1 electrical technology et 201 define series impedances and analyze series ac circuits using...
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ELECTRICAL TECHNOLOGY ET 201
Define series impedances and analyze series AC circuits using circuit techniques.
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14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current (review)
FIG. 15.46 Reviewing the frequency response of the basic elements.
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(CHAPTER 15)
SERIESSERIESAC CIRCUITSAC CIRCUITS
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15.3 Series Impedances• The overall properties of series AC circuits are
the same as those for DC circuits.
• For instance, the total impedance of a system is the sum of the individual impedances:
[Ω]
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Example 15.7
Draw the impedance diagram and find the total impedance.
15.3 Series Impedances
84
90021
j
jXR
XR
L
L
T
ZZZ
34.6394.8 TZ
Solution
6
26
12106
90900321
j
jj
jXjXR
XXR
CL
CL
T
ZZZZ
Example 15.8
Draw the impedance diagram and find the total impedance.
15.3 Series Impedances
43.1832.6 TZ
Solution
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15.3 Series AC Circuit• In a series AC configuration having two
impedances, the current I is the same through each element (as it was for the series DC circuit)
• The current is determined by Ohm’s Law:
21 ZZZ T
????, 21 VV
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• Kirchhoff’s Voltage Law can be applied in the same manner as it is employed for a DC circuit.
• The power to the circuit can be determined by:
Where
E, I : effective values (Erms, Irms)
θT : phase angle between E and I
15.3 Series Configuration
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14.5 Power Factor
• For a purely resistive load;
Hence;
• For purely inductive or purely capacitive load;
Hence;
TpF cos factor Power
0T 1cos TPF
TrmsrmsIEP cos
rmsrmsTrmsrms IEIEP cos
90T
0cos TrmsrmsIEP
0cos TPF
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14.5 Power Factor
• Power factor can be lagging or leading.– Defined by the current through the load.
• Lagging power factor: – Current lags voltage– Inductive circuit
• Leading power factor:– Current leads voltage– Capacitive circuit
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R-L1. Phasor Notation
te sin4.141 0V 100 E
15.3 Series Configuration
Series R-L circuit Apply phasor notation
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R-L2. ZT
Impedance diagram:
43
)904()03(21
j
T
ZZZ
13.53 5 TZ
15.3 Series Configuration
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R-L
3. I
13.53 5
0V 100
TZ
EI
13.53A 20 I
15.3 Series Configuration
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R-L
4. VR and VL
Ohm’s Law:
)0 3)(13.53A 20(
RR ZIV
V 13.5360 RV
)90 4)(13.53A 20(
LL ZIV
V 87.3680 LV
15.3 Series Configuration
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R-LKirchhoff’s voltage law:
Or;
In rectangular form,
0 LR VVEV
LR VVE
15.3 Series Configuration
V; 483613.53V 60 jR V
V 486487.36V 80 jL V
LR VVE
0V 100
0100)4864()4836(
jjj
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R-LPhasor diagram:
I is in phase with the VR and lags the VL by 90o.
I lags E by 53.13o.
15.3 Series Configuration
13.53A 20 I
V 13.5360 RV
V 87.3680 LV
0V 100 E
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R-L
Power: The total power delivered to the circuit is
Where
E, I : effective values;
θT : phase angle between E and I
Or;
W1200
13.53cos)20)(100(
cos
TT EIP
W120032022 RIPT
15.3 Series Configuration
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R-L
Power factor:
13.53cos
cos
TpF
lagging 6.0pF
15.3 Series Configuration
TZ
R
IE
R
E
IR
EI
RI
EI
P
EIP
2
cos
cos
TTP Z
RF cos
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R-C1. Phasor Notation
A 13.53sin07.7 ti A 13.535 I
15.3 Series Configuration
Series R-C circuit Apply phasor notation
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R-C2. ZT
Impedance diagram:
86
)908()06(21
j
T
ZZZ
13.5310 TZ
15.3 Series Configuration
21
R-C
3. E
)13.5310)(13.535(
TIZE
V 050 E
15.3 Series Configuration
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R-C
4. VR and VC
Ohm’s Law:
)06)(13.535(
RR ZIV
V 13.5330 RV
)908)(13.535(
CC ZIV
V 87.3640 CV
15.3 Series Configuration
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R-CKirchhoff’s voltage law:
Or;
15.3 Series Configuration
0 CR VVEV
CR VVE
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R-CPhasor diagram:
I is in phase with the VR and leads the VC by 90o.
I leads E by 53.13o.
15.3 Series Configuration
A 13.535 I
V 050 E
V 13.5330 RV
V 87.3640 CV
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R-CTime domain: V 050 E
V 13.5330 RV
V 87.3640 CV
15.3 Series Configuration
V sin7.70 te
V 13.53sin42.42 tvR
V 87.36sin56.56 tvC
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R-C
Power: The total power delivered to the circuit is
Or;
W150
13.53cos)5)(50(
cos
TEIP
W1506522 RIP
15.3 Series Configuration
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R-C
Power factor:
Or;
13.53cos
cos
TpF
leading 6.0pF
15.3 Series Configuration
TTP Z
RF cos
leading 6.010
6PF
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R-L-C1. Phasor Notation
TIME DOMAIN
PHASOR DOMAIN
15.3 Series Configuration
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R-L-C Impedance diagram:
2. ZT
43
373
90900321
j
jj
XXR CL
T
ZZZZ
15.3 Series Configuration
13.535TZ
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R-L-C
3. I
13.535
050
TZ
EI
15.3 Series Configuration
A 13.5310 I
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R-L-C
4. VR , VL and VC
Ohm’s Law:
V 13.5330 RV
15.3 Series Configuration
V 13.14330 CV
)907)(13.5310( LL IZV
)03)(13.5310( RR IZV
)903)(13.5310( CC IZV
V 87.3670 LV
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R-L-CKirchhoff’s voltage law:
Or;
15.3 Series Configuration
0 CLR VVVEV
CLR VVVE
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R-L-CPhasor diagram:
I is in phase with the VR , lags the VL by 90o, leads the VC by 90o
I lags E by 53.13o.
