Cct component - Universiti Sains Malaysia
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Transcript Cct component - Universiti Sains Malaysia
Single Phase System
I
R-L parallel circuit
IR
IL
VL = VR = V
V p2
V
V
IL
p
XL XL 2
XL
IR
V
R
IL lags V by p/2
IR and V are in phase
IR
V
From the phasor diagram
p/2
I T (I R I L )
2
2
IT
IL
(V / R)2 (V / XL )2
IT
1
2
2
(1 / R) (1 / X L ) Y
V
Z
IL
1 R
1 R
tan
tan
tan
IR
XL
L
1
or
IR
1 Z
cos
cos
IT
R
1
Admittance triangle
For parallel circuit we look at admittance
G
1
Y admittance [ S ]
Z
V
1
G conductanc e[ S ]
R
1
BL susceptance[ S ]
XL
Y G 2 BL (1 / R) 2 (1 / L) 2
2
= tan-1 (BL/G) = tan-1 (1/LG)
cos = G/Y = Z/R
-p/2
Power factor
BL
Y
R-L parallel circuit
I
IR
VC = VR = V
V p2
V
V
IL
p
XC
XC 2
XC
V
IR
R
IC leads V by p/2
IR and V are in phase
IC
IC
IT
From the phasor diagram
p/2
IT ( I R I C )
2
2
(V / R) 2 (V / X C ) 2
I
Y (1 / R) 2 (1 / X C ) 2
V
(1 / R ) (C )
2
IR
V
Admittance triangle
G
V
2
-p/2
(G) 2 (BC ) 2
= tan-1 (BC/G) = tan-1 (C/G)
cos = G/Y = Z/R power factor
BL
G 1/ R
Y
BC C
A circuit consists of a 115 resistor in parallel
with a 41.5 mF capacitor and is connected to a
230 V, 50Hz supply. Calculate:
(a) The branch currents and the supply current;
I
(b) The circuit phase angle;
(c) The circuit impedance
V
IR
230V
V 230
IR
2. 0 A
50Hz
R 115
115
XC
1
1
76.7
2p f C 2p 50 41.5 106
V
230
IC
3.0 A
X C 76.7
IC
41.5mF
continue
IT ( I R I C )
2
cos
1
2
2.0
2
3.02 3.6 A
IR
1 2.0
cos
56.3o
I
3.0
V 230
Z
63.956.3o
IT 3.6
Three branches, possessing a resistance of 50W, an inductance
of 0.15H and a capacitor of 100mF respectively, are connected
in parallel across a 100V, 50Hz supply. Calculate:
(a) The current in each branch;
(b) The supply current;
(c) The phase angle between the supply current and the supply
I
voltage
V
100V
50Hz
IR
50
IC
mF
IL
0.15H
IC
solution
IT
IC-IL
V 100
IR
2.0 A
R 50
V
100
IL
2.12A
X L 2p 50 0.15
IR
IL
V
IC
100 2p 50100106 3.14A
XC
I T I R I C I L 2.02 3.14 2.12 2.24A
2
2
IR
2
cos
0.893
I
2.24
2
26o 45'
V
Parallel impedance circuits
Impedance sometime has the magnitude and phase . For
example we combine the resistance and inductance such in
inductor. In practical inductor has resistance and inductance.
If we have two impedances in parallel the current passing
through impedance 1 will be I1 and in impedance 2 will be
I2. To solve this, these current components can be resolve
into two components, i.e active and reactive , thus
I I1 I 2
I cos I1 cos1 I 2 cos2
(active)
I sin I1 sin 1 I 2 sin 2
(reactive)
I ( I1 cos 1 ) 2 ( I 2 sin 2 ) 2
I1 sin 1 I 2 sin 2
tan
I1 cos1 I 2 cos2
1
cos
1
I1 sin 1 I 2 sin 2
I
A parallel network consists of branches A,B and C. If
IA=10/-60oA, IB=5/-30oA and IC=10/90oA, all phase angles,
being relative to the supply voltage, determine the total
supply current.
IC
I
V
IB
IC
IA
o
o
10/-60oA 5/-50 A 10/90 A
C
A
I I A IB IC
I cos I A cosA I B cosB I C cosC
I sin 10sin A 5 sin B 10sin C
B
IB
V
I
IA
IA +I
B
I cos 10cos(60) 5 cos(30) 10cos(90) 9.33A
I sin 10sin(60) 5 sin(30) 10sin(90) 1.16A
I ( I cos 2 I sin 2 )
9.33 1.16
2
2
9.4 A
I sin
1 1.16
tan
tan
7.1o
I cos
9.33
1
I 9.4 7.1
o
The current is 7.1o lags the voltage
A coil of resistance 50W and inductance 0.318H is connected in parallel
with a circuit comprising a 75W resistor in series with a 159mF
capacitor. The resulting circuit is connected to a 230V, 50Hz a.c
supply. Calculate:
(a) the supply current;
(b)the circuit impedance; resistance and reactanceI.
