Unformatted text preview: 31 Chapter 3
Review of
Basic Electric Circuits Exit 2001 by N. Mohan Print TOC " ! 32 Conventions
! MKS (SI) Units
! lower case v and i for instantaneous quantities
! upper case V and I for average and rms
! voltage and current subscripts
+ a
+ vab
i − b
+ va vb − − ! voltage polarities and current directions
Exit 2001 by N. Mohan TOC " ! 33 Phasor Domain Representation for
Sinusoidal Steady State AC
φ Imaginary v
ωt =0 i positive
angles ˆ
V = V ∠0 Real ωt −φ
ˆ
I = I∠ −φ ˆ
v(t ) = V cos(ω t )
ˆ
i (t ) = I cos(ω t − φ ) Exit 2001 by N. Mohan ⇔
⇔ ˆ
V = V ∠0
ˆ
I = I ∠ −φ TOC " ! 34 TimeDomain Analysis
i( t ) v( t )
ˆ
=V cos( ω t ) + L − R
C di (t ) 1
ˆ
Ri (t ) + L
+ ∫ i (t ) ⋅ dt = V cos(ω t )
dt
C Exit 2001 by N. Mohan TOC " ! 35 Phasor Domain Analysis
I + ˆ
V =V ∠0 − Im jω L = j X L R 1 − j
= − j XC
ωC − jX c jX L Z
Re R
0 Z = R + j X L − j X C = Z ∠φ
Z = 1 R2 + ω L − ωC ˆ
V
i (t ) = cos(ω t − φ )
Z
Exit 2001 by N. Mohan 2 ; ⇔ 1 ω L − ωC −1 φ = tan R ˆ
V V
I = = ∠ −φ
Z
Z
TOC " ! 36 Instantaneous Power
i (t )
+
Subcircuit 1 ˆ
v(t ) =V cos(ω t + φv )
Subcircuit 2 v (t )
− ˆ
i (t ) = I cos(ω t + φi )
p (t ) = v (t ) i (t ) φ /ω average
power p (t ) 0 t v (t ) p (t ) average
power 0 t v(t )
i (t ) x v and i in phase (φv = φi )
x power flows in one direction
x maximum average power
for given V and I
Exit 2001 by N. Mohan i (t ) x v and i out of phase (φv ≠ φi )
x power flow reverses periodically
x average power lower than
maximum possible
TOC " ! 37 Real Power, Reactive Power and
Power factor
I " Complex Power S = V I ∗ (S is a complex number)
= V I ∠( φv − φi ) = V I ∠φ
S = P + jQ = S ∠φ " Real Power (average power)
P = V I cos φ " Reactive Power
Q = V I sin φ [W ] P 2 + Q 2 = VI " Power Factor =
Exit 2001 by N. Mohan Subcircuit 1 [VA]
P
P
=
= cos φ
S
VI Subcircuit 2 V
− S = P + jQ ˆ
V = V ∠φv Im φv − φi [VAR ] " Apparent Power
S = + Re ˆ
I = I ∠φi
Im
Q S φ = φv − φi
P Re
TOC " ! 38 Inductive Load
❏ The impedance is Z = Z ∠φ , where φ is positive ❏ The current lags the voltage by the impedance
angle φ ❏ Corresponds to a lagging power factor of operation
❏ In the power triangle, the same angle φ relates P,Q and S ❏ An inductive load draws positive reactive power (VARs)
❏ Most loads are inductive, particularly motors and
transformers Exit 2001 by N. Mohan TOC " ! 39 Three Phase Circuits
(Transmission and Distribution)
❏ Oneline diagram of power systems
Step up
Transformer Generator
Transmission
line 13.8 kV Feeder Load Exit 2001 by N. Mohan TOC " ! 310 Three Phase Circuits
❏ Wyeconnection Ia ˆ
Van = Vs ∠0 o
ˆ
Vbn = Vs ∠ − 120 o a + V an
V cn
+ − n ˆ
Vcn = Vs ∠ − 240 o − V bn
+ N
c Ib Ic Van + Vbn + Vcn = 0 van (t ) + vbn( t) + vcn ) = 0
(t ZL − a−b−c van (t ) vbn (t ) vcn (t )
