Predictive Digital Average Current Control In Dc Dc Converters

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Predictive Digital Average Current Control In Dc Dc Converters

Byadmin ianuarie 1, 2024

PREDICTIVE DIGITAL AVERAGE CURRENT CONTROL IN DC-DC CONVERTERS

Contents

Introduction

Chapter 1. PWM modulation techniques

General considerations

PWM modulation techniques

Trailing-edge modulation

Leading-edge modulation

Trailing triangle modulation

Leading triangle modulation

Chapter 2. Predictive Average Current Control

2.1. Trailing average control

2.2. Leading average control

2.3. Trailing triangle average control

2.4. Leading triangle average control

Chapter 3. Predictive digital average current control employing trailing-edge modulation

3.1. Average current control employing trailing-edge modulation (control law)

3.2. Stability analysis

3.3. Analysis of discrete space-state model for TA control

3.4. Verification by simulating the circuit

Chapter 4. Predictive digital average current control employing leading-edge modulation

4.1. Average current control employing leading-edge modulation (control law)

4.2. Stability analysis

4.3. Analysis of discrete space-state model for LA control

4.4. Verification by simulating the circuit

Chapter 5. Predictive digital average current control employing trailing triangle modulation

5.1. Average current control employing trailing triangle modulation (control law)

5.2. Stability analysis

5.3. Analysis of discrete space-state model for TTA control

5.4. Verification by simulating the circuit

Chapter 6. Predictive digital average current control employing leading triangle modulation..

6.1. Average current control employing leading triangle modulation (control law)

6.2. Stability analysis

6.3. Analysis of discrete space-state model for LTA control

6.4. Verification by simulating the circuit

Chapter 7. Conclusions

Bibliography

Annexes

Introduction

The present paper describes the predictive digital average current control in dc-dc converters using following types of modulation: leading-edge, trailing-edge, leading triangle edge, trailing-triangle edge. The techniques employed in the design include a combination the reference voltage, with a carrier signal, usually a saw-tooth signal. The control laws are derived for the boost converter.

The study is focused on the improvement of the digital control performance using the predictive technique. In one switching period the duty cycle for the next switching is calculated based on the sensed and observed state and input/output information, such that the error of the controlled variable is cancelled out or minimized in the next cycle or in the next several cycles.

For each type of modulation it is described the principle of control and is determined the control law. After this, the theoretical considerations are verified using both MATLAB and finally with the CASPOC circuit simulation package. Some simulation aspects regarding the new developed dedicated blocks are also presented.

The paper is organized as follows. Pulse width modulation techniques are first introduced. In section II, predictive average current control are presented, following leading average control, trailing average control,

Chapter 1

PWM modulation techniques

General considerations

The purpose of this chapter is to review the current stage of the predictive digital current controls and to establish the main interests for the author for bringing his contributions.

Digital controls [1], [2], [3], [4], [5], [6], [7] offer advantages in terms of lower sensitivity to variation of parameters and the improvement of the performances using more sophisticated control schemes. The microprocessors and DSP are already widespread in motor controls of single phase or three phase applications. Thanks to the progress in microprocessors and DSP technology, especially on increasing processing capacity and lower costs, digital control becomes more viable even for the high frequency converters, of lower and medium power [8].

In some applications based on digital controls, the sampling and the processing of information bring some delays which can compromise the performance of the control. The improvement of the performance of digital control consists on using predictive techniques [9], [10], [11], [12]. These techniques are based on the next principle: in a switching period the duty cycle for the next switching period is calculated considering the current duty cycle, the current stage and the current input/output dimensions so the error of the controlled dimension to be minimized or eliminated in the next switching period or in the next few periods. The predictive techniques began also to be used in the controls of DC-DC converters, based on the applications of the control of three phase systems with low frequency.

Today, there are 3 predictive control in current (valley, average, peak) which can be applied on the main converters (buck, boost, buck-boost) [10]. It is known that the problem of instability also exists in the case of predictive digital controls. Avoiding the instability can be achieved by the right correlation of a modulation technique with the predictive control used. There are known 4 types [10] of modulation techniques: trailing-edge, leading-edge, trailing triangle, leading triangle).Associating each modulation technique with each control method, results 12 types of predictive digital current control: trailing valley, trailing peak, trailing average, leading valley, leading peak, leading average, trailing triangle valley, trailing triangle peak, trailing triangle average, leading triangle valley, leading triangle peak, leading triangle average.

PWM modulation techniques

DC-DC converters are power electronic equipment which including semiconductor devices working in switching mode. The switching process is a dynamic process by which the semiconductor devices are tilted between two stable stages: the total conduction state and total lock state. Using control signals moments in which the switching process should be launched are fixed.

Pulse width modulation (PWM) technique is the main technique used in the generation of control signals in power electronics. The leading and falling time equal to zero, in the case of an ideal PWM, represents a desideratum which provide a perfect control modality of semiconductor devices. The presence of leading and trailing edges very sharp ensure the input, respectively the output, from conduction of semiconductors devices as fast as possible, reducing the time of switching of transition and such the losses of associating switching. Taking into account the considerations regarding to the electromagnetic interference (EMI), in practical applications may be imposed a superior limit of the input/output speed from conduction of electromagnetic devices, resulting a trailing and leading time different from zero which, in most of cases, can be ignored in the analyze of PWM processes and signals.

Pulse width modulation can be realized in many ways. Pulse frequency is one of the most important parameters when the type of PWM modulation is defined. Pulse frequency can be constant or variable. In the analyzed application the PWM frequency is fixed, for the switching frequency of the converter to be fixed. The switching function q(t), may be obtained by comparing a control signal, vc(t), with a carrier signal, vcarrier(t), and can be mathematical written as:

(1.1)

where sgn(.) represent sign function (signum).

Depending on the type of the carrier signal, may be defined 4 types of modulation: trailing edge modulation, leading edge modulation, trailing triangle modulation, leading triangle modulation.

Trailing edge modulation

Trailing edge modulation – T is realized by using a rising saw-tooth carrier type, as shown in fig.1.1. The switching function q(t) is obtained by comparing the control voltage, vc(t), with the carrier signal, vsaw(t). In trailing edge modulation technique, the transistor is bring in the state of conduction at the beginning of each switching period Ts and is locked after DTs time. The value of duty cycle D can be modified by increasing or decreasing the value of the reference voltage vc(t). The transistor remains in the locked state for the rest of switching period, namely (1-D)Ts. The leading edges of the switching function of transistor are equidistant appearing after the same time, while trailing edges can appear sooner or later in a switching period, depending on the value of the reference voltage. In other words, the leading edge appear on multiples of switching period Ts, while the trailing edge is modulated in rhythm of control signal. A switching time contains two switches, a bringing in conduction and a lock.

Fig.1.1. Trailing edge modulation – T – stationary state

Leading edge modulation

Leading edge modulation- L is realized by using a falling saw-tooth carrier type, as shown in fig.1.2. The triangular signal q(t) is obtained by comparing the control voltage, vc(t), with a falling saw-tooth carrier signal, vsaw(t). In leading edge modulation technique, the transistor is locked at the beginning of each switching period Ts and is brought in the state of conduction after (1-D)Ts time. The transistor remains in conduction for the rest of switching period, namely DTs. The trailing edges of the switching function of transistor are equidistant appearing after the same time, while leading edges can appear sooner or later in a switching period, depending on the value of the reference voltage. In conclusion, the leading edge of the triangular signal is modulated in rhythm of the control signal. A switching time contains, like in the case of T modulation, two switches, a lock and a bringing in conduction.

Fig.1.2. Leading edge modulation – L – stationary state

Trailing triangle modulation

Trailing triangle modulation – TT is realized using a triangular carrier which starts leading. In case of this modulation technique switching function q(t) is obtained by comparing the reference voltage vc(t) with a triangular carrier signal vtri(t), (fig.1.3), so being replaced the saw-tooth carrier signal type. In case of this type of modulation, during of a switching period Ts, the transistor has two conduction intervals, at the beginning and at the ending of the period. The transistor leads from the beginning to the moment (D/2)Ts locked and is brought back in conduction at the moment (1-D/2)Ts. In conclusion, during a period, the transistor leads (D/2)Ts from the time, (1-D/2)Ts is locked and again it leads (D/2)Ts. The two edges, of the switching function, from the inside of a period have variable location in time, in depending on the value of control voltage vc(t). A period contains two switches, a lock and a bringing in conduction.

