Analysis of an AC-DC Full-controlled Converter … of an AC-DC Full-controlled Converter Supplying Separately Excited DC Motor Parallel with ... Eng. Sci., vol. 13 no. 2, pp ... 44

Analysis of an AC-DC... 43

Analysis of an AC-DC Full-controlled Converter Supplying Separately Excited DC Motor Parallel with

Series DC Motor Loads

YUSUF A. Al-TURKI

MOHAMMED M. AL-HINDAWI and OBAID T. AL-SUBAIE

Department of Electrical and Computer Engineering,King AbdulAziz University, Jeddah, Saudi Arabia

ABSTRACT. Controlled rectifiers are widely used as a source of DCmotors power supply. It is important to study motor characteristicswhen fed from such converters. Early work has studied single motorcharacteristics when fed from a converter. This paper is concernedwith the study of an AC-DC full-controlled converter supplying separ-ately excited DC motor parallel with series DC motor loads. Continu-ous and discontinuous converter currents are considered. The criticalfiring angle is deduced. The margin firing angle that separates motoroperation from generator operation of the separately excited machineis investigated. The performance characteristics have been derivedand studied for each of: constant firing angle, constant torque of theseparately excited motor and constant horsepower of the series motor.Waveforms for each load current and converter current are investigat-ed for different modes of operation.

KEY WORDS: Controlled rectifiers, AC-DC converters, Full-controlledconverters, Series motor, separately excited motor, Paral-lel loads.

1. Introduction

DC motors are found in many applications in our daily life. Such motors areheavily used in industrial processes, traction and elevators, entertainment “in-dustry”, toys and home appliances. Separately excited DC motor is used in ap-plications requiring independent field control where field supply is separatefrom the armature supply. On the other hand, the series motor has both field and

43

JKAU: Eng. Sci., vol. 13 no. 2, pp. 43-71 (1421 A.H. / 2001 A.D.)

Yusuf A. Al-Turki, Mohammed M. Al-Hindawi & Obaid T. Al-Subaie44

armature windings connected in series. In this manner, armature supply controlsboth field and armature circuits. It is common in industry to fix more than oneDC motor connected in parallel and fed from the same source. At present, solidstate converters with high ratings are available. This led to the use of such con-verters in supplying DC motors from AC source. The most common method ofconverter control is the phase control. In this method, thyristor commutation isachieved by line commutation and, therefore, no additional circuitry is needed.This makes it simple, less expensive and it requires minimal maintenance[1].

Some researchers have studied converter performance connected to a passiveload. Chung[2] presented a steady-state analysis of a two-branch resistance-inductance parallel circuit controlled by an AC switch. Al-Juhani[3] also madean analysis of an AC-DC converter supplying two-parallel inductive loads. Re-garding the active load, Doradla and Sen[4] presented one of the early attemptsto study solid state series motor drive. They made an evaluation of controlschemes for thyristor-controlled DC motor[5]. Harmonics presented by a con-verter supplying a single-motor have also been studied[6]. Most of the work re-ported in the literature is related to converter supplying a single DC motor ortwo parallel passive loads.

In this paper, analysis of a phase-controlled (AC-DC full-controlled) convert-er feeding separately excited DC motor parallel with series DC motor loads ofdifferent parameters and speeds will be investigated. The steady-state analysisof this system will be obtained for each of the two modes of operation, (i.e. con-tinuous and discontinuous converter current). An expression for the criticalfiring angle (at which the converter current changes from one mode to the other)will be deduced. The margin firing angle (that separates motor operation fromgenerator operation of the separately excited machine) will be investigated. Per-formance characteristics such as input power factor, supply current distortionfactor, torque-speed, and motor current ripple factors have been derived andstudied for: constant firing angle, constant torque of the separately excited mo-tor and constant horsepower of the series motor.

2. Steady State Analysis

The circuit under consideration is shown in Fig. 1, where S1, S2, S3, S4 arefour power thyristors connected as a full converter. The sinusoidal voltagesource is considered to have an rms value of V volts, frequency of ω/2π Hz andzero internal impedance. The load consists of a separately excited DC motorparallel with a series DC motor. The motor currents are controlled by changingthe thyristors-firing angle. The supply current may be continuous or discontinu-ous depending on the firing angle, the load parameters, and motor speeds. Eachof the two modes will be treated separately.


2.1 Continuous Mode

In this mode, the converter current il has a positive value at any instant oftime. However, as the firing angle α increases, the value of il at ω t = α decreas-es until it reaches zero value at the so-called critical firing angle αc , Beyondαc , the converter current will be discontinuous.

Throughout the period of α ≤ ω t ≤ π + α , the voltage across each motor inthe continuous mode is equal to the supply voltage assuming zero drops acrossthyristors. This means that for the separately excited DC motor:

and for the series DC motor,

Where ic1 is the armature current of separately excited DC motor (ia1) and ic2is the armature current of series DC motor (ia2) during the continuous mode.

During this mode, the converter current icl equals the supply current ics and isgiven by:

ics = icl = ic1 + ic2 , α ≤ ω t ≤ π + α (3a)

– ics = icl = ic1 + ic2 , π + α ω t ≤ 2 π + α (3b)

Separately Excited DC Motor Current

If steady-state speed and magnetic linearity are assumed, the solution ofequation (1) is of the form:

(1)

(2)

2 1 1 1 1 1V t i R L

ddt

i K N tc a a c asin , ω α ω π α= + + ≤ ≤ +Φ

2 2 2 2 2 2 2 2 2 2V t i R L

ddt

i K i N K N tc a a c af c ressin , ω α ω π α= + + + ≤ ≤ +

FIG. 1. Full controlled converter supplying separately excited DC motor parallel with series DCmotor loads.


Where:

R1 = Ra1 , and Ka Φ is the constant of the separately excited DC motor,

and Ic1 is the initial value of ic1 at ω t = α .

Under steady state operation, ic1 at ω t = α equals that at ω t = π + α and isgiven by:

The ripple factor of ic1 is given by:

Where Ic1av and Ic1rms are the average and the rms values of the motor cur-rent respectively.

