MODELING OF CONTROLLED DC MOTOR WITH A CHOPPER CIRCUIT

To design a control system for any system, it is required to model the system's dynamic behaviour. The modeling and analysis of these systems can be either numerical or analytical. This paper studies the analytical methods of modeling DC motors mainly continuous-time averaging technique and discrete-time approach. It shows that the discrete-time approach provides clear expressions representing the operation of DC motors and thus can help designers gain a better understanding of the performance of the motor. The results are shown using numerical simulations.


Introduction
The electrical motor is a motor that convert electrical enegy into mechanical energy.In general there are two types of motors which are AC motors and DC motors.A simple motor uses electricity and magnetic field for producing torque which rotate the motor.DC motors outperform to AC motors in terms of providing better speed control on high torque loads, use in wide indestrial applications, and more useble as it designed to use with batteries and solar cells energy sources which provide protabillity where it is required and thus provide cost effective solution as it is not possible to have an AC power supply in every place.
Usually DC motors show responses at both voltage and current, the applied voltage describes the speed of the motor while the current in the armature windings shows the torque.Generally it can be said that the applied voltage effect the speed of the motor while the torque is controlled by the current.One of the most effective method of controlling the speed of the motor is using a chopper circuit which has been used in this paper.
The analysis of controlled DC motor with a chopper circuit, which are nonlinear systems, requires the use of nonlinear mathematical techniques.However, unlike linear systems where the solutions can be written in closed-form in terms of the system's eigenvalues and eigenvectors, most nonlinear systems are known to be very hard to solve analytically i.e. there is no general method for obtaining a closed-form solution for these systems.To work around this problem, power electronics engineers normally use the technique of averaging for determining the stability and dynamic behaviour of switching systems [5][6][7].In this method, the actual nonlinear system is linearised around a steady-state operating point to yield a linear model.This averaging method gives a simple and accurate model at low operating frequencies.Another approach to study the stability of switching systems is based on the construction of a sampled data model [7][8][9][10][11].Stability can be achieved by explicitly deriving a discrete map that describes the evolution of the state from one clock instant to the next and then locally linearising the map around its steady-state operating point.
In this paper, the controlled DC motor with a chopper circuit will be modeled using two approaches, the continuous-time averaging approach and discrete-time iterative map.During the interval when the switch is ON, the diode is reverse biased and the input voltage provides energy to the load and the inductor.During the second interval when the switch is OFF, the inductor current flows through the diode transferring some of its stored energy to the load.

Basic operation of the DC motor
During the first interval ton, when the switch is closed, the equations that represent the DC motor operating in continuous conduction mode are: These equations can be written in state space form as: x is the state vector, and is the input vector.A and B are the system matrices that contain the system parameters, defined as: During the interval t off , when the switch is open, the equations that represent the DC motor operating in continuous conduction mode are: These can be described in state space form as: where The ratio of t on to T is the duty cycle d.It is defined as is the period of one cycle.
In this paper, the following parameters of the dc motor are used to investigate the behaviour of the circuit.L = 53.7mH,, R = 2.8Ω , B=0.000275 Nms, J=0.000557 Kg m 2 /s 2 , T L =0.38N.m , K e =0.1356 V/rad, and K t =0.1324 Nm/Amp

Modeling of DC MOTORS
To design a control system for a DC motor, it is necessary to model the motor's dynamic behaviour.The modeling and analysis of these systems can be either numerical or analytical.In numerical techniques, various algorithms are used to produce quantitative results.These methods are accurate, powerful and easy to use, however, they fail to provide the design insight needed to understand the behaviour of the converters i.e. they are not ideal for tasks such as stability analysis or controller design [3][4][5][6][7].On the other hand, analytical techniques provide clear expressions representing the operation of dc motors and thus can help designers gain a better understanding of the performance of the system.
Analytically, there are two main approaches to model and analyze switching systems.
The first approach is the state space averaging model proposed by Middlebrook and Ćuk [5], also known as the continuous-time averaging model.This method is widely used to analyze and design circuits at low operating frequencies.
The sampled data model or discrete-time iterative map, first reported by Lee [8] and Verghese [10], is another approach to the modeling of switching systems.In this analysis, the switching continuous system is replaced by a discrete system that describes the states of the system at the switching frequency.

Continuous-time averaging approach
The conventional way of modelling any switching system is to take an average of the state variables over one switching cycle.In this model, the system is linearised around a steadystate operation to yield a linear model.This makes it possible to use a Laplace transform domain analysis which is useful for control theory.In addition this method approximates the original circuits by continuous-time systems which make it easier for the designer to design a feedback controller.
In the DC motor, the state space averaged model can be derived by taking the average of the states during ON and OFF intervals.This can be made by multiplying Equation (3) by dT (the time when the switch is ON) and Equation ( 7) by T d (the time when the switch is OFF) and then summing the result to take the average over one cycle.
This yields the following time-varying continuous system:  Designing a suitable feedback controller using any standard linear technique such as the Nyquist criterion or root-locus method.
The simulation results of the open loop DC motor modeled using the averaging approach are shown in Figure 2 and Figure 3. From the figures it can be seen that the state space averaging model smoothed out the ripple and captures only the basic dynamics.This approach discards the switching details and hence it fails to predict the fast-scale dynamics that are higher than the switching frequency of the system and can capture only the slow-scale dynamics that are much slower than the switching frequency of the system [8,12].Therefore a more comprehensive method of analysis is needed to obtain all necessary information about the system's behaviour.

Discrete-time iterative map approach
In order to explore the nonlinear phenomena such as sub-harmonic and chaos which may appear across a wide range of frequencies, an alternative modeling approach which is based on deriving the exact discrete model must be used [3,16].The aim of this approach is to transform a continuous-time system into a discrete-time system by sampling its states at chosen instants.This modeling technique can predict the fast-scale dynamics as well as the slow-scale dynamics.Different types of maps have been proposed in the literature for switched systems, but the most common maps are the stroboscopic map where the states are sampled at the beginning of the switching cycle [11] and the impact map where the states are sampled at the switching instants [12].
In this paper, the stroboscopic map is used to obtain the discrete model i.e. the system states are periodically sampled at fixed time instants t= nT as shown in   (15) Equation (11) describes the dynamic of the DC motor however, it is complex and transcendental in form; therefore, there is no easy method for analyzing the behaviour of the system [14][15][16][17][18][19][20].
The output waveforms of the open loop DC motor modeled by the sampled data approach are shown in Figure 5 and Figure 6

Figure 1 Figure 1
Figure 1 shows the schematic diagram of the open loop controlled DC motor.It contains of a source of voltage E, a chopper circuit which consists of a diode D and a switch S controlled by a pulse-width modulated signal.The required output voltage of the chopper circuit is achieved by setting the switch's duty cycle d., and an equivalent circuit of DC motor which is connected to the chopper circuit.The simple DC motor has the following a physical parameters: d  is the ratio of t off to T and related to the duty ratio as d  = 1-d.The model of the converter becomes a nonlinear time invariant model.Perturbing the above system around the steady-state point system around that point, a small signal model can be obtained.The small signal model is very useful for: Studying the dynamic behaviour of the system.

Figure 2 Figure 3
Figure 2 Inductor current waveform at input voltage of 20V and d =0.5;

Figure 4 .Figure 4
Figure 4 Discrete model of a periodic dynamical system

ФФ
ON and Ф OFF are the state transition matrices during ON and OFF intervals, given by: ON and Ф OFF can be calculated by the series:

Figure 5
Figure 5Inductor current waveform at input voltage of 20V and