Fig.1.3. Trailing triangle modulation – TT – stationary state

Leading triangle modulation

Leading triangle modulation – LT is realized using a triangular carrier which starts trailing, (fig.1.4.). During a switching period, Ts, the transistor is locked twice. The transistor enters in conduction at the moment [(1-D)/2]Ts and is locked at the moment [(1+D)/2]Ts. In other words, during a switching time, the transistor is locked [(1-D)/2]Ts time, dTs time leads and it is locked again [(1-D)/2]Ts time. In this case the two edges of switching function, from the inside of a period can also have variable location in time, depending on the value of control voltage vc(t). Like the TT modulation, a period has two switches, a bringing in conduction and a lock, with variable location in time.

Fig.1.4. Trailing triangle modulation – LT – stationary state

Chapter 2

Predictive Average Current Control

Predictive digital current controls are the result of the combinations of PWM modulation techniques with the control current methods. They are known, [10], the following digital current control methods:

Peak current control – P;

Valley current control – V;

Average current control – A;

By combe ending of the period. The transistor leads from the beginning to the moment (D/2)Ts locked and is brought back in conduction at the moment (1-D/2)Ts. In conclusion, during a period, the transistor leads (D/2)Ts from the time, (1-D/2)Ts is locked and again it leads (D/2)Ts. The two edges, of the switching function, from the inside of a period have variable location in time, in depending on the value of control voltage vc(t). A period contains two switches, a lock and a bringing in conduction.

Fig.1.3. Trailing triangle modulation – TT – stationary state

Leading triangle modulation

Leading triangle modulation – LT is realized using a triangular carrier which starts trailing, (fig.1.4.). During a switching period, Ts, the transistor is locked twice. The transistor enters in conduction at the moment [(1-D)/2]Ts and is locked at the moment [(1+D)/2]Ts. In other words, during a switching time, the transistor is locked [(1-D)/2]Ts time, dTs time leads and it is locked again [(1-D)/2]Ts time. In this case the two edges of switching function, from the inside of a period can also have variable location in time, depending on the value of control voltage vc(t). Like the TT modulation, a period has two switches, a bringing in conduction and a lock, with variable location in time.

Fig.1.4. Trailing triangle modulation – LT – stationary state

Chapter 2

Predictive Average Current Control

Predictive digital current controls are the result of the combinations of PWM modulation techniques with the control current methods. They are known, [10], the following digital current control methods:

Peak current control – P;

Valley current control – V;

Average current control – A;

By combining the 4 PWM modulation techniques with the 3 digital current control methods it will result 12 types of predictive digital current control. Each of the 12 types of predictive control is based on inductive current and input/output current samples, obtaining the value of the duty cycle for the next switching period. The value of controlled sample (point) target (peak, valley or average current) should be equal to the value of the reference current. The sampling is done at equidistant time intervals, equals to switching period, Ts.

Further will be presented the stability domain indicated by the literature, [19], for the predictive digital average current control, showing the variable linear form with two slopes of the inductive current of a generic converter working in CCM. The values corresponding to a switching period will be noted with the period index. By notations of type i[n], i[n+1], i[n+2] will be understood the value of the sampled current at the beginning of the switching period i[n], i[n+1], respectively i[n+2].

2.1. Trailing average control

Trailing average control (TA) represent the result of the correlation of the trialing edge modulation (T) with average current control method (A). The purpose of this control type is to prognosis the duty cycle for the next switching period, so as to ensure in the next period an average current equal to the reference current Iref, fig.2.1. TA control is done to be stable for a value of the duty cycle smaller than 0.5, while for a value higher than 0.5 is done to be instable.

Fig.2.1. Inductive current for TA control. [19]

2.2. Leading average control

Leading average control (LA) is obtained by combining of the leading edge modulation (L) with average current control method (A). The duty cycle proposed for the next switching period should assure in the next period an average current equal to the reference current Iref, fig 2.2. LA control should be instable for a value of duty cycle smaller than 0.5, while for a value higher than 0.5 should be stable.

Fig.2.2. Inductive current for LA control. [19]

2.3. Trailing triangle average control

Trailing triangle average control (TTA) results by associating the trailing triangle modulation technique (TA) with the average current method (A). The purpose of the duty cycle for the next switching period is to ensure the average current to be equal to the reference current Iref, fig.2.3. TTA control should be unconditionally stable.

Fig.2.3. Inductive current for TTA control. [19]

2.4. Leading triangle average control

Leading triangle average control (LTA) result from the combine of the leading triangle modulation (LT) with the average current control (A). The purpose of the duty cycle is to ensure the average current to be equal to the reference current Iref, fig 2.4. LTA control should be unconditionally stable.

Fig.2.4.Inductive current for LTA control. [19]

Chapter 3

Predictive digital average current control employing trailing-edge modulation

The purpose of this chapter is to study the predictive digital average current control named trailing average control (TA) in DC-DC converters. This control uses the trailing-edge modulation (T) technique in correlation with the average current control (average – A). The study is focused on finding the recurrence relation of the duty cycle, after that the stability analysis is done. It is demonstrated that the TA control is stable for every value of duty cycle, simplifying the design problems. The analysis is done in a general manner, independently from the converters topology, following that the obtained results can be applied on converters (buck, boost, buck-boost, etc.). Using a boost converter, the theoretical results are confirmed both on the space state model using Matlab program, and on the development and simulation of circuit in Caspoc simulator program.

3.1. Average current control employing trailing-edge modulation (control law)

The sample value at the beginning of the switching period will be notated with i[n] because the current through the coil is sampled at the beginning of each switching period. The purpose is to obtain a relation for the duty cycle from the n+1 period, dn+1, in function of the duty cycle dn, from the n switching period, so the average current in n+1 period to be equal to the reference current value Iref. The target point from the average current at the end of the n+1 period, iave n+1 in fig.3.3, is evaluated in function of i[n]. This point should be equal to the reference current Iref.

Fig.3.3. Inductive current in dynamic regime for TA control.

Also it is known that in stationary state the duty cycle D is done by the relation:

(3.1)

The slope ratio is:

(3.2)

The two relation can be demonstrated by expressing the inductive current variations in each topological state and equalizing the results.m1 and m2 slopes depend on the input and output voltages. These voltages should be sampled. For example, in a boost converter the inductive current slope values are:

(3.3)

(3.4)

Firstly the sampled value, i[n+1], from the inductive current will be computed in function of the i[n] value from the previous switching period and the dn duty cycle, determined by the slopes m1 and m2 which are known.

(3.5)

(3.5) relation could be extended to the next switching period, making n -> n+1:

(3.6)

The average current target point through the coil at the end of the n+1 period is equal to:

(3.7)

Using (3.5) and (3.6) relations, the average inductive current point can be expressed as follows:

(3.8)

Imposing, it results:

(3.9)

From (2.9) equation, the value of the duty cycle is:

(3.10)

This represents the general recurrence relation of the duty cycle in TA control case. The relation can be applied on any type of converter substituting the m1 and m2 slopes corresponding to the converter topology. For example, the duty cycle for the boost converter is:

(3.11)

A modality to verify (3.10) relation is to obtain the duty cycle in a stationary state depending on M1 and M2 slopes.

(3.12)

(3.13)

(3.14)

where: I0 – trailing current in stationary state; D – duty cycle in stationary state

Substituting i[n], dn, dn+1 and Iref from (3.12), (3.13) and (3.15) in (3.10) relation and doing the computing results: . This expression represent the (3.1) relation, which confirms the truthfulness of (3.10) relation.

3.2. Stability analysis

In this subchapter it is presented the study of the stability of TA control using the geometric model, [13]. It is known that the variation of the inductive current in a DC-DC converter is exponentially amortized, with big time constants reported to the switching period. In the geometrical model approach the approximation is done considering the exponential to be linear, and the inductive current slopes are considered the same in both stationary state and in small perturbations presence.

Fig.3.4. Inductive current in TA control in presence of perturbations.

In fig.3.4., it is presented the inductive current waveform in both situations. The continuous line represents the current variation when the converter is in stationary state, and with dotted line is presented the mode in which a little perturbation appeared at the beginning of the switching period n is spreading in current, taking into account the compulsion of this type of control regarding to the average current value. Until t=nTs, the converter had worked in stationary state and at t=nTs the perturbation appears. As follows, the duty cycle from the n switching period will be equal with the duty cycle D, from stationary state because before the n period there were no perturbations.