Full expressions for Ic1av and Ic1rms are given in Appendix [A].

Series DC Motor Current

If steady-state speed and magnetic linearity are assumed and the armature cir-cuit resistance Ra2 and inductance La2 include the resistance and inductance ofthe series field winding, the solution of equation (2) is of the form:

Where:

R2 = Ra2 + Kaf2 N2 , Kaf2 and Kres2 are constants of the series motor,

xxxsxxx xxxxxxxxxxxφ2 = tan–1 (wLa2 / R2) , Q2 = ωLa2 / R 2 ,Kc2 = Ic2 – xxx

(V / V2) sin (α – φ2), and Ic2 is the initial value of ic2 at ωt = α.

Under steady state operation, ic2 at ω t = α equals that at ω t = α + α and isgiven by:

The ripple factor of ic2 is given by:

Where Ic2av and Ic2rms are the average and the rms values of the motor cur-rent respectively.

Full expressions for Ic2av and Ic2av are given in Appendix [A].

Z R La2 22

22= + ( ) ,ω 2

(4) i V Z t K e K N R ecl ct a Q

at Q= +2 11 1 1 1 1

1 1( / )sin( – ) – ( / )( – )(–( – ) / (–( – / )ω φ ω ω αΦ

Z R L L R Q L R K I V Za a a c c1 12

12

11

1 1 1 1 1 1 1 1 12= + = = =( ) , tan ( / ) , / , – ( / )sin( – ) ,–ω φ ω ω α φ

(5) I V Z e e K N RclQ Q

a= +– ( / )[ ) /( – )]sin( – ) – ( / ).(– / ) (– / )2 1 11 1 1 11 1π π α φ Φ

K I Icr c rms c av1 12

12 1= / – .

(7) i V Z t K e K N R ec c

t Qres

t Q2 2 2 2 2 2 22 12 2= +( / )sin( – ) – ( / )( – )(–( – ) / ) (–( – ) / )ω φ ω α ω α

(6)

(8) I V Z e e K N RcQ Q

res2 2 2 2 2 22 1 12 2= +– ( / )[( ) /( – )]sin( – ) – ( / ) .(– / ) (– / )π π α φ

(9) K I Icr c rms c av2 2

22

2 1= / – .


Critical Firing Angle

The critical firing angle αc , which is the highest firing angle, for the continu-ous mode satisfies the condition of:

Ic1 + Ic2 = 0 (10)

Therefore, the critical firing angle αc is given from:

Where:

Supply Current

The rms value of the fundamental component of the supply current is:

And the phase angle of that component is given by:

θ = tan–1 (ac1 / bc1) (13)

Where ac1 and bc1 are the amplitudes of the cosine and sine fundamentalcomponents of ics respectively in the continuous mode.

The supply power factor is:

Pf = (Icf / Icsrms) cos θ (14)

And the supply current distortion factor is:

Df = Icf / Icsrms (15)

Where Icsrms is the rms value of the supply current.

Full expressions for Icsrms , ac1 , and bc1 are given in Appendix [A].

2.2 Discontinuous Mode

The discontinuous mode of operation occurs at a firing angles greater thanαc. In this mode, the converter current il increases from zero at ω t = α to amaximum value and then decreases to become zero again at ω t = β which is re-ferred to as the extinction angle. The angle β is, of course, less-than π +α.Thus, the current id1 is zero for β ≤ ω t ≤ π + α .

(11) αc A C A C A B B C A B= ± + + +sin [( ( )( – )) /( )]–1

1 1 12

12

12

12

12

12

12

12

A F F F V Z e e

F V Z e e B F F

C K N R K

Q Q

Q Q

1 1 1 2 2 1 1

2 2 1 1 1 2 2

1 1 1

2 1 1

2 1 1

1 1

2 2

= + = +

= + = +

= +

cos cos , – / ) ( ) /( – ) ,

–( / ) ( ) /( – ) , sin sin , and

( / ) (

(– / ) (– / )

(– / ) (– / )

φ φ

φ φ

π π

π π

αΦ rres N R2 2 2/ ) .

(12) I a bcf c c= +1

212 2/


Detailed derivations of the currents in this mode of operation (conduction andnon conduction periods) could be carried out in a manner similar to the continu-ous mode. Appendices [B&C] give such details.

The developed power and torque of the series DC motor are given by:

Pmotor = Kaf N2 I 2motorrms

, (16)

Tmotor = Kaf I2motorrms

(17)

The developed power and torque of the separately excited DC motor are giv-en by:

Pmotor = Ka Φ N1 Imotorava , (18)

Tmotor = Ka Φ N1 Imotorava , (19)

3. System Performance

In order to study the performance characteristics of the system under consid-eration, a computer program has been developed based on the equations derivedin Section 3. The following data of load parameters are used[1]:

Separately exited Ra1 = 0.6 Ω. Series DC motor : Ra2 = 1.0 Ω.

M1 La1 = 0.006 h. M2 La2 = 0.012 h.

Ka Φ = 0.55 V.sec/rad. Kaf2 = 0.027 h.

V = 120 volts Kres2 = 0.0273 V/rad/s.

3.1 Critical Firing Angle

It is noted from Fig. 2 that, the critical firing angle αc decreases as the speedof any motor or both is increased. The effect of changing M1 field current (orKaΦ) is shown in Fig. 3. The figure reveals that the critical firing angle αc de-creases as the value of KaΦ is increased. Fig. 4 shows the operating regions ofthe separately excited DC motor. In this figure, there are two modes of opera-tion (continuous and discontinuous). For the continuous mode there is a marginangle αmg that separates motor operation from generation operation of the ma-chine. For firing angle α higher than αmg the machine acts as a generator. It isnoticed that αmg is constant in the continuous mode. This could be realized inview of equation (A-1), which shows that I1av is independent of M2 parametersin the continuous mode. Since both Ka Φ and N1 are constants in Fig 4 then αmg(at which I1av = zero) is also constant. For the discontinuous mode, I1av dependson the parameters of both motors as seen in equation (C-1). Therefore, changingN2 would effect αmg leading to a variable αmg in the discontinuous mode. The


intersection of αc and αmg curves divides the operation range into three distinc-tive modes normally:

1. Motor in the continuous mode.2. Motor in the discontinuous mode.3. Generator mode.

FIG. 2. Critical firing angle versus series DC motor speed N2 for different values of separately ex-cited DC motor speed N1.

