Calculation Algorithm For Single Phase Bridge Rectifier With Capacitor Filter [PDF]

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Jelgava, 24.-25.05.2012.

APPROXIMATE CALCULATION ALGORITHM FOR SINGLE PHASE BRIDGE RECTIFIER WITH CAPACITOR FILTER Armands Grickus, Agris Treimanis

Research Center of Liepaja branch of Riga Technical University, Latvia [email protected], [email protected] Abstract. In the paper approximate calculation algorithm for single phase rectifier with C filter is described. The offered algorithm contains formulas which allow determining capacitor current, total power factor as well as capacitor rated ripple current and equivalent series resistance without referring to datasheet. Although the formulas used in this algorithm are based on simplified mathematical models, there are influential circuit parameters observed such as transformer leakage inductive reactance, diode slope resistance and diode commutation interval. The algorithm corresponds to the analyzer style. It means that there are completely given single phase transformer, diodes, capacitor battery parameters and load resistance, and the analyzer finds out voltages, currents and powers. The algorithm contains only evaluated formulas so there is no need for advanced mathematics operations. It is tested on MS Excel. Calculation in Excel is not fully automated: there are two manually operable cells for leading the iteration process to its convergence. Keywords: capacitor filter, rectifier, power factor, ripple current, calculation algorithm.

Introduction The authors of the paper had a target to discover a full and easy-to-use analyzing algorithm for a single phase bridge rectifier with a capacitor filter. As readily available calculation tool software MS Excel was chosen. So, the formulas in the algorithm had to answer to Excel functionality. It means that the formulas must not contain advanced mathematical operations. As a base for the new development a simple algorithm from the reference was taken [1]. Reference analysis One of the key formulas which were taken from the reference [1] is formula (12) in its modified form – peak value of secondary winding current. The second formula in its modified form is (15) – conduction interval of secondary current. The third formula in its modified form is (16) – average output voltage. Formulas (11), (13), (36), (38) are taken from the reference [2]. Formula (31) is common in several capacitor manufacturer datasheets. Structure of the algorithm The introduced formulas are (5), (19), (20), (24), (27), (28), (33), (48). The rest of the formulas are basic electrical engineering formulas. In the formula evaluating process online calculation tool Wolphram Alpha was used. As a circuit simulator 5SPICE was used. Due to the algorithm complexity there is a necessity for the iterative process. Since software EXCEL cannot automatically calculate such a problem without specially programmed macros, there must be break points introduced to avoid “Circular Reference” error. In EXCEL spreadsheet values from formulas (13) and (17) are manually operated to get the iteration process to its convergence. Consequently the algorithm is semi-automatic, however, it can be supplemented with specially programmed macros which allow obtaining fully automated operation. The input values are presented as the following physics exercise. The given single phase transformer connected to f = 50 Hz AC grid. No load voltage at the secondary winding is equal to V20 = 25V, short circuit resistance RT = 0.26 Ω, short circuit reactance XT = 0.23 Ω. Also the given single phase bridge rectifier with one diode threshold voltage V(T0) = 0,78V and slope resistance rT = 0.055 Ω. At the rectifier output a capacitor battery is connected which contains NCP = 1 capacitors in parallel and NCS = 1 capacitors in series. Each capacitor capacitance C = 16500 µF and rated voltage VCR = 50 V. Load resistance Rload = 5.91 Ω. All formulas operate correctly with SI units. Formula series which form the mentioned algorithm follow here. AC grid oscillation period T:

456

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Jelgava, 24.-25.05.2012.

T=

1 . f

(1)

Total resistance of transformer and diode RΣ:

RΣ = RT + rT .

(2)

Total impedance of transformer and diode ZΣ:

Z Σ = RΣ2 + X T2 .

(3)

Equivalent capacitance of capacitor battery Cekv:

C ekv = C ⋅

N CP . N CS

(4)

Equivalent series resistance of single capacitor ESR:

ESR =

0.02 . C ⋅ VCR

(5)

Equivalent series resistance of capacitor battery ESRekv:

ESRekv = ESR ⋅

N CS . N CP

(6)

Capacitive reactance of capacitor battery XC:

XC =

1 . 2π ⋅ f ⋅ C

(7)

Total resistance in circuit RΣC:

RΣC = RΣ + ESRekv .