(3.15)

The perturbations are equals to the difference between the disrupted signal and the value of the signal in stationary state. Starting with the n+1 period the duty cycle is disrupted. In the representation the size of this perturbation is exaggerated for the clarity of illustration. ∆i[n], ∆i[n+1] and ∆i[n+2] represent the perturbations at the beginning of n, n+1 and n+2 switching period. Because of this small perturbation, the converter will work almost in stationary state, such that m1 and m2 slopes will be considered unchanged and equals to their values in stationary state, M1 and M2. The purpose of this analysis is finding a relation between the perturbation ∆i[n+2] from the beginning of n+2 switching period, and the perturbation ∆i[n] from the beginning of the n switching period. Based in this recurrence relation the stability can be easily determined analyzing if and in which conditions the ∆i[n+2] perturbation converges to zero when n→∞. From fig.2.4. the ∆i[n] perturbation is :

(3.16)

Because of the equality of the current slopes in stationary state and perturbation state in switching period n and the duty cycle equal to the duty cycle from stationary state, it results that the perturbation value at the beginning of n+1 period is unchanged, so:

(3.17)

From the (2.9) relation, dn+1 represents the first duty cycle after the period n which takes into account that the current shape is disrupted. It results that the perturbation at the beginning of the n+2 period will be different from the perturbation at the beginning of n period. From (2.16) we have:

(3.18)

From what it was said previously and taking into account the relation (3.18), it results:

(3.19)

Based on the relation (3.18), the recurrence relation (3.10) becomes:

(3.20)

Saying that the duty cycle from the n+1 period is ∆dn+1, it is obvious:

(3.21)

Substituting dn from relation (3.15), dn+1 from (3.21), Iref from (2.14), D from (2.1) and i[n] from (2.18), the relation (2.20), after the computing becomes:

(3.22)

The sampled inductive current at the beginning of n+2 switching period is:

(3.23)

By using simple geometrical considerations, and with relations (3.15) and (3.16), the current i[n+2] value is:

(3.24)

From relations (3.23) and (3.24) it is obtained:

(3.25)

Next, by substituting dn+1 from the relations (3.21) and (3.25), it results:

(3.26)

Substituting in (3.26) relation ∆dn+1 from (3.22) and D from (2.1), it is found:

(3.27)

Using the ratio M1/M2 given by the relation (3.2), the right member of the (3.27) can be rewritten depending on the duty cycle in stationary state:

(3.28)

Relation (3.28) represents the recurrence relation. Changing n→n+2, it results:

(3.29)

Replacing ∆i[n+2] from (2.28) in (2.29), the value of ∆i[n+4] is:

(3.30)

By induction it is shown that, after 2k switching periods, the perturbation becomes:

(3.31)

When k→∞, the perturbation converges to 0, if and only if the absolute value of –D/2-D, is smaller than 1. The stability condition becomes:

(3.32)

How 0<D<1 it is easy to observe that:

(3.33)

It follows the stability condition to be equivalent with:

(3.34)

Solving the inequality it is obtained D<1, which is always true. In conclusion, in the first phase, TA control is unconditionally stable (without oscillations) for the entire domain of the duty cycle.

3.3. Analysis of discrete space-state model for TA control

With the purpose of validation of the previous theoretical considerations looking the TA control, firstly it will be done a check using the state-space model. These will be studied on a boost converter, represented in fig.3.5.

Fig.3.5. Boost converter

Converter parameters has the next values:

Vg=10V; L=500µH; C=100µF; RL=1mΩ; fs=40kHz (3.35)

State vector is chosen as:

(3.36)

In CCM working, the converter can be modeled considering next equations:

(3.37)

where:

(3.38)

(3.39)

(3.40)

It is known that the discrete model of the converter, in a trailing-edge modulation type conditions, it is described by the equation:

(3.41)

where:

(3.42)

(3.43)

(3.44)

(3.45)

The converter will be simulated by using the (3.41) equation, with the duty cycle, obtained after the computing of the predictive control given by the (3.11) relation. It will be chosen an arbitrary duty cycle and the simulation will run to pass the initial transient regime. If the functionality is stable, then the results in stationary state will be a sequence of constant discrete values. How the instability appears at D<0.5 or D>0.5, there are chosen two values for the reference current: a value which impose the functionality of the system at D<0.5, respectively a value which force the functionality at D>0.5.

The results of the simulation for a reference current Iref=2.5A, (D<0.5) are presented in fig.3.6. and fig.3.7. while in fig.3.8. are detailed the last ten switching periods of the inductive current for the same value of the reference current. The functionality is stable, the duty cycle becomes constant after the initial transient regime and the inductive current reaches o typical periodical shape with a period equal to the switching period.

The results of the simulation for the reference current Iref=11A, (D>0.5) are represented in fig.3.9. and fig.3.10. for the duty cycle respectively for the inductive current. The last ten switching periods of the inductive current have the same value with the reference current and are detailed in fig.3.11. In this case the functionality is also stable.

The simulation was realized in MATLAB program, the source code can be found in Annex 1. In fig.3.7. respectively fig.3.10 are represented the samples of the inductive current, namely the trailing points.

Fig.3.6. Duty cycle depending on time for Iref=2.5A, (D<0.5)

Fig.3.7. Sampled inductive current depending on time for Iref=2.5A, (D<0.5)

Fig.3.8. Ten periods of the inductive current after the stationary state install for Iref=2.5A, (D<0.5)

Fig.3.9. Duty cycle depending on time for Iref=11A, (D>0.5)

Fig.3.10. Sampled inductive current depending on time for Iref=11A, (D>0.5)

Fig.3.11. Ten periods of the inductive current after the stationary state install for Iref=11A, (D>0.5)

In conclusion, the obtained results after by the mathematical simulation of the discrete state-space model confirm that the predictive average current control (TA) employing the trailing-edge modulation (T) is stable for any value of the duty cycle.

3.4. Verification by simulating the circuit

For the simulation of the boost converter with TA control, it was used the CASPOC package [14]. General scheme of simulation is presented in fig.3.12.

Fig.3.12.CASPOC scheme for the simulation of the boost converter using TA control

The average inductive current should follow the reference Iref. The necessary duty cycle for the next switching period is calculated using the inductive current samples, input voltage and output voltage from the current switching period, (2.10).

For the simulation scheme was necessary the implementation of the next blocks: sampling block and front memory (“edge”-E) (SPH_E), the shift register (SHIFT_REG), the slopes computation block (M1_M2_BOOST), the block for the computation of the recurrence relation for the duty cycle (TA), the PWM modulator on trailing edge (PWM_TE). Also it was used dedicated block from the internes libraries of CASPOC program: PWM modulator (PWM_FS), used for the clock of the entire circuit, the limiter block (LIM) which limits the value of the duty cycle in the [0.1, 0.9] interval, the probe for the inductive current reading (CP), two probes for reading the input/output voltages (VP).

The logic functionality of the simulation scheme is:

-the input voltage, output voltage and the inductive current are sampled on the leading edge at a frequency of 40KHz; the value of each sample is memorized in the SPH_E block.

-the sampled values are introduced in shift registers for the computation of M1 and M2 slopes, and also in the computation of the recurrence relation of the duty cycle.

-the samples from the input and output voltage are used in the computation of slopes in M1_M2_BOOST block regarding the (3.3) and (3.4) relations.

-the recurrence relation of the duty cycle is implemented using the (3.10) relation with the help of the TA block. At the output of this block the value of the duty cycle dn+1 is found.

-the dn+1 duty cycle is limited using the LIM block in the 0.1 and 0.9 interval.

-the transistor is controlled with the frequency of 40KHz using the PWM_TE block, having at the input the value of the duty cycle dn+1 obtained from the recurrence relation.

-for the implementation of the recurrence relation, dn+1 will be delay with the help of a SHIFT_REG block becoming dn, the good value for the TA block, namely for the computation of the dn+1 duty cycle.

The purpose is to simulate the boost converter controlled TA, both in the case of a duty cycle bigger than 0.5, and in the case of a duty cycle smaller than 0.5. The first results of the simulation for the duty cycle, inductive current and output voltage are presented in fig.3.13, fig.3.14, fig.3.15 and fig.3.16.

Fig.3.13. The duty cycle depending on time, for Iref=2.5A (D<0.5)

The simulation was done with a value of 2.5A for the reference current, which forces the functionality with a duty cycle smaller than 0.5. It can be observed the obtaining of a stable functionality.

Fig.3.14. Inductive current, for Iref=2.5A (D<0.5).

Fig.3.15. presents a detail of 12 periods of the inductive current. At each 25µs (fs=40KHz), the transistor drives, the inductive current raise with the m1 slope for dTS. After the transistor is locked, the current decrease with m2 slope through the end of the switching period. The prescribed value of 2.5A for the reference current it is at the middle of the iL waveform, which is confirmed in the simulation.