FIG. 3. Critical firing angle versus motor speed N2 for different values of Ka Φ and motor speed N1.


FIG. 4b. Operating regions of separately excited DC motor (τ1 = 10 m.sec) for different values ofmotor speed N1.

FIG. 4a. Operating regions of separately excited DC motor.

Figs 4b and 4c are given for different time constant of M1. It is clear that in-creasing τ1 changes αc which leads to an extended range of operation in thecontinuous mode.


3.2 Current and Voltage Waveforms

The currents il , i1 and i2 are obtained using equations (3), (4) and (7) for thecontinuous mode and (B-3), (B-4), (B-6), (B-9) and (B-10) for the discontinu-ous mode. The continuous mode, critical firing angle, and discontinuous modeare shown in Figs. (5-7).

3.3 System Performance Parameters for Constant Torque Operation

It is seen from Fig. 8 that, the firing angle α decreases as the motor speed N1is increased. At constant motors’ speeds, the firing angle α increases as thetorque level is decreased. The regions of discontinuous converter current arerepresented by dotted lines.

3.3.1 Supply Power Factor

When the separately excited DC motor is used for constant-torque applica-tions, Fig. 9 shows that the supply power factor decreases with a decrease ineither motor speed N1 or load torque.

FIG. 4c. Operating regions of a separately excited DC motor (τ1 = 46 m.sec) for different values ofmotor speed N1.


FIG. 5. Current and voltage waveforms in continuous converter current mode for N1 = 500 rpm,N2 = 1000 rpm, and α = 30º.

FIG. 6. Current and voltage waveforms in continuous converter current mode for N1 = 500 rpm,N2 = 1000 rpm, and α = αc = 58.2134º.


FIG. 7. Current and voltage waveforms in discontinuous converter current mode for N1 = 1500rpm, N2 = 1000 rpm, and α = 45º.

FIG. 8. Firing angle versus motor speed N1 for different values of torque level of M1, for N2 =500 rpm.


FIG. 9. Supply power factor versus motor speed N1 for different values of torque level of M1, forN2 = 500 rpm.

3.3.2 Supply Current Distortion Factor

Fig. 10 shows that for continuous converter current mode, the distortion fac-tor decreases as motor speed N1 is increased. And for constant motors’ speeds,the distortion factor increases as the load torque is decreased. Opposite conclu-sions have been obtained for the discontinuous converter current mode.

It is noted that the best value of the distortion factor is obtained when thefiring angle is closest to the critical firing angle.

3.3.3 Motor Current Ripple Factor

It is evident from Fig. 11 that, the current ripple factor of M1 increases as theload torque is decreased keeping motors’ speeds constant.

3.4 System Performance Parameters for Constant- HP Operation

Series motors are normally used for constant-horsepower applications. It isseen from Fig. 12 that the firing angle increases as the horsepower level of M2is decreased keeping motors’ speeds constant. The critical firing angle is alsoindicated.


FIG. 10. Supply current distortion factor versus motor speed N1 for different values of torquelevel of M1, for N2 = 500 rpm.

FIG. 11. Current ripple factor of motor one versus motor speed N1 for different values of torquelevel of M1, for N2 = 500 rpm.


FIG. 12. Firing angle versus motor speed N2 for different values of horsepower level of M2, forN1 = 500 rpm.

3.4.1 Torque-speed characteristics

Fig. 13 shows that, the torque-speed characteristics of the series motor aredrooping in nature. The system produces high torque at low speed and vice versa.

3.4.2 Supply Power Factor

It is evident from Fig. 14 that, the supply power factor decreases as the loadhorse power level of M2 is decreased keeping motors’ speeds constant.

3.4.3 Motor Current Ripple Factor

According to Fig. 15, the current ripple factor of the series motor increases asthe load horsepower level is decreased keeping motors’ speeds constant.

3.5 System Performance Parameters for Constant Firing Angle, αααα , Operation

3.5.1 Separately Excited DC Motor Performance Parameters

a. Torque-Speed Characteristics: In Fig. 16 the dotted lines indicate dis-continuous converter current. This figure shows that, as expected, the convertercurrent is discontinuous at high values of the firing angle, α, and low values oftorque.


FIG. 13. Torque-speed characteristics (N2 versus T2) for different values of horsepower level ofM2, for N1 = 500 rpm.

FIG. 14. Supply power factor versus motor speed N2 for different values of horsepower level ofM2, for N1 = 500 rpm.


FIG. 15. Current ripple factor of motor two versus motor speed N2 for different values of horse-power level of M2, for N1 = 500 rpm.

FIG. 16. Torque-speed characteristics (N1 versus T1) for different values of firing angle for N2 =1000 rpm.


b. Supply Power Factor: Fig. 17 shows that, the supply power factor de-creases as the firing angle is increased keeping motors’ speeds constant.

FIG. 17. Supply power factor versus motor speed N1 for different values of firing angle for N2 = 1000.

c. Motor Current Ripple Factor: It is evident from Fig. 18 that, the cur-rent ripple factor of M1 increases with the increase of either the firing angle orthe motor speed N1.

FIG. 18. Current ripple factor of motor one versus motor speed N1 for different values of firingangle, for N2 = 1000 rpm.


3.5.2 Series DC Motor Performance Parameters

a. Torque-Speed Characteristics: Fig. 19 shows that the torque-speedcharacteristics of the series motor are drooping in nature. As expected, the con-verter current is discontinuous at high values of the firing angle α, high speedsand low values of torque.

