Frequency converter energetic performances analysis

Frequency Converter Energetic Performances Analysis 1

Horia Balan1, Mircea Chindris1, Ioan Vadan1, Aurel Botezan1, Victor Proca2

Technical University of Cluj Napoca, 15th Daicoviciu Street, 400020 Cluj Napoca, Romania 2 Researches, Development and Testing National Institute for Electrical Engineering 144th Calea Bucuresti Street, 200515 Craiova, Romania

Abstract – The issue of non-conventional energy sources now is correlated with energy efficiency and clean energy. An application researched by the authors is vibration stress relieving, which consist in vibration of stress reliving piece with electromagnetic or electro-dynamic vibrator to determine frequencies of resonance. Technology is energy efficient and involves a single-phase inverter with variable frequencies, which generate a load current with minimal distortion factor in vibrator coil. The studied and experimental solution used by the authors is a classic inverter with LC turn-off circuit, who use sinusoidal modulation as command principle. This solution is compared from energetic point of view with un-modulated inverter and hybrid commutation inverter. For energy efficiency point of view power factor is interested. Active power P and total power S is analytic determined and is showed their dependence on quality factor Q. Paper emphasis dependence between angle of modulation α and power factor cosφ, for different values of resonance frequencies who making stress relieving.

I.

INTRODUCTION

This paper proposes the study of three cases of dc-ac converters concerning power factor problems: • LC turn-off converter; • single modulation inverter; • hybrid commutation inverter. At any electro-mechanic devices for electrical energy conversion is putting the problem of efficiency. Systems which have on base static switching follow through conversion to obtain wave shape appropriate by ideal shape form (sinusoidal) and this to obtain high efficiency, and low losses. In fig. 1 is presented an electromagnetic vibrator used for vibration stress relieving system. The coil of this electromagnetic vibrator will be supplied with variable

Fig. 1 Electromagnetic vibrator This work was supported by Romanian Ministry of Education and Research, the CEEX project 153/2006

1-4244-0891-1/07/$20.00 2007 IEEE

voltage and frequency from an inverter. II. STUDY OF INVERTER – ELECTROMAGNETIC VIBRATOR SYSTEM, FOR UN-MODULATED VOLTAGE INVERTER For determination of vibrator coil current, we consider inductive load (electromagnetic vibrator), running with an instantaneous displacement of mobile armature described by time harmonic function, with double radian frequency as the radian frequency of supplying voltage ua. Hypothesis above is in concordance with reality because the system “electromagnetic vibrator – mechanic receiver”, has a high rigidity mechanical factor for vibration stress relieving technology and is selective for harmonics of active force which actuate the mobile armature. Sinusoidal movement approximation is in connection with condition that oscillating system remains almost conservative [1]. Mobile coil inductivity expression for electromagnetic vibrator is:

La =

N a2 ⋅ µ 0 ⋅ A L0 L0 ⋅ = = l0 1 − x / l0 1 − x / l0

L0 = 1 − ε ⋅ sin( 2 ⋅ ω ⋅ t )

,

(1)

where: La is mobile coil inductivity; µ0 is free space magnetic permeability; Na is number of turns for vibrator coil; A is magnetic pole surface area; l 0 is the initial length of air-gap; x is mobile armature displacement; ω is supply voltage radian frequency; ε is ratio between vibration amplitude of x and l0. In vibration stress relieving technology experimental results show that the high value of inductivity, specific to reactive electromagnetic vibrators, and high values of load currents, make possible, in this situation, the neglecting of current coil resistance in report with reactance. Starting from the zero initial conditions, from the equilibrium equation of voltages are obtained extreme current values in first voltage semi period i0 and i1:

i0 =

U0 π ⋅ , i1 = 0, L0 ω

and load current expression is:

(2)

ia (ω ⋅ t ) =

U0 L0

 ω ⋅ t − (n − 1) ⋅ 2π  ω   cos 2 ⋅ ωt − 1 +ε ⋅ 2⋅ω 

For k≥2, initial phase γk can be approximated with π/2. RMS value of voltage ua, is giving by expression:



+ ,

(3)

  

U aef = U 0 ⋅ 1 −

ω ⋅t 

− n ⋅ 2π

ω

  , −1  

+

cos 2 ⋅ ωt + ε ⋅ 2 ⋅ω 

(4)

(12)

iaef =

U0 π2 ε2 ⋅ + , ω⋅L 3 4

(13)

Active power can be calculated with known expression [2]:

for (2n-1)π≤ωt≤2nπ, identically with current from alternation n. The current shape is quasi-triangle; the deviation from this form is determinate by work frequency and value of ratio between vibration maxim amplitude and value of work air gap l0. Functions definite by expressions (3) and (4) admit development in Fourier series with following coefficients:

π

U A0 = ⋅ 0 , 2 ω ⋅ L0 2 ⋅ Bk =

,

where modulation angle α is zero. RMS value of load current is obtained from expressions (3) and (4):

for (2n-1)π≤ωt≤(2n-1)π, and

U ia (ω ⋅ t ) = − 0  L0 

α π

(5)