Fig.3.15.Inductive current detailed, for Iref=2.5A (D<0.5).

Fig.3.16.Output voltage, for Iref=2.5A (D<0.5).

The reference current put on 11A implies a functionality of a duty cycle bigger than 0.5. In this conditions, the simulations results are presented in fig.3.17, fig.3.18, fig.3.19 and fig.3.20. It is also obtained a stable functionality of the converter.

Fig.3.17. The duty cycle depending on time, for Iref=11A (D>0.5)

Fig.3.18. Inductive current, for Iref=11A (D>0.5).

Like in the previous case, for Iref=2.5A, it wants to be highlight the theoretical results, by a detailed of 12 periods of the inductive current shown in fig.3.19. In this case, it can also be observed that at each 25µs (fs=40KHz), the transistor drives, and the inductive current rise with the m1 slope for dTS, after that the transistor is locked, the current decrease with the –m2 slope until the end of the switching period. The 11A value of reference current is exactly the average value of iL.

Fig.3.19..Inductive current detailed, for Iref=11A (D>0.5).

Fig.3.20.Output voltage, for Iref=11A (D>0.5).

Chapter 4

Predictive digital average current control employing leading-edge modulation

The purpose of this chapter is to investigate the predictive average current control in DC-DC converters using a leading-edge modulation type. This control named leading average control (LA) is the result of the correlation between the leading-edge modulation (L) with the average current control method (average-A). Like in the previous chapter, the purpose is to find the recurrence relation of the duty cycle, which is exactly the law for this type of control. Analyzing the stability, it is demonstrated that this type of control is stable for every duty cycle value. The study is done on a boost converter. The theoretical results are confirmed on the state-space model using the Matlab program, and also by developing and simulating the circuit in Caspoc program.

4.1. Average current control employing leading-edge modulation (control equation)

The duty cycle is obtained for the duty cycle from the n+1 period depending on the dn duty cycle from the n switching period, so the average current from the n+1 period to be equal to the reference current Iref. To obtain the control law regarding the duty cycle, the average point from the inductive current at the end of the n+1 period will be evaluated depending on i[n]. At the end, this point will be equal with the value of the reference current. Firstly, it will be computed the sampled value i[n+1] from the inductive current depending on the value of the sample i[n] from the previous switching period and the dn duty cycle determined by the m1 and m2 slopes.

Fig.4.1. Inductive current in dynamic regime for LA control

Like in the previous chapter, TA case, the duty cycle, D, and the slopes, m1 and m2, have the same values:

(4.1) (4.2) (4.3)

The inductive current shape can be written as follow:

(4.4)

For the next switching period, n → n+1, the relation becomes:

(4.5)

At the end of the n+1 period, the average inductive current point is:

(4.6)

As in the TA case, by expressing iave n+1 and imposing that iave n+1 = Iref, the formula for the duty cycle is:

(4.7)

The relation represents the law control of the predictive duty cycle în LA control case. The recurrence relation for the predicitve duty cycle can be applied on every converter substitutuing the m1 and m2 slopes. Replacing the values of the slopes, the duty cycle for the boost converter becomes:

(4.8)

The relation can be verified obtaing the duty cycle în stationary state depending on M1 and M2 slopes. Using the follow relations, it can be rewritten:

(4.9)

(4.10)

(4.11)

Where Io is the leading current in stationary state. Replacing i[n], dn+1, dn and Iref , it will result. This expression is the duty cycle in stationary state, which confirms the (4.8) relation.

4.2. Stability analysis for the LA control

Fig. 4.2.. Inductive current in LA control in presence of perturbations.

For the LA control the stability condition will be demonstrated using the linearity of the function. Starting from the duty cycle relation (4.8) and also from the relation between two consecutives samples in the inductive current, which is the same for all the control methods:

(4.12)

the perturbations of the inductive current and of the duty cycle will be highlighted as follows:

– represents the duty cycle in stationary state

– the perturbation of the duty cycle

– – perturbation of the inductive current

from (4.8) and (4.12) it will results:

(4.13)

(4.14)

From (4.14) it is clear that:

(4.15)

The stability condition will be obtained after the computing of, disrupted. Vo and Vg are replaced depending on the M1 and M2 slopes.

(4.16)

In the end, after the replacements and computation, the equation will be:

(4.17)

By noting the left member with Q (z) and the right one with P (z), will be:

(4.18)

After the computation, the equation roots z1 and z2 will meet the conditions:

– 0<z1*z2<1 and 0<z1+z2<1 if the roots have real values;

– z1*z2=ρ2 => ρ<1 if the roots are complex numbers.

After this conditions, it results that D>0 which means that the LA control is unconditionally stable.

4.3. Analysis of discrete space-state model for LA control

With the purpose of validation of the previous theoretical considerations looking the LA control, firstly it will be done a check using the state-space model. These will be studied on a boost converter, represented in fig.4.3.

Fig.4.3. Boost converter

Converter parameters has the next values:

Vg=10V; L=500µH; C=100µF; RL=1mΩ; fs=40kHz (4.19)

State vector is chosen as:

(4.20)

In CCM working, the converter can be modeled considering next equations:

(4.21)

where:

(4.22)

(4.23)

(4.24)

It is known that the discrete model of the converter, in a leading-edge modulation type conditions, it is described by the equation:

(4.25)

where:

(4.26)

(4.27)

(4.28)

(4.29)

The converter will be simulated by using the (4.25) equation, with the duty cycle, obtained after the computing of the predictive control given by the (4.8) relation. Like in the TA control case, the purpose of the simulation is to show if the functionality of the control is stable. Two values will be chosen for the duty cycle, D<0.5 and D>0.5, and two values for the reference current, 2.5A and 11A.

The results of the simulation for a reference current Iref=2.5A, (D<0.5) are presented in fig.4.4 and fig.4.5 while in fig.4.6 are detailed the last ten switching periods of the inductive current for the same value of the reference current. The functionality is stable, the duty cycle becomes constant after the initial transient regime and the inductive current reaches o typical periodical shape with a period equal to the switching period.

The results of the simulation for the reference current Iref=11A, (D>0.5) are represented in fig.4.7 and fig.4.8 for the duty cycle respectively for the inductive current. The last ten switching periods of the inductive current have the same value with the reference current and are detailed in fig.4.9. In this case the functionality is also stable.

The simulation was realized in MATLAB program, the source code can be found in Annex 2. In fig.4.5 respectively fig.4.7 are represented the samples of the inductive current, namely the trailing points.

Fig.4.4. Duty cycle depending on time for Iref=2.5A, (D<0.5)

Fig.4.5. Sampled inductive current depending on time for Iref=2.5A, (D<0.5)

Fig.4.6. Ten periods of the inductive current after the stationary state install for Iref=2.5A, (D<0.5)

Fig.4.7. Duty cycle depending on time for Iref=11A, (D>0.5)

Fig.4.8. Sampled inductive current depending on time for Iref=11A, (D>0.5)

Fig.4.9. Ten periods of the inductive current after the stationary state install for Iref=11A, (D>0.5)

In conclusion, the obtained results after by the mathematical simulation of the discrete state-space model confirm that the predictive average current control (LA) employing the leading-edge modulation (T) is stable for any value of the duty cycle.

3.4. Verification by simulating the circuit4

For the simulation of the boost converter with LA control, it was used the CASPOC package [14]. General scheme of simulation is presented in fig.4.10.

Fig.4.10.CASPOC scheme for the simulation of the boost converter using LA control

The average inductive current should follow the reference Iref. The necessary duty cycle for the next switching period is calculated using the inductive current samples, input voltage and output voltage from the current switching period, (4.8).

For the simulation scheme was necessary the implementation of the next blocks: sampling block and front memory (“edge”-E) (SPH_E), the shift register (SHIFT_REG), the slopes computation block (M1_M2_BOOST), the block for the computation of the recurrence relation for the duty cycle (LA), the PWM modulator on trailing edge (PWM_LE). Also it was used dedicated block from the internes libraries of CASPOC program: PWM modulator (PWM_FS), used for the clock of the entire circuit, the limiter block (LIM) which limits the value of the duty cycle in the [0.1, 0.9] interval, the probe for the inductive current reading (CP), two probes for reading the input/output voltages (VP).

The logic of the functionality is the same like in the case of the TA control.

The purpose is to simulate the boost converter controlled LA, both in the case of a duty cycle bigger than 0.5, and in the case of a duty cycle smaller than 0.5. The first results of the simulation for the duty cycle, inductive current and output voltage are presented in fig.4.11, fig.4.12, fig.4.13 and fig.4.14.