FIG. 19. Torque-speed characteristics (N2 versus T2) for different values of firing angle, for N1 =500 rpm.

b. Supply Power Factor: It is evident from Fig. 20 that, the supply powerfactor decreases as the firing angle is increased keeping motors’ speeds constant.

c. Motor Current Ripple Factor: It is evident from Fig. 21 that, the motorcurrent ripple factor of M2 increases with the increase of either the firing angleor the motor speed N2. Comparison of ripple factors of Fig. 18 with those ofFig. 21 reveals that, for any value of firing angle and when N1 = N2, the ripplefactor of the separately excited DC motor current is greater than the ripple fac-tor of the series motor current.

4. Conclusion

In this paper, an AC-DC full-controlled converter supplying a separately ex-cited DC motor parallel with a series DC motor has been investigated forsteady-state operation. The margin firing angle, αmg, that separates motor oper-


FIG. 20. Supply power factor versus motor speed N2 for different values of firing angle, for N1 =500 rpm.

FIG. 21. Current ripple factor of motor two versus motor speed N2 for different values of firingangle, for N1 = 1000 rpm.


ation from generator operation of the separately excited machine has been stud-ied. The following conclusions have been inferred:

4.1 Critical Firing Angle ααααc

The critical firing angle αc decreases as the speed of any motor or both isincreased. However αc increases as the field current of the separately excitedmotor is decreased. Increasing the time constant of any machine or both leads toan extended range of operation in the continuous mode.

4.2 Margin Firing Angle, ααααmg

For the continuous converter current mode, αmg is independent of the seriesmotor parameters. For this mode, decreasing the separately excited motorspeed, N1, leads to increase αmg.

For the discontinuous converter current mode, αmg depends on the parame-ters of both motors. For this mode, αmg increases either by increasing the seriesmotor speed, N2, or by decreasing the separately excited motor speed, N1.

4.3 Supply Current Distortion Factor

The best supply current distortion factor DF is obtained at α which is closestto αc.

4.4 Constant Torque, T1, Operation

During this situation, α decreases and the input power factor PF increases asthe motor speed N1 is increased.

4.5 Constant HP Operation, P2

During this situation, α decreases , the input PF increases and the motor cur-rent ripple factor RF2 decreases as the motor speed N2 is increased. Oppositeconclusions have been obtained for very small values of N2.

4.6 Constant Speeds (N1 and N2) Operation

The increase in the value of torque, T1, leads to a lower α, higher power fac-tor PF and a lower motor current ripple factor RF1. Decreasing the firing angleα increases the input power factor PF and decreases the motor current ripplefactor. The decrease in the horsepower level of M2 leads to a higher α, lowerinput PF, and a higher motor current ripple factor RF2.


(A – 2)

I V Z K K N R VQ Z Q Q

e VK N Z R

Q K K

c rms c a

Qa

c a

12

12

1 1 1 12

1 12

1 1

1 1 1 1 1

1 1

2 2 1 1

1 4 2

2

1

= + + + −

+ − + − −

+ +

−

[( / ) ( ( / ))( /( ( ))[( / )sin( )

cos( )]( ) ( / )cos( )

( / )( (

( / )

Φ

Φ

Φ

π α φ

α φ π α φ

π

π

NN R e Q K N R K

K N R e K N R

Qa c

aQ

a

1 12 2

1 1 1 1

1 1 1 12 1 2

1 2

1

1

1

/ )) ( ) – ( / )(

( / ))( ) ( / ) ]

( / )

( / ) /

−

+ − +

−

−

π

π

πΦ

Φ Φ

4.7 Constant Firing Angle αααα Operation

The increase in any motor speed leads to decrease of the torque level of thatmotor, small variation of PF for continuous converter current mode and increasethe motor current ripple factor of the same motor. For any value of firing angleα and when N1 = N2, the ripple factor of the separately excited DC motor cur-rent is greater than the ripple factor of the series motor current.

References

[1] Sen, P.C., “Thyristor DC Drives”, John Wiley, 1981.[2] Yung-Chung, Li, “Steady-State Analysis of a Two Branch Resistance-Inductance Parallel

Circuit Controlled by a Forced-Commutation Bidirectional AC Switch”, IEEE Transactionson Industrial Electronics and Control Instrumentation, Nov. 1978.

[3] Al-Johani, A.H., “Analysis of an AC-DC Converter Supplying Two Parallel InductiveLoads”, Master Thesis, King Abdulaziz University, S.A., 1987.

[4] Doradla, S.R. and Sen, P.C., “Solid State Series Motor Drive”, IEEE Transactions on Indus-trial Electronics and Control Instrumentation, May, 1975.

[5] Sen, P.C. and Doradla, S.R., “Evaluation of Control Schemes for Thyristor-Controlled DCMotors”, IEEE Transactions on Industrial Electronics and Control Instrumentation, August,1978.

[6] Kelley, A.W. and Yadusky, W.F., “Phase-Controlled Rectifier Line-Current Harmonics andPower Factor as a Function of Firing Angle and Output Filter Inductance”, IEEE Transac-tions on Industrial Electronics, 1990.

[7] Sen, P.C., “Electric Motor Drives and Control – Past, Present, and Future”, IEEE Transac-tions on Industrial Electronics, 1990.

Appendix [A]

Continuous Mode

The avarage value of the separately excited DC motor current is obtained from: Ic1av = (1/π)

xxxxxxxxxxand is given by:

And the rms value of the separately excited DC motor current is obtained from: Ic1rms =

xxxxxxxxxxxx xxx and is given by:

(A – 1)

i d tc1 ω

α

π α+∫

I V Z K Q e

K N R Q e

c av cQ

aQ

1 1 1 1 1

1 1 1

2 2 1

1 1

1

1

= + −

− − −

( / )cos( – ) ( / )( )

–( / )[ ( / )( )]

(– / )

( / )

π α φ π

π

π

πΦ

( / )1 1

2π ωα

π αi d tc

+∫


The average value of the series DC motor current is obtained from:

and is given by:

And the rms value of the series DC motor current is obtained from:

and is given by:

The rms value of the supply current is obtained as:

It could be shown that rms value of supply current is given by:

