ANALYSIS, CALCULATION AND SIMULATION OF THE
SWITCHED-MODE POWER SUPPLY WITH ANALOG
AND DIGITAL CONTROLLERS
Denisenko D. Y., Denisenko M. E., Finaev V. I., Ignatyev V. V., Ivanov Y. I.,
Pushnina A. A. and Spiridonov O. B.
Southern Federal University, Bolshaya Sadovaya str., Rostov-on-Don, Russia E-Mail: [email protected]
ABSTRACT
The article describes a switched-mode power supply in the form of the automatic control system. The block diagram of the automatic control system is adduced. Input and output signals of the automatic control system and the concept of "stability system" are defined. The functional flow block diagram of a buck converter comprising a proportional-integral-derivative controller and pulse-width modulator are adduced. A pulse-width modulator sets the mode of operation of the switched-mode power supply. Transfer functions and models of analog and discrete automatic control systems are considered. The problem of synthesis the automatic control system model consists in choosing the controller parameters. The analysis of the switched-mode power supply power unit is realized. The systems of difference equations to describe the change in voltage and current in the circuit depending on the input voltage are founded. Solutions of the systems of equations with respect to the input and output voltages are founded and the transfer function of the switched-mode power supply is determined. The studies of the switched-switched-mode power supply with an analog controller are done. The form of the transfer function, the formula for determining the frequency of the pole, the attenuation pole and gain at zero frequency are founded. The modeling in the program MicroCap-11 is done. The form of the transfer function and the error - the difference between the predetermined voltage and the output voltage of the stabilization system is studied. Similar studies are made for switched-mode power supply with digital control. The proposed method of calculation allows selecting the parameters of controllers and providing the required dynamic characteristics of the switched-mode power supply.
Keywords: switched-mode power supply, analog controller, digital controller, automatic control system, transfer characteristic, buck converter.
INTRODUCTION
Controlled electronic switches, inductors and capacitors are used in the design of the circuit of the electric energy statistical transformers’ output stage (power units) [1-7]. These transformers have dynamic characteristics. Dynamic characteristics are described in the form of the system of differential equations and discrete transfer functions [8-13]. Differential equations are defined in the time domain and the discrete transfer functions are defined in the frequency domain. Transformers (in the design) should have the desired (predetermined) dynamic characteristics.
The output stages of transformers include the voltage and (or) current feedback, to solve this problem [18, 14]. Feedback loops are implemented using discrete components or specialized chips - controllers of the switched-mode power supply [15, 16, 22], and also on the base of the microcontrollers and digital signal processors [17, 19].Any specific implementation of the transformer’s power unit control circuit has its own features in designing the switched-mode power supply.
The switched-mode power supply can be regarded as a closed automatic control system with analog or digital controller. In the article the analysis of the power unit of the switched-mode power supply is regarded on the example of the buck converter. In the article the calculation and selection of the analog and digital controller’s parameters are also given.
MODELOFTHESWITCHED-MODEPOWER SUPPLY
The switched-mode power supply can be regarded as an automatic control system. Figure-1 shows the block diagram of the automatic control system. The diagram consists of the control object, the controller and the comparator [14].
g(t) e(t) u(t) y(t)
ОУ
_ +
Figure-1. A block diagram of the automatic control system.
Reference signal g(t) and controlled (output) value y(t) are fed to the comparator inputs. The error signal e(t) is fed from the comparator output to controller. The error signal is processed by the controller. Control action u(t) for the control object appears on the controller output.
system, if g(t) is predetermined as a program, then it’s a system of numerical program control.
Figure-2 shows a functional diagram of a buck converter. Figure-2 introduced the notation: VR is a voltage reference defining the reference signal g(t); Σ is a comparator implemented in the form of the error amplifier; is a controller (for example, a proportional -integral-derivative (PID)), the PWM is a pulse width modulator, PU is a control object (a power unit of the converter).
Figure-2. A functional diagram of the buck converter.
The transfer functions are determined by the dynamic characteristics of the automatic control system. Uninterrupted (analog) automatic control system are described in the time domain by functions of uninterrupted time (differential equations) and by Laplace transfer functions with variable "s" in frequency domain. Discrete (digital and pulse) automatic control systems are described by discrete time functions (difference equations) in the time domain and by Laplace transfer functions with variable "z" in frequency domain.
