Instruction


FACTA UNIVERSITATIS  

Series: Electronics and Energetics Vol. 31, N
o
 1, March 2018, pp. 41 - 50 

https://doi.org/10.2298/FUEE1801041P 

INVESTIGATING THE STEADY STATE MODE OF POWER 

INVERTERS FOR INDUCTION HEATING 

 

Evgeniy Popov, Nikolay Hinov 

Faculty of Electronic Engineering and
 
Technologies, Department of Power Electronics, 

Technical University of Sofia, Sofia, Bulgaria

 

Abstract. This article formulates the new unified interpretation of the analysis of 

electromagnetic processes in the autonomous (usually resonant) inverters with power 

circuits having a serial RLC configuration either with or without free wheeling diodes. 

The investigation starts with clarifying the parameters of the inverter circuit by 

bringing the fourth order power network into such of a second order in a normalized 

form. On this basis the novel compendious relationships between the most important 

internal inverter parameters are given. A MATLAB program calculates and displays 

the frequency characteristics of both types of inverters and simulates their steady state. 

The results from characteristics and simulation confirm each other in many ways. They 

were also proved experimentally. The whole processed information helps better 

understanding and organizing intelligent design, measurement and control of the 

inverters for technological applications (induction heating). 

Key words: modeling, electromagnetic processes, frequency characteristics, 

RLC inverters, steady state, unified interpretation.  

1. INTRODUCTION 

The voltage fed RLC inverter (Fig. 1), its dual counterpart the current fed RLC 

inverter and the serial RLC inverter without free–wheeling diodes (Fig. 2) cover a very 

wide range of practical autonomous inverter circuits generally applied in electronic 

technology [1]-[4]. In this case important and complex mutually connected problems are 

the accurate design of the power circuit, appropriate adjustment between the inverter and 

the load and adequate control providing stable, reliable and efficient operation of the 

converter when wide variations of the load are expected [5]-[7]. 

The mathematical relationships between the parameters of the mentioned inverters are 

rather complicated [5], [8]-[10]. It has been proved that all the quantities in these second order 

                                                           
Received September 15, 2016; received in revised form September 29, 2017 

Corresponding author: Nikolay Hinov  

Faculty of Electronic Engineering and Technologies, Department of Power Electronics, Technical University of 

Sofia, Sofia, Bulgaria 

(E-mail: hinov@tu-sofia.bg) 



42 E. POPOV, N. HINOV 

topologies depend on two variables - the ratio between the controlling angular frequency and 

the generalized angular frequency (frequency coefficient) In n /      on one hand, and 

the ratio between the damping coefficient and the generalized angular frequency 

I
n n /
 

    of the power circuit, on the other. At the same time engineering practice in 

the area of the power electronics needs fast and accurate means for simultaneous solution of the 

already stated problems. 

VT1

VT4

U
d

VD1

VD4

VT2

VT3

VD2

VD3

R
i

L
i Ci

i u

i
d

u
i

 

Fig. 1 Voltage fed RLC inverter 

When, for oscillatory mode i i iR 2 L / C of the inverter circuit (Fig 1, Fig. 2) 

coefficient is c 1 , for over damped mode ( i i iR 2 L / C ) c 1   and for critical 

mode (
i i i

R 2 L / C ) c 0 .  

The parameters that determine the development of the electromagnetic processes in the 

steady state in the power inverter circuits are 

          
2 2

cos
( .; );1 ( )

(4 sin cos )

I I
I

I I I I

B
n n osc over damped critical

c B B
 


  

   
     (1) 

For resonant inverters (oscillatory mode) the coefficient of hesitation is 

                                  
0

1/ [1 exp( / )] 1/ [1 exp( )]
I

k n


                               (2) 

The frequency coefficient is 

2 2

2 2
( .; ); ( )

cos(4 sin cos )
I

I II I I I

n n osc over damped critical
Bc B B

 


  

   
  (3) 

VS3

VS2

C
i Rii

u

U
d

VS1

VS4

L
i1

i
d

L
i2

or

L
i
=L

i1
+L

i2

b
 

Fig. 2 A serial RLC inverter without free–wheeling diodes 



 The Steady State Mode of Power Inverters for Electro Technology Applications 43 

2. THE INVERTER ANALYSIS IN THE STEADY STATE MODE 

 

In contrast to the approaches used in [10]-[15] provided herein is a unified approach to 

the analysis, which alleviates the description of the behavior of the power circuit. A 

constant ci reflecting the type of the inverter is introduced, having values ci = +1 for the 

RLC inverter with free wheeling diodes (Fig. 1) or ci = 1 for the RLC inverter without 

free wheeling diodes (Fig. 2). 