Where:

And ac1 and bc1 are the amplitudes of the cosine and the sine fundamental components of ics re-

spectively and are given by: xxxxxxxxxxxxxxxxxxxxxxxand is obtained from: a i td tc cs1

21=

+∫( / ) cos ,π ω ωα

π α

I V Z K K N R VQ Z Q Q

e VK N Z R

Q K

c rms c res

Qres

c

22

22

2 2 2 2 22

2 22

2

2 2 2 2 2 2

2 2 2

2 2 1 1

1 4 2

2

2

= + + +

− + − + −

− + +

−

[( / ) ( ( / ))( /( ( ))[ / )

sin( ) cos( )]( ) ( / )

cos( ) ( / )(

( / )

π

α φ α φ π

α φ π

π

(( / )) ( )

( / )( ( / ))( )

( / ) ]

( / )

( / )

/

K N R e

Q K N R K K N R e

K N R

resQ

res c resQ

res

2 2 22 2

2 2 2 2 2 2 2 2

2 2 22 1 2

1

2 1

2

2

−

− + −

+

−

−

π

ππ

(A – 4)

(A – 3)

I i d tc av c2 21=

+∫( / )π ωα

π α

I V Z K Q e

K N R Q e

c av cQ

resQ

2 2 2 2 2

2 2 2 2

2 2 1

1 1

2

2

= − + −

− − −

−

−

( / )cos( ) ( / )( )

( / )[ ( / )( )]

( / )

( / )

π α φ π

π

π

π

I i d tc rms c2

21 2=+

∫( / )π ωα

π α

(A –5) I i d t I I i i d tcsrms cs c rms c rms c c= = + +

+ +∫ ∫( / ) ( / )1 1 22

12

22

1 2π ω π ωα

π α

α

α α

A V Z Z

B Q Q Q Q K K N R K K N R e

C VQ Z Q K K

c a c resQ Q Q Q

c re

=

= + + + −

= + +

− +

( / )cos( – ) ,

( / ( ))( ( / ))( ( / ))( ) ,

( / ( ))( (

( ( ) / )

2

2 1

2 2 1

21 2 2 1

1 2 1 2 1 1 1 2 2 2 2

22

1 22

2

1 2 1 2

φ φ

π

π

πΦ

ssQ

c aQ

a

N R Q e

D VQ Z Q K K N R Q e

E V Z K

2 2 2 2 1 1

12

2 12

1 1 1 1 2 2

2

1 1

2 2 1 1 1

4 2

2

1

/ ))(( / )sin( ) cos( ))( ) ,

( / ( ))( ( / ))(( / )sin( ) cos( ))( ) ,

(– / )(

( / )

( / )

α φ α φ

π α φ α φ

π

π

π

− + − +

= + + − + − +

=

−

−Φ

ΦΦ

Φ

N R

F V Z K N R

G Q K K N R K N R K N R e

H Q K

res

c res a resQ

c

1 1 2

1 2 2 2 1

1 1 2 2 2 1 1 2 2 2

2 2

4 2

2 1

2

1

/ )cos( ) ,

( / )( / )cos( ) ,

( / )( ( / ) ( / )( / ))( ) ,

(– / )(

( / )

α φ

π α φ

π

π

π

−

= − −

= − + −

=

−

(( / ) ( / )( / ))( ) , and

( / )( / )

( / )K N R K N R K N R e

J K N R K N R

a a resQ

a res

Φ Φ

Φ

1 1 1 1 2 2 2

1 1 2 2 2

1

2

2+ −

=

−π

(A –6) I I I A B C D E F G H I Jcsrms c rms c rms= + + + + + + + + + + +1

22

2


(A –7)

a V Z Q Q K K N R Q e

K N R V Z Q Q K

K

c c aQ

a c

res

1 1 1 12

12

1 1 1

1 1 2 2 22

22

2

2

2 2 1 1 1

4 2 2 1

1= − + + + − +

+ − + +

+

−( / )sin ( /( ) )( ( / ))(( / )cos sin )( )

( / )sin ( / )sin ( /( ) )(

(

( / )φ π α α

π α φ π

πΦ

Φ

NN R Q e K N RQres2 2 2 2 2 21 1 42/ ))(( / )cos sin )( ) ( / )sin(– / )α α π απ− + +

And XX XXxX XxXXXXXXXX and is obtained from:

Appendix [B]

Discontinuous Mode

Conduction Period of the Discontinuous Mode

Throughout the period α ≤ ω t ≤ β, the load voltage is equal to the supply voltage. This meansthat for the separately excited DC motor:

and for the series DC motor:

Where: idl is the armature currents (ia1) of the separately excited DC motor and id2 is the arma-ture currents (ia2) of the series DC motor during the discontinuous mode.

During α ≤ ω t ≤ β, of the converter current idl equals the supply current and is given by:

ids = idl = id1 + id2 α ≤ ω t ≤ β (B – 3(a))

– ids = idl = id1 + id2 π + α ≤ ω t ≤ π + β (B – 3(b))

Separately Excited DC Motor Current

The solution of equation (B – 1) is of the form:

Where:

And id1 is the initial value of id1 at ω t = α.

Series DC Motor Current

The solution of equation (B – 2) is of the form:

b V Z Q Q K K N R Q

e K N R V Z Q Q K

K

c c a

Qa c

res

1 1 1 12

12

1 1 1

1 1 2 2 22

22

2

2

2 2 1 1

1 4 2 2 11

= + + + +

+ − + + + +−

( / )cos ( /( ) )( ( / ))(( / )sin cos )

( ) ( / )cos ( / )cos ( /( ) )(

(

( / )

φ π α α

π α φ ππ

Φ

Φ

NN R Q e K N RQres2 2 2 2 2 21 1 42/ ))(( / )sin cos )( ) ( / )cos( / )α α π απ+ + −−

(A – 8)

b i td tc cs1

21=

+∫( / ) sin ,π ω ωα

π α

2 1 1 1 1 1V t i R L

ddt

i K Nd a a d asin ,ω = + + Φ (B – 1)

2 2 2 2 2 2 2 2 2 2V t i R L

ddt

i K i N K Nd a a d af d ressin ω − + + + (B – 2)

i V Z t K e K N R e td dt Q

at Q

1 1 1 1 1 12 11 1= − + ≤ ≤− −( / )sin( ) – ( / )( – ),(–( – ) / ) ( ( ) / )ω φ α ω βω α ω αΦ (B – 4)

K I V Zd d1 1 1 12= – ( / )sin( – )α φ (B – 5)

i V Z t K e K N R e td dt Q

rest Q

2 2 2 2 2 2 22 12 1= + ≤ ≤

− −( / )sin( – ) – ( / )( – ),(–( – ) / ) ( ( ) / )

ω φ α ω βω α ω α(B – 6)


Where:

And Id2 is the initial value of id2 at ω t = α .

