useful trigometric identities

7/30/2019 Useful Trigometric Identities

1/15

UI: EE422/NTU: PS 512-S

Power Systems Analysis

Session 13; Page 1/15

Spring 2004

Useful Trigometric Identities in AC Machine Analysis

1. sin

2

+ cos

2

= 12. sin = sin cos cos sin

3. cos = cos cos sin sin

4. Acos + Bsin = A2 B2 sin , = tan 1 A B

5. Acos + Bsin = A2 B2 cos , = tan 1 B A

6. sin cos = 12

[sin + sin

7. cos cos =12 [cos + cos

8. sin sin = 12

[cos - cos

9. cos + cos 23

+ cos 23

= 0

10. sin + sin 23

+ sin 23

= 0

11. sin cos + sin 23

cos 23

+ sin 23

cos 23

0

12. cos2

+ cos2

23 + cos

2

23

3

2

13. sin2 + sin2 23

+ sin2 23

= 32

14. cos cos 23

+ cos 23

cos 23

+ cos 23

cos 34

15. sin sin 23

+ sin 23

sin 23

+ sin 23

sin 34

16. sin cos 23

+ sin 23

cos + sin 23

cos 23

3 34

17. sin cos 23

+ sin 23

cos 23

+ sin 23

cos = 3 34

18. cos cos + cos 23

cos 23

+ cos 23

cos 23

= 32

cos

19. sin sin + sin 23

sin 23

+ sin 23

sin 23

= 32

cos

20. sin cos + sin 23

cos 23

+ sin 23

cos 23

= 32

sin


2/15




Spring 2004

Synchronous Machine Windings

G

C

C

B

B

A

Quadrature Axis

Direct

Axis

Reference

D

F

Q

Q

F

D

A

Transient Model for a Synchronous Machine

Generator Convention

G


3/15




Spring 2004

Notation

f can represent i, v, .

f

= r rotating reference frame fixed on the rotor

s stationary reference frame fixed on the stator

= s stator quantity

r rotor quantity

a A-Phase quantityb B-Phase quantity

c C-phase quantity

d direct-axis quantity

q quadrature-axis quantity

F field quantity

D direct-axis damper winding quantity

Q quadrature-axis damped winding quantity

g eddy current winding quantity

l leakage quantitym mutual quantity

= rotor quantities referred to the stator

(superscript on the subscript)

fabc

fa

fbfc

; fr

dq0

frd

fr

qfr0

F denotes that the quantity F is a complex vectorF denotes that the quantity F is a phasor


4/15




Spring 2004

Inductances

Stator Flux Linkage Equations:

abc Lsiabc LsriFDgQ

Rotor Flux Linkage Equations:

FDgQ LTsriabc LriFDgQ

Combined Stator and Rotor Flux Linkage Equations:

abcFDgQ

Ls LsrLTsr Lr

iabciFDgQ

a

b

c

F

D

g

Q

Laa Lab Lac LaF LaD Lag LaQ

Lba Lbb Lbc LbF LbD Lbg LbQ

Lca Lcb Lcc LcF LcD Lcg LcQ

LFa LFb LFc LFF LF D LF g LFQ

LDa LDb LDc LDF LDD LDg LDQ

Lga Lgb Lgc LgF LgD Lgg LgQ

LQa LQb LQc LQF LQD LQg LQQ

ia

ib

ic

iF

iD

ig

iQ

where


5/15




Spring 2004

Stator Self-inductances

Laa

Ls

Lm

cos2d

Lbb Ls Lmcos2 d 2 3

Lcc Ls Lmcos2 d 2 3

Stator Mutual-inductances

Lab Lba Ms Lmcos2 d 6

Lbc Lcb Ms Lmcos2 d 2

Lca Lac Ms Lmcos2 d 5 6

Rotor Self-inductances

LF

LD

Lg

LQ

Rotor Mutual-inductances

LFD

LDF

Mrd

LgQ LQg Mrq

LFg LgF LDg LDg 0

LFQ LQF LDQ LDQ 0

Stator-rotor Mutual-inductances

LaF LFa MFcosd

LbF LFb MFcos d 2 3

LcF LFc MFcos d 2 3

LaD LDa MDcosd

LbD LDb MDcos d 2 3

LcD LDc MDcos d 2 3


6/15




Spring 2004

Lag

Lga

Mgcos

d

Lbg Lgb Mgcos d 2 3

Lcg Lgc Mgcos d 2 3

LaQ LQa MQsind

LbQ LQb MQsin d 2 3

LcQ LQc MQsin d 2 3


7/15




Spring 2004

Parks Transformation

frodq R r P 0 fabc

where r r2

Coordinate axis transformation

P 0 :2

3

1

2

1

2

1

2

1 1

2

1

2

0 323

2

abc odq

Transformation to rotating reference frame

R r :