The following designations are applied: 
S

f (x) sin x , 
C

f (x) cos x , 
i i

R / (2L )  , 
2

0 i
1/ (L Ci)      for oscillatory mode; Sf (x) sinh x , Cf (x) cosh x , i iR / (2L )  , 

2

i i
1/ (L C )    for over damped mode; 

S
f (x) x , 

C
f (x) 1 , 

i i
R / (2L )  ,    

for critical mode. 

Then the inverter current and the voltage across the capacitor Ci can be written in the 

following manner 

                           0
0

( ) ( ) ( )
t td

S i C S

U U
i e f t c I e f t f t

L

   
        

                        (4) 

               0
0

( ) ( ) ( ) ( )
t t

d d C S i S

I
u U U U e f t f t c e f t

C

  
          

                (5) 

The angle 
2 0

/ / n


         corresponding to the half period is determined 

from the controlling angular frequency 2 f  and the generalized frequency  . 
The parameters of the inverter circuit (Fig. 1, Fig. 2) can be determined taking into 

account the initial conditions for the steady state 

0

2

(0) . ( )

(0) ( )

i
i c i

u u

  

  
    (6) 

and 

                                                        
1

( ) 0i                                           (7) 

Then the determination of the parameters follows. The parameter 
0 0

/ ( )
d

a I L U U    is 

                                          
2

2

.

2 2

( )

. ( ) . . ( )

S

n

i C i S

f
a

e c f c n f







   
                                       (8) 

    For the inverter in Fig. 1 only the angle 
1
  is determined from 

1 1

1 1

( ) / ( )

1 . ( ) / ( )

S C

S C

f f
a

n f f


 


  
    (9) 

The generalized coefficient of hesitation is 

                        
2. 2

2 2

1

1 [( . . ) ( ) ( )]
n

i i S C

K
e c ca n c a n f f
 

 


     

                           (10) 



44 E. POPOV, N. HINOV 

It should be underlined that when calculating a  and K  for discontinuous inverter current 

mode for Fig. 2 
0 2
   must be equal to   in the already given expressions (8) and (10). 

The initial capacitor voltage is 

                                                 0
0
' 2. 1

d

U
U K

U
                                      (11) 

The maximal voltage across the capacitor 
i

C  
m

U  for Fig. 2 is also given by (11). For 

Fig. 1 it is given by 

                                      
1

' 2( ) 1m K
m K

d

U
U K

U
                        (12) 

The expression for the coefficient 
1

K  is the same as (10), but the variable 
2

  is 

exchanged with 
1
  (

2 1
  ). 

The normalized inverter current and capacitor voltage are respectively 

                         
( )

'( ) 2 (1 . . ) ( ) . . ( )
n

i S i C

d

i L
i Ke c a n f c a f

U
 



 
                              (13) 

                    
2( )

'( ) 1 2 ( . . . . ) ( ) ( )
n

i i S C

d

u
u Ke c c a n c a n f f

U
 

 


                            (14) 

The average value of the input current (all values are normalized) is 

                                
0

2

0 00

1 1 2(2 1)
' '( ) .d
d

d

I L K
I i d

U n c





 
    

  
                (15) 

The RMS value of the inverter current is 

                                           ' / ' / (2. )
d d

I I L U I n


                          (16) 

The output characteristic (Fig. 1 and Fig. 2) is 

                                                     
'/ '

d
OCH I I           (17) 

The input characteristic is 

                                                          1/ ( . ' )
d

ICH n I


           (18) 

The characteristic of the coefficient of nonlinear distortion (klir – factor) of the inverter 

current is 

                                              
2 2

(1) (1)
[%] 100. ' ' / 'kf I I I           (19), 

where I'(m) (m = 1, 3, 5, 7...) is the m - th harmonic component of the inverter current. 

(The mathematical expressions for calculation of the harmonic components in the 

different circuits and modes of operation are disparate, and rather complicated. They have 

been found and published in earlier authors’ publications.) 

From that point on the analysis of the power inverter may be continued without 

problems in a normalized or in a non normalized form. 



 The Steady State Mode of Power Inverters for Electro Technology Applications 45 

3. AN EXAMPLE WITH A SERIAL RESONANT INVERTER WITHOUT FREE WHEELING DIODES 

A half bridge circuit (Fig. 3) is under study but it can be easily converted into the 

bridge one. The power losses in the inverter are neglected. The commutation of the power 

semiconductor devices is instantaneous. A parallel equivalent circuit represents the 

induction heater. The quality factor of the load circuit is sufficiently high that the voltage 

across the load has a close to the sine wave shape. 

