Small-signal AC modeling Technique of Buck Converter with DSP ...

2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia

Small-signal AC modeling Technique of Buck Converter with DSP Based Proportional-IntegralDerivative (PID) Controller Mohammad Faridun Naim Tajuddin School of Electrical Systems Engineering Universiti Malaysia Perlis Kangar, Malaysia

Nasrudin Abdul Rahim Department of Electrical Engineering University of Malaya Kuala Lumpur, Malaysia

Abstract—Control applications of switched mode power supplies have been widely explored. The main objective of research and development (R&D) in this field has always been to find the simplest method to analyze and model the DC/DC converter and the most suitable control method to be implemented in various DC/DC converter topologies. This paper presents a simple and systematic approach to the design of a practical Digital Signal Processing (DSP) based Proportional-Integral-Derivative (PID) voltage controller for Buck converters. A simple and easy-to-follow design procedure is demonstrated. Experimental results are presented to illustrate the design procedure.

for buck converter, to bridge the gap between the control principle and circuit implementation. Moreover, the approach is presented in a manner involving only a standard digital PID buck converter model and some simple guided steps and design equations. This allows designer to skip through laborious preliminary derivations when performing the controller’s design. II.

The inductor current and capacitor voltage of smallsignal AC model of the Buck converter are given in (1) and (2).

Keywords—Buck converter; continuous conduction mode; PID controller

̂

I. INTRODUCTION One of the challenges in controlling the DC/DC converter is compensation of the converter’s nonlinear lightly damped dynamics, which is a function of load parameters. Advances in signal processing technology have spurred research in new control techniques to improve converter control. Traditionally, regulation of the output voltage of DC/DC converters has been achieved through the use of analog control techniques. An analog control system operates in real time and can have a high bandwidth. In addition, the voltage resolution for an analog system is theoretically infinite [1]. However, an analog system is usually composed of discrete hardware that must be modified to change controller gains or algorithms. In addition, the implementation of advanced control algorithms requires an excessive number of components. On the other hand, the complexity of a digital control system is contained mostly in software. Once it is working properly, software is more consistent and reliable than a complex analog system. Digital processors also have the advantage of being less susceptible to aging and environmental or parameter variations. In addition, the processor can monitor the system, perform self diagnostics and tests, and communicate status to a display or a host computer [2-3]. This paper provides the introduction of AC small signal analysis and presents the programming algorithm in implementing PID controller using DSP so that engineers may conveniently adopt it. Nevertheless, this paper is still principally focused on introducing a simple approach that is easily applicable in the development of a PID controller

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THE BASIC MODELLING APPROACH

904

(1) (2)

The AC output voltage variation can be expressed as the superposition of the terms arising from those two inputs (3) The first term represents the control to output transfer function while the second term represents the line to output transfer function. The transfers function of and can be defined as (4) and (5) An algebraic approach to drive these transfer functions is by taking the Laplace transform and letting the initial condition be zero. We can find the general transfer function in (6). The line to output transfer function contains DC gain , while the control to output transfer function has . The salient features of the line to output a DC gain

2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia

and control to output transfer functions of the basic Buck converter are summarized in Table I.

B. PID Controller Design Steps The standard procedure of designing the controller for the Buck converter is focuses on how to overcome the weak points of the converter such as poor phase margin at high frequency and low DC gain at low frequency. Therefore, at this point and with the aid of (6) and Table I, it is possible to design the Buck converter with standard procedure focusing on these weak points and improving these parameters by addition of PID control. Our discussion here starts with the assumption that the converter’s parameters are known and they are given in Table II. These parameters are calculated on the basis that the converter is to be operated in CCM for 20 V to 30 V input supply at 0.5 A to 2 A. The maximum peak to peak ripple voltage is 50 mV.

1 1

(6) 1 1

TABLE I. SILENT FEATURES OF THE SMALL-SIGNAL CCM TRANSFER FUNCTION FOR DC/DC BUCK CONVERTER [5].

TABLE II. SPECIFICATIONS OF DC/DC BUCK CONVERTER Parameter name D Buck Converter Silent Features

Symbol

Nominal Value / Spec. no 30V

Input voltage

20V

1

12V

Output voltage

√

(Quality factor)

Output Capacitor

220μF

Inductor

2.12mH

Load resistance

18 Ω

Switching

III.

20kHz

frequency

STANDARD DESIGN PROCEDURE

A standard PID buck converter module is proposed in this section, along with a step by step design procedure for practical implementation. A design example is provided for illustration.

