Control of the Power in Induction Heating Systems with L-LC Resonant Voltage Source Inverters

Mihaela Popescu, Alexandru Bitoleanu, Vlad Suru Faculty of Electrical Engineering

University of Craiova Craiova, Romania

mpopescu@em.ucv.ro, alex.bitoleanu@em.ucv.ro, vsuru@em.ucv.ro

Abstract—This paper is a synthesis of the control circuit in an induction heating system with three-phase fully-controlled rectifier, single-phase voltage source inverter and L-LC resonant load. The attention is mainly directed on the control of the power transmitted to the induction coil. It is highlighted that the analytical design of a proportional integral derivative or proportional integral controller is practically impossible when the direct control of the current through inductor is desired. The two possibilities of power control taken into consideration are the inverter output current control and the inverter input voltage control. The design of the current and voltage controllers of proportional integral derivative type was carried out successfully based on Modulus Optimum criterion in Kessler variant. The performances of the control system in both variants of the power control were tested and validated by simulation under the Matlab-Simulink environment by using real parameters of induction coils and heated pipes from a leading Romanian manufacturer.

Keywords—converter control; induction heating; resonant converter; voltage source inverters

I. INTRODUCTION In practice today, the induction heating is one of the fastest,

more controllable and precise heating method applied in industrial field. To implement an induction heating system, a high frequency electrical power source and a work coil (inductor) to generate the alternating magnetic field are required.

In the practical implementation, the work coil is incorporated into a series or parallel resonant tank circuit, so that losses in the supply resonant converter are minimized. Basic resonant converters associated to these tanks are DC-link converters and they can be either current source parallel resonant inverters or voltage source series resonant inverters [1]–[3].

For a long time, based on control system capability to enable attaining the maximum power level, the parallel resonant converters with fully controlled line-commutated rectifiers were the preferred systems for induction heating [4]. The three-phase AC/DC controlled converters for DC power regulation are also usual adopted solutions in the control of series resonant converters [5], [6].

When the power regulation is achieved by the inverter control, the most common methods are pulse-frequency modulation, phase shift and pulse density modulation [7]–[9].

Several drawbacks have been highlighted over the years on classical structures of resonant converters used in induction heating. For instance, the overvoltage protection systems are needed in the case of current source parallel resonant inverters and a matching transformer between inverter and load is needed in the case of voltage source series resonant inverters [1].

By the newer class of resonant tanks with three elements (L-LC) supplied by voltage source inverters, the main limitations of the previous structures are overcome and an increased short-circuit immunity is obtained [4], [10]–[13].

In the induction heating system taken into consideration in this paper, the three-phase fully-controlled bridge rectifier supplies the single-phase H-bridge voltage source inverter (VSI) via a voltage DC-link circuit. At the inverter output, a matching inductance is needed to achieve the maximum transfer of the power from the power supply to the work coil (Fig. 1) [14].

The structure of the paper is as follows. Section II introduces the configuration of the induction heating system with VSI and L-LC resonant load. The possibility of direct control of the current through inductor is discussed in Section III. In Section IV, the control of the inverter output current is analyzed and the design of a proportional integral derivative (PID) current controller is performed based on Modulus Optimum criterion in Kessler variant. It is followed by tests on the control system performance. Section V is concerned with the control of the inverter input voltage. The PID voltage controller is tuned and the voltage control loop behavior is analyzed. Some conclusions are finally given in section VI.

II. SYSTEM CONFIGURATION The induction heating system of seamless pipes is

considered below. As a principle, the inverter operating frequency is controlled by an auto-adaptive loop, in order to be the resonant frequency of the parallel circuit formed by the equivalent inductor and the resonant capacitor. As regards the control of the power transmitted to the load, the following three ways can be identified: the direct control of the current through

inductor, the inverter output current control and the inverter input voltage control.

In addition, the transferred power increases when the inverter frequency increases over the resonant value without exceeding a certain limit which depends on the load.

It must be specified that the control of the current through inductor and the control of the inverter input voltage are equivalent provided that the loop frequency is fast enough.