A MATLAB program processes all the mathematical information describing the 

steady state operation of the inverter in the allowed frequency range. The frequency 

characteristics of the inverter are obtained and graphically displayed in Fig. 4. These 

particular characteristics correspond to a practically implemented inverter (PT1-100-

2400) with the following data: Ud=500 V; LK=45 H; CK=84 F; Rlr= 0.4739599 , flr=2083 

Hz, Llr=8.8717 H, Cl,=657.88 F,n1=0, n2=1, f=1600 -2600 Hz. The first graphic shows: the 
average input current Id [A] (solid line). The second graphic shows: the maximal device voltage 

UVSm (solid line). Other parameters of the circuit can also be calculated and displayed. 

U
d 
/2

R
l

C
K

L
K1

i

VS1

VS2

i
d

L
l

C
l L

K2

u
l

U
d 

/2

or

 

Fig. 3 A practical serial resonant inverter without free–wheeling diodes 

1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600
0

500

1000

X: 2196

Y: 200.3

Frequency (Hz)

T
h
e
 a

v
e
ra

g
e
 i
n
p
u
t 

c
u
rr

e
n
t 

(A
)

1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600
0

1000

2000

3000

4000

X: 2196

Y: 1005

Frequency (Hz)

T
h
e
 m

a
x
im

a
l 
th

y
ri
s
to

r 
v
o
lt
a
g
e
 (

V
)

 

Fig. 4 The inverter frequency characteristics 



46 E. POPOV, N. HINOV 

The stability and efficiency of the inverter can be studied from the characteristics. In 

general there is a minimum of the input current and power, load voltage, serial capacitor 

voltage and device voltage around the resonance of the load circuit. If the parameters of 

the inductor heater vary during the induction heating process it is advisable to maintain 

almost constant input current (power) in a slight capacitive detuning of the load circuit 

Rl,Ll,Cl, where the operation is stable, by exercising an influence on the controlling 

frequency. The calculated slope of the frequency characteristic of the input current helps 

for determining the parameters of the closed loop automatic control system. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10
-3

-1000

-500

0

500

1000

Time (s)

T
h
e
 i
n
v
e
rt

e
r 

c
u
rr

e
n
t 

(A
)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10
-3

-1000

0

1000

2000

X: 0.000386

Y: 1045

Time (s)

T
h
e
 d

e
v
ic

e
 v

o
lt
a
g
e
 (

V
)

 

Fig. 5 The detailed simulation results at f 2195 Hz  

The steady state is simulated by a MATLAB program based on the method described 

in [16] for direct determination of the steady state mode (in the case for the discontinuous 

inverter current mode) that is experimentally confirmed. The results from the frequency 

characteristics, from the simulation and from experiments are in good agreement according 

to Table I. That confirms the correctness of the whole study. The detailed simulation results 

of the power inverter are graphically displayed (Fig. 5) for f=2195 Hz. The first diagram 

shows: the inverter current iCK [A] (solid line). The second diagram shows: the voltage 

across a device uVS1 [V] (solid line). The simulation results for the same circuit at f=694 Hz 

displaying an interesting case – a non typical but possible operation in an inductive 

detuning of the load generally with the third harmonic component of the inverter current 

(a case that is difficult to be studied analytically) are given in Fig. 6. 



 The Steady State Mode of Power Inverters for Electro Technology Applications 47 

Table 1 Results for fig. 3 

f[Hz] 694  1823   2018 2083 

Study Sim. Fr.ch. Sim. Fr.ch. Exper. Sim. Fr.ch. 

Load Ind.  Ind.   Ind. Res. 

Id [A] 169.5 400.7 388.7 200 200 196.5 178.2 

Ul[V] 200.4 304.4 303.5 215.4 217 215.8 201.7 

tq.c.[S] 604 113.6 125.9 61 65 68.3 63.8 

UCKm[V] 1456 1308 1270 590 585 580 509 

UVSm[V] 1927 1342 1342 866 870 875 863 

 

f[Hz] 2083 Recom.  2195  2394 

Study Sim. Fr.ch. Exper. Sim. Fr.ch. Sim. 

Load Res.   Cap.  Cap 

Id [A] 180.1 199.8 200 208.2 400.6 416.3 

Ul[V] 206.6 211.9 218 222.1 302.2 314.1 

tq.c.[S] 72.5 82.9 85 89.9 97.3 99.6 

UCKm[V] 515 542 560 565 996 1034 

UVSm[V] 886 1003 1030 1045 1636 1689 

4. AN EXAMPLE WITH A SERIAL RESONANT INVERTER WITH FREE WHEELING DIODES 

A real circuit is given in Fig. 7. The inverter frequency characteristics are graphically 

displayed in Fig. 8. They correspond to an inverter (PT2-50-4000) with: Ud=500 V; 

LK=0.3 mH; CK=4 F; Rlr=4 , flr=4000 Hz, coslr=0.24254, n1=0, n2=1, f=3000 – 
4800 Hz. The graphics show: the average input current Id [A] (solid line); the load voltage 

ULlm [V] (solid line). For each frequency there is a check whether the shape of the load 

voltage is close to the sine wave. 