Non Conduction Period of the Discontinuous Mode

During the period β ≤ ω t ≤ π + α, the converter current is equal to zero, and id1 = – id2. Themotors are connected in parallel. Thus:

Solving (B – 8), one obtains:

(B – 9)

Similarly,

(B – 10)

Where: Q = ω (La1 + La2) / (R1 + R2) , and I ′d1 is the initial value of id1 at ω t = β and is given

by:

and I ′d2 is the initial value of id2 at ω t = β and is given by:

At ωt = π +α , id1 equals Id1 , and id2 equals Id2 . Therefore, from equations (B – 5), (B – 9),and (B – 11) one gets:

Similarly: from equations (B – 7), (B – 10), and (B – 12) one gets:

The extinction angle β is determined by the solution of the equation Id1 = – Id2 . (B – 15)

Full expressions for Id1av , Id1rms , Id2av , and Id2rms are given in Appendix [C].

Where Id1av and Id1rms and the average and the rms values of the separately excited DC motorcurrent respectively, and Id2av and Id2rms are the average and the rms values of the series DC mo-tor current respectively.

The ripple factor of Idl is given by: (B – 16)

K I V Zd d2 2 2 22= – ( / )sin( – )α φ (B – 7)

R i L

ddt

i K N R i Lddt

i K N ti d a d a d a d res1 1 1 1 2 2 2 2 2 2+ + = + + ≤ ≤ +Φ , β ω π α (B – 8)

′ = − + − −− − − −I V Z K e K N R ed dQ

aQ

1 1 1 1 1 12 11 1( / )sin( ) ( / )( )( ( ) / ) ( ( ) / )β φ β α β αΦ (B – 11)

′ = − + − −− − − −I V Z K e K N R ed dQ

resQ

2 2 2 2 2 2 22 12 1( / )sin( ) ( / )( )( ( ) / ) ( ( ) / )β φ β α β α (B – 12)

I K N K N R R e V Z

e e K N R

e

d res aQ

Q Qa

Q

1 2 2 1 1 2 1

1 1 1 1

1 2

1

1

1

= − + − +

− − − −

−

− + −

− − − + −

− −

[(( ) /( ))( ) ( / )

(sin( ) sin( ) ) ( / )

( )

( ( ) / )

( ( ) / ) ( ( ) / )

( ( ) / )

Φ

Φ

π α β

β α π α β

β α

β φ α φ

ee e eQ Q Q( ( ) / ) ( ( ) / ) ( ( ) / )] /( )− + − − − − + −−π α β β α π α β1 1

(B – 13)

I K N K N R R e V Z

Q e e K N R

e

d a resQ

Q Qres

Q

2 1 2 2 1 2 2

2 2 2 2 2

1 2

1

2

2

= − + − +

− − − −

−

− + −

− − − + −

− −

[(( ) /( ))( ) ( / )

(sin( ) sin( ) ) ( / )

(

( ( ) / )

( ( ) / ) ( ( ) / )

( ( ) /

Φ π α β

β α π α β

β α

β φ α

)) ( ( ) / ) ( ( ) / ) ( ( ) / )) ]/( )e e eQ Q Q− + − − − − + −−π α β β α π α β1 2

(B – 14)

K I Idr d rms d av1 12

12 1= ( / ) –

i K N K N R R I K N K N R R ed res a d res at Q

1 2 2 1 1 2 1 2 1 1 2= − + + ′ − − + − −( ) /( ) ( ( ) /( )) ( ( ) / )Φ Φ ω β

i K N K N R R I K N K N R R ed a res d a rest Q

2 1 2 2 1 2 2 1 2 2 1 2= − + + ′ − − + − −( ) /( ) ( ( ) /( )) ( ( ) / )Φ Φ ω β


The ripple factor of id2 is given by: (B – 17)

The rms value of the fundamental component of the supply current is

And the phase angle of that component is given by:

θ = tan–1 (ad1 / bd1) (B – 19)

Where: ad1 and bd1 are the amplitudes of the cosine and sine fundamental components of ics re-spectively in the conscontinous mode.

The supply power factor is:

Pf = (Idf / Idsrms) cos θ (B – 20)

And the supply current distortion factor is:

Df = Idf / Idsrms (B – 21)

Full expressions for Idsrms , ad1 , and bd1 are given in Appendix [C].