15.3 Series Configuration
A 13.5310 I
V 13.5330 RV
V 13.14330 CV
V 87.3670 LV
V 050 E
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R-L-CTime domain:
15.3 Series Configuration
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R-L-C
Power: The total power delivered to the circuit is
Or;
Power factor:
W30013.53cos)10)(50(cos TT EIP
W30031022 RIPT
13.53coscos TpF
15.3 Series Configuration
lagging 6.0pF
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• The basic format for the VDR in AC circuits is exactly the same as that for the DC circuits.
Where
Vx : voltage across one or more elements in a series that have total impedance Zx
E : total voltage appearing across the series circuit.
ZT : total impedance of the series circuit.
15.4 Voltage Divider Rule
EZ
ZV
T
xx
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Example 15.11(a)
Calculate I, VR, VL and VC in phasor form.
15.3 Series Configuration
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Example 15.11(a) - Solution
Combined the R’s, L’s and C’s.R T L T C T
10 0.1 H 100 mF
202sin377tv
i
H 1.005.005.021
LLLT
104621 RRRT
21
111
CCCT F 100
200200
200200
21
21 m
CC
CCCT
15.3 Series Configuration
e
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Example 15.11(a) – Solution (cont’d)
Find the reactances.
1. Transform the circuit into phasor domain.
7.37
)1.0(377TL LX
53.26)10100(377
116
TC CX
V 377sin220 te V 020 E
i I
15.3 Series Configuration
R T X L X C
10 37.7 26.53
200 VV
IE
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Example 15.11(a) – Solution (cont’d)
2. Determine the total impedance.
3. Calculate I.
17.1110
53.267.3710
j
jj
jXjXR CLTT
Z
16.4815 TZ
15.3 Series Configuration
R T X L X C
10 37.7 26.53
200 VV
I
16.4815
020
TZ
EI A 16.481.33 I
E
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Example 15.11(a) – Solution (cont’d)
4. Calculate VR, VL and VC
V 16.483.13 RV
)010)(16.4833.1(
RR IZV
15.3 Series Configuration
R T X L X C
10 37.7 26.53
200 VV
I
V 84.4114.50 LV)907.37)(16.4833.1(
LL IZV
V 16.13828.35 CV)9053.26)(16.4833.1(
CC IZV
E
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15.3 Series ConfigurationExample 15.11(b)
Calculate the total power factor.
Solution
Angle between E and I is
16.48coscos TpF
lagging 667.0pF
A 16.481.33 IV 020 E16.48
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Example 15.11(c) Calculate the average power delivered to the circuit.
Solution
15.3 Series Configuration
16.48cos)33.1)(20(cos TT EIP
W74.17TP
A 16.481.33 IV 020 E
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Example 15.11(d) Draw the phasor diagram.
Solution
15.3 Series Configuration
A 16.481.33 I
V 16.483.13 RV
V 84.4114.50 LV
V 16.13828.35 CV
V 020 E
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Example 15.11(e) Obtain the phasor sum of VR, VL and VC and show that it equals the input voltage E.
Solution
15.3 Series Configuration
V 933.9894.8V 16.483.13 jR V
V 446.33355.37V 84.4114.50 jL V
V 534.23284.26V 16.13828.35 jC V
534.23446.33933.9284.26355.37894.8 jjjCLR
VVVE
V 020020021.0965.19 jjE
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Example 15.11(f) Find VR and VC using voltage divider rule.
Solution
15.3 Series Configuration
R T X L X C
10 37.7 26.53
200 VV
IE
16.4815 TZ
)020(16.4815
010
E
Z
ZV
T
RR V 16.483.13 RV
)020(16.4815
9053.26
E
Z
ZV
T
CC V 16.13837.35 CV
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15.6 Summaries of Series AC Circuits
For a series AC circuits with reactive elements:
• The total impedance will be frequency dependent.
• The impedance of any one element can be greater than the total impedance of the network.
• The inductive and capacitive reactances are always in direct opposition on an impedance diagram.
• Depending on the frequency applied, the same circuit can be either predominantly inductive or predominantly capacitive.
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15.6 Summaries of Series AC Circuits
(continued…)
• At lower frequencies, the capacitive elements will usually have the most impact on the total impedance.
• At high frequencies, the inductive elements will usually have the most impact on the total impedance.
• The magnitude of the voltage across any one element can be greater than the applied voltage.
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15.6 Summaries of Series AC Circuits
(continued…)
• The magnitude of the voltage across an element as compared to the other elements of the circuit is directly related to the magnitude of its impedance; that is, the larger the impedance of an element , the larger the magnitude of the voltage across the element.
• The voltages across an inductor or capacitor are always in direct opposition on a phasor diagram.
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15.6 Summaries of Series AC Circuits
(continued…)
• The current is always in phase with the voltage across the resistive elements, lags the voltage across all the inductive elements by 90°, and leads the voltage across the capacitive elements by 90°.
• The larger the resistive element of a circuit compared to the net reactive impedance, the closer the power factor is to unity.