Pulse-Width Modulated Amplifier for DC Servo System and its Matlab Simulation

: This paper introduces a mathematical model for Pulse-Width Modulated Amplifier for DC Servo Motor. The relationship between pulse-width modulated (PWM) signal and reference rotation speed is specified, and a general model of motor represented by transfer function is also put forward. When the input signal changes, the rotation speed of the servo motor will change accordingly. By changing zeros and poles, transient performance of this system is discussed in detail, and optimal ranges of the parameters is recommended at the end of discussion.


INTRODUCTION
DC servo system plays an important role in the daily life.For example, they can be applied to spark machines, manipulators and other accurate machines (Tal, J. et al, 1999).Besides, reduction box can be added to DC servo system for the purpose of high accuracy and torque (Dai et al, 2014).Since DC servo system has the advantage of fast dynamic responses and strong anti-interference ability, it has been applied broadly in many fields.Meanwhile, pulse-width modulation's principle is that the average value of voltage (and current) fed to the load is controlled by turning the switch between supply and load on and off at a fast rate.The longer the switch is on compared to the off periods, the higher the total power supplied to the load.The main advantage of pulse width modulation is that power loss in the switching devices is very low.When a switch is off there is practically no current, and when it is on and power is being transferred to the load, there is almost no voltage drop across the switch.Power loss, being the product of voltage and current, is thus in both cases close to zero.PWM also works well with digital controls, which, because of their on/off nature, can easily set the needed duty cycle (M.Barr, 2003).
In normal motor speed (RPM) control system, resistors are used to control the speed (RPM).However, it may generate much heat, which people don't like to see.Instead, we can use pulse width modulation to achieve the goal of controlling the motor speed (RPM) [1].Specifically, the motor speed (RPM) can be controlled by short pulses, and these pulses vary in duration (duty cycle) to change the speed (RPM) of the motor.The longer the pulses, the faster the motor turns, and vice versa (Milosavljevic et al, 2013).

CONVERSION FROM MOTOR SPEED TO PWM SIGNAL
The following block diagram Fig. (1) depicts how the input (desired input speed) relates to the output (speed of the DC servo motor): The reference rotation speed is proportional to the effective value of a rectangular pulse, so we can use a series of rectangular voltage pulse with certain duty cycle to model it.The relationship is: where stands for the rotation speed of the reference motor, stands for the effective voltage of the impulse rectangular train, and d stands for the duty cycle.In this context, we set the coefficient = 1 for convenience.
Once we receive the rectangular voltage pulse, its effective voltage can be used as the input of the DC servo motor.Considering the duty cycle is not constant [2][3][4] (because the rotation speed of the reference motor can change from time to time), we can model the turning points using combination of unit step inputs, which showed in Fig. (2).
The magenta line is the effective input voltage of the DC servo motor, and we hope the output of the DC servo motor can follow this line (because the proportionality between the rotation speed and the voltage is 1, as declared in the previous part (Table 1).
When only considering the turning point at 5 second, as shown in Fig. (2), the function can be written as: (3) And it can be written in unit step function : Similarly, when only considering the turning point at 10 second, the function can be written as: (5)  And it can be written by using combinations of : (6) In summary, the model can be established as:

THE TRANSFER FUNCTION
For a typical DC servo motor model, the dynamic characteristic equation for closed-loop servo mechanism is equation ( 9): According to Eq (9), the rotation speed is controlled by , and is affected by .
Using Laplace transform, the transfer functions of the system can be written as:   .By adding more and more poles and zeros, we can obtain higher order systems.

Modeling of the Input
As what we have discussed in the previous paragraphs, the input can be modeled as:

Modeling of the Motor
When considering add poles or zeros to Eq(10): with Eq(16), the relationship between the input and the output can be written as:

SIMULATION AND TRANSIENT PERFORMANCE ANALYSIS Parameter Settings
In this paper, we do transient performance analysis based on the following premises: 1. .

2.
, as declared in the previous part.
4. The input is the unit step function.
5. The settling time is defined as the time required for the system to settle within 2% of the amplitude.

Transient Performance Analysis
There are three criteria when considering estimate and design a control system.Firstly, the system must be stable.Secondly, the control should be accurate.Thirdly, the response should be quick-acting.What's more, there are four criteria in transient performance measures which are rise time, peak time, settling time and percent overshoot.In the below Fig.   2 contains more results.
From the table above, we find that: the second-order system response with zero differs a lot from the systems without zero.In the system without zero point, [5][6][7] the rising time only depends from system damping and oscillating angular frequency .However, in the system response with zero points, the rising time is also related to the real component.Which can be shown in figure that the more zero point is closed to the imaginary axis, the less the rising time would be.Also, we can infer from that the more r is, the more oscillatory the system would be.The percentage overshoot also relates to the position of zero point.The less zero is, the more φ would be, which would make decrease.
Another variable that affects the output signal is the poles.Now we set the zeros back to 0, and set poles to -300, the figure will be: Fig ( 5a) shows the condition in red lines when we set the pole to -300; In Fig ( 5b), the green line shows the condition when we the pole to -800.More results are shown in Table 3.   From our analysis, we infer that the output signal will be most stable when the zero is set to -828～-869 and pole is set to about -300.This conclusion is calculated according to the following steps: According to data above, we can get zero point, pole point, the values of settling time and percentage overshoot.If the settling time and percentage overshoot are smaller, the control system is more stable.Therefore, based on simulation, we need to find a best point for getting the most stable system.For convenience, we narrow the coordinates of zeros and poles 100 times smaller, and enlarge the coordinates of the settling time 1000 times.
According to the diagram, we can get a conclusion is that the position of poles is set to the -3 to -4 if the settling time is the shortest [8,9].
Secondly, according to the data of the percentage overshoot, poles and zeros, we can get the relationship between percentage overshoot, poles and zeros, which is threedimensional diagram and Contour Plots and Color Mapping Figs.(7)(8)(9).
C g and C mThe electromotive constant and torque constant, which has something with motor structure Φ Air-gap flux L T load torque

Fig. ( 3 ).
Fig. (3).Block diagram of the transfer function.We can expand the order of the transfer function by adding poles and zeros.When adding zeros, we need to multiply to the transfer function, and the zero is .When adding poles, we need to multiply to the transfer function, and the pole is speed of the reference motor, is the effective voltage of the impulse rectangular train, is the maximum voltage of the impulse rectangular train, d is the duty cycle.Also, (4) -Fig.(7), x-axis represents time while y-axis represents RPM.

First, according to
the data of the settling time, poles and zeros, we can obtain the relationship of the settling time, poles and zeros.Using the method of interpolation, we can draw the three dimensional diagram and Contour Plots and Color Mapping.The figure is shown in Fig. (6).

Table 1 . Definition of symbols. Symbols Definitions
..The rotor velocity of DC servo motor