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European Journal of Applied Sciences – Vol. 11, No. 3

Publication Date: June 25, 2023

DOI:10.14738/aivp.113.14636.

Audu, E. E., & Nwazor, N. O. (2023). A Theoretical Study of Solenoid Inductive Effect on Propagation Delay in Closed Feedback

Control. European Journal of Applied Sciences, Vol - 11(3). 422-435.

Services for Science and Education – United Kingdom

A Theoretical Study of Solenoid Inductive Effect on Propagation

Delay in Closed Feedback Control

Eliazar Elisha Audu

Research, Technology, and Innovation (RTI), FCT, Abuja, Nigeria and

Centre for Information and Telecommunication Engineering,

University of PortHarcourt, PortHarcourt, Nigeria

Nkolika Ogechukwu Nwazor

Department of Electrical and Electronics Engineering,

University of PortHarcourt, Nigeria

ABSTRACT

Solenoid is widely and commonly used electro-mechanical devices in industrial

processes and engines for hydraulic, fuel and pneumatic controls. The speed,

reliability and response time of solenoids can be used to characterize their

performance especially where propagation delay can cause instability in the

system. In this paper, we model the dynamic characteristics of a proportional

solenoid, and investigate its effect on the overall response time in closed loop. The

transfer function of the proportional translational motion solenoid at different

positions of the plunger is derived. Appropriate boundary conditions are applied to

solve transients and stead state current and deduce the response time of the

system. At different resistance and inductance, the results show that the rising and

settling times of a solenoid system is inversely to the values of Resistance- Inductance (R-L) pair. The lower the values of R-L pair, the higher the rising and

settling times. The transient current is shown to be inversely proportional to the

values of R-L pair. Solenoid with lowest resistance has the highest transient current

flowing through it. These indicates that the selection of solenoid R-L parameters are

important consideration in closed loop control system designs.

Keywords: Solenoids, Propagation delay, Resistance-Inductance (R-L), Transients, Closed

loop, Kirchoff’s law, Transfer Function.

INTRODUCTION

Feedback control systems are widely used class of control systems with wide area of

applications from home and offices to industries and manufacturing processes. Control systems

can be either open loop system or a closed loop system [1],[2] (Canete et al., 2018; Phillips et

al., 1990). The performance of a controller (digital or analogue) is the response time between

when a command is issued and when the control action is taken place. Although, there are many

indicators that can be used to measure controller’s performance (Sternfield and Gates, 1950)

[3]. Time delay affects controller performance and stability of the control system. Timely

information is crucial in modern process plants, digital controllers widely used for their

flexibility, speed and efficiency to deliver precise control signals to actuators to alter the

behaviour of a plant or processes. Discrete-time processors are computers that sample

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Audu, E. E., & Nwazor, N. O. (2023). A Theoretical Study of Solenoid Inductive Effect on Propagation Delay in Closed Feedback Control. European

Journal of Applied Sciences, Vol - 11(3). 422-435.

URL: http://dx.doi.org/10.14738/aivp.113.14636.

analogue data through its input and discretized into a finite but quantifiable values for digital

processing. The results of the computations are converted back to the analogue form to correct

or maintain the position of an actuator.

Digital control system works on difference equation and discrete-time processing. The study of

physical and engineering systems usually requires the use of mathematical model that

describes the static and dynamical behaviour of any physical systems.

Different mathematical models have been proposed for modelling the dynamic response of

control systems closed loop control systems. Many physical systems can be modelled using the

linear time-invariant (LTI) approach in both continuous and discrete time domains (Canete et

al., 2018; Phillips et al., 1990) [1],[2]. LTI combines the properties of linearity and time- invariants to simplify mathematical modelling, analysis and synthesis.

Linearity imposes the condition on the control system that the output is linearly related to the

output (Dinuzza, 2013; Moudgalya, 2007) [4],[5]. LTI has been applied analyze stability and

performance requirements in absolutely distinguishable discrete dynamic systems (Rosa and

Silvestre, 2011) [6]. In digital control theory, modeling, designing, analysis and implementation

is done in discrete time domain (Tahir et al., 2018; Moudgalya, 2007) [5],[7]. Linear time- invariant (LTI) model has been used for synthesis of automated digital controller to guarantee

safety and stability in automatic control systems (Abate et al., 2017) [8]. LTI model of analyzing

controllers and control systems represents a class of wide range of dynamical system that has

applications in different domains especially in continuous time LTI. However, the use of LTI in

digital control system design and synthesis present a new challenge as a result of the effect of

finite representation of signal and quantization noise during digital to analogue or vice versa

conversions (Abate et al., 2017) [8]. Khargonekar et al. (1985), The analysis of robust control

of linear time-invariant plant using both time-varying and time-invariant, stated that time

varying controller provides better performance over time-invariant ones for large classes of

control problems [9]. In process plant and manufacturing industries, digital controls such as

the programmable logic controller (PLC) drive electro-mechanical actuators to effect control

characteristics. Actuators are electromechanical devices that converts electrical energy into

mechanical energy using magnetic field as an intermediary force.

Fig.1: Discrete time closed loop automatic control system [1] (Canete et al., 2018).