Fig.4.11. The duty cycle depending on time, for Iref=2.5A (D<0.5)

The simulation was done with a value of 2.5A for the reference current, which forces the functionality with a duty cycle smaller than 0.5. It can be observed the obtaining of a stable functionality.

Fig.4.12. Inductive current, for Iref=2.5A (D<0.5).

Fig.4.13. presents a detail of 12 periods of the inductive current. At each 25µs (fs=40KHz), the transistor drives, the inductive current raise with the m1 slope for dTS. After the transistor is locked, the current decrease with m2 slope through the end of the switching period. The prescribed value of 2.5A for the reference current it is at the middle of the iL waveform, which is confirmed in the simulation.

Fig.4.13.Inductive current detailed, for Iref=2.5A (D<0.5).

Fig.4.14.Output voltage, for Iref=2.5A (D<0.5).

The reference current put on 11A implies a functionality of a duty cycle bigger than 0.5. In this conditions, the simulations results are presented in fig.3.17, fig.3.18, fig.3.19 and fig.3.20. It is also obtained a stable functionality of the converter.

Fig.4.15. The duty cycle depending on time, for Iref=11A (D>0.5)

Fig.4.16. Inductive current, for Iref=11A (D>0.5).

Like in the previous case, for Iref=2.5A, it wants to be highlight the theoretical results, by a detailed of 12 periods of the inductive current shown in fig.3.19. In this case, it can also be observed that at each 25µs (fs=40KHz), the transistor drives, and the inductive current rise with the m1 slope for dTS, after that the transistor is locked, the current decrease with the –m2 slope until the end of the switching period. The 11A value of reference current is exactly the average value of iL.

Fig.4.17..Inductive current detailed, for Iref=11A (D>0.5).

Fig.4.18.Output voltage, for Iref=11A (D>0.5).

Chapter 5

Predictive digital average current control employing trailing triangle modulation

The purpose of this chapter is to study the predictive digital average current control named trailing average control (TA) in DC-DC converters. This control uses the trailing triangle modulation (TT) technique in correlation with the average current control (average – A). The study is focused on finding the recurrence relation of the duty cycle, after that the stability analysis is done. It is demonstrated that the TTA control is stable for every value of duty cycle, simplifying the design problems. The analysis is done in a general manner, independently from the converters topology, following that the obtained results can be applied on converters (buck, boost, buck-boost, etc.). Using a boost converter, the theoretical results are confirmed both on the space state model using Matlab program, and on the development and simulation of circuit in Caspoc simulator program.

5.1. Average current control employing trailing triangle modulation (control law)

The sample value at the beginning of the switching period will be notated with i[n] because the current through the coil is sampled at the beginning of each switching period. The purpose is to obtain a relation for the duty cycle from the n+1 period, dn+1, in function of the duty cycle dn, from the n switching period, so the average current in n+1 period to be equal to the reference current value Iref. The target point from the average current at the end of the n+1 period, iave n+1 in fig.4.1, is evaluated in function of i[n]. This point should be equal to the reference current Iref.

Fig.5.1. Inductive current in dynamic regime for TTA control.

Also it is known that in stationary state the duty cycle D and the slope ratio are the following:

(5.1)

(5.2)

The two relation can be demonstrated by expressing the inductive current variations in each topological state and equalizing the results.m1 and m2 slopes depend on the input and output voltages. These voltages should be sampled. For example, in a boost converter the inductive current slope values are:

(5.3)

(5.4)

Firstly the sampled value, i[n+1], from the inductive current will be computed depending on the i[n] value from the previous switching period and the dn duty cycle, determined by the slopes m1 and m2 which are known.

(5.5)

(5.5) relation could be extended to the next switching period, making n -> n+1:

(5.6)

The average current target point through the coil at the end of the n+1 period is equal to:
(5.7)

Using (5.5) and (5.6) relations, the average inductive current point can be expressed as follows:
(5.8)

Imposing, it results:
(5.9)

From (5.9) equation, the value of the duty cycle is:
(5.10)

This represents the general recurrence relation of the duty cycle in TTA control case. The relation can be applied on any type of converter substituting the m1 and m2 slopes corresponding to the converter topology. For example, the duty cycle for the boost converter is:

(5.11)

5.2. Stability analysis

In this subchapter it is presented the study of the stability of TTA control using the geometric model, [37]. It is known that the variation of the inductive current in a DC-DC converter is exponentially amortized, with big time constants reported to the switching period. In the geometrical model approach the approximation is done considering the exponential to be linear, and the inductive current slopes are considered the same in both stationary state and in small perturbations presence.

Fig.5.2. Inductive current in TTA control in presence of perturbations.

Following the same steps like in the TA control case, namely using the geometrical model, it will be demonstrated that the TTA control is unconditionally stable. Starting from the (5.10) relation, and making the same replacements, dn, dn+1, Iref , D and i[n], the relation will become:

(5.12)

Using the inductive current from the beginning of the n+2 switching period and making the substitutions and it will be obtained:

(5.13)

From the previous relations it results:

(5.14)

Substituting in (5.14) relation and results:

(5.15)

Using the ratio , the right member of the (5.15) can be rewritten depending on the duty cycle in stationary state:

(5.16)

This relation represents the recurrence relation. Changing n→n+2, the result will be also 0. For any switching period n+2k the result will be 0.

(5.17)

It results that the TTA control is unconditionally stable (without oscillations) for the entire domain of the duty cycle.

5.3. Analysis of discrete space-state model for TTA control

With the purpose of validation of the previous theoretical considerations looking the TTA control, firstly it will be done a check using the state-space model. These will be studied on a boost converter, represented in fig.4.3.

Fig.5.3. Boost converter

Converter parameters has the next values:

Vg=10V; L=500µH; C=100µF; RL=1mΩ; fs=40kHz (5.18)

State vector is chosen as:

(5.19)

In CCM working, the converter can be modeled considering next equations:

(5.20)

where:

(5.21)

(5.22)

(5.23)

It is known that the discrete model of the converter, in a trailing triangle modulation type conditions, it is described by the equation:

(5.24)

where:

(5.25)

(5.26)

(5.27)

(5.28)

The converter will be simulated by using the (5.24) equation, with the duty cycle, obtained after the computing of the predictive control given by the (5.11) relation. It will be chosen an arbitrary duty cycle and the simulation will run to pass the initial transient regime. If the functionality is stable, then the results in stationary state will be a sequence of constant discrete values. How the instability appears at D<0.5 or D>0.5, there are chosen two values for the reference current: a value which impose the functionality of the system at D<0.5, respectively a value which force the functionality at D>0.5.

The results of the simulation for a reference current Iref=2.5A, (D<0.5) are presented in fig.5.4 and fig.5.5 while in fig.5.6 are detailed the last ten switching periods of the inductive current for the same value of the reference current. The functionality is stable, the duty cycle becomes constant after the initial transient regime and the inductive current reaches o typical periodical shape with a period equal to the switching period.

The results of the simulation for the reference current Iref=11A, (D>0.5) are represented in fig.5.7 and fig.5.8 for the duty cycle respectively for the inductive current. The last ten switching periods of the inductive current have the same value with the reference current and are detailed in fig.5.9 In this case the functionality is also stable.

The simulation was realized in MATLAB program, the source code can be found in Annex 3. In fig.5.5 respectively fig.5.8 are represented the samples of the inductive current, namely the trailing points.

Fig.5.4. Duty cycle depending on time for Iref=2.5A, (D<0.5)

Fig.5.5. Sampled inductive current depending on time for Iref=2.5A, (D<0.5)

Fig.5.6. Ten periods of the inductive current after the stationary state install for Iref=2.5A, (D<0.5)

Fig.5.7. Duty cycle depending on time for Iref=11A, (D>0.5)

Fig.5.8. Sampled inductive current depending on time for Iref=11A, (D>0.5)

Fig.5.9. Ten periods of the inductive current after the stationary state install for Iref=11A, (D>0.5)

In conclusion, the obtained results after by the mathematical simulation of the discrete state-space model confirm that the predictive average current control (TTA) employing the trailing triangle modulation (TT) is stable for any value of the duty cycle.

5.4. Verification by simulating the circuit

For the simulation of the boost converter with TTA control, it was used the CASPOC package [14]. General scheme of simulation is presented in fig.5.10.

Fig.5.10.CASPOC scheme for the simulation of the boost converter using TA control

The average inductive current should follow the reference Iref. The necessary duty cycle for the next switching period is calculated using the inductive current samples, input voltage and output voltage from the current switching period, (5.10).