(8)

2 Z ΣC = RΣC + X T2 .

(9)

Total impedance in circuit ZΣC:

No load peak voltage of secondary winding V20p:

V2PK = 2 ⋅ V20 .

(10)

Commutation interval Tcom:

Tcom =

 I ⋅Z T ⋅ arccos1 − d Σ 2π U 20P 

  , 

(11)

where Id – load current. Peak value of secondary winding current I2PK:

I 2PK =

Vd0

.

(12)

2 2 ⋅ RΣC + ( X T − X C ) 2 ⋅ Rload

Peak forward voltage on single rectifier diode VF:

VF = V(T0) + I 2PK ⋅ Rd .

(13)

No load voltage on capacitor battery Vd0:

Vd0 = V2PK − 2 ⋅ VF .

457

(14)

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Conduction interval of secondary current TCI:

Z ΣC T T ⋅4 + com . 2 2 ⋅ Rload 2

TCI =

(15)

Average output voltage at load VdAVG:

 ZΣ VdAVG = Vd0 ⋅ 1 − 2 ⋅ Rload 

 .  

(16)

Load current Id:

Id =

VdAVG . Rload

(17)

TCI ⋅ 2π . T

(18)

Conduction angle of secondary current θCI:

θ CI = Short circuit peak current I2SPK ():

I 2SPK =

I 2PK  π − θ CI 1 − sin   2

  

.

(19)

Output voltage increasing angle θCU:

I   π − θ CI  + d .  2  I 2SPK 



θ CU = π − 2 ⋅ arcsin sin 

(20)

Output voltage increasing interval TCU:

T . 2π

(21)

T − TCU . 2

(22)

I d TDU . C ekv

(23)

TCU = θ CU Output voltage decreasing interval TDU:

TDU = Output voltage ripple swing ∆Vd:

∆V d = Output voltage minimum Vdmin:

3 T  Vdmin = VdAVG − ∆Vd ⋅  − CU  . 4 T 

(24)

Maximum output voltage Vdmax:

Vdmax = Vdmin + ∆Vd .

(25)

∆V d ⋅ 100% . Vd

(26)

Output voltage relative ripple ∆Vd%:

∆Vd% = Secondary RMS current I2RMS:

458

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I 2RMS =

1 2π

Jelgava, 24.-25.05.2012.

θ CI   2 2 2 θ CI 3I 2SPK + 4 I 2SPK I 2PK + 2 I 2 PK + I 2SPK sin θ CI + 8I 2SPK I 2 PK − I 2SPK sin 2  . (27)  

(

)

(

)

Secondary average current I2AVG:

I 2AVG =

θ  1 θ CI I 2PK − I 2SPK + 2 I 2SPK sin CI  .  2  π

(

)

(28)

Secondary RMS current in diode IVDRMS:

I 2 RMS

I VDRMS =

.

(29)

I 2AVG . 2

(30)

2

Secondary average current in diode IVDAVG:

I VDAVG = Single capacitor leakage current ILEAK:

I LEAK = 0.03 ⋅ C ⋅ VCR .

(31)

Capacitor battery leakage current ILEAKΣ:

I LEAKΣ = N CP ⋅ I LEAK .

(32)

Capacitor rated ripple current IRIPPLE: 0.5 I RIPPLE = 20.3 ⋅ C 0.79 ⋅ VCN .

(33)

Capacitor battery RMS current ICRMSΣ:

I CRMSΣ = I 22RMS − I d2 .

(34)

∆PTR = I 22RMS ⋅ RT .

(35)

Power losses in transformer ∆PTR:

Power losses in single diode ∆PVD1: 2 ∆PVD1 = V(T0) ⋅ I VDAVG + I VDRMS ⋅ rT .

(36)

Total power losses in four diodes ∆PVD4:

∆PVD4 = 4 ⋅ ∆PVD1 .

(37)

2 ∆PC = VdAVG ⋅ I LEAKΣ + I CRMS Σ ⋅ ESR EKV .

(38)

Power losses in capacitor ∆PC:

Secondary winding RMS voltage at given load:

V2RMS = V20 − I 2RMS ⋅ RT .