The increase of the controlling frequency leads to increase of the input current 

(power), load voltage, peak serial capacitor voltage and RMS value of the inverter current 

and to decrease of the circuit turn-off time. But around the load resonant frequency the 

character of most functions is opposite (inflexed points) and the changes of parameters 

are not so large. Therefore, if the load Rl,Ll varies during the induction heating, it is 

advisable to maintain a resonance of the load circuit Rl,Ll,Cl  by  influencing  the 

controlling frequency. The results from the frequency characteristics, from the simulation 

of the steady state mode and from experiments are compared in Table 2. They are in good 

agreement confirming the correctness of the results. 

The steady state results for f=4000 Hz are shown in Fig. 8. They are: iCK [A] (solid 

line), ul [V] (solid line). 



48 E. POPOV, N. HINOV 

0 0.5 1 1.5 2 2.5 3

x 10
-3

-4000

-2000

0

2000

4000

Time (s)

T
h
e
 i
n
v
e
rt

e
r 

c
u
rr

e
n
t 

(A
)

0 0.5 1 1.5 2 2.5 3

x 10
-3

-2000

-1000

0

1000

2000

X: 0.00129

Y: 1926

Time (s)

T
h
e
 d

e
v
ic

e
 v

o
lt
a
g
e
 (

V
)

 

Fig. 6 The detailed simulation results for f 694 Hz  

U
d

R
l

C
K

L
K

i

VD1

VD4

VS1

VS4

VS2

VS3

VD2

VD3

i
d

L
l

C
l


0


0

 

Fig. 7 A real resonant inverter with free–wheeling diodes 

3000 3200 3400 3600 3800 4000 4200 4400 4600 4800
0

50

100

150

200

X: 4000

Y: 74.92

Frequency (Hz)

T
h
e
 a

v
e
ra

g
e
 i
n
p
u
t 

c
u
rr

e
n
t 

(A
)

3000 3200 3400 3600 3800 4000 4200 4400 4600 4800
0

200

400

600

800

X: 4000

Y: 385.7

Frequency (Hz)

T
h
e
 l
o
a
d
 v

o
lt
a
g
e
 (

V
)

 

Fig. 8 The inverter frequency characteristics 



 The Steady State Mode of Power Inverters for Electro Technology Applications 49 

Table 2 Results for Fig. 8 

Hz f= 3200 f= 3600 Recom. f= 4000 

Par. Fr.ch. Sim. Fr.ch. Sim. Fr.ch. Exper. Sim. 

Id ,A 16.74 16.62 88.53 88.39 74.92 75 74.34 

Vl ,V 182.3 182.3 420.2 420.5 385.7 387 385.6 

tq.c,S. 69.83 70.31 35.98 36.88 22.33 22 23.19 

VCKm,V 1683 1700 2129 2125 1320 1325 1320 

ICK ,A 94.20 94.58 137.7 137.8 96.77 97 96.75 

 

Hz f = 4400 f= 4800 

Par. Fr.ch. Sim. Fr.ch. Sim. 

Id ,A 75.81 75.45 182.0 181.7 

Vl ,V 388.7 388.5 603.1 602.8 

tq.c,S. 31.13 31.31 24.28 24.42 

VCSm,V 1548 1551 3120 3121 

IS ,A 122 122 268 268 

0 50 100 150 200 250 300 350 400 450 500
-200

-100

0

100

200

X: 288.8

Y: 145.8

Time (0.000001 s)

T
h
e
 i
n
v
e
rt

e
r 

c
u
rr

e
n
t 

(A
)

0 50 100 150 200 250 300 350 400 450 500
-1000

-500

0

500

1000

X: 291

Y: 546.1

Time (0.000001 s)

T
h
e
 l
o
a
d
 v

o
lt
a
g
e

 

Fig. 9 The detailed simulation results at f 4000 Hz  

5. CONCLUSIONS 

Serial RLC inverters with or without free wheeling diodes for induction heating are 

investigated. A novel unified interpretation of the electromagnetic processes is applied 

based on the previously calculated inverter network parameters in a normalized form. The 

frequency characteristics, the steady state simulation parameters and the experimental 

results are obtained and mutually confirmed. That proves the correctness of the whole 

study. The requirements for the control system are defined, which is useful for research on 

the type of offline simulation, hardware in the loop and rapid prototyping. The main 

contribution of the work is an approach to analysis of series RLC DC/AC converters for 



50 E. POPOV, N. HINOV 

induction heating. This allows the creation of methods for engineering design that are 

simple, such as mathematical ones, but with sufficient precision for engineering practice. 

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