Power MOSFET

M

IRF 530

Power diode

D

MUR 820

1) Step 1: The dc gain, corner frequency and -factor of the converter are calculated by equating like coefficients of (6) and Table I.

A. Standard Buck Converter with PID controller Model Fig. 1 shows the proposed buck converter with DSP based PID controller. Note that in the DSP based system, the voltage sensing, compensator and pulse width modulator functions are still present but appear under different names. Similar to conventional schemes, the feedback sensing network for is provided by the voltage divider circuit.

24 0.5 1 2

(7)

12

24 2 2.12 220 233.05

2 √

18

220 2.12

5.7985

(8)

(9)

2) Step 2: Substituting the control to output part of (5) yields the equation for the loop gain, . 1 1

(10)

3) Step 3: Equate 0 in (10) and with unity gain compensator, the uncompensated DC gain of the loop system is:

Figure 1. Buck Converter with DSP Based PID Controller

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2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia

1

24

1

7) Step 7: The low-frequency compensator gain of 15.27 dB at 4 kHz is obtained as follow:

(11)

27.6042

1

4) Step 4: With the help of Matlab®, bode plot of the overall uncompensated system is shown in Fig. 2. The overall system has a poor phase margin of 2.06 degree at a frequency equal 1.16 kHz.

(15)

4.2267

8) Step 8: To further improve the low frequency regulation, an addition of inverted zero (lag compensator) is introduced. Hence a PID compensator is built up, and to let the integral control not affecting the phase margin, we choose to be one tenth of . 1 10

(16)

400

9) Step 9: This lead compensator will increase the gain at frequency below 100 Hz. From previous calculations the PID control will have the following transfer function (17), and bode plot of the Buck converter with PID control is shown in Fig. 3. 1

(17)

1

Figure 2. Bode diagram of uncompensated loop gain, T s

It can be seen that the phase of , is approximately equal to 520 over the frequency range of 2 kHz to 20 kHz. Hence the variations in the component value, which cause the crossover frequency to deviate somewhat from 4 kHz, should have little impact on the phase margin.

5) Step 5: A common practice in the design of switch mode power supplies is to choose the crossover between 10 - 20% of the switching frequency, frequency of the power converter [6]. Here a of 20% of the switching crossover frequency, frequency is selected. 20 100

1

(12)

4

6) Step 6: The gain at is equal to -15.27 dB. To obtain a unity gain at , the compensator should have a gain of 15.27 dB at . This compensator should improve the phase margin. Therefore a PD compensator is needed. The phase margin of 52 degree (assumption) is chosen based on the common practice for the switch mode power supplies where the phase margin, should be somewhere around 45° and 70° [6]. The lead compensator will have the following pole and zero:

4

1 1

sin 52 sin 52

1.3773

(13)

4

1 1

sin 52 sin 52

11.617

(14)

Figure 3. Bode diagram of compensated loop gain,

,

C. Implementation of Digital Based PID Controller Implementing digital PID controller requires the difference equation that represents PID controller in analog (17) to be converted into digital form. There are a number of methods available for this task [7]. The following difference equation can be produced from the discrete time transfer function [8].

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2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia

Interrupt Subroutine Starts

(18)

Delay 3 μs

1

Wait for the end of ADC conversion

In this equation, is the new duty cycle calculated sample, and is the error of the from the sample. The error is calculated as , where is the converted digital value of sample, and Ref is the digital value corresponding the to the desired output voltage. The second term in the equation is the sum of the errors, and 1 is the difference between the error of the sample and the error of the 1 sample. Equation (18) is implemented on the TMS320F2812 evaluation module, which produces the gating signal applied to the MOSFET of the buck converter. The output voltage of the buck converter is measured using a TDS220 oscilloscope during the steady state and a startup transient. The complete real-time controller implementation in DSP is interrupt driven. The PWM module loads the new value of the duty cycle at the beginning of every switching cycle. All calculations regarding the duty cycle are implemented in the ADC interrupt service routine. The implementation of digital PID controller in DSP is represented by the flow charts shown in Figs. 4 and 5.

e(k) = Vref - ADC(k)

up(k) = Kp * e(k)

ui(k) = u(k-1) + Ki *[e(k-1) + e(k)] Yes

ud(k) = Kd *[ e(k-1) - e(k)]

ui(k) > uimax No

ui(k) = uimax

Yes

ui(k) < uimax

ui(k) = -uimax

No

u(k) = up(k) + ui(k) + ud(k)

Yes

u(k) > umax No

u(k) = umax

u(k) < -umax

Yes

No

u(k) = -umax

u(k) = up(k) + ui(k) + ud(k)

Yes

u(k) < 40% N o

u(k) = 40%

u(k) > 60%

No

Yes u(k) = 60%

Duty cycle = u(k)

Update the controller history e(k-1) = e(k) u(k-1) = u(k)

Clear the interrupt mask

Return

Figure 5. Flowchart of interrupt service routine implementing PID control algorithm on DSP

IV. SIMULINK® SIMULATION MODEL The Simulink® software is used to validate the PID controller in previous section. This program is used to simulate the steady-state and transient behaviours of the circuit. The simulation results can then be used to compare with the experimental results later. Fig. 6 shows the buck converter with PID controller (closed loop). The experimental component values shown in Table I are used for the simulation.