(39)

RMS current in single capacitor:

I CRMS =

I CRMSΣ . N CP

(40)

Vdmax . N CS

(41)

Maximum voltage on single capacitor:

VCmax =

459

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Load power:

Pd = VdAVG ⋅ I d .

(42)

P1 = Pd + ∆PTR + ∆PVD4 + ∆PC .

(43)

Total consumed active power:

Total from grid consumed apparent power:

S1 = V20 ⋅ I 2 RMS .

(45)

Total from transformer consumed apparent power:

S 2 = V2RMS ⋅ I 2 RMS .

(46)

Total power factor:

χ1 =

P1 . S1

(47)

Displacement power factor:

T  cos ϕ1 = cos com ⋅ 2π  .  T 

(48)

Distortion power factor:

ν1 =

χ1 . cos ϕ1

(49)

Pd . P1

(50)

Efficiency:

η=

Calculation result comparison 5SPICE was used to compare some of the calculation results. SPICE circuit component parameters are visible in Figure 1, while the corresponding simulation graph is visible in Figure 2. The red line is I2 curve, orange line is Vd curve while the green line is V2 curve. In the circuit diode bridge GPBC1510 was used. Approximate SPICE parameters of one diode of this bridge are given in Table 1. Most useful datasheet parameters of single diode of the bridge GBPC1510 are as follows: VF = 1.23 V, V(T0) = 0.78V, rT = 55 mΩ, IFAV = 15 A, IFSM – 300 A.

Fig. 1. Circuit schematic in 5SPICE circuit editor 460

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Fig. 2. SPICE simulation graph Table 1 Diode SPICE parameters Parameter IS RS N EG CJO VJ M FC TT BV IBV AF KF XT1

Unit A Ω eV F V

s V A

SKN20 170n 0.009 2 1.55 400p 1 0.35 0.5 10n 400 4m 1 0 3

GBPC1510 6.5n 0.028 1.45 1.11 300p 1 0.494 0.5 10n 1000 5u 1 0 3

Excel spreadsheet with the corresponding values is visible in Figure 3 where the green cells contain the given values; gray cells are manually operable ones while the yellow cells display key values for comparison. Calculation result comparison between Excel algorithm and SPICE simulation is given in Table 2. Table 2 Result comparison between Experiment, Excel algorithm and SPICE model Parameter TCI Vd0 I2PK VdAVG Id TCU TDU ∆Vd I2RMS I2AVG ∆PVD4 V2RMS

Unit ms V A V A ms ms V A A W V

Experiment 5.5 33.4 12.3 25.7 4.35 4.4 5.6 1.45 6.6 4.3 13 23.3 461

Excel 5.0 32.9 15.2 26.6 4.5 4.7 5.3 1.6 7.8 5.0 14.4 23.0

SPICE 5.2 – 14.3 – – 4 6 1.50 – – – –

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Table 2 (continued) Parameter ICRMS Vdmax Vdmin Pd P2

Unit A V V W W

Experiment 4.9 – – 112 125

Excel 6.3 27.3 25.8 120 135

SPICE – 27.3 25.8 – –

Analyzing the calculation results we can note that significant calculation error (18 %) is introduced in formula (27) – I2RMS. Formula (27) is evaluated on the simplified mathematical model assuming that ∆Vd = 0, besides, the result is calculated using the values of I2PK and I2SPK, which are also calculated through approximated models. The rest or the formulas which are not after I2RMS give the value within 4% tolerance range.

Fig. 3. Excel spreadsheet example Conclusions As a result of the research new series of formulas suitable for full analysis of the single phase bridge rectifier with a capacitor filter using widely available MS Excel software were introduced. The given series of formulas written into the spreadsheet (without macros) form the semi-automatic analysis algorithm which gives the voltage values within 4 % tolerance range, current values in 18 % tolerance range while the power values in 8 % range. References 1. Титце У., Шенк К. Полупроводниковая схемотехника (Advanced electronic circuits). Перевод с немецкого под редакцией д-ра технических наук А.Г. Алексенко. Москва: Мир, 1982., стр. 253. – 256. (In Russian). 2. Ivars Raņķis, Inna Buņina (Rodionova), Energoelektronika, trešais atkārtotais izdevums. Rīga: RTU, 2007., 186 lpp. (In Latvian).

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