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Formula sheet – Final Exam – IGEE 401 / ELEC 6411
Average and rms values
The instantaneous power is:
Sinusoidal steady-state and phasors Power, reactive power and power factor
S = VI* = Ve j Ie jΦ = VI e jΦ = Se jΦ
Apparent Power (VA) = S = VI
Real Power (W) = P = Re[S] = VI cosΦ
Reactive Power (VAr) = Q = Im [S] = VIsinΦ
PF = P = cosΦ
S
In phase ( IP ) and out-of-phase ( IQ ) components:
Three-phase systems
Transformers (ideal): NpIp = NsIs Vp/Np = Vs/Ns VpIp = VsIs
Line current distortion
Average power, total power factor, and displacement power for a nonlinear load:
Thyristor based ac controllers
Phase controlled
Fundamental current for resistive load:
Fundamental component for inductive load:
On-off or integral half-cycle control: n cycles on and m cycles off.
T
o
rms
T
o
T
o
AV i dt T
vidt I I T
pt dt T
p t vi P
1 1 ()
1 ( )
2
− , tan ( )
2 sin( )
2 sin
1
2 2
X L X R
I V Z Z R X
i I t
v V t
L L
L
ω φ
ω φ
ω
φ φ φ
j j j
j j e Ie Z
V
Ze
Ve Ve
− − = = = = =
0 0
Z
V
V andI
I (^) P = Icos φ→P=VIP, IQ=Isin φ→Q=VI Q
) 6
) ( 6
( ab a b
3
2
c
3
2
b
0 Va ,V ,V V V V 3
π π π π j LL
j j j j = Ve =Ve =Ve = − = Ve =V e
−
s
s
h
dis s s sh s
dis
h
sh
T
s s h
s s sh I
I I I I I I
I i t dt I THD T
i t i t i t I
,peak
1
2 2 1
2
1 1
2
12
0
2
1
1 () , (%)^100 ,where ,Crestfactor
1 ( ) () (), = = = − = =
≠ = ≠
DPF THD
DPF PF I
I
I
I
VI
VI V ti dt VI S VI PF T
P
i s
s
s
s
s s
s s s s s s
T
o
AV s s 2
1 1
1 1 1 1 1 1
1 cos ,
cos 2 sin cos , ,
1
φ ω φ
2 2 2 S ,
S cos sin
P jQ S P Q
VI jVI
= φ + φ
2 2
I n=V 0 _n Zn where Zn = R +(n ωL)
S (^) φ VLIL Pφ Sφ θ Qφ S φ θ VL ω C
2 3 =^3 , 3 = 3 cos , 3 = 3 sin =^3
( ) ( α π α) π
α π
sin 2 2 2 2
2 cos 2 1 , 2
2 ˆ (^) , 1 1 2 1
2 1 =^1 + = − = + − R
V b R
V I a b a S S
1 (sin^2 α^2 π^2 α)^ (rms) π ω
= + − L
V I S L
Single-phase diode bridge rectifier (considering a highly inductive load):
Line voltage distortion:
Three-phase full bridge rectifiers:
Input current components considering a highly inductive load:
Single-phase thyristor ac-dc converters:
Three-phase thyristor ac-dc converters considering a highly inductive load:
SVC (TCR+FC):
Voltage regulation
Tuned harmonic filters
Buck dc-dc converter:
Boost dc-dc converter:
∑ ∑ ∑^ (^ )
∞
≠ ≠ ≠
1
2
1
_ 1
1 _ 1 1 1
h
sh Lsh h
sh PCC dist s
s PCC s s h
s s sh
s PCC s s I X dt
di v L dt
di i i i v v L dt
di v v L
- 9 , and 0. 9 , cos 1 , 0. 9
1 0 = = 1 = = 1 = = = s
s d s s d s d I
I
V V I I I I DPF φ PF DPF
v d vPn vNn Vd 2 VLLcos td t,Vd 1. 35 VLL 2. 34 V s
/ 6
/ 6
= − 0 = ∫ = = −
π
π
1 = 1 = = DPF= PF= h
I
I I I I I
s s d s d sh
s s s s s s s s S dc
d S S dc dc d
S V I S V I P V I Q V I I I
V V V P I V
, , cos , sin , 0. 9
cos 0. 9 cos ,
α α
α π
φ α π
α α cos
, cos cos ,
- 35 cos ,
1 = = = 1 = = = 1 = = =
s
s d LL s d d s d s I
I
V V I I I I I I DPF PF DPF
h h
h grid C L
f h f 2 π
1 = ⋅ =
2
2 , ( )
_
2
1 _
2 I I X
V
X
V Q Q Q Th rms C
s
L eff
s SVC = TCR− FC= − = α
V S =jX SI+VT,VS=VS∠ δ,VT=VT∠ 0 ,I=I∠− φ,VT=VS−jXS I
( ) ( ) V D
V
I
I
V
v D T
t
V
V V V t V t
o
d
d
o
st
ctrl
S
on
d
o d o on o off
1 , ˆ
− = − , = = = = =
( ) (^) oB
S o d o
S
LB Lpeak D I
L
T V
V V
L
DT
I = I = − = 1 − =
,
LC
f
f
f
D
LC
T D
V
V
c s
S c
o
o
π
π
2 (^22)
( ) ( ) ( D)
I
I
V D
V
V t V V t
d
o
d
o
d on d o off = −
2 , (^1 ) 2
D D
L
V T
D D I
L
V T
L
V t I I
o S oB
d on o S LB = Lpeak= = − = − τ
S S
o
o T
D
RC
DT
V
V
S = 3 V LL Is,S 1 = 3 VLLIs 1 ,P 1 = 3 VLLIs 1 cosα ,Q 1 = 3 VLLIs 1 sin α
UPF diode rectifier:
Switch-mode bi-directional grid interface:
HVDC:
Vdc _ B = − 1. 35 VLLcosα B,γB= 180 −αB,extinctionangle,Vdc_B= 1. 35 VLLcos γ B
o
PdB = 1. 35 VLLIdcosγ B,QdB= 1. 35 VLLIdsin γ B
cos θ sin , sin 1 cos
1
2
1
1
2
1 s
conv
s
s s s s
conv
s
s s s V
V
L
V
Q VI
V
V
L
V
P VI
[ ( )] dc
a conv s s s V
m V V L I 2
20.^5 1
2
1 = +^ ω =
s d
d rip d
d c d
dripple
d
s s
d
s s in s s s s s s d d d d
f L
V
I
C
I t i dt C
v t
V
V I t
V
V I
p t V tI t V I V I t p t V i t i t
,Forconstantfrequencycontrol: 2
1 sin 2 ()
cos 2 () ˆ sin ˆ sin cos 2 , () (), ()
