒘𝒏
𝟐
𝟏
𝐬
𝑹 𝒔 =
𝟏
𝒔
Unit Step
Function
𝑪 𝒔
𝑹(𝒔)
=
𝒘𝒏
𝟐
(𝒔 + 𝜻𝒘𝒏 + 𝒘𝒏 𝜻𝟐 − 𝟏)(𝒔 + 𝜻𝒘𝒏 − 𝒘𝒏 𝜻𝟐 − 𝟏)
𝑹 𝒔 =
𝟏
𝒔
Unit Step
Function
𝑪(𝒔) =
𝒘𝒏
𝟐
𝒔(𝒔 + 𝜻𝒘𝒏 + 𝒘𝒏 𝜻𝟐 − 𝟏)(𝒔 + 𝜻𝒘𝒏 − 𝒘𝒏 𝜻𝟐 − 𝟏)

13. ) STABILITY OF A MASS-SPRING-DAMPER System

14. B) STABILITY OF A MASS-SPRING System

15. Expression of 𝒄 𝒕 in all
the cases
Case 1: Underdamped || 𝟎 <
𝜻 < 𝟏
𝑪(𝒔) =
𝒘𝒏
𝟐
𝒔 + 𝜻𝒘𝒏 + 𝒋𝒘𝒅 𝒔 + 𝜻𝒘𝒏 − 𝒋𝒘𝒅
𝟏
𝐬
Using Partial Fraction
𝑪 𝒔 =
𝟏
𝒔
−
𝒔 + 𝟐𝜻𝒘𝒏
𝒔𝟐 + 𝟐𝜻𝒘𝒏𝒔 + 𝒘𝒏
𝟐
=
𝟏
𝐬
−
𝐬 + 𝜻𝒘𝒏
𝒔 + 𝜻𝒘𝒏
𝟐 + 𝒘𝒅
𝟐
−
𝜻𝒘𝒏
𝒔 + 𝜻𝒘𝒏
𝟐 + 𝒘𝒅
𝟐

16. Important Observations of
Case 2
Case 2: Critically Damped
|| 𝜻 = 𝟏
𝑪(𝒔) =
𝒘𝒏
𝟐
𝒔 + 𝒘𝒏
𝟐
𝟏
𝐬
Case 3: Overdamped ||
𝜻 > 𝟏
𝑪(𝒔) =
𝒘𝒏
𝟐
𝒔(𝒔 + 𝜻𝒘𝒏 + 𝒘𝒏 𝜻𝟐 − 𝟏)(𝒔 + 𝜻𝒘𝒏 − 𝒘𝒏 𝜻𝟐 − 𝟏)

17. Fig.: Unit-step response curves of the
system
A family of unit-step response curves 𝒄(𝒕) with various values of
𝜻 w.r.t. 𝒘𝒏𝒕

18. Second Order System and Its Transient Response Specifications
Salient Features:
 Frequently, the performance characteristics of a control system are specified in
terms of the transient response to a unit-step input, since it is easy to
generate and is sufficiently drastic.
 The transient response of a system to a unit-step input depends on the initial
conditions. For convenience in comparing transient responses of various systems,
it is a common practice to use the standard initial condition that the system is
at rest initially with the output and all-time derivatives thereof zero.
 Then the response characteristics of many systems can be easily compared. The
transient response of a practical control system often exhibits damped
oscillations before reaching steady state.
 In specifying the transient-response characteristics of a control system to a
unit-step input, it is common to specify the following:
1) Delay Time 𝒕𝒅
2) Rise Time (𝒕𝒓)
3) Peak Time 𝒕𝒑
4) Maximum Overshoot 𝑴𝒑
5) Settling Time (𝒕𝒔)

19. Definitions:
1) Delay Time 𝒕𝒅 The delay time is the time required for the response to reach half
the final value the very first time.
𝑐 𝑡𝑑 =
1
2
𝑐 ∞ =
1
2
1
2
= 1 −
𝑒−𝜁𝑤𝑛𝑡𝑑
1 − 𝜁2
sin 𝑤𝑑𝑡𝑑 + tan−1
1 − 𝜁2
𝜁
Example: For underdamped second-order
system:

20. 2) Rise Time 𝒕𝒓
• The rise time is the time required for the response to rise from
10% to 90%, 5% to 95%, or 0% to 100% of its final value.
• For underdamped second-order systems, the 0% to 100% rise time
is normally used.
• For overdamped systems, the 10% to 90% rise time is commonly
used.
𝑐 𝑡𝑟 = 100% 𝑐 ∞ = 1
1 = 1 −
𝑒−𝜁𝑤𝑛𝑡𝑟
1 − 𝜁2
sin 𝑤𝑑𝑡𝑟 + tan−1
1 − 𝜁2
𝜁
Example: For underdamped second-order
system:
Where;
𝑤𝑑 = 𝑤𝑛 1 − 𝜁2, 𝜁𝑤𝑛 = 𝜎
And 𝛽 is shown here

21. 3) Peak Time
𝒕𝒑
The peak time is the time required for the response to reach the
first peak of the overshoot.
Example: For underdamped second-
order system:

22. 4) Maximum % Overshoot
𝑴𝒑
The maximum overshoot is the maximum peak value of the
response curve measured from unity. If the final steady-state
value of the response differs from unity, then it is common
to use the maximum percent overshoot.
Example: For underdamped second-
order system:
(A)

23. 4) Settling Time (𝒕𝒔)
The settling time is the time required for the response curve to
reach and stay within a range about the final value of size
specified by absolute percentage of the final value (usually 2% or
5%).
• The settling time is related to the largest time constant of the
control system. Which percentage error criterion to use may be
determined from the objectives of the system design in question.
Example: For underdamped second-
order system:

24.  The relative dominance of closed-loop poles is determined by the ratio of the real
parts of the closed-loop poles, as well as by the relative magnitudes of the
residues evaluated at the closed-loop poles.
 The magnitudes of the residues depend on both the closed-loop poles and zeros.
 If the ratios of the real parts of the closed-loop poles exceed 5 and there are no
zeros nearby, then the closed-loop poles nearest the 𝒋𝒘 axis will dominate in the
transient-response behaviour because these poles correspond to transient-response
terms that decay slowly.
Those closed-loop poles that have dominant effects on the transient-response
behaviour are called dominant closed-loop poles.
 Quite often the dominant closed-loop poles occur in the form of a complex-
conjugate pair. The dominant closed-loop poles are most important among all
closed-loop poles.
 Note that the gain of a higher-order system is often adjusted so that there will
exist a pair of dominant complex-conjugate closed-loop poles. The presence of such
poles in a stable system reduces the effects of such nonlinearities as dead zone,
DOMINANT CLOSED LOOP POLES