Section 2 - DC Motor Drives

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1 Section 2 - DC Motor Drives eview of DC motors nd chrcteristics Switched mode PWM converters. Single nd three phse thyristor converter circuits. Anlysis of converter nd DC motor circuits. Effects of discontinuous conduction on drive. 1

2 Section 2 - DC Motor Drives eview of DC motors nd chrcteristics (3105) Switched mode PWM converters. Single nd three phse thyristor converter circuits. Anlysis of converter nd DC motor circuits. Effects of discontinuous conduction on drive. 2

3 eview of DC Motors Control of seprtely excited DC motor is very strightforwrd, vi i nd i f, which re de-coupled from ech other. The commuttor-brush ssembly provides for this simplicity. AC mchines strive to emulte such control vi mchine-model bsed controllers which re rther complex. The commuttor-brush hs mny limittions nd mintennce issues. 3

4 Working Principle Field is either from electro or permnent mgnets The field circuit is sttionry - in the sttor The rmture crries conductors in slots nd rottes with the rotor 4

5 DC Motor Electric Circuits For lrge DC mchines, the field is from electro mgnets, there re two circuits which cn both be controlled Field circuit Armture circuit For smll DC mchines (< 20kW), the field is from permnent mgnets, there is No field circuit; llows no field control Only rmture current control 5

6 The Field Circuit i f f B f f L f di f vf fif Lf dt When field excittion is constnt, The ir-gp B field (in Tesl) is constnt for constnt field current I f, s is the flux per pole,, in Weber. For liner mgnetic circuit, K I. f f f I f f f A I f 6

7 Seprtely Excited DC Mchine Equtions Torque Bck emf T 2N lrbi k i K i ' r T f T e 2N lrb k K ' r m E f m E m K T in Nm/A = K E in /rd/sec. v i L di e dt di f vf fif Lf dt L i i f L f f v e v f 7

8 Stedy stte Torque Speed Chrcteristic with rible v In the stedy stte, + m, rd/sec I E ' K E f m I I I I m ' ' KE f KEK f I f T T em em ' ' KT f KT K fi f m m m K K I K K K I ' ' ' 2 E f f E T f f T em -T em T em A BT m em m

9 Stedy stte Torque Speed Chrcteristic with rible T K I K T K T T E T m K T E T m K K K K E ω-xis intercept Constnt Slope Here K k k k I ' ' T T f T f f k K ' E f E in SI units (, rd/s, Nm,, A). T stll is the stll torque when rted is pplied i.e., = rted 9

10 T ω Chrcteristic with rible Φ (=o, ω> ωb) b I o, rted K E = 0.3 pu m K o ' ' ' 2 E KEKT T em m, rd/sec =0.5 pu = 1 pu Here, ' T K,K ' E CONST T e, Nm m 0 ' stll T o T K Stll torque is proportionl to field Te 0 ' m o KE No-lod speed is inversely proportionl to field 10

11 , I f nd I Boundries, ted m, d/sec I f, min T CONST Field Control Armture oltge Control T, Nm I, ted T, Nm Envelop for mx rm. current m, d/sec 11

Exmple 1: Stedy stte torque speed chrcteristic An ppliction requires continuous torque of 0.86 Nm (7.6 lb-in) t speed of 2750 PM. The pek torque required for ccelertion is 6.25 Nm (56.8 lb-in).

12 Exmple 1: Stedy stte torque speed chrcteristic An ppliction requires continuous torque of 0.86 Nm (7.6 lb-in) t speed of 2750 PM. The pek torque required for ccelertion is 6.25 Nm (56.8 lb-in). Will M-3358-C work in this ppliction with = 100 nd 50? D M-3358-C B Point A (0.75 Nm / 2750 rpm) is in the continuous opertion re (OA). Point B (6.25 Nm) is in the intermittent OA. DC motor rtings: o = 100, Nr = 5000 rpm, I,rted = 4.9 A, Trted = 0.86 Nm. K E C A /rd/s, 1.4 Ohm If = 50, the mximum speed t full lod (0.86Nm) is 2375 rpm; The pek toque is only chieved t lower speed. KEm I /

13 Series excited DC mchine F+ f L f L I = I f F- A+ Field circuit Armture circuit E A- ' I E I K f f E f m m, rd/sec increses ' I K K I f E f m T, Nm T K K e ' e f ' E f ' m KK t f KK t f T T, Nm T e K increses m, rd/sec 13