If the power unit of the converter operates in a pulsed mode, the converter is described by a discrete transfer function [1, 20, 24-26].
The controller’s transfer function can be either uninterrupted or discrete, depending on the implementation of the controller.
Structural model of uninterrupted automatic control system is shown in Figure-3 a. Figure-3 b shows a structural model of discrete automatic control system.
G(s) E(s) U(s) Y(s)
Fp(s)
_ +
Foy(s)
(a)
G(z) E(z) U(p) Y(z)
F
p(z)
_+
F
oy(z)
(b)
Figure-3. Mathematical models of uninterrupted and discrete automatic control systems.
The transfer function of uninterrupted automatic control system is determined by the formula
(1)
The transfer function of discrete automatic control system is determined by the formula
(2)
In the theory of automatic control it’s supposed that the coefficients of the control object’s transfer function Fco are constants. Consequently, the nature of the transient response the entire system is defined by the coefficients of the controller transfer function Fc. The problem of the system synthesis (the coefficients of the transfer functions (1) and (2)) is reduced to the selection of controller parameters, i.e., the order of controller transfer function and its coefficients.
ANALYSISOFTHESWITCHED-MODEPOWER SUPPLYPOWERUNIT
The task of finding the transfer function of the power unit (PU) is set. The state of the diagram in Figure-2 in two phases [1] is considered. In the phase "a" a switch S1 is closed and switch S2 is open. The PU circuit in the phase "a" is shown in Figure-4.
Uвых(n) Rн
E(n) +
R
Uc(n-1)
а
Figure-4. The PU circuit in the phase "a".
The system of difference equations that describe the state of the circuit in the phase "a" takes the form for the circuit in Figure-4:
(3)
The first equation (3) defines the value of the
current in the capacitor C at the n-th period, in the Uвых
Rн R1
R2 +
-E
+ S1
S2
И
R
-+
И +
-Σ
)
(
)
(
1
)
(
)
(
)
(
)
(
)
(
s
F
s
F
s
F
s
F
s
G
s
Y
s
W
oy p
oy p
.
)
(
)
(
1
)
(
)
(
)
(
)
(
)
(
z
F
z
F
z
F
z
F
z
G
z
Y
z
W
oy p
oy p
). ( ) 1 ( ) (
); ( ) (
); 1 ( 1 1 ) 1 ( ) ( ) (
n U n
U n U
n I C n U
n U R R R
n U n E n I
a C C
C
C a
C
C H
C C
C
phase "a". UC(n-1) is the voltage at the capacitor C at the end of the previous switching period of switches S1and S2 (see the Figure-2).
The second equation of (3) defines the voltage increment on the capacitor C during the time at the n-th period, in the phase "a".
The third equation of the system (3) defines the voltage on the capacitor C at the n-th period, by the end of phase "a".
In phase "b" the switch S1 is open and the switch S2 is closed. PU circuit in the phase "b" is shown in Figure 5.
Uвых(n) Rн
E(n) +
R
Uc(n)а
Figure-5. PU circuit in the phase "b".
The system of the difference equations describing the state of the circuit in the phase "b" takes the following form:
(4)
There are the similar notations in the system of equations (4) as in the system of equations (3).
In formulas (3) and (4) there are the following notations: is the circuit location time in the phase "a", (
T-) is the circuit location time in the phase "b", n is the number of ticks (n = 1,2,3, ...), T is the switching period of the keys.
Let’s find the solution of systems of equations (3) and (4) relative to the input and output voltages. The decision will give the following result
(5)
If the discrete Laplace transform to the last equation is applied, the transfer function will be defined as:
(6)
THESWITCHED-MODEPOWERSUPPLYWITH ANANALOGCONTROLLER
Let’s define the automatic control system’s transfer function with the analog controller (look Figure-3 a and formula (1)). Let’s convert the discrete transfer function of the power unit and get its continuous equivalent. Then let’s carry out a change of variable in expression (6) using the Maclaurin expansion
𝑧− = 𝑒−𝑠𝑇≈ 1 − 𝑠𝑇 +𝑠2𝑇2− ⋯ (7)
And neglecting the expansion of the second-order terms and more, the form of the transfer function is founded:
(8)
Where is the mark-to-space ratio,
is the time constant of the control object, is the transfer coefficient of the control object in DC.