Two possible approaches can be taken into consideration in the calculation of the induction coil current set value. The former is based on the assumption that the heated piece moves through the inductor at a preset speed and the preset current depends on the required temperature gradient. The latter is intended to maximize the inverter productivity by imposing the maximum voltage at the inverter output and adjusting the heated piece speed in order to obtain the required gradient of the temperature.

When the first approach is adopted, the control system requires two control loops, which are practically independent (Fig. 2).

The following blocks are highlighted in the global block diagram shown in Fig. 2: Ci(u) – current or voltage controller; CR – controlled rectifier; DC – DC-link circuit; VSI – single-phase voltage source inverter; MI – matching inductor; HI – heating inductor; RC – resonant capacitor; FAB – frequency adapting block; RMS – rms value calculation; PB - protection block.

The main task of the frequency loop is to achieve the permanent and dynamic observance of the frequency, so that it is equal to or higher than the resonant frequency of the parallel circuit consisting of the equivalent inductor and the resonant capacitor, to facilitate the switching process of inverter’s power semiconductors.

As the parameters of the circuit are not constant, the dynamic self adaptation of the frequency is required, by using only quantities provided by system. It results that the frequency loop cannot be controlled by external signals.

Based on the phasorial diagram of the inverter load circuit, the resonant frequency can be obtained at the output of a proportional integral controller by identifying the error that cancels when the resonance condition is met.

On the other hand, in order to obtain the zero-current switching of the inverter’s IGBTs, the operating frequency must be slightly higher than the resonant frequency of the load circuit. Consequently, the frequency control loop must be able to achieve this second requirement too.

III. DIRECT CONTROL OF THE CURRENT THROUGH INDUCTOR

In the block diagram shown in Fig. 3, the forward path includes the transfer functions of the current controller (GCi(s)), three-phase controlled rectifier (GCR(s)), voltage DC-link circuit (GDC(s)), single phase VSI (GVSI(s)), adapting circuit (GMC(s)), and equivalent inductor (GI(s)). Eib(s) is the Laplace transform of the current error εib.

Fig. 1. Block diagram of the induction heating system with L-LC resonant voltage source inverter.

Fig. 2. Basic block diagram of the induction heating control system.

The associated expressions are [15]:

( )μsT

KsG R

CR +=

1; (1)

( )22 1

11

1sTTsTsCLsCR

sGedemdeddddd

DC++

=++

= ; (2)

( ) VSIVSI KsG == π2 ; (3)

( ) ( ) ( )( )

;1

1111

33

22

1 TsTssTRRsT

RR

sCsLRsLRsG

bb

emb

a

b

bbaaMC

+++++

⋅=

=++⋅++

= (4)

( )sT

RsLR

sGemb

b

bbI +

=+

=1

11 ; (5)

where: dded CRT = ; ddemd RLT = ; (6)

CRT deb = ; bbemb RLT = ; (7)

aaema RLT = ; (8)

ebemaabemd TTRRTT ++⋅=1 ;; (9)

( )embemaeb TTTT +⋅=2 ; (10)

embemaeb TTTT ⋅⋅=3 ; (11)

KTi is the proportional constant of the current transducer; KR and Tμ are the proportional and integral time constants of the rectifier given by the rms secondary voltage and the average dead-time associated of the firing circuit.

After some processing stages, the transfer function of the fixed part of the unity feedback system can be expressed as:

( ) ( ) ( )

( ) .TsTssTRR

sTTsTsTRKKK

sG

bb

edemded

aTiVSIRFb

33

22

1

2

11

111

++++⋅

⋅++⋅+

⋅⋅⋅=

μ (12)

It is found that the transfer function in (12) is very complicated, which makes it impossible the analytical approach to synthesize a proportional integral or a proportional integral derivative controller.

In addition, it must be also taken into consideration that, given the proper operation of the frequency control loop, the control of the inverter current or of the DC voltage determines univocally the control of the current through inductor.

Fig. 3. Transfer functions based block diagram of the inductor current closed loop control system.