Appendix [C]

Average and rms values of the currents in the discontinuous mode

The average value of the separately excited DC motor current is given by:

Therefore:

The rms value of the separately excited DC motor current is found to be:

Therefore:

K I Idr d rms d av2 22

22 1= ( / ) –

I a bdf d d= +( ) /12

12 2 (B – 18)

I i d td av d1 11=

+∫( / ) .π ωα

π α

I a bdf d d= +( ) /12

12 2 (B – 18)

I V Z Q K

K N R e K N R

K N K N R R

Q

d av d

aQ

a

res a

1 1 1 1 1 1

1 1 1 1

2 2 1 1 2

2

1 1

= − − − +

+ − + −

+ − + + −

+

− −

( / )[cos( ) cos( )] ( / )(

( / ))( ) ( / )(( ) / )

(( ) /( ))(( ) / )

( /

( ( ) / )

π α φ β φ π

α β π

π α β π

β αΦ Φ

Φ

ππ π α β)((( ) /( )) )( )( ( ) / )K N K N R R I eres a dQ

2 2 1 1 2 1 1− + − ′ −− + −Φ

(C – 1)

I i d td rms d1 1

21=+

∫( / ) .π ωα

π α

I V Z

V Z K N R

V Z Q Q K K N R e

d rms

a

d a

12

12

1 1

1 1 1 1 1

1 12

12

1 1 1

2 2 2 2 4

2 2

2 2 1

= − + − − −

+ − − −

+ + + − −

( / )[( ) / [sin ( ) sin ( )]/

( / )( / )[cos( ) cos( ]

( / )( /( ))( ( / ))[ ( ( )

π β α α φ β φ

π β φ α φ

π β α

Φ

Φ // )

( ( ) / )

( ( / )

sin( ) cos( )) (( / )sin( ) cos( ))]

( / )( ( / )) ( ) ( / )( ( / )

Q

d aQ

d a

Q

Q

Q K K N R e Q K K N R

1

1

1

1

2 1 2

1

1 1 1 1 1

1 1 1 12 2

1 1 1 1

−− − − + − + −

− + − +− −

β φ β φ α φ α φ

π πβ αΦ Φ


+ − + −

+ − + + −

+ − + − ′

− −2 1

2

1 12

1 12

2 2 1 1 22

2 2 1 1 22

1

1( / ) )( ) ( / ) ( ) /

(( ) /( )) (( ) / )

( / )[(( ) /( )) ((

( ( ) / )K N R e K N R

K N K N R R

Q K N K N R R I

aQ

a

res a

res a d

Φ Φ

Φ

Φ

β α β α π

π α β π

π KK N K N

R R e Q K N K N

R R I e

res a

Qres a

dQ

2 2 1

1 2 2 2 1

1 2 12 2 1 2

1 2

1

−

+ − − −

+ − ′ −

− + −

− + −

Φ

Φ

)

/( ))]( ) ( / )((( )

/( )) ) (

( ( ) / )

( ( ) / ) /

π α β

π α β

π

(C – 2)

The average value of the series DC motor current is given by:

Therefore:

The rms value of the series DC motor current is found to be:

Therefore:

The value of the supply rms current is obtained from:

I i d td av d2 21=

+∫( / ) .π ωα

π α

I V Z Q K

K N R e K N R

K N K N R R

d av d

resQ

res

a res

2 2 2 2 2 2

2 2 2 2 2 2

1 2 2 1 2

2

1 2

= − − − +

+ − + −

+ − + + −

+

−

( / )[cos( ) cos( )] ( / )(

( / ))( ) ( / )(( ) / )

(( ) /( ))(( ) / )

(

(–( ) / )

π α φ β φ π

α β π

π α β π

β α

Φ

QQ K N K N R R I ea res dQ/ )((( ) /( )) )( )( ( ) / )π π α βΦ 1 2 2 1 2 2 1− + − ′ −− + −

(C – 3)

I i d td rms d2 2

21=+

∫( / ) .π ωα

π α

I V Z V Z

K N R V Z Q Q K

K N R e

d rms

res d

res

22

22

2 2 2

2 2 2 2 2 2 22

22

2

2 2 2

2 2 2 2 4 2 2

2 2 1

= − + − − − +

− − − + +

+

( / )[( ) / [sin ( ) sin ( )]/ ] ( / )

( / )[cos( ) cos( )] ( / )( /( ))(

( / ))[ (

π β α α φ β φ π

β φ α φ π

−− −

− −

− − − −

+ − + − − +

− +

( ) / )

( ( ) / )

( ( / )sin( ) cos( ))

( / )sin( ) cos( ))] ( / )( ( / ))

( ) ( / )( ( / )

β α

β α

β φ β φ

α φ α φ π

π

Q

d res

Qd res

Q

Q Q K K N R

e Q K K N R

2

2

1

1 2

1 2

2 2 2

2 2 2 2 2 2 2 22

22 2 2 2 ++

− + − + −

+ + − + − +

− ′

− −

2

1

2

2 2 22

2 2 22

1 2 2 1

22

1 2 2 1 22

2

( / ) )

( ) ( / ) ( ) / (( ) /

)) (( ) ( / )[(( ) /( ))

( ( ) / )

R N R

e K N R K N K N R

R Q K N K N R R

I

res

Qres a res

a res

β α β α π

π α β π

Φ

Φ

dd a resQ

a

res dQ

K N K N R R e Q K N

K N R R I e

2 1 2 2 1 2 1

2 2 1 2 22 2 1 2

1 2

1

(( ) / ))]( ) ( / )(((

) /( )) ) ( )

( ( ) / )

( ( ) / ) /

Φ Φ− + − −

− + − ′ −

− + −

− + −

π α β

π α β

π

(C – 4)

I i d tdsrms ds= ∫( / ) .1 2π ω

α

β(C – 5)


Thus:

Then rms value of the supply current is found to be: I i d t i d t i i d tdsrms d d d d= + + ∫∫∫( / ) ( / ) ( / )1 1 1 21

22

21 2π ω π ω π ω

α

β

α

β

α

β (C – 6)

I V Z V Z

K N R V Z Q Q K

K N R e

dsrms

a d

a

= − + − − − +

− − − + +

+ − −

( / )[( ) / [sin ( ) sin ( )] / ] ( / )

( / )[cos( ) cos( )] ( / )( /( ))(

( / ))[ ( (

2 2 2 2 4 2 2

2 2 1

212

1 1 1

1 1 1 1 1 12

12

1

1 1

π β α α φ β φ π

β φ α φ π

β

Φ

Φ αα

β α

β φ β φ

α φ α φ π

π

) / )

( ( ) / )

( ( / ) sin( ) cos( ))

(( / ) sin( ) cos( ))] ( / )( ( / ))