Control action (input) for the power unit in DC / DC converter is the mark-to-space ratio q. The transfer function of the power unit on this parameter is finding from the formula (8)
(9)
The voltage from the controller output is the input signal for the PWM. The output parameter is the mark-to-space ratio. PWM circuit realization can be considered as the proportional gain, and its transfer function is defined as the conversion factor
(10)
Where UPWM is an amplitude of PWM voltage.
From automatic control theory it’s known that the order of the controller transfer function should be equal to the order of the control object transfer function [10, 14,
). ( ) ( ) ( ) ( ); ( ) ( ); ( 1 1 ) ( n U n U n U n U n I C T n U n U R R n I C a C C ВЫХ C C a C H C
R R C T n U n U n U RC n E H c c c 1 1 ) 1 ( ) 1 ( ) ( ) ( R R C T z z RC z E z U z F H ых 1 1 1 ) ( ) ( ) ( 1 1 , 1 1 1 1 1 1 ) ( ) ( ) ( oy oy oy H ых s qK R R C s T RC s E s U s F q =
T
у RR = C R R у R K =R R
21]. If this condition is fulfilled, then all the coefficients of the automatic control system transfer function can be selected (defined) in an arbitrary manner.
Thus the transfer function (9) of the power unit has the first order; the controller can be created with PI control law. The transfer function takes the following form:
(11)
Where K is the transfer coefficient of the controller’s proportional part, is the time constant of the controller’s integral part.
Substituting formula (9) - (11) into the formula (1), the transfer function in an analog form of the pulsed buck voltage converter is founded:
(12)
where is the transfer coefficient of the
resistive divider in the circuit in Figure-2, is the conversion factor.
Pole’s frequency is defined from the transfer function by the formula
(13)
The attenuation pole is defined by the formula
(14)
The transfer coefficient at zero frequency is calculated by the formula
(15)
Figure-6 shows a simulation circuit of the converter, which is made in the program MicroCap-11 [19]. The circuit has a feedback on voltage.
Figure-6. The buck converter with an analog controller.
In the circuit in Figure-6 transfer coefficient of the resistor divider Kd is chosen equal to one, therefore resistors R1 and R2aren’t taken into account in the circuit in Fig. 6 (see. Figure-2). Frequency of the keys’ switching (in the simulation was selected to be 10 kHz) is defined in the PWM unit. Controller’s integrator is implemented on an operational amplifier, capacitor C2 and resistor R3. Controller’s proportional part amplifier transfer coefficient Kn. Inverse keys’ switching occurs when given the output voltage from the PWM on the "+" (control input of key S1) and "-" (input of key S2).
Simulation of the circuit is made in Figure-6. The transfer characteristics are obtained (see Figure-7) of the circuit (output Out) and the same characteristic for modeling the transfer function (12). The transfer function (12) was defined by unit of setting a uninterrupted transfer function LF (output Out1)of the stabilization system.
Figure-7. Transfer characteristics of the circuit Figure-6.
The lower graph in Figure-7shows the changing of the error. The error is the difference between the defined value and the voltage obtained at the output of the stabilization system. The graph shows that with time the error approaches zero.
Analysis of transfer characteristics’ graphs of the simulated circuit and the transfer function shows a slight deviation of one characteristic from another. This is primarily due to the fact that the approximate equation (7) was used in the transition from the discrete transfer function to the uninterrupted transfer function.
,
s
1
K
)
s
(
F
p
K K K ,K K K K 1 1 s s
1 K K 1 K K sK
) s ( W
p oy
d oy d
oy oy
2
p oy oy oy
oy
2 1
2 d
R
R
R
K
PWM U
E K
p oy
d oy p
K
K
K
oy p
d oy
d oy p
K
K
K
K
K
K
K
1
d
.
1
d
Analysis of the expression (13) - (15) shows that the overshooting of the overshoot (the transient response nature) in the circuit can be changed by changing the parameters of a proportional-integral controller.