X L 2p f L 2p 50 0.318 100
V
R1
I1
50
L
0.318H C
R2
I2
75
Z1 R1 X L 50 2 100 2 112
2
2
V 230
I1
2.05A
Z1 112
1 cos1
R1
50
cos1
cos1 0.447 63.5o
Z1
112
I1 2.05 63.5o Lags the voltage
VL
V
1
IR1
1
VR1
159mF
VR2
1
1
XC
20
6
2p f C 2p 5015910
2
I2
Z 2 R2 X C 752 202 77.7
2
2
V
V
230
I2
2.96A
Z 2 77.7
2 tan1
XC
20
tan1
tan1 0.267 15o
R2
75
VC
Leads the voltage
I 2 2.9615o
I I1 I 2
I cos I1 cos1 I 2 cos2
I2
2
1
V
I
2.05cos(63.5o ) 2.96cos(15o ) 3.77A
I1
I sin I1 sin 1 I 2 sin 2
2.05sin(63.5o ) 2.96sin(15o ) 1.07A
I
I cos 2 I sin 2
3.77 2 1.07 2
3.9 A
(b)
Z
V 230
58.7
I 3.9
I cos
3.77
R Z cos Z .
58.7.
56
I
3.9
I sin
1.07
X Z sin Z .
58.7.
16
I
3.9
More capacitive
Resonance circuits
I
From phasor diagram, since the voltage
VL (BO) and VC (CO) are in line, thus the
resultants for these two component is DO
(BO-CO) which is not involved in phase.
AO is the voltage across R. Thus
EO2= AO2 + DO2
I
2
2
V RI 2p f LI
2p f C
I
V
R
V
L
C
B
2
V
2
Z
1
R 2 2p f L
2p f C
D
O
C
E
A
I
Therefore the impedance is
V
1
2
Z R 2p f L
I
2p f C
Resultant reactanceX 2p f L -
X L XC
tan
R
When resonance
1
r L
r C
and Z=R
2
1
X L XC
2p f C
X
sin
Z
RI R
cos
ZI Z
X L XC
1
r
LC
therefore
I=V/R ;
or
fr
=0
1
2p LC
Current (I) and impedance (Z) vs frequensi (f) in
R, L & C serial circuit
Z []
I [A]
Z
R
I
0
fr
f [Hz]
At f=fr
increasing of voltage occured where
VL > V
or
VL = QV
VC > V
or
VC = QV
Q = VL/V = VC/V
In serial circuit:
VL = IXL; VC = IXC and V = IR
Q = IXL/IR = IXC/IR
Q = XL/R
= XC/R
Substitute XL and XC
Q = rL/R
= 1/rCR
where r = 2pfr
1
Q
R
L
C
Q is a circuit tuning quality
A circuit having a resistance of 12 , an inductance of 0.15H and a
capacitance of 100 mF in series, is connected across a 100V,
50Hz supply. Calculate:
(a)The impedance;
(b)The current;
(c)The voltage across R, L and C;
(d)The phase difference between the current and the supply voltage
(e)Resonance frequency
(a)
1
2
Z R 2p f L
2p f C
2
1
122 2p 500.15
2p 50100106
144 47.1 31.85 19.4
2
2
note
XL=47.1
XC=31.85
(b)
I
V 100
5.15 A
Z 19.4
VL=242.5
V=100
(c) VR IR 5.1512 61.8V
VL-VC
=78.5
VL IX L 5.15 47.1 242.5
VC IXC 5.15 31.85 164V
(d)
(e)
cos 1
fr
VR
61.8
cos 1
51o50'
V
100
I
VR=61.8
VC=164
1
1
41.1Hz
6
2p LC 2p 0.1510010
IL
IC
Z L R 2 X L2
tan
1
XL
R
ZL
XL
sin
ZL
ZL=ZL/
V
V
IL
Z L Z L
ZC X C p / 2
V
V
IC
p / 2
X C p / 2 X C
Red notation for impedance triangle
XL
ZL
V VC
VL
IC
R
VR
IL
Not at resonance
Y YL YC
YL
IC
1
1
Z L Z L
I
1
1
YC
p / 2
X C p / 2 X C
V
1
1
1
G Y cos
cos
cosp / 2
cos
ZL
XC
ZL
B Y sin
1
1
1
1
sin
sin p / 2
sin
ZL
XC
ZL
XC
Y (G B )
2
2
B
tan
G
1
VC
Y Y
IL
At resonance
At resonance =0 , thus sin =0
1
1
1 XL 1
B0
sin
ZL
XC ZL ZL XC
2
ZL X L X C 0
IC
I
V
R2 X L2 X L X C 0
IL
R (ωr L) L/C 0
2
2
L
r L R 2
C
2 2
2
1
R
r
2
LC
L
1
fr
2p
2
1
R
2
LC L
When R is so small we can put R=0 then
fr
1
2p LC
Previously we have tan
1
XL
R
X L r L
tan
Q
R
R
Q factor of the circuit