14 Shunt excited DC mchine I f f I L B f L f E I f ' E f m I K I I m ' ' ' KE f KEf KEf I T e f ' t f K K K I K f f f f f 2 f f ' ' ' 2 2 E f KEKtK f m T K K e 14

15 Losses in DC mchine P I in Pdev EI mtdev P T out m shft P Field loss (shunt) I 2 f f f Commuttorbrush losses (contct drop shortcircuiting) Armture copper loss P I 2 Core losses (hysteresis, eddy current) Mechnicl losses Windge nd frictions T shft T P dev P out out P P P P b in out losses 100% 15

16 Exmple 2: Losses in DC mchine The prmeters of Kollmorgen U9D-B servo DC motor re given in the following tble. Wht is the developed torque of the motor t rted output power? Performnce specifictions Symbol Unit vlue ted power output P Wtts 133 ted continuous current I Amps 8.64 Bck EMF constnt KE /kpm 6 Torque constnt KT N-cm/Amp 5.7 ted continuous torque T N-cm 42.4 ted speed N PM 3000 T P / Ncm shft out m Tdev KT I Ncm> T P dev EI W> P shft out KE /rd/s = K T 16

17 Section 2 - DC Motor Drives eview of DC motors nd chrcteristics Switched mode PWM converters. Single nd three phse thyristor converter circuits. Anlysis of converter nd DC motor circuits. Effects of discontinuous conduction on drive. 17

18 Converters for DC motor drives Power Supply Power Supply v c Power Converter Armture v e i F Field Power Converter Armture voltge control up to rted ; field control bove bse speed Two types of converters for DC motor drive: 1.PWM converters for smll DC motors 2.Thyristor converters for medium nd lrge DC motors 18

19 PWM switching pulses e c PWM SW signls D Power Electronic Converter,vrg v Ds e / c tri s v c v tri t Comprtor output: High for T ON ; Low for T OFF t 19

20 PWM DC-DC converter in continuous conduction mode (CCM) I I I s L E D s L E D s L E D T 0~T on T T ~T on S T S v D S 0 i E I mx 00 t I min 0 T on DT S off T 1 D T S I t T S 20

21 PWM DC-DC converter in continuous conduction mode (CCM) s v s I L T E D 0 I 0 i I mx T on = DT s T off = (1-D)T s E I min t t T s Ts Ton DT T s off 1 1 DT v (t)dt dt 0dt D s s s s s Ts T 0 s T 0 DT s 21

22 From Fourier nlysis v cosn t b sinnt 0 n n 2 n1 T s 0 1 v (t)dt D 2 T s s 0 2 2D 1 1 n vcos n t d t S cos n t d t sin 2n D 0 0 n S 2 1 bn v sin n t d t 1 cos 2n D 0 n S 2 2 2s cn ˆ n n bn sinn D n ipple mplitude, not MS 22

23 0 f s 2f s 3f s The ripple voltge is mximum for D = f s 1 dt D DT s 2 MS s s Ts ˆ ˆ ˆ ; ; ; where The DC voltge develops i, torque nd useful output power. The ripple voltges cuse ripple currents in the rmture dditionl loss in the mchine. 23

24 Exercise 1: Mx. ripple voltge in PWM DC-DC converter in CCM Find the MS vlue of the AC voltge cross the rmture nd the duty cycle when the MS ripple voltge is mximum? 1 dt D DT s 2 MS s s Ts cms cms MS S D(1 D) dcms 1 2D S 0 dd 2 D(1 D) D cms The ripple voltge is mximum for D =0.5. The DC voltge develops I, torque nd useful output power. The ripple voltges cuse ripple currents in the rmture, resulting in dditionl loss in the mchine. 24 S

25 Anlysis of i t constnt speed. Continuous Conduction Mode (CCM) S I L 0 D s E I min v E i I mx 0 T Ton DT T S off 1DT T S di During 0 t T on s i L E dt s E Prticulr solution: i1 s 1 i E Homogeneous solution: di 2 02 i L dt i C C e t L D S 00 S t t t C i 1 1 t 0C1C2 Imin t s L i 1 e I e E min t L 25