( ) ( / )( ( / ) (

Q

d a

Qd a a

Q

Q Q K K N R

e Q K K N R K N

1

1

1

1 2

1 2 2

1 1 1

1 1 1 1 1 1 12

21 1 1 1

− − − −

+ − + − − +

− + +− −

Φ

Φ Φ 11 12

1 12 2

22

2 2 2 2 2 2 2

2

1 1 2 2

2 2 4 2 2

/ ) )

( ) ( / ) ( ) / ( / )[( ) /

[sin ( ) sin ( )] / ] ( / )( / )[cos( )

cos( )] (

( ( ) / )

R

e K N R V Z

V Z K N R

Qa

res

− − − + − + −

+ − − − + −

− − +

β α β α π π β α

α φ β φ π β φ

α φ

Φ

22 2 1

1 1

2

2 22

22

2 2 2 2

2 2 2 2 2

2 2 2 2 2 22

2

V Z Q Q K K N R

e Q Q

Q K K N R e

d res

Q

d res

/ )( /( ))( ( / ))

[ (–( / ) sin( ) cos( )) (( / ) sin( )

cos( ))] ( / )( ( ( / )) (

( ( ) / )

π

β φ β φ α φ

α φ π

β α

+ +

− − − + −

+ − − + =

− −

(( ( ) / )

( ( ) / )

)

( / )( ( / ) ( / ) )( )

( / ) ( ) / ( / ) cos( )( ) (

− −

− −

−

+ + −

+ − + − − −

2

2 2 2 2 2 2 2 22

2 2 22 2

1 2 2 1

2

2

1

2 2 1

2

β α

β απ

β α π π φ φ β α

Q

d res resQ

res

Q K K N R K N R e

K N R V Z Z V 221 2

2 1 2 1 2 1 1 2

2 1 2 2 2 1 1

2 12

12

2 2 2 2

2 2

2 2 1

/ )

[sin( ) sin( )] ( / )( / )[cos( )

cos( )] ( / )( / )[cos( ) cos( )]

( / )( /( ))(

π

β φ φ α φ φ π β φ

α φ π β φ α φ

π

Z Z

V Z K N R

V Z K N R

V Z Q Q K

a

res

− − − − − + −

− − + − − −

+ +

Φ

dd aQ

d resQ

K N R e Q

Q Q

V Z Q Q K K N R e

1 1 1 1

2 2 1 2 2

1 22

22

2 2 2 2

1

2

1

1

2 2 1

+ −

− − − + − + −

+ + +

− −

− −

( / ))[ ( ( / )

sin( ) cos( )) (( / ) sin( ) cos( ))]

( / )( /( ))( ( / ))[ (

( ( ) / )

( ( ) / )

Φ β α

β α

β φ β α φ α φ

π −−

− − + − + −

+ + + +

− +− + −

1 1

1

2

1 2

2

1 1 2 1 1

1 2 1 2 1 1 1 2 2 2 2

1 2 1 2

( / )

sin( ) – cos( )) (( / ) sin( ) cos( ))]

( / ( ))( ( / ))( ( / ))

( ) (( ( )( ) / )

Q

Q

Q Q Q Q K K N R K K N R

e Q

d a d res

Q Q Q Q

β φ β φ α φ α φ

π

β α

Φ

22 1 1 2 2 2 2

1 2 2 2 1 1 1

1 1 2 2

2

1

1 2

1 2

/ )( / )( ( / ))

( ) ( / )( / )( ( / ))

( ) ( / )( / )(

( ( ) / )

( ( ) / )

π

π

π

β α

β α

K N R K K N R

e Q K N R K K N R

e K N R K N

a d res

Qres d a

Qa res

Φ

Φ

Φ

+

− + +

− +

− −

− − // )( ) /R21 2β α−

(C – 7)


And ad1 and bd1 are the amplitudes of the cosine and sine fundamental components of ids re-spectively and are given by:

and

a V Z

Q Q K K N R e Q

Q K N R

d

d aQ

a

1 1 1 1 1

12

12

1 1 1 1

1 1 1

2 2 2 2 2

2 1

1 2

1

= − − − − −

+ + + −+ + − −

− −

( / )[cos( ) cos( ) ( )sin ]

( /( ( )))( ( / ))[ ( ( )cos

sin ) (( / )cos sin )] ( /

( ( ) / )

π α φ β φ β α φ

π ββ α α π

β αΦΦ ))(sin))(sin sin )

( / )[cos( ) cos( ) ( )sin ]

( /( ( )))( ( / ))[ ( ( / )cos

sin ) (( / )cos sin )] (

( ( ) / )

β α


π ββ α α

β α

−

+ − − − − −

+ + + −+ + − −

− −

2 2 2 2 2

2 1 1

1 2

2 2 2 2

22

22

2 2 2 2 2

2

2

V Z

Q Q K K N R e Q

Qd res

Q

KK N Rres2 2 2/ ))(sin sin ) ,π β α−

(C – 8)

b V Z

Q Q K K N R e Q

Q K N

d

d aQ

a

1 1 1 1 1

12

12

1 1 1 1

1 1

2 2 2 2 2

2 1 1

1 2

1

= − − − + −

+ + + − −+ + +

− −

( / )[sin( ) sin( ) ( )cos ]

( /( ( )))( ( / ))[ ( ( / )sin cos )

(( / )sin cos )] ( /

( ( ) / )


π β βα α π

β αΦΦ RR

V Z

Q Q K K N R e Q

Qd res

Q

1

2 2 2 2

22

22

2 2 2 2 2

2

2 2 2 2 2

2 1 1

1

2

))(cos cos )

( / )[sin( ) sin( ] ( )cos ]

( /( ( )))( ( / ))[ ( ( / )sin cos )

(( / )sin cos )]

( ( ) / )

β α


π β βα α

β α

−

+ − − − + −

+ + + − −+ + +

− −

(( / ))(cos cos )2 2 2 2K N Rres π β α−

(C – 9)


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Documents

Analysis of an AC-DC Full-controlled Converter … of an AC-DC Full-controlled Converter Supplying Separately Excited DC Motor Parallel with ... Eng. Sci., vol. 13 no. 2, pp ... 44