For the simulation scheme was necessary the implementation of the next blocks: sampling block and front memory (“edge”-E) (SPH_E), the shift register (SHIFT_REG), the slopes computation block (M1_M2_BOOST), the block for the computation of the recurrence relation for the duty cycle (TTA), the PWM modulator on trailing edge (PWM_TRIANGLE_TE). Also it was used dedicated block from the internes libraries of CASPOC program: PWM modulator (PWM_FS), used for the clock of the entire circuit, the limiter block (LIM) which limits the value of the duty cycle in the [0.1, 0.9] interval, the probe for the inductive current reading (CP), two probes for reading the input/output voltages (VP).

The logic functionality of the circuit is the same like in TA and LA cases.

The purpose is to simulate the boost converter controlled TTA, both in the case of a duty cycle bigger than 0.5, and in the case of a duty cycle smaller than 0.5. The first results of the simulation for the duty cycle, inductive current and output voltage are presented in fig.5.11, fig.5.12, fig.5.13 and fig.5.14.

Fig.5.11. The duty cycle depending on time, for Iref=2.5A (D<0.5)

The simulation was done with a value of 2.5A for the reference current, which forces the functionality with a duty cycle smaller than 0.5. It can be observed the obtaining of a stable functionality.

Fig.5.12. Inductive current, for Iref=2.5A (D<0.5).

Fig.5.13. presents a detail of 12 periods of the inductive current. At each 25µs (fs=40KHz), the transistor drives, the inductive current raise with the m1 slope for dTS. After the transistor is locked, the current decrease with m2 slope through the end of the switching period. The prescribed value of 2.5A for the reference current it is at the middle of the iL waveform, which is confirmed in the simulation.

Fig.5.13.Inductive current detailed, for Iref=2.5A (D<0.5).

Fig.5.14.Output voltage, for Iref=2.5A (D<0.5).

The reference current put on 11A implies a functionality of a duty cycle bigger than 0.5. In this conditions, the simulations results are presented in fig.5.15, fig.5.16, fig.5.17 and fig.5.18. It is also obtained a stable functionality of the converter.

Fig.5.15. The duty cycle depending on time, for Iref=11A (D>0.5)

Fig.5.16. Inductive current, for Iref=11A (D>0.5).

Like in the previous case, for Iref=2.5A, it wants to be highlight the theoretical results, by a detailed of 12 periods of the inductive current shown in fig.3.19. In this case, it can also be observed that at each 25µs (fs=40KHz), the transistor drives, and the inductive current rise with the m1 slope for dTS, after that the transistor is locked, the current decrease with the –m2 slope until the end of the switching period. The 11A value of reference current is exactly the average value of iL.

Fig.5.17..Inductive current detailed, for Iref=11A (D>0.5).

Fig.3.20.Output voltage, for Iref=11A (D>0.5.

Chapter 6

Predictive digital average current control employing leading triangle modulation

The purpose of this chapter is to study the predictive digital average current control named leading average control (LA) in DC-DC converters. This control uses the leading triangle modulation (LT) technique in correlation with the average current control (average – A). The study is focused on finding the recurrence relation of the duty cycle, after that the stability analysis is done. It is demonstrated that the LTA control is stable for every value of duty cycle, simplifying the design problems. The analysis is done in a general manner, independently from the converters topology, following that the obtained results can be applied on converters (buck, boost, buck-boost, etc.). Using a boost converter, the theoretical results are confirmed both on the space state model using Matlab program, and on the development and simulation of circuit in Caspoc simulator program.

6.1. Average current control employing leading triangle modulation (control law)

The sample value at the beginning of the switching period will be notated with i[n] because the current through the coil is sampled at the beginning of each switching period. The purpose is to obtain a relation for the duty cycle from the n+1 period, dn+1, in function of the duty cycle dn, from the n switching period, so the average current in n+1 period to be equal to the reference current value Iref. The target point from the average current at the end of the n+1 period, iave n+1 in fig.6.1, is evaluated in function of i[n]. This point should be equal to the reference current Iref.

Fig.6.1. Inductive current in dynamic regime for LTA control.

Also it is known that in stationary state the duty cycle D and the slope ratio are the following:

(6.1)

(6.2)

The two relation can be demonstrated by expressing the inductive current variations in each topological state and equalizing the results.m1 and m2 slopes depend on the input and output voltages. These voltages should be sampled. For example, in a boost converter the inductive current slope values are:

(6.3)

(6.4)

The sampled value, i[n+1], from the inductive current will be computed depending on the i[n] value from the previous switching period and the dn duty cycle, determined by the slopes m1 and m2 which are known.

(6.5)

Relation (6.5) can be written for the next switching period, n+1, as follows:

(6.6)

The average current target point through the coil at the end of the n+1 period is equal to:
(6.7)

Using (6.5) and (6.6) relations, the average inductive current point can be expressed as follows:
(6.8)

Imposing, it results:
(6.9)

From (6.9) equation, the value of the duty cycle is:
(6.10)

It is easy to observe that the value of the duty cycle is the same as for TTA control.

This represents the general recurrence relation of the duty cycle in LTA control case. The relation can be applied on any type of converter substituting the m1 and m2 slopes corresponding to the converter topology. For example, the duty cycle for the boost converter is:

(6.11)

6.2. Stability analysis for the LTA control

Fig.6.2. Inductive current in LTA control in presence of perturbations.

For the LTA control the stability condition will be demonstrated using the geometric model. Starting from the. Starting from the (6.10) relation, and making the same replacements, dn, dn+1, Iref , D and i[n], the relation will become:

(6.12)

Having the same duty cycle, the same recurrence relation as in the TTA case, it is obvious that the stability analysis for the LTA case is identical with the stability analysis for the TTA control.

Starting from the inductive current of the n+2 switching period, we have:

(6.13)

(6.14)

(6.15)

(6.16)

This relation represents the recurrence relation. Changing n→n+2, the result will be also 0. For any switching period n+2k the result will be 0.

(6.17)

It results that the LTA control is unconditionally stable (without oscillations) for the entire domain of the duty cycle.

6.3. Analysis of discrete space-state model for TTA control

With the purpose of validation of the previous theoretical considerations looking the LTA control, firstly it will be done a check using the state-space model. These will be studied on a boost converter, represented in fig.6.3.

Fig.6.3. Boost converter

Converter parameters has the next values:

Vg=10V; L=500µH; C=100µF; RL=1mΩ; fs=40kHz (6.18)

State vector is chosen as:

(6.19)

In CCM working, the converter can be modeled considering next equations:

(6.20)

where:

(6.21)

(6.22)

(6.23)

It is known that the discrete model of the converter, in a leading triangle modulation type conditions, it is described by the equation:

(6.24)

where:

(6.25)

(6.26)

(6.27)

(6.28)

The converter will be simulated by using the (6.24) equation, with the duty cycle, obtained after the computing of the predictive control given by the (6.11) relation. It will be chosen an arbitrary duty cycle and the simulation will run to pass the initial transient regime. If the functionality is stable, then the results in stationary state will be a sequence of constant discrete values. How the instability appears at D<0.5 or D>0.5, there are chosen two values for the reference current: a value which impose the functionality of the system at D<0.5, respectively a value which force the functionality at D>0.5.

The results of the simulation for a reference current Iref=2.5A, (D<0.5) are presented in fig.6.4 and fig.6.5 while in fig.6.6 are detailed the last ten switching periods of the inductive current for the same value of the reference current. The functionality is stable, the duty cycle becomes constant after the initial transient regime and the inductive current reaches o typical periodical shape with a period equal to the switching period.

The results of the simulation for the reference current Iref=11A, (D>0.5) are represented in fig.6.7 and fig.6.8 for the duty cycle respectively for the inductive current. The last ten switching periods of the inductive current have the same value with the reference current and are detailed in fig.6.9 In this case the functionality is also stable.

The simulation was realized in MATLAB program, the source code can be found in Annex 4. In fig.6.5 respectively fig.6.8 are represented the samples of the inductive current, namely the trailing points.

Fig.6.4. Duty cycle depending on time for Iref=2.5A, (D<0.5)

Fig.6.5. Sampled inductive current depending on time for Iref=2.5A, (D<0.5)

Fig.5.6. Ten periods of the inductive current after the stationary state install for Iref=2.5A, (D<0.5)

Fig.6.7. Duty cycle depending on time for Iref=11A, (D>0.5)

Fig.6.8. Sampled inductive current depending on time for Iref=11A, (D>0.5)

Fig.6.9. Ten periods of the inductive current after the stationary state install for Iref=11A, (D>0.5)

In conclusion, the obtained results after by the mathematical simulation of the discrete state-space model confirm that the predictive average current control (LTA) employing the leading triangle modulation (LT) is stable for any value of the duty cycle.