1 2 ⋅U 0 ⋅ ε k 2 + 2 ⋅ (cos k ⋅ π − 1), (6) π ω ⋅ L0 k (k 2 + 4) 1 2 ⋅U 0 1 2 ⋅ Ck = ⋅ (cos k ⋅ π − 1), (7) π ω ⋅ L0 k 2

P = U0 ⋅ I0 +

∑U ∞

k

⋅ I k ⋅ cos γ k ,

(14)

k =1

After successive transformations result:

P=

1

π2

⋅

∑

8 ⋅ U 02 ∞ 1 ⋅ ⋅ ω ⋅ L0 k =1 k 2

k2 +2 1 ⋅ ε2 ⋅ 2 + ⋅ cos ϕ k k +4 k2

,

(15)

Or with approximation from expression (11):

P=

1 8 ⋅ U 02 ⋅ π 2 ω ⋅ L0

3 ε + 1. , 5

(16)

2 ⋅ U 0 cos kπ − 1 k2 +2 1 Total power, take into account expressions (12) and (13), ⋅ ⋅ ε2 ⋅ 2 + 2 , (8) is: π ω ⋅ L0 k k +4 k U 02 π2 ε2 and load current expression is: (17) S= + , 4 ω ⋅ L0 3 1 4 ⋅U 0 π U0 ia (ωt ) = ⋅ − ⋅ ⋅ Respective, power factor expression: 2 ω ⋅ L0 π ω ⋅ L0 Ak =

1

∑

1 k2 + 2 1 ⋅ ( ε2 ⋅ 2 + sin( k ⋅ ωt + γ k ) k + 4 k2 k =1 k ∞

,

(9)

cos ϕ =

including continuous component and odd harmonics, and harmonic initial phase of range k is:

γ k = arctg

k2 +4 , ε ⋅ k ( k 2 + 2)

16

π2

⋅

3 3⋅ε 2 + 5 , ⋅ 5 3⋅ε 2 + 4 ⋅π 2

Power factor cosφ variation, varying with inductivity modulation degree is shown in fig. 2.

(10)

The effect of inductivity modulation L0 is manifesting through existing Bk components for odd harmonics and under unit value of ε coefficient allows simplification of (9):

ia (ωt ) =

π

U0 1 4 ⋅U 0 − ⋅ ⋅ 2 ω ⋅ L0 π ω ⋅ L0 ⋅

∑

∞ 3 1 ⋅ ( ε 2 + 1 sin(ωt + γ 1 ) + ⋅ cos kωt ) 2 5 k =2 k

,

(18)

(11) Fig. 2 Power factor variation of reactive electromagnetic vibrator supplied to inverter

III.

LOADS SUPPLIED WITH SINGLE MODULATED VOLTAGE

Existing applications of converters, for example vibration sources commanded with adjustable amplitude and frequency [1], where for first is putting problem of harmonic response. Energetic performances must not be neglected and for this pain focused to find most suitable compromise “harmonic response-energetic performances”. Output voltage of autonomous inverter with LC turn off circuit presented in fig. 3, who running with pulse width modulation, it is usual referred to square output voltage [3]. Ratio between maxim value of fundamental output voltage time modulated and fundamental value of square voltage, define inverter voltage efficiency:

η u = cos

α 2

,

(19)

who is decreasing with increasing of command angle α . In case of multiple modulations, the ratio between frequency of modulator and modulated signal, affects the value of fundamental output voltage. In the case of high frequency the keeping of a small ratio between frequency of modulator and modulated signal, as low as possible is compulsory; but at low frequencies the effect of ratio on fundamental value is smaller. Total power of the system is defined as well as in sinusoidal regime, through product of load current and voltage RMS values:

S=

I M ⋅ KU ⋅U 0

π

⋅ α

− Q ⋅ (π − α ) ⋅ (π − α ) 2 − ⋅ (1 + e Q ) KU

, (20)

I aef

U aef = U 0 ⋅ 1 −

1 (U 0 + T 0

∑ u ) ⋅ (I + ∑ i ) ⋅ dt ∞

∞

k

k

0

k =1

,

(22)

k =1

where:

u k = 2 ⋅ U k ⋅ sin( kωt + γ uk ) = 2 2U 0 kα kα , ⋅ cos ⋅ sin( kωt − ) 2 2 π ⋅k 2U 0 ⋅ ik = 2 ⋅ I k ⋅ sin( kωt + γ ik ) = 2 ⋅ π ⋅ Ra

(23)

= 2⋅

⋅

1 − cos kπ k ⋅ k 2 ⋅Q2 +1

⋅ sin( kωt −

⋅ cos

kα ⋅ 2

,

(24)

kα π +γk − ) 2 2

Uk and Ik results:

2 2 ⋅U 0 kα ⋅ cos ; πk 2 , (25) kα 2 1 − cos kπ ⋅ cos Ik = ⋅ π ⋅ Ra k k 2 ⋅ Q 2 + 1 2

Uk =

with expression (21) and (23), through integration of expressions from (22), active power is:

P=

∑

∞ 2U 02 (1 − cos kπ ) ⋅ (1 + cos kα ) ⋅ ⋅ 2 π ⋅ Ra k =1 , (26) k 2 ⋅ k 2 ⋅ Q2 +1

⋅ sin γ k ,

with Iaef and Uaef, defined by expressions:

(π − α ) ⋅ K U − Q ⋅ (1 + e = I M ⋅ KU ⋅ π ⋅ KU

∫

T

P=

Or consider expression tgγ k = tg (1 /( k ⋅ Q)) : −

α Q

)

P=

; , (21)

α π

∑

∞ 2U 02 (1 − cos kπ ) ⋅ (1 + cos kα ) , ⋅ 2 π ⋅ Ra k =1 k 2 ⋅ k 2 ⋅Q2 +1

(27)

Value of active power depends on quality factor Q. At high value of Q, effect of upper harmonics is neglected, and expression of active power is:

4U 02 (1 + cos α ) , P= 2 ⋅ π ⋅ Ra Q2 +1

Active power which represents average value on a period of instantaneous power is [4]:

(28)

and power factor is:

cos ϕ =

Fig. 3 Single phase inverter with LC turn off circuit

4 1 + cos α ⋅ 2 π (π − α ) Q + 1 α

,

(29)

− Q 1− ⋅ (1 + e Q ) K U ⋅ (π − α )

The dependence of power factor to command angle α , the work frequency being a parameter, is presented in fig. 4.

respectively the power factor is:

cos ϕ =

2 ⋅ Ra I aM 2 ⋅ Ra2 + (ω ⋅ La ) 2 I aM

= ,

(33)

1

1 = ≈ 2 Q 1+ Q

Fig. 4 Power factor of inductive load supplied with single modulated voltage

Power factor has a maximum value for command angle equal to π / 4 , easy decreasing with increasing of work frequency and not much different to these obtained for canceled modulation ( α =0), total different to the values obtained when α → π , which converge to zero. Maximum values of power factor are decreasing with frequency increasing. IV.

In certain applications, the authors used an inverter with hybrid commutation, forced commutation and commutation from load (fig. 5). From the analysis made to capacitive and inductive loads [1], the authors draw the conclusion that inverter with hybrid commutation, can running at resonance frequency of series circuit, if are carried out some conditions [1], case when the load current has a variation of the form: (30)

Voltage drop at load terminals is:

u a (t ) = u La (t ) + u Ra (t ) =

= I aM (Ra sin ωt + ωLa cos ωt ) = ,

2 ⋅ Ra Ph I aM π = = > 1, (34) Pr ((U 0 ⋅ I aM ⋅ 2 ⋅ cos ϕ 2 ) / π ) 2 ⋅ cos ϕ 2

Or ratio between the active power Ph and Pf the power generated by autonomous inverter with LC turn off circuit:

INVERTER WITH HYBRID COMMUTATION

ia (t ) = I aM ⋅ sin ωt ,

the approximation being valid for strong reactive loads and starting from some values of work frequency. Therefore if work frequency is fixed and is used an inverter with hybrid commutation, the load must be designed realizing the most adequate compromise with expression (33). The ratio between the active power, Ph generated by inverter with hybrid commutation and Pr, generated by inverter with commutation from load [1], is:

2 ⋅ Ra Ph I aM = = 2 Pf 4 ⋅ U 0 (1 + cos α ) / π 2 ⋅ Ra Q 2 + 1

(

π 2 ⋅ (Q 2 + 1) = >1 4 ⋅ (1 + cos α )

= I aM Ra2 + (ωLa ) 2 sin(ωt + α 3 ) where: α 3 = arctg (ω ⋅ La / Ra ) = arctgQ, and developed active power is: 2 2 P = I aM ⋅ Ra2 + (ω ⋅ La ) 2 ⋅ cos α 3 = I aM ⋅ R, (32)

(

))

, (35)

These expressions put in evidence the superior operation, from energetic point of view, of inverter with hybrid commutation comparative with inverter with commutation from load, respectively forced commutation. V.

(31)

)(

CONCLUSIONS

This study shows the superior operation, from energetic point of view, of inverter with hybrid commutation comparative with inverter with commutation from load, respectively forced commutation. The using of thyristors inverter instead IGBT and MOS inverter has some advantages: their high limit value parameters (current and voltage) supported by these devices as well as high capacity in overload regime. These advantages make the thyristors inverter suitable for HVDC systems (High Voltage Direct Current), with high values of voltages (hundred kilovolts) and currents (tens of kilo amperes). REFERENCES [1] Balan, H., Vibration sources electronic commanded in non-conventional

Fig. 5 Inverter with hybrid commutation

technologies, PhD Thesis, Technical University of Cluj Napoca, 1994; [2] Magureanu, R., Micu, D., Frequency Static in Drive with Asynchronous Motors. Technical Publishing House, Bucharest, 1988; [3] Erickson, R. W., Maksimovic, D., Fundamentals of Power Electronicsecond edition. Kluwer Academic Publishers, Norwell, Massachusetts, 2004; [4] Zhang, Y., Spen, P.C., D-Q models for resonant converters, Proceedings 35th Annual IEEE Power Electronics Specialists Conference, Aachen, Germany, 2004, pp. 1749-1753;