26 Exercise 2: Anlysis of i t constnt speed. (CCM, DT S t T S ) I L s E T D S 0 I min 0 v E i I mx I T on DT S T S D T 1D T off S S t t E I 1e Imine At t = DT s, i = I mx, s DT / mx During DT S t T s, i 1 e I e E t'/ where t' t T t DT on di 0 i L E dt' mx s t'/ nd L DT/ s s (13) t' C E t' 0 C C I, the electricl time constnt mx 26

27 E s DT s / DT/ s (13) Imx 1 e Imine I min occurs t t = T s DT s =(1 D)T s E I min 1e I e (1D)T s / (1D)T s / mx (16) From 13 nd 16 I I DT s / s 1 e mx T s / 1 e DT / s min T / s e 1 s e 1 E DT / DT / S S ripple mx min T/ T/ Î I I E S S 1 e e 1 S S 1 e e 1 (17) (18) (19) 27

28 T- chrcteristic with CCM Ds I E ' E f m s E K D I I D s E D I D I s s m ' ' KE f KEK f I f E or m Disc D = 1.0 D = 0.75 Boundry of CCM nd DCM Imin = 0 D = 0.5 D = min I or T em 28

29 PWM DC-DC converter discontinuous conduction mode (DCM) s v E i=0 s I L D T on = DT s T off = (1-D)T s E i T I t T s 29

30 During 0 t DT s v During DT s t t v = 0 s During t t T s v =E T 1 s t v dt D 1 E s Ts 0 Ts DTs T 1 s t dt E dt D 1 E MS s s Ts 0 t Ts s E t n sin2nd sin2n n n Ts s bn 1 cos 2 nd 1 cos ; n n Ts E 2 nt c ˆ b 2 2 n n n n 30

31 Anlysis of i t constnt speed. Discontinuous Conduction Mode (DCM) s During 0 t DT t/ s i 1e s E I mx 1e s E DT / During freewheeling (i.e., diode conducting) E t'/ t'/ E i 1 e Imxe 1 e (tdt s )/ The rmture current becomes zero t t, given by E t lne 1 1e E DT / s DT / s s s E 1 e DT s / e (tdt)/ s (2.2.33) 31

32 T chrcteristics with discontinuous conduction For given speed (E ), the boundry between CCM nd DCM (when I min =0) occurs for duty cycle D for which i =0tT s. Thus D'T / s E e 1 T s / e 1 s (2.2.34) E or m D = 1.0 D = 0.75 D = 0.5 DCM CCM D > D implies opertion in CCM (Imin > 0). D = 0.25 D = I or T em Additionl inductnce in the rmture my be required to reduce the power loss due to ripple current in the rmture nd to prevent DCM opertion. The required minimum inductnce L min for CCM cn be from or

33 Determine the boundry on motor chrcteristics (Mtlb) E For given D t substitute t into I E / t = T for boundry s I is function of E E,I DCM Chrcteristics I

34 During 0 t DT s v During DT s t t v = 0 s During t t T s v =E 1 v dt 1 dt E dt D 1 t E Ts DTs Ts s s Ts 0 Ts 0 t Ts DTs T 1 s t dt E dt D 1 E MS s s Ts 0 t Ts s E t n sin2nd sin2n n n Ts s bn 1 cos 2 nd 1 cos ; n n Ts E 2 nt c ˆ b 2 2 n n n n 34

35 Armture current ripple vi Fourier nlysis I n ˆ / 2 n 2 nl 2 Why is there no E in the eqution? MS I I I I I... The input power to the rmture, ignoring other losses, T 1 s P v i dt I E I 2 in MS Ts 0 Only DC component contributes to the developed power. DC power input to the motor, neglecting core losses, 2 in,dc P I E I 35

36 Opertion in qudrnts 1 & 2 I s L E D Q1 T s D E L I + Q2 T 36

37 egenertive PWM DC-DC converter D E L s I + T s v 0 T s E t T on = DT s T off = (1-D)T s 0 I i D is ON I mx D is OFF I min t 37

38 Anlysis of i in Q2 t constnt speed Note: Diode D is on during DT s, switch T is on during (1 - D)T s. During 0 t T on s E di L i dt di During t - T on t T s, 0 i L E dt The differentil equtions re the sme s those in Q1 I DT s / s 1 e mx T s / 1 e E I DT / s min T / s e 1 s e 1 E Îripple I mx I min 38