If the resistive voltage divider is set on output, according to formula (15), the output voltage can be defined by selecting a reference voltage source (V8 in Figure-6) with the voltage (Vin in the circuit) and the dividing coefficient of the voltage divider.
THESWITCHED-MODEPOWERSUPPLYWITHA DIGITALCONTROLLER
To define the transfer function of the switched-mode power supply with digital controller the transfer function of the power unit (6) is transformed to the following form
(16)
Or with respect to the control action to the form
(17)
Substituting the formula (7) into the formula (11), the transfer function of a digital proportional-integral controller is found
(18)
Sampling frequency (controller operation) and frequency of keys’ switching in the power unit of the converter may not be the same in general. Frequency of keys’ switching is denoted by fs=1/T. The clock frequency of the digital controller denoted by fT=1/T. The transfer function of the control object, taking into account adopted notations, takes the form
(19)
Controller transfer function takes the form
(20)
Substituting formula (19), (20) and (10) into the formula (2), the transfer function of the buck converter with the digital controller is found:
(21)
Where:
,
Figure-8 shows a simulation circuit of the converter with a feedback loop and a digital voltage regulator.
Figure-8. The buck converter with the digital controller.
Parameters of the circuit with digital controller for simulation are the same as in the simulation of the circuit with an analog controller.
Key switching frequency in the power unit of the converter and the clock frequency of the digital controller were chosen equal to fs= fT=10kHz.
Standard element "T" is taken for the implementation of the delay elementz-1in the circuit. Standard element "T" in the program Micro Cap is the element of the digital filter.
Figure-9 shows the results of simulation the buck converter with a digital controller. As the result of simulation are obtained transient characteristics of the circuit (output Out) and a similar characteristic obtained by modeling a discrete transfer function, which was set by unit Z (output Out_D) in accordance with the expression (21).
Changes of the transient response and the transfer function, obtained by circuit simulation, are quite close as with the analog controller. The nature of the changes is the same with changes the results of buck converter with an analog controller simulation (look Figure-6).
T z z K q z E z U z F oy oy ых
) 1 ( ) ( ) ( ) ( 1 1 T z z K z E z q z U z F oy oy ых oy ) 1 ( ) ( ) ( ) ( ) ( 1 1 oy pz
T
K
z
F
)
1
(
)
(
1
S oy oy ых oy f z z K z E z q z U z F
) 1 ( ) ( ) ( ) ( )( 1 1
oy T p z f K z F
) 1 ( 1 ) ( 1 , 1 1 1 1 1 ) ( 1 2 oy 1 oy oy A B z f z f K K K z f f K K f K K K z W oy s oy s p oy s T oy s oy s oy s d f f K K K K B
1 1 2 oy . 1 11 oy oy
Figure-9. Transfer functions of the circuit in Figure-8.
Differences of the buck converter with digital controller circuit: the power unit of the converter is defined by the transfer function of the pulse system; the controller is defined y the transfer function of the digital system. Both transfer functions are shown as discrete functions.
If on the circuit in Figure-2 resistor R is replaced by an inductor L, the buck converter circuit with output low pass second order filter is obtained [1, 23]. It’s necessary to use the controller with the second order transfer function [14] to provide the necessary dynamic characteristics of the switched-mode power supply when this replacement is done. The method of calculation and choosing the parameters of the circuit doesn’t change.
CONCLUSIONS
For the power unit of the converter and the switched-mode power supply circuit with analog and digital controller is received the same method of calculation with the same analytical formulas. Application of the calculation method allows you to select the parameters of controllers and provide the required dynamic characteristics of the switched-mode power supply. The problem is solved by determining the coefficients of the transfer functions.
Comparison and analysis of the transient characteristics for modeling the diagrams and transfer functions shows that the founded analytical formulas with sufficient accuracy describe the nature of the transient response in the switched-mode power supply.
Thus the coefficients of the transfer function depends on the converter load, the transient response nature of the entire system is also dependents on the load.
The materials of articles are prepared in compliance with the plan of scientific-research work 213.01-07-2014/02 PCHVG "Development of multi criteria optimization methods of the hybrid adaptive intellectual regulates parameters by the hard-formalized objects" fulfillment.
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