Transcript Document

Chapter
21
Inductive Circuits
Topics Covered in Chapter 21
21-1: Sine-Wave iL Lags vL by 90°
21-2: XL and R in Series
21-3: Impedance Z Triangle
21-4: XL and R in Parallel
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Topics Covered in Chapter 21
 21-5: Q of a Coil
 21-6: AF and RF Chokes
 21-7: The General Case of Inductive Voltage
McGraw-Hill
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
21-1: Sine-Wave iL Lags vL by 90°
 When sine-wave variations of current produce an induced voltage,
the current lags its induced voltage by exactly 90°, as shown in Fig.
21-1.
 The phasors in Fig. 21-1 (c) show the 90° phase angle between iL
and vL.
 The 90° phase relationship between iL and vL is true in any sinewave ac circuit, whether L is in series or parallel.
Fig. 21-1
21-1: Sine-Wave iL Lags vL by 90°
 The phase angle of an inductive circuit is 90° because
vL depends on the rate of change of iL.
 The iL wave does not have its positive peak until 90°
after the vL wave.
 Therefore, iL lags vL by 90°.
 Although iL lags vL by 90°, both waves have the same
frequency.
21-2: XL and R in Series
 When a coil has series resistance, the current is
limited by both XL and R.
 This current I is the same in XL and R, since they are
in series.
 Each has its own series voltage drop, equal to IR for
the resistance and IXl for the reactance.
21-2: XL and R in Series
Fig. 21-2:
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21-2: XL and R in Series
 Instead of combining waveforms that are out of phase, they can be added
more quickly by using their equivalent phasors, as shown in Fig. 21-3 (a).
 These phasors show only the 90° angle without addition.
 The method in Fig. 21-3 (b) is to add the tail of one phasor to the arrowhead
of the other, using the angle required to show their relative phase.
Fig. 21-3:
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21-3: Impedance Z Triangle
 A triangle of R and XL in series, as
shown in Fig. 21-4, corresponds to a
voltage triangle.
 The resultant of the phasor addition
of R and XL is their total opposition in
ohms, called impedance, with the
symbol ZT.
 The Z takes into account the 90°
phase relation between R and XL.
Fig. 21-4:
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21-3: Impedance Z Triangle
Phase Angle of a Series RL Circuit
40 Ω
I=2A
50 Ω
R = 30 Ω
VA = 100
q
XL = 40 Ω
VL
VA
Θ= Tan-1
53°
I
XL
R
= Tan-1
VA leads I by 53°
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
40
= 53°
30
30 Ω
21-4: XL and R in Parallel
Currents in a Parallel RL Circuit
IT = 5 A
IR
VA = 120
R = 30 Ω
XL = 40 Ω
IL
VA
120
=
= 4A
IR =
R
30
IT =
IR2 + IL2 =
IL =
VA
XL
=
120
= 3A
40
42 + 32 = 5 A
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IT
21-4: XL and R in Parallel
Phase Angle in a Parallel RL Circuit
IT = 5 A
q
VA = 120
XL = 40 Ω
R = 30 W
3A
Θ = Tan
−1
−
4A
IL
IR
= Tan
−1
−
3
= −37°
4
The total current lags the source voltage by 37°.
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5A
21-4: XL and R in Parallel
 Phasor Current Triangle
 Fig. 21-6 illustrates a phasor
triangle of inductive and resistive
branch currents 90° out of phase
in a parallel circuit.
 This phasor triangle is used to
find the resultant IT.
Fig. 21-6:
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21-4: XL and R in Parallel
Impedance of XL and R in Parallel
IT = 5 A
4A
VA = 120
R = 30 W
XL = 40 W
3A
ZEQ=
VA
IT
120
=
= 24Ω
5
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5A
21-4: XL and R in Parallel
In a parallel circuit with L and R:
 The parallel branch currents IR and ILhave individual
values that are 90° out of phase.
 IR and IL are added by phasors to equal IT, which is the
main-line current.
 The negative phase angle −Θ is between the line
current IT and the common parallel voltage VA.
 Less parallel XL allows more IL to make the circuit
more inductive, with a larger negative phase angle for
IT with respect to VA.
21-5: Q of a Coil
 The ability of a coil to produce self-induced voltage is
indicated by XL, since it includes the factors of
frequency and inductance.
 A coil, however, has internal resistance equal to the
resistance of the wire in the coil.
 This internal resistance ri of the coil reduces the
current, which means less ability to produce induced
voltage.
 Combining these two factors of XL and ri , the quality
or merit of a coil is, Q = XL/ri.
21-5: Q of a Coil
 Figure Fig. 21-7 shows a
coil’s inductive reactance XL
and its internal resistance ri.
 The quality or merit of a coil
as shown in Fig. 21-7 is
determined as follows:
Q = XL/ri
Fig. 21-7:
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21-6: AF and RF Chokes
 In Fig. 21-9, XL is much greater than R for the frequency of the ac source
VT.
 L has practically all the voltage drop with very little of VT across R.
The inductance here is used as a choke to prevent the ac signal from
developing any appreciable output across R at the frequency of the
source.
Fig. 21-9
21-7: The General Case of
Inductive Voltage
 The voltage across any inductance in any circuit is




always equal to L(di/dt).
This formula gives the instantaneous values of vL
based on the self-induced voltage produced by a
change in magnetic flux from a change in current.
A sine waveform of current I produces a cosine
waveform for the induced voltage vL, equal to L(di/dt).
This means vL has the same waveform as I, but vL and
I are 90° out of phase for sine-wave variations.
The inductive voltage can be calculated as IXL in sinewave circuits.