V cn
0 ωt b positive
sequence
120° 120°
120° V an V bn
2π
Exit 2001 by N. Mohan 3 2π 3
TOC " ! 311 Per phase Analysis of Balanced Three
phase Circuits
ˆ
Van
Vs
Ia =
=
∠ −φ
ZL
ZL
ˆ
Vbn
Vs
2π
Ib =
=
∠−
−φ
ZL
ZL
3 Ia
V an
V cn
+ ˆ
Vcn
Vs
4π
Ic =
=
∠−
−φ
ZL
ZL
3 a + −
− n − V bn
+ Ic ZL In
N
c b Ib I n = ( I a + I a + I c ) = 0 ⇒ in ( t ) =[ ia ( t ) + ib ( t ) + ic ( t ) = 0 a
+ Ia V cn V an (Hypothetical)
2001 by N. Mohan φ Ib −
n Exit Ic a N V an Ia
V bn
TOC " ! 312 LinetoLine Voltages
Ia a +
V an
V cn
+ − Vc
+
+ − n − V bn
+ ZL N c Ib Ic ˆ
Vab =Va − Vb = 3Vs ∠30 Va
−
− o ˆ
Vbc =Vb − Vc = 3Vs ∠ − 90 o
ˆ
Vca = Vc − Va = 3Vs ∠ − 210 o
Exit 2001 by N. Mohan Vca −Vb 30 o V ab
Vb
+ Vab Va
−
b Vb Vbc VLL = 3V ph ∠30 o TOC " ! 313 Delta Connection
Phase Currents in Delta Load
Ia I a = I ab − I ca
I b = I bc − I ab
I c = I ca − I bc V an a
I ca +
− V cn − n
+ − V bn
+ Z∆
Ib c b I ca 1
I∆ =
Il ∠30 o
3 I ab
30 o Ib 2001 by N. Mohan Z ∆ I ab Ic Ic Exit Ibc Z∆ Ibc " I ca "
Ia
TOC " ! 314 ∆ − Y Transformation
❏ Allows per phase analysis
Ia equivalent
to external
circuit a I ca
c Z∆
I ab Ibc Delta connected load Ia
a ZΥ ⇔ b c ZΥ ZΥ
b Wye connected load These circuits are indistinguishable to the external circuit,
when
Z∆
ZΥ = Exit 2001 by N. Mohan 3
TOC " ! 315 Summary
❏ Why is it important to always indicate the directions of
currents and the polarities of voltages?
ˆ
❏ What are the meanings of i, I, I, and I ?
❏ In a sinusoidal waveform voltage, what is the
relationship between the peak and the rms values?
❏ How are currents, voltages, resistors, capacitors, and
inductors represented in the phasor domain? Express
and draw the following as phasors, assuming both φv
and φi to be positive:
ˆ
ˆ
v(t ) =V cos(ω t + φv ) and, i (t ) = I cos(ω t + φi )
Exit 2001 by N. Mohan TOC " ! 316 Summary
❏ How is the current flowing through impedance Z ∠φ
related to the voltage across it, in magnitude and phase?
❏ What are real and reactive powers? What are the
expressions for these in terms of rms values of voltage
and current and the phase difference between the two?
❏ What is complex power S? How are real and reactive
powers related to it? What are the expressions for S, P,
and Q, in terms of the current and voltage phasors?
What is the power triangle? What is the polarity of the
reactive power drawn by an inductive/capacitive circuit?
❏ What are balanced threephase systems? How can their
analyses be simplified? What is the relation between
linetoline and phase voltages in terms of magnitude
and phase? What are wye and delta connections?
Exit
TOC
"
2001 by N. Mohan ! ...
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 Fall '06
 Scalzo
 Power factor, N. Mohan

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