Figure 4. Flowchart of PID controller main algorithm implemented on DSP

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2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia

waveforms obtained for input voltage of 24 V under nominal load condition. Channel 1 shows the output voltage while Channel 2 gives the duty cycle. It can be seen that when the input voltage changes, the controller adjusts the duty cycle to a new value, leading to a new steady-state duty cycle that provides zero steady-state error. Table 3 shows the converter output voltage (experimental) for the three different input voltages. The deviation and the percentage of deviation of the output voltage and the current are also shown. The output voltage deviation is calculated from the required voltage of 12V.

1>

Figure 6. Simulink® model of DC/DC Buck Converter with PID controller 2> Inductor current

2

iL Sw itching signal 1

inductor current g

m

d

s

1) Ch 1: 2) Ch 2:

Output current

Inductor, L = 2.12 mH i + -

Mosfet

i

Io

-

output current

5 Volt 25 us 2 Volt 25 us

+

Figure 7. Waveforms of Output Voltage & Current at steady-state operation under nominal operating conditions V = 24V and R 12Ω.

1 Output voltage

DC Input voltage Diode Capacitor, C =220uF

LoadLoad

+

Vo

v -

Output voltage1

Output Voltage

Figure 6(a). Simulink® model for internal structure of the DC/DC Buck Converter with PID controller given in Fig. 3 1>

1 Output voltage

Controller output

1

>

v oltage error

12 Reference voltage

Sw itching Signal

PID controller

Relational Ope rator

2>

PWM generator

2 1) Ch 1:

2) Ch 2:

Figure 6(b). Simulink® model for internal structure of the PID

5 Volt 25 us 2 Volt 25 us

Figure 8. Output voltage and Duty cycle of Buck converter with PID controller for input voltage of 20V. New duty cycle is 60%.

V. EXPERIMENTAL RESULTS This section evaluates the performance of the PID controller for Buck converter designed using the procedure described in previous section.

TABLE III. SILEOUTPUT VOLTAGE VARIATIONS FOR CLOSED LOOP OPERATION USING PID CONTROLLER

A. Steady-State Performance Fig. 7 displays the output voltage and current waveforms of the converter operating with the input voltage of 24V at half load. From the simulation results shown in Fig. 7, it can be observed that the system with PID controller produces the required output voltage and current. These results are in well agreement with the theoretical values. To analyse the performance of the PID controller in steady-state condition, the input voltage is varied from 20 to 30V to see whether the desired output voltage of 12V can be maintained. Fig. 8 displays the steady-state

908

Input Voltage,

Output Voltage,

20

Deviation ∆

Deviation %∆

11.997

-0.003

0.025

24

11.999

-0.001

0.008

30

12.002

+0.002

0.017

2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia

REFERENCES

B. Transient Response Characteristics of Buck Converter with PID Controller The evaluation of the transient response is done to analyse the stability of the model and its speed response. In order to evaluate the transient response, the behaviour of the model during the start up transient is analysed. The start up transient response of the buck converter for input voltage of 24 V at nominal load is shown in Fig. 9. From the experimental results, it is seen that the settling time is about 2ms with very little overshoot.

[1] [2] [3]

[4]

[5] [6] [7]

[8]

1>

1) Ch 1:

5 Volt 5 ms

Figure 9. Start up transient response of Buck Converter

VI. CONCLUSION A detailed analysis of the design principle of a Buck converter with PID controller is presented. The design takes into consideration the practical aspects of the converter. The overall system is tested where the stability can be assessed using phase margin test. Compensator is added in the forward path of feedback loop to shape the loop gain such that the desired performance is obtained. To facilitate implementation, a standard Buck converter with PID controller module is introduced. Design guidelines are provided in a simple step by step manner. The overall model is implemented using Matlab/Simulink®. A prototype of the simulated model is built-up. The experimental results are presented to verify the converter design and procedure. The paper shows that with standard mathematical procedure and a simple effective control technique, it is possible to design a switch power supply with good performance.

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