EE30372 - Electric Machinery and Power Systems Analysis
Miscellaneous formulae
Physics:
!
!
Work(W) = F dr = τ dθ, Power = dW
dt
i
Flux density: B = µN
for
i
amperes
in
coil of N turns and mean flux path length lc
! lc
Total flux(φ): φ = B·dA
Faraday’s and Lenz’s laws for N turns around flux φ: eind = −N dφ
dt
Force on wire carrying i amperes in field of flux density B: F = i(l × B)
Voltage induced in conductor moving at velocity v in flux density B: eind = (v × B) · l
Torque between two flux fields BR and BS = kBR ×BS .
General:
Complex power: S = VI∗ = P + jQ
p
Turns ratio of transformer: a = N
Ns
Referencing of impedance through transformer (ref. to primary): Z"L = a2 ZL
+NC
Autotransformer apparent power rating advantage: NSE
NSE
2
)
base
Per-unit analysis: Zbase = (VSbase
= VIbase
, Sbase = Vbase Ibase .
base
Transformer phasor diagrams (ref. to secondary): VaP = Vs + Req IS + jXeq IS
3(V
)2
base
Per-unit in 3-phase: Zbase = LN,base
, Ibase = 3VSLN,base
Sbase
Voltage induced in coils of P-pole stator windings with NC turns enclosing total flux φtot in
P-pole field, and rotating at speed ωm relative to magnetic field: NC φtot ωe cos(ωe t), where
ωe = ωm P/2.
Mechanical power in rotating machine: Pconv = τind ωm .
Power output by rotating machine: Pout = τload ωm .
γ
−γf ull load
× 100%
“Regulation” of quantity γ: no load
γf ull load
Parallel generators: P = sP (fnl − fsys ), Q = sQ (Vnl − Vf l )
Motors:
−nm
120∗fe
Induction motor speed: nsync = no.
, s = nsync
poles
nsync
Induction rotor internal frequency: fr = sfe .
Power transferred to induction rotor: PAG = Pconv + PRCL = τind ωsync
3V 2 R /s
Three-phase induction motor torque: τind = ωsync [(RT H +RT2H/s)22+(XT H +X2 )2 ]
DC motor converted power: Pconv = EA IA
Back EMF in DC motor: EA = Kφω.
Induced Torque in DC motor: τind = KφIA
RA
Vt
Shunt DC motor speed: ω = Kφ
− (Kφ)
2 τind
V
+RS
T
Series DC motor speed: ω = √Kc√τ − RAKc
ind