6.4. Verification by simulating the circuit

For the simulation of the boost converter with LTA control, it was used the CASPOC package [14]. General scheme of simulation is presented in fig.6.10.

Fig.6.10.CASPOC scheme for the simulation of the boost converter using LTA control

The average inductive current should follow the reference Iref. The necessary duty cycle for the next switching period is calculated using the inductive current samples, input voltage and output voltage from the current switching period, (5.10).

For the simulation scheme was necessary the implementation of the next blocks: sampling block and front memory (“edge”-E) (SPH_E), the shift register (SHIFT_REG), the slopes computation block (M1_M2_BOOST), the block for the computation of the recurrence relation for the duty cycle (LTA), the PWM modulator on trailing edge (PWM_TRIANGLE_TE). Also it was used dedicated block from the internes libraries of CASPOC program: PWM modulator (PWM_FS), used for the clock of the entire circuit, the limiter block (LIM) which limits the value of the duty cycle in the [0.1, 0.9] interval, the probe for the inductive current reading (CP), two probes for reading the input/output voltages (VP).

The logic functionality of the circuit is the same as in TA, LA and TTA cases.

The purpose is to simulate the boost converter controlled TTA, both in the case of a duty cycle bigger than 0.5, and in the case of a duty cycle smaller than 0.5. The first results of the simulation for the duty cycle, inductive current and output voltage are presented in fig.6.11, fig.6.12, fig.6.13 and fig.6.14.

Fig.6.11. The duty cycle depending on time, for Iref=2.5A (D<0.5)

The simulation was done with a value of 2.5A for the reference current, which forces the functionality with a duty cycle smaller than 0.5. It can be observed the obtaining of a stable functionality.

Fig.6.12. Inductive current, for Iref=2.5A (D<0.5).

Fig.6.13. presents a detail of 12 periods of the inductive current. At each 25µs (fs=40KHz), the transistor drives, the inductive current raise with the m1 slope for dTS. After the transistor is locked, the current decrease with m2 slope through the end of the switching period. The prescribed value of 2.5A for the reference current it is at the middle of the iL waveform, which is confirmed in the simulation.

Fig.6.13.Inductive current detailed, for Iref=2.5A (D<0.5).

Fig.6.14.Output voltage, for Iref=2.5A (D<0.5).

The reference current put on 11A implies a functionality of a duty cycle bigger than 0.5. In this conditions, the simulations results are presented in fig.6.15, fig.6.16, fig.6.17 and fig.6.18. It is also obtained a stable functionality of the converter.

Fig.6.15. The duty cycle depending on time, for Iref=11A (D>0.5)

Fig.5.16. Inductive current, for Iref=11A (D>0.5).

Like in the previous case, for Iref=2.5A, it wants to be highlight the theoretical results, by a detailed of 12 periods of the inductive current shown in fig.3.19. In this case, it can also be observed that at each 25µs (fs=40KHz), the transistor drives, and the inductive current rise with the m1 slope for dTS, after that the transistor is locked, the current decrease with the –m2 slope until the end of the switching period. The 11A value of reference current is exactly the average value of iL.

Fig.5.17..Inductive current detailed, for Iref=11A (D>0.5).

Fig.3.20.Output voltage, for Iref=11A (D>0.5)

Chapter 7

Conclusions

The paper had the purpose of studying the predictive digital average current control in dc-dc converters using the leading-edge, trailing-edge, leading triangle edge and trailing triangle edge modulations. The subject was chosen starting from two articles and a brevet elaborated by Boulder University from Colorado. The author proposed a general form for all the 4 types of modulations, demonstrating the unconditionally stability of them.

The personal contributions are related, in generalized form, for all the 4 chapters where the predictive average current controls were studied.

For each control the control law was determinate. Also it was developed theoretical considerations regarding the stability of each type of control.

The analysis was done in a general manner, being true for every type of converter.

It is demonstrated that for any duty cycle, bigger or smaller than 0.5, all 4 predictive average current controls are unconditionally stable.

Realization of the Matlab program for each control. The purpose is to verify of TA, LA, TTA, LTA control by simulating the discrete state-space model.

The results obtained from the Matlab simulation are presented in detail by graphs.

Simulation of the circuit scheme in the power circuit simulator Caspoc for all 4 types of controls, TA, LA, TTA and LTA, using the blocks developed for each control and the PWM modulation. The blocks were done by the Simulation Research, which developed the Caspoc program.

BIBLIOGRAPHY

Bibian S., Jin H. "Digital control with improved performances for boost power factor correction circuits", in Proc. IEEE APEC’01 Conf., 2001, pp. 137–143.

Buso S., Mattavelli P., "Digital Control in Power Electronics", Morgan&Claypool Publishers, First Edition, 2006.

Buso S., Mattavelli P., Rossetto L., and Spiazzi G., "Simple digital converter improving dynamic performance of power factor preregulators", IEEE Trans. Power Electron., vol. 13, pp. 814–823, Sept. 1998.

Franklin G., Powell J.D., Workman M., "Digital Control of Dynamic Systems", Addison- Wesley, 3rd ed., 1997.

Kim S. and Enjeti Dr. P., "Digital Control of Switching Power Supply – Power Factor Correction Stage", Power Electronics and Power Quality Laboratory Department of Electrical Engineering Texas A&M University College Station, TX – 77843-3128.

Obais A. M., Pasupuleti J., "Design of a Continuously Controlled Linear Static Var Compensator for Load Balancing and Power Factor Correction Purposes", International Review on Modelling and Simulations, (IREMOS), vol. 4, no.2, April 2011 (Part B), pp. 803-812.

Peng H., Maksimović D., Prodic A., Alarcon E., “Modeling of quantization effects in digitally controlled DC-DC converters,” IEEE PESC 2004, pp: 4312 – 4318.

Choudhury S., Harrison M., "DSPs simplify digital control implementation of SMPS", Texas Instruments, Dallas, Power Electronics Technology, July 1, 2003.

Bibian S., and Jin H., "High performance predictive dead-beat digital controller for dc power supplies" in Proc. IEEE APEC’01 Conf., 2001, pp. 67–73.

Chen J., Prodić A., Erickson R. W. and Maksimović D., "Predictive Digital Current Programmed Control", IEEE Transactions on Power Electronics, Vol. 18, No. 1, January 2003, pp. 411-419.

Maksimović D., Chen J., Prodic A., Erickson R. W. "Predictive digital current controllers for switching power converters", United States patent, Patent No. US 7,148,669 B2, Dec. 12, 2006

Shen Z., Chang X., Wang W., Tan X., Yan N., Min H., "Predictive digital current control of single-inductor multiple-output converters in CCM with low cross regulation", IEEE Trans. Power Electron., vol. 27, no. 4, April 2012, pp. 1917-1925.

Erickson R. W. and Maksimović D., "Fundamentals of Power Electronics, 2nd Ed.", Chapman and Hall, 2001.

CASPOC, user manual, http://www.simulation-research.com

Annex 1

%%boost trailing average

clear all; close all; clc;

Vg=10; R=10; L=500e-6; RL=1e-3; C=100e-6; fs=40e3; Ts=1/fs;

Iref=11;

Tsim=12e-3; Nmax=Tsim/Ts;

A1=[-RL/L 0;

0 -1/(R*C)];

B1=[1/L; 0];

E1=[0 1; 1 0]; F1=[0; 0];

A2=[-RL/L -1/L;

1/C -1/(R*C)];

B2=[1/L; 0];

E2=[0 1; 1 0]; F2=[0; 0];