IV. CONTROL OF THE INVERTER OUTPUT CURRENT In the inverter current control loop (Fig. 4), a PID controller

is adopted. Its transfer function expression is written as:

( )sT

sTTsTKsG

ii

diiiiipiCi

21 ++= , (13)

where the proportional constant (Kpi) and integral and derivative time constants (Tii and Tdi) can be determined by using the Modulus Optimum criterion in Kessler variant [16], [17].

Based on the Kirchhoff's laws in the Laplace domain applied to the circuit consisting in the matching inductance and the resonant capacitor in parallel with the equivalent inductor, the transfer function of the matching and resonant circuit can be expressed as follows:

( )( )

( )( )

( ) ( ) ( ).

sTTsTsTsTRR

sTTsTR

sLRsLRsC

sLRsUsI

)s(G

embebebemaemba

b

embebeba

aabb

bbi

iMR

2

2

111

11

1

1

++⋅+++⋅

++⋅=

=++

+++

==

(14)

Assuming that ebembeb TTT << and 1<<ab RR , expression (14) becomes:

sT

R)s(G

ema

aMR +

=1

1. (15)

Fig. 4. Transfer functions based block diagram of the inverter current closed loop control system.

A. Inverter Current Controller Design In order to use the Modulus Optimum criterion, the open-

loop unity feedback transfer function is expressed as:

( ) ( ) ( )( ) ( )

.sTTsT

sTTsTK

sTsTsTRKK

sG

edemded

diiiiipi

emaii

aTiRdi

2

2

1

1

1112

++

++⋅

⋅+⋅+⋅⋅⋅⋅

=μ

π

(16)

The conditions of eliminating the time constants of the DC-link circuit in (16) lead to following two equations:

edemddiiiediipi TTTT;TTK == . (17)

The integral time constant of the controller is provided by the condition of canceling the denominator term which contains a difference in the modulus square of the closed-loop unity feedback system transfer function [15]:

( ) ( ) ( )emaaTiRii TTRKKT +⋅⋅⋅⋅⋅= μπ 122 . (18)

Thus, based on (17) and (18), the other two parameters of the current controller are:

( ) ( ) ( )emaaTiR

edpi TTRKK

TK

+⋅⋅⋅⋅=

μπ 14; (19)

( ) ( ) ( )emaaTiR

emdeddi TTRKK

TTT

+⋅⋅⋅⋅⋅

=μπ 14

. (20)

B. Performance of the Current Control System The performance of the control system was tested by

simulation under Matlab-Simulink environment by using the parameters of real induction coils and heated pipes from a leading Romanian manufacturer which produces seamless pipes for industrial applications (Table 1).

The line-to-line supply voltage provided by the power transformer is 660 Vrms and the supply frequency is 50 Hz. The DC-link circuit parameters are Cd=2000µF, Ld=0.1mH and Rd=0.1Ω and the matching inductor is characterized by La=45µH and Ra=0.1Ω.

TABLE I. PARAMETERS OF THE RESONANT LOAD

Φ 170/7 Φ 170/14 Rb (mΩ) Lb (mH) C (µF) Rb (mΩ) Lb (mH) C (µF)

25.6 0.0121 185 12.2 0.0140 920

Φ 210/10 Φ 210/20 Rb (mΩ) Lb (mH) C (µF) Rb (mΩ) Lb (mH) C (µF)

13.93 0.0148 198 6.55 0.0170 3000

When the inverter current of 800 A rms is step prescribed to heat a 170 mm diameter pipe, the system response in terms of the rms inverter fundamental current highlights a very small overshoot of about 0.08 % in the case of the wall thickness of 7 mm (pipe Φ170/7) (Fig. 5a).

As shown in Fig. 5b, by increasing the wall thickness from 7 mm to 14 mm, the overshoot increases but it is below 3 %. For both cases, the transient regime ends in about 0.02 seconds.

As it can be seen in Fig. 6a, the behavior of the inverter current control loop when the heated pipe is Φ210/10 is very similar to that of pipe Φ170/14 (Fig. 5b).

Even in the case of a pipe twice as thick, the overshoot does not exceed 5% (Fig. 6b).