1 0 0

0 cosr sin r0 sinr cos r

0dqs 0dqr Rotation

Combine into one step

P r R r P 0

fr0dq P r fabc

fr0

frd

frq

2

3

1

2

1

2

1

2

cosr cos r23 cos r

23

sinr sin r23 sin r

23

fa

fb

fc


8/15




Spring 2004

P r :2

3

1

2

1

2

1

2

cosr cos r23 cos r

23

sinr sin r23 sin r

23

P 1 r PT r

P 1 r2

3

1

2cosr sinr

1

2cos r

23 sin r

23

1

2cos r

23 sin r

23

P rdP 1 r

dt

0 0 0

0 0

0 0

: x


9/15




Spring 2004

Synchronous Machine Equations

1. Stator Voltage Equations: (Note: p = d/dt)

vabcs rsiabcs pabcs

v0dqs rsi0dqs p0dqs x0dqs

2. Rotor Voltage Equation:

vFDgQr RriFDgQr pFDgQr

3. Stator Flux Linkage Equations:

abcs Lsiabcs LsriFDgQr

odqs Lsiodqs LsriFDgQr

4. Rotor Flux Linkage Equations:

FDgQr LTsriabcs LriFDgQr

FDgQr LT

sriabcs LriFDgQr


10/15




Spring 2004

0s

ds

qs

Fr

Dr

gr

Qr

L0

0 0 0 0 0 0

0 Ld 0 kMF kMD 0 0

0 0 Lq 0 0 kMg kMQ

0 kMF 0 LF MD 0 0

0 kMD 0 MD LD 0 0

0 0 kMg 0 0 Lg MQ

0 0 kMQ 0 0 MQ LQ

i0s

ids

iqs

iFr

iDr

igr

iQr

k3

2

TE idq iqd

p3 t i0 v0 id vd iq vq

Ea0MFiFe

j

2

Ea

Ea0MFiF

2


11/15




Spring 2004

Synchronous Machine Parameters

Xd

direct axis reactance

Xq quadrature axis reactance

Xd direct axis transient reactance

Xq quadrature axis transient reactance

Xd direct axis subtransient reactance

Xq quadrature axis subtransient reactance

X2 negative sequence reactance

X0 zero sequence reactance

rsdc stator dc resistance

rsac stator ac resistance

rf r field resistance referred to the statorr2 negative sequence resistance

Td0 direct axis open-circuit transient time-constant

Td direct axis short-circuit transient time-constant

Td direct axis short-circuit subtransient time-constant

Ta armature short-circuit (d.c.) time-constant


12/15




Spring 2004

Machine Turbo Hydro Synchronous Synchronous

Constant Generator Generator Condensor Motor

Xd 1.1 1.15 1.80 1.20

Xq 1.08 0.75 1.15 0.90

Xd 0.23 0.37 0.40 0.35

Xq 0.23 0.75 1.15 0.90

Xd 0.12 0.24 0.25 0.30

Xq 0.15 0.34 0.30 0.40

X2 0.13 0.29 0.27 0.35X0 0.05 0.11 0.09 0.16

rsdc 0.003 0.012 0.008 0.01

rsac 0.005 0.012 0.008 0.01

r2 0.035 0.100 0.05 0.06

Td0 5.6 5.6 9.0 6.0

Td 1.1 1.8 2.0 1.4

Td 0.035 0.035 0.035 0.035Ta 0.16 0.15 0.17 0.15


13/15




Spring 2004

Hydro Generator (salient pole rotor)

Xmd 0kMD Xmq 0kMQ

Xd 0Ld Xmd Xld s

Xq 0Lq Xmq Xlqs

XdXmdXl f r

Xmd

Xl f r

Xld s

Xq Xmd Xld s

XdXmdXld r Xl f r

XmdXl f r Xl f r Xld r Xld r XmdXld s

XqXmqXlqr

Xmq XlqrXlqs

X2Xd Xq

2

X0Xld Xlq

2


14/15




Spring 2004

Td01

rf r

Xmd Xl f r

b

TdXdXd

Tdo

TdXdXd

Tdo

Tq

Xq

Xq Tqo

Ta1

rsb

2XdXq

Xd Xq


15/15




Spring 2004

Three Phase Short Circuit of a Synchronous Machine

ias t 2 Ea1

Xd

1Xd

1Xd

et

Td

1X

d

1X

de

t

Td sin et

2 Ea12

1Xd

1Xq

et

Ta sin

2 Ea12

1Xd

1Xq

et

Ta sin 2et

Name Magnitude Frequency T

Steady Ea1

XdFundamental

Transient Ea1

Xd

1Xd

Fundamental Td

Subtransient Ea1

Xd

1Xd

Fundamental Td

Asymmetrical Ea12

1X

d

1Xq

sin Zero Ta

Second Harmonic Ea12

1Xd

1Xq

Double Fundamental Ta

useful trigometric identities

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