I=eye(2);

n=1;

x(1,n)=0; x(2,n)=1e-6; i(n)=x(1,n); M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; d(n)=0.1;

while n<Nmax

M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; i(n)=x(1,n);

phi1=expm(A1*d(n)*Ts); psi1=A1\(phi1-I)*B1;

phi2=expm(A2*(1-d(n))*Ts); psi2=A2\(phi2-I)*B2;

x(:,n+1)=phi2*phi1*x(:,n)+(phi2*psi1+psi2)*Vg;

d(n+1)=-2*(M1(n)+M2(n))/(2*M1(n)+M2(n))*d(n)-2/((2*M1(n)+M2(n))*Ts)*(i(n)-Iref)+3*M2(n)/(2*M1(n)+M2(n));

if d(n+1)<0.01

d(n+1)=0.01;

elseif d(n+1)>0.99

d(n+1)=0.99;

end

n=n+1;

end

index_d=0:1:length(d)-1;

time_d=Ts*index_d*1000;

plot(time_d,d,'-'); xlabel('time [ms]'); ylabel('d');

index_i=0:1:length(i)-1;

time_i=Ts*index_i*1000;

figure; plot(time_i,i,'-'); xlabel('time [ms]'); ylabel('iL');

p=10;

k=1;

h=20e-9;

contor=0;

t=0; tsim(1)=0;

m=1; xsim(:,m)=x(:,length(d)-p); phi=expm(A1*h); psi=A1\(phi-I)*B1;

while k<=p

if((k-1)*Ts<t)&&(t<=(k-1+d(Nmax-k+1))*Ts)

phi=expm(A1*h); psi=A1\(phi-I)*B1;

elseif((k-1+d(Nmax-k+1))*Ts<t)&&(t<=k*Ts)

phi=expm(A2*h); psi=A2\(phi-I)*B2;

end

xsim(:,m+1)=phi*xsim(:,m)+psi*Vg; tsim(m+1)=tsim(m)+h;

m=m+1; t=t+h; contor=contor+1;

if contor==Ts/h

k=k+1; contor=0;

else

end

figure; plot(tsim*1e3+(Tsim-p*Ts)*1e3*ones(length(Tsim)),xsim(1,:));xlabel('time [ms]');

ylabel('iL');

Annex 2

%boost leading average

clear all; close all; clc;

Vg=10; R=10; L=500e-6; RL=1e-3; C=100e-6; fs=40e3; Ts=1/fs;

Iref=2.5;

Tsim=12e-3; Nmax=Tsim/Ts;

A2=[-RL/L -1/L;

1/C -1/(R*C)];

B2=[1/L; 0];

E2=[0 1; 1 0]; F2=[0; 0];

A1=[-RL/L 0;

0 -1/(R*C)];

B1=[1/L; 0];

E1=[0 1; 1 0]; F1=[0; 0];

I=eye(2);

n=1;

x(1,n)=0; x(2,n)=1e-6; i(n)=x(1,n); M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; d(n)=0.1;

while n<Nmax

M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; i(n)=x(1,n);

phi2=expm(A2*(1-d(n))*Ts); psi2=A2\(phi2-I)*B2;

phi1=expm(A1*d(n)*Ts); psi1=A1\(phi1-I)*B1;

x(:,n+1)=phi1*phi2*x(:,n)+(phi1*psi2+psi1)*Vg;

d(n+1)=-2*(M1(n)+M2(n))/(M1(n)+2*M2(n))*d(n)-2/((M1(n)+2*M2(n))*Ts)*(i(n)-Iref)+4*M2(n)/(M1(n)+2*M2(n));

if d(n+1)<0.01

d(n+1)=0.01;

elseif d(n+1)>0.99

d(n+1)=0.99;

end

n=n+1;

end

index_d=0:1:length(d)-1;

time_d=Ts*index_d*1000;

plot(time_d,d,'-'); xlabel('time [ms]'); ylabel('d');

index_i=0:1:length(i)-1;

time_i=Ts*index_i*1000;

figure; plot(time_i,i,'-'); xlabel('time [ms]'); ylabel('iL');

p=10; % ultimele p perioade sunt reprezentate

k=1; % contorul de perioade

h=20e-9; % pasul de simulare

contor=0; % contorul care vede cand s-a terminat o perioada de comutatie

t=0; tsim(1)=0;

m=1; xsim(:,m)=x(:,length(d)-p); phi=expm(A2*h); psi=A2\(phi-I)*B2;

while k<=p

if ((k-1)*Ts<t)&&(t<=(k-1+1-d(Nmax-k+1))*Ts)

phi=expm(A2*h); psi=A2\(phi-I)*B2;

elseif ((k-1+1-d(Nmax-k+1))*Ts<t)&&(t<=k*Ts)

phi=expm(A1*h); psi=A1\(phi-I)*B1;

end

xsim(:,m+1)=phi*xsim(:,m)+psi*Vg; tsim(m+1)=tsim(m)+h;

m=m+1; t=t+h; contor=contor+1;

if contor==Ts/h

k=k+1; contor=0;

else

end

figure; plot(tsim*1e3+(Tsim-p*Ts)*1e3*ones(length(Tsim)),xsim(1,:)); xlabel('time [ms]'); ylabel('iL');

Annex 3

%boost trailing triangle average

clear all; close all; clc;

Vg=10; R=10; L=500e-6; RL=1e-3; C=100e-6; fs=40e3; Ts=1/fs;

Iref=11;

Tsim=12e-3; Nmax=Tsim/Ts;

A1=[-RL/L 0;

0 -1/(R*C)];

B1=[1/L; 0];

E1=[0 1; 1 0]; F1=[0; 0];

A2=[-RL/L -1/L;

1/C -1/(R*C)];

B2=[1/L; 0];

E2=[0 1; 1 0]; F2=[0; 0];

I=eye(2);

n=1;

x(1,n)=0; x(2,n)=1e-6; i(n)=x(1,n); M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; d(n)=0.1;

while n<Nmax

M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; i(n)=x(1,n);

phi1=expm(A1*1/2*d(n)*Ts); psi1=A1\(phi1-I)*B1;

phi2=expm(A2*(1-d(n))*Ts); psi2=A2\(phi2-I)*B2;

x(:,n+1)=phi1*phi2*phi1*x(:,n)+(phi1*phi2*psi1+phi1*psi2+psi1)*Vg;

d(n+1)=-d(n)+2*(M2(n)/(M1(n)+M2(n)))+(Iref-i(n))/((M1(n)+M2(n))*Ts);

if d(n+1)<0.01

d(n+1)=0.01;

elseif d(n+1)>0.99

d(n+1)=0.99;

end

n=n+1;

end

index_d=0:1:length(d)-1;

time_d=Ts*index_d*1000;

plot(time_d,d,'-'); xlabel('time [ms]'); ylabel('d');

index_i=0:1:length(i)-1;

time_i=Ts*index_i*1000;

figure; plot(time_i,i,'-'); xlabel('time [ms]'); ylabel('iL');

p=10;

k=1;

h=20e-9;

contor=0;

t=0; tsim(1)=0;

m=1; xsim(:,m)=x(:,length(d)-p); phi=expm(A1*h); psi=A1\(phi-I)*B1;

while k<=p

if(((k-1)*Ts<t)&&(t<=(k-1+1/2*d(Nmax-k+1))*Ts))

phi=expm(A1*h); psi=A1\(phi-I)*B1;

elseif((k-1+1/2*d(Nmax-k+1))*Ts<t)&&(t<=(k-1+1-1/2*d(Nmax-k+1))*Ts)

phi=expm(A2*h); psi=A2\(phi-I)*B2;

elseif(((k-1+1-1/2*d(Nmax-k+1))*Ts<t)&&(t<=k*Ts))

phi=expm(A1*h); psi=A1\(phi-I)*B1 ;

end

xsim(:,m+1)=phi*xsim(:,m)+psi*Vg; tsim(m+1)=tsim(m)+h;

m=m+1; t=t+h; contor=contor+1;

if contor==Ts/h

k=k+1; contor=0;

else

end

figure; plot(tsim*1e3+(Tsim-p*Ts)*1e3*ones(length(Tsim)),xsim(1,:)); xlabel('time [ms]'); ylabel('iL');

Annex 4

%boost leading triangle average

clear all; close all; clc;

Vg=10; R=10; L=500e-6; RL=1e-3; C=100e-6; fs=40e3; Ts=1/fs;

Iref=11;

Tsim=12e-3; Nmax=Tsim/Ts;

A1=[-RL/L 0;

0 -1/(R*C)];

B1=[1/L; 0];

E1=[0 1; 1 0]; F1=[0; 0];

A2=[-RL/L -1/L;

1/C -1/(R*C)];

B2=[1/L; 0];

E2=[0 1; 1 0]; F2=[0; 0];

I=eye(2);

n=1;

x(1,n)=0; x(2,n)=1e-6; i(n)=x(1,n); M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; d(n)=0.1;

while n<Nmax

M1(n)=Vg/L; M2(n)=(x(2,n)-Vg)/L; i(n)=x(1,n);

phi2=expm(A2*1/2*(1-d(n))*Ts); psi2=A2\(phi2-I)*B2;

phi1=expm(A1*d(n)*Ts); psi1=A1\(phi1-I)*B1;

x(:,n+1)=phi2*phi1*phi2*x(:,n)+(phi2*phi1*psi2+phi2*psi1+psi2)*Vg;