39 i s T1 s T 1 T 2 I D 1 D 2 L E T2 s D1 v i T1 D2 T2 I D1 i s D = 1.0 E or m D = 0.75 D = 0.5 D = 0.25 Q2 Q1 0 I or T em The stright-line T-ω chrcteristics of Q1 CCM extends into Q2 with the sme slopes nd intercept 39

40 4Q PWM DC-DC converter drive +v, + m T1 D1 I L T3 D3 FB Q2 FM Q1 s T4 D4 T 2 D2 -i, -T Q3 M Q4 B +i, +T -v, - m Unipolr: (s/0; 0/-s) Only one switch is controlled for given polrity of the output voltge, the current freewheels through the other switch nd diode during Toff ; Bipolr (±s) Two digonl switches re lwys switched together. 40

41 4Q PWM DC-DC converter drive (Unipolr) Unipolr (>0) s T1 D1 I L T3 D3 s T1 D1 I L T3 D3 T4 D4 T 2 D2 T4 D4 T 2 D2 During T on During T off S v T1&T2 T2&D4 T1&T2 0 i t 0 T on T off I t T S 41

42 4Q PWM DC-DC converter drive (Unipolr) Unipolr (<0) s T1 D1 I L T3 D3 s T1 D1 I L T3 D3 T4 D4 T 2 D2 T4 D4 T 2 D2 During T on During T off 0 S 0 i v t t T on T off 42

43 Exercise 3: 4-Q DC-DC converter drive Sketch v nd i in bipolr switching mode, indicting the conduction pths of i through the switches nd diodes. s T1 D1 T3 L I D3 s T1 D1 I L T3 D3 T4 D4 T 2 D2 T4 D4 T 2 D2 T1 D1 T3 L I D3 T1 D1 I L T3 D3 s s T4 D4 T 2 D2 T4 D4 T 2 D2 v S 0 S i t S 0 S 0 i v t t 0 T on T off t T on T off 43

44 Switching scheme nd PWM switching frequency Unipolr versus bipolr switching in 4-qudrnt converter (SW signls, ripple) The PWM switching frequency is selected from the following considertions: 1. for the switching frequency, 2fsL >> (from current ripple considertion) 2. High switching frequency reduces the current ripple nd motor losses. It lso voids discontinuous conduction. 3. f s should be much higher thn the speed control bndwidth. Thus f s >10 speed control bndwidth. 4. f s should be higher thn ny significnt resonnt frequencies 5. f s should be sufficiently high to void udible noise (> = 5kHz) 6. Too high switching frequency will result in excessive switching losses in the switching devices (trnsistors). 7. Too high switching frequency limits the rnge of output nd introduces offset into the power converter input-output chrcteristics. At high switching frequencies the finite dely times of gte switching circuits nd ded-times for device protection my become comprble to the switching period. 44

45 Section 2 - DC Motor Drives eview of DC motors nd chrcteristics Switched mode PWM converters. Single nd three phse thyristor converter circuits. Anlysis of converter nd DC motor circuits. Effects of discontinuous conduction on drive. 45

46 Thyristor converter drive for DC motor A controlled diode, turned on by gte current pulse when forwrd bised. It continues to conduct while the voltge cross it is not reversed, even when the current into the gte stops. It will be turned off when the node current flls to zero.

47 Single phse hlf-wve Thyristor AC-DC converter L i v c F C C mx sint v e vs v E i = 45 di 0 dt di 0 dt vs E i 0 47

48 1 2 mx E mx sintd( t) Ed( t) cos cos di v sint i L e dt s mx t mx E L i sin t Ae ; 2 L At t=, i =0, 2 L 1 tn mx E L 0 sin Ae 2 2 L mx E L A sin e 2 2 L mx E mx E L i sint sin e L L t 48

49 At t, i 0, thus, mx E mx E 0 sin sin e L L When is found, i nd v wveforms re completely known, nd then I nd cn be determined. L, m 1 2 I, T 49

50 Single-phse fully-controlled thyristor bridge converter drive i p is mx sint T1 T3 i L T4 T2 E i p is mx sint T1 T3 i L i p is mx sint T1 T3 i L T4 T2 E T4 T2 E T1 nd T2 re triggered v = v s, i s = i T3 nd T4 re triggered v = -v s, i s = -i 50

51 Single-phse fully-controlled thyristor bridge converter drive v 45 (Q1) -v s v s 1 mxsin td( t) 2 mx cos v 135 (Q 4) v s -v s -i s i s 51

52 2 2mx mx cos I cos T ' L KT m ' ' KE KE 1 I E 1 I E 52

53 b n n Armture voltge nd current ripples mx cos n 1 cos n 1 2 n1 n1 mx sin n 1 sin n 1 2 n1 n1 2 2 n n n v b n = 2, 4, 6,... Only even-order hrmonics v 1 n L n tn n i n sin n t n ; 2 2 nl i I E 2mx E i ; n n2,4,6,... I cos I ipples in rmture current cuse dditionl losses 2 n 53

54 I n 2 n n L 2 n nmx 2 I I I I I MS Armture Form Fctor = IPF I MS Converter input Power Fctor (idel converter) = I I I cos I I I I MS MS MS MS MS MS (A mesure of motor heting) cos Note tht 1 is the power fctor ngle ssocited with the fundmentl (i.e., hrmonic order n = 1) of input voltge nd current wveforms. It is lrgely determined by the firing ngle. 1 Distortion fctor 54

55 Anlysis of i in CCM Solving mx sin di t i L E for t dt mx E i sint Ae Z 2 2 Z L L t 1 L tn In the stedy-stte, rmture current flls to its minimum vlue t t =,, Imin i( ) i( ) A 55

56 A L 2 mx e sin Z L e 1 L mx e 1 E Imin i ( ) sin Z L e 1 1 I i( t) dt Complicted 2mx E or, I ( E) / cos Simple I ' KE It is lso stright line for certin firing ngle.

57 The criticl (required minimum) rmture inductnce For opertion t the boundry of CCM nd DCM,. The condition for minimum L min is given by L 2 2 mx L e 1 E sin L e 1 57 Note tht the criticl (minimum required) rmture inductnce is found from this trnscendentl eqn [2.3.36]

58 Exercise 4: Find in DCM nd mx E L i sint Ae Z t i flls to zero t t=. mx sin Z mx Z sin E E Ae Ae i ( ) i ( ) 0 L L cos Z e E cossin( ) E cossin( ) / L mx / L mx e

59 Exercise 4: Find in DCM nd 1 mxsintd( t) Ed( t) mx E coscos 59

60 m rd/sec ω T chrcteristics in DCM 1 mxsintd( t) Ed( t) mx E coscos e E cos sin( ) E cos sin( ) / L mx / L mx e = 0 = 60 T, Nm = 150 = 170 for given E (2.3.25). DCM Substitute into I E DCM A series of ω-t curves for different firing ngles cn be drwn. At the boundry 60

61 Effect of source inductnce on speed i p i L s T 1 T 3 I mx sint v si L T 2 T 4 + E 2mx 2Ls I cos voltge droop fctor m 2mx 2 L cos K E s I A further droop in ωt Chrcteristics (CCM) 61

62 Effect of source inductnce on speed v µ All four thyristors conduct during commuttion overlp (µ) due to the source inductnce; sin di mx t Ls dt The input current through Ls chnges by 2I,the missing voltge is I 2 sin t d t Ldi L I mx I s s 2mx 2LsI cos 62

63 Single-phse hlf-controlled thyristor bridge converter drive L L s T1 T2 mx sintdt mx sint D1 D2 I D f E 1 mx 1cos s i i T1 or m Q1 i Df i s 0 2 I or T 63

64 T - ω chrcteristic under HC drive The input current through Ls chnges by I due to the freewheeling pth. 1 1 I L s sintdt L di I 0 mx s mx 1 cos L I s m mx L K s 1cos T / K E T 64

65 T - ω chrcteristic under HC drive HC drive vs. FC drive: HC hs higher IPF, becuse the lgging component of input current is freewheeled loclly rther thn fed bck to input. Lower ripples in rmture current nd voltge nd less likely to be DCM. Low cost. First qudrnt only, becuse the freewheel diode prevents the rmture voltge to become negtive. 65

66 Three-phse fully-controlled thyristor bridge converter drive v n v bn v cn i i b i c T 1 T 3 T 5 T 4 T 6 T 2 1n i L + E 2n mxl l 2 3 mxll 3 1 sin td t /3 3 cos Q1 Q4 I 66

67 FC Converter wveforms T1 on, 1n=vn; T3 on, 1n=vbn; T5 on, 1n=vcn; T4 on, 2n=vn; T6 on, 2n=vbn; T2 on, 2n=vcn; =1n-2n; T6,T1 on, =vb; T1,T2 on, =vc; T2,T3 on, =vbc; T3,T4 on, =vb; T4,T5 on, =vc; T5,T6 on, =vcb. 67

68 FC Converter wveforms (Ls) The voltge pulse is missing during µ: mxl l 2 0 sin t d t L di L I I s s 68

69 With CCM nd negligible source inductnce 3mxll cos E I ; m 3 mxll cos K E I With CCM nd source inductnce L S m 3 mx l l 3L K s cos I E 69

70 n b n Output voltge ripples 2sinn1 cosn1 2sinn1 cosn1 n1 n1 3mxl l sin n 1 sin n 1 2 sin n 1 sin n 1 n1 n1 3mx ll mx ll cos 2 cn nmx n bn 2 2 n1 n1 n 1n 1 for n = 6, 12, 18,... Multiples of 6 th hrmonics 70

71 Criticl inductnce for continuous conduction sin sin e 1 Z 3 L 3 E mxll = 0 m rd/sec = 60 T, Nm = 150 =

72 Three-phse vs. single-phse converter drive The output wveforms of 3-ph converter re smoother. 3-ph: multiples of 6 th order; 1-ph: multiples of 2 nd order. The lower current ripple clls for smller inductnce required for CCM. The effective converter switching frequency: 3-ph: 300 Hz; 1-ph: 100 Hz. The input current wveform hs better distortion fctor. This clls for reduced filter requirement t the input AC side. 3-ph: 6k ± 1; 1-ph: 2k ±1. Complexity, cost nd power hndling cpcity of 1-ph converter is lower thn 3-ph converter. 72

73 Hlf-controlled three-phse thyristor bridge converter driven DC motor + D /2 v v n i T 1 T 3 T 5 i L n v bn v cn i b i c D 4 D 6 D 2 D f d L e 3 2 mxl l D /2 1cos Q1 Firing ngle I 73

HC converter wveforms 1 2 1 2 2 3 3 3 3mxl

74 HC converter wveforms mxl l mxllsin t dt mxl lsintdt 1 cos

75 Converter oltge Gin (3-ph FC) Assuming CCM 3 3 cos cos v v mx ll 1 mx ll c c 3 mx l l v c 1 cos v c cos -1 Firing Control Circuit v The firing ngle is mde equivlent to the control voltge, vc, to the firing controller. Between the firing controller nd motor terminl, the converter behves s voltge gin of 3 mxll 75

76 Assuming CCM Four-Qudrnt Converter C1 I L C2 E = Suppressed hlf control mode: (either C1 or C2 is enbled) C1, α1: Q1 (1>0, I>0, α1<90) nd Q4 (1<0, I>0, α1>90); C2 disbled. C2, α2: Q2 (2>0, I<0, α2>90) nd Q3 (2<0, I<0, α2<90); C1 disbled. Crossover dely for smooth trnsfer of current between motoring nd generting. 76

77 Circulting current mode C1 I L C2 E = Both converters re operted together, = 180. Due to the instntneous output voltge differences of C1 nd C2, circulting current flows, which is normlly limited by center-tpped inductor. Zero trnsfer dely. 77

78 Motor de-rting due to ripple current The rmture current includes DC vlue nd ripples. DC vlue produces the developed torque Kt I ; n 2 ipples produce ripple torque nd extr heting I Motor output power is proportionl to I. Copper loss is proportionl to I rms squred. P I 2 copper rms Becuse of the ripple current I n, the motor is to be derted by the fctor I /I rms. A 50kW DC motor is regrded s 40kW mchine if the rtio is 0.8, extr heting loss is 10kW; P 1 E I n dev

Solution of Tutorial 2 Converter driven DC motor drive

Solution of Tutorial 2 Converter driven DC motor drive chool of Electricl Engineering & Telecommunictions, UNW olution of Tutoril Converter driven DC motor drive Question 1. T V s D V I L E V 50 V,.5, I 0 A rted rted f 400 Hz, 0 rev/ min s rted (i) 0 6.8 rd