V. CONTROL OF THE INVERTER INPUT VOLTAGE If the operating frequency is constant and each inverter’s

IGBT is closed during a half-cycle, the output current is determined directly by the inverter output voltage, which depends on the voltage provided by the controlled rectifier. Thus, a voltage control loop can be adopted too (Fig. 7).

A. Voltage Controller Design According to the MO criterion, the expression of the open-

loop unity feedback transfer function, i.e.

( )( )

( ) ( ) ,sTTsTsTsT

sTTsTKKK

K)s(G)s(G)s(GsG

edemdediu

duiuiupuTuR

TuDCCRCudu

2

2

11

1

++⋅⋅+

++⋅⋅=

=⋅⋅⋅=

μ

(21)

allows removing the dominant time constants (Ted and Temd) of the control system when a controller of PID type is chosen.

Fig. 5. Response of the inverter current control loop for two pipes of 170 mm diameter: a) Φ170/7; b) Φ170/14.

Fig. 6. Response of the inverter current control loop for two pipes of 210 mm diameter: a) Φ210/10; b) Φ210/20.

Fig. 7. Transfer functions based block diagram of the voltage closed loop control system.

Thus, the following two conditions are imposed:

edemdduiuediupu TTTT;TTK == . (22)

The third condition used in the controller design is related to the canceling of terms containing differences in the denominator of the modulus square of the closed loop unity feedback transfer function (M 2(ω)). Thus, taking into consideration (22),

( ) ( )( )

( )( )

( )

( ) .TjTjKK

KK

TjTjKKKK

jGjG

jGjG

M

iuTuR

TuR

iuTuR

TuR

du

du

du

du

μ

μ

ωω

ωω

ωω

ωωω

−−⋅

⋅++

=

=−+

−⋅

+=

1

1

112

(23)

By processing expression (23), the following form is obtained:

( ) ( ) 224222

222

2 μμ ωωω

TTTTKKTKKKKM

iuiuTuRiTuR

TuR

+−−= . (24)

So, the simple condition of canceling the second term in the denominator gives the expression of the integral time constant:

μTKKT TuRiu 2= . (25)

Now, by using (25), the conditions expressed in (22) provide the derivative time constant and the proportional constant of the voltage controller:

μTKK

TTT

TuR

emdedd 2

= ; (26)

μTKK

TKTuR

edp 2

= . (27)

B. Performance of the Voltage Control System As shown in Fig. 8 – Fig. 10, the performance of the

control system when a step DC voltage is prescribed is very good. Specifically, irrespective of the set voltage value, only a half-oscillation in the system response appears, the overshoot is between 4% and 5% and the transient regime duration is between 16 ms and 18 ms.

VI. CONCLUSIONS After analyzing the possibilities of power control in an

induction heating system with L-LC resonant voltage source inverter supplied by a fully controlled rectifier, the main conclusions that can be drawn are:

Fig. 8. Response of the voltage control loop when the prescribed DC voltage is 300 V.

Fig. 9. Response of the voltage control loop when the prescribed DC voltage is 600 V.

Fig. 10. Response of the voltage control loop when the prescribed DC voltage is 850 V.

- both a current loop to regulate the output inverter current and a voltage loop to regulate the input inverter voltage can be adopted and designed by analytical way;

- the tuning of the PID current and voltage controllers was performed by using the Modulus Optimum criterion in Kessler variant;

- the determination of controllers’ parameters is unique and leads to elimination of inertia introduced by the DC-link circuit;

- the parameters of the inverter input voltage controller do not depend on the load;

- the simulation results illustrate a very good behavior of the control system in both variants;

- the inverter input voltage control loop has superior performance;

- the use of an inverter input voltage control loop for regulating the power transmitted to the load is the best option, with the mention that, for each heating pipe, the prescribed value is different, so that the allowable current of the inverter is not exceeded.

Therefore, based on these results, there are all prerequisites to successfully complete the next stage of our research by implementing a high performance control system in the experimental setup.

ACKNOWLEDGMENT This work is the result of research activity within Grant

352/21.11.2011, POSCCE-A2-O2